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)
HYDROSTATIC LUBRICATlON
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
HYDROSTATIC BEARINGS
1.1
INTRODUCTION
Hydrostatic lubrication consists in pushing a lubricant between the surfaces of a kinematic pair by means of an external pressurization system. This lubrication mechanism has now a well-defined collocation in the large field of lubrication 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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC BEARINGS
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
HYDROSTATIC LUBRICATION
- 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
HYDROSTATIC BEARINGS
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
HYDROSTATIC LUBRICATlON
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
HYDROSTATIC BEARINGS
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC BEARINGS
-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
HYDROSTATIC LUBRICATION
-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
HYDROSTATIC BEARINGS
-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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC BEARINGS
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
HYDROSTATIC LUBRICATION
Figure 1.16 shows a slide sustained by a system of opposed-pad bearings in the vertical and horizontal directions. Figure 1.17 shows a special hybrid bearing in which a rolling bearing is 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
COMPENSATING DEVICES
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
COMPENSATING DEVICES
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
HYDROSTATIC LUBRlCATlON
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).
COMPENSATING DEVICES
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
HYDROSTATIC LUBRICATION
-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
HYDROSTATIC LUBRICATION
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 .
COMPENSATING DEVICES
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
HYDROSTATIC LUBRICATION
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.
COMPENSATING DEVICES
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
HYDROSTATIC LUBRICATION
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.
COMPENSATING DEVICES
27
Certain bearings have been proposed which have a built-in controlled restrictor. Let us consider the bearing in Fig. 2.19 (ref. 2.16). 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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATlON
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
HYDROSTATIC LUBRICATlON
-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.
COMPENSATING DEVICES
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATlON
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
HYDROSTATIC LUBRICATlON
-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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRlCATlON
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATlON
(4.15) where we have stated U=Ul+U2. Equation 4.14 is still valid when the lubricant film is not flat, on the condition that its thickness is much smaller than the curvature radius of the bounding 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 h
BASIC EQUATIONS
61
After having substituted Eqns 4.16 and the film thickness:
h = C [l-E C O S ( ~ $ ) ] ; we finally obtain
In the last equation, the term proportional to sin2(6-@)may also be disregarded, since, on average, it is much smaller than the others.
4.3.2
Cylindrical coordinates
Let us consider a thin plane clearance: when we state the usual simplificative hypotheses above, the Navier-Stokes equations (Eqns 4.7), are reduced to the following:
(4.19)
In the first equation the inertia term pv2/r has been retained, since i t may prove not to be negligible at high turning velocity, as will be shown in sect. 6.2.1. Integrating the second of Eqns 4.19, with the boundary conditions v(O)=O, v(h)=V,the tangential velocity of the fluid is obtained: (4.20)
Substituting it in the &st of Eqns 4.19 and integrating for u(O)=O, u(h)=U,the radial velocity turns out to be (after further simplification): (4.21)
Let us now integrate the continuity equation (Eqn 4.8) along the film thickness:
HYDROSTATIC LUBRICATION
62 h
Substituting in it Eqn 4.20 and Eqn 4.21, the Reynolds equation i s obtained:
+ 1 2 p W + 3~ ; z~( h 3 V 2 ) (4.22) where W=h=w(h)-w(O)is the squeezing velocity of the surfaces. If the upper surface does not slide, but rotates around the z axis at a constant speed 0, it is U=Oand V=r; Eqn 4.22 now becomes:
Equation 4.22 may also be used for a conical film, provided the thickness of the film is much smaller than the minimum curvature radius, i.e. all the points of the domain are far enough from the vertex of the cone. With reference to the coordinate system in Fig. 4.5, Eqn 4.22 may be rewritten as:
If we now assume that the axes of the conical surfaces bounding the lubricant film always remain parallel (i.e. the thickness of the film does not depend on r), and that the turning velocity of the journal is Ckconst., it is possible to obtain simple expressions for U,V , W. The velocity of each point of the surface of the journal is the vectorial sum of the velocity of its axis, plus the turning velocity around the same axis; hence, the velocity components are:
u = E c sina cos(++) + E 4 c sina sin(+@)- R t V = - E C sin(+@ + E
~ cos(+$) C
+0
ah
g a s + V , cosa
r sina
w = - E C cosa cos(99) - E 4 C cosa sin(+$) + O ah,x + V , sina
(4.25)
BASIC EQUATIONS
63
Fig. 4.5 Thin fluid film between conical surfaces.
In Eqns 4.25 V, is the axial velocity of the journal (squeeze velocity) and C& t g a is the radial play. Let us now introduce Eqns 4.25 in Eqn 4.24.Taking into account that yl=&ina, hn=C cosafl-e cos(&$)], and that for all the domain being considered i t is C<
+ 12 p V, sina + 35;P hnlZ 3 sina [lZ r sina - C & sin(+@)- C E 4 cos(+$)]
(4.26)
As for the journal bearings, the term that is proportional to sins(&-$)may be discarded, since, on average, i t is much smaller than the others.
64
HYDROSTATIC LUBRICATION
4.3.3
Spherical coordinates
Let us write the Reynolds equation for a thin film between two spherical surfaces, whose diameters are D and D-2C, respectively, with C
(4.27)
where we have made the coordinate change r=D/2-6. Furthermore, we have assumed that the rotational velocity of the surfaces are small enough to allow cancellation of the inertia terms. If we assume that the outer sphere is stationary, the boundary velocities are:
In this case, we use U,V, W to indicate the radial and tangential (in the cp and 19 directions, respectively) components of the velocity of a generic point of the inner spherical surface. From Eqns 4.27 it is easily obtained:
(4.28)
Substituting velocities u and w in the continuity equation
and integrating it along the film thickness, the Reynolds equation is obtained:
For the sake of simplicity, we shall limit ourselves to the steady-state case, in which the velocity of the points of the inner surface is due solely to rotation around a
65
BASIC EQUATIONS
diameter that is assumed to remain parallel to the axis z=(p=O), whatever the displacement of its center is. Hence the components of the velocity of a point of the journal are:
Substituting them in Eqn 4.29, it is reduced to:
(4.30)
4.4
THE LAPLACE EQUATION
It is well known that, if no external pressurization of the lubricant is provided, the film of lubricant cannot sustain any load when one of the following circumstances occurs: - no squeeze nor sliding relative velocity of the surface exists (V=U=O); - a sliding velocity exists, but thickness h is a constant all over the film of lubricant. In this case, Eqn 4.15 is reduced to: (4.31)
-a-
-
-b-
Fig. 4.6 Hydrostatic bearings: a- without recess; b- with recess. Pressure profiles are also represented
66
HYDROSTATIC LUBRICATION
which is the so-called Laplace equation, for which, as is well known, only one solution exists when the value of p on the boundary is assigned. Furthermore, if p is a constant on the entire boundary (e.g. the atmospheric pressure), it takes on the same value on all the inner points as well. So, if the boundary pressure is the atmospheric pressure, a source of pressurized lubricant must be provided to ensure a lubricant film between the surfaces, as in Fig. 4.6.a; this is precisely the principle of hydrostatic lubrication. In Fig. 4.6.b a recess is widened around the source, in which the pressure is constant. Of course, both the hydrostatic and the hydrodynamic pressure fields may exist a t the same time, a s in the so-called "hybrid" bearings.
4.5
LOAD CAPACITY, FLOW RATE, FRICTION
4.5.1
Load capacity
The load capacity W of a film of lubricant is defined as the resultant of the pressure field. Hence, if A is the surface on which the pressure acts, and n is its normal direction. it is:
4.5.2
Flow rate
Integrating the velocity components (Eqn 4.12 and Eqn 4.13) between y=O and y=h, the components of the flow rate per unit length q are obtained:
(4.32)
(it should be borne in mind that Eqn 4.12 and Eqn 4.13, and consequently Eqns 4.32, refer to the case of two planes sliding one on the other in the x direction). For a system of polar coordinates, Eqns 4.32 are best substituted by the following:
(4.33)
where it has been assumed that the upper surface (z=h)of the pair rotates around the z axis.
BASIC EQUATIONS
67
To obtain the volume flow rate of lubricant crossing a certain contour S, whose external normal direction is n,it is necessary to integrate the scalar product qn:
Q = Jq.n dS S
4.5.3
Friction
In connection with bearings, it is important to evaluate the friction forces on the surfaces. The shearing stresses in the lubricant can be found by means of the constitutive equations (Eqn 4.2). On the basis of the assumptions stated in section 4.3, they are reduced to:
i.e. to the well-known Newton formula of the shear stress (see also sect. 3.2). Substituting Eqn 4.12 and Eqn 4.13 in these equations, we obtain:
(4.34)
When the pressure field is known, (that is after the relevant Reynolds equation has been solved), Eqns 4.34 directly give the shearing stresses in the film of lubricant between two surfaces sliding with a relative velocity U=U2-U1 in the x direction. The stresses on the surface y=O are, therefore,
On the opposite surface (y=h)we have:
68
HYDROSTATIC LUBRICATION
Integrating shear stress z, one obtains the tangential force on the surfaces. Hence in the direction of the sliding velocity, the total drag is: (4.35)
the upper sign referring to the fixed surface (y=O), the lower to the moving surface (y=h). In a very similar way, i t is possible to evaluate the friction moment which acts on two surfaces rotating around a common axis, with a relative velocity Lk (4.36)
4.6
THE ENERGY EQUATION
As has been already pointed out, the characteristics of a fluid depend on temperature T.In particular, for the most common lubricants, the variation in viscosity is notable, even for temperature changes of few degrees. Thus, in order to carry out a rigorous analysis, ,u should be considered a s a function of the temperature, and a new equation is required to compensate for the introduction of the new unknown.
The energy equation is obtained by imposing the energy balance for a n infinitesimal control volume. Assuming that heat radiation is negligible and that no heat is generated in the fluid (except for the dissipation of mechanical work due to shearing), we obtain (ref. 4.2): (4.37)
In the equation above, the left-hand side is the rate of change in internal energy of the fluid (c being its specific heat). On the right-hand side, the first term accounts for the power due to the expansion in volume of the fluid; the second term accounts for the heat conduction in the fluid (k being the coefficient of thermal conductivity); the last term is the dissipation of mechanical power. Since the energy equation must be applied to the thin films typical of hydrostatic and hydrodynamic lubrication, some further simplification can be introduced: - the specific heat is a constant (since no great temperature change is expected), as well as coefficient k;
69
BASIC EQUATIONS
-the fluid is considered to be incompressible (hence Eqn 4.6 is valid); - only steady-state conditions are considered (i.e. aT/dt=O). Introducing Eqn 4.2 in Eqn 4.37, we obtain:
As in section 4.3, it is possible to disregard all the velocity gradients, except for au& and awlay. Furthermore, heat conduction in the x and z direction is small, compared to the convective term; thus Eqn 4.38 becomes: (4.39) Heat conduction may sometimes be totally disregarded (i.e. k=O) as well as the heat transfer to the boundaries (adiabatic flow).
A similar equation may be obtained in cylindrical coordinates. Limiting ourselves to axial-symmetric configurations, we find: (4.40)
4.7
LAMINAR FLOW THROUGH CHARACTERISTIC CONFIGURATIONS
4.7.1
Parallel surfaces
Let us consider a plane surface of infinite length moving on another with velocity U (Fig. 4.7.a). The velocity of the lubricant is obtained from Eqn 4.12: (4.41) When U=O, the velocity profile is a parabola, whose axis of symmetry is y = h / 2 (pressure flow, or "Poiseuille flow"). In general, the linear term Uy / h (shear flow or "Couette flow") has to be added to the parabola. In Fig. 4.7.b the velocity profile is plotted for certain values of the non-dimensional pressure gradient
P*
=& (-g)
When p*
70
HYDROSTATICLUBRICATlON
-a-
-b-
Yt
L
a
J
Fig. 4.7 Flow between infinite-length parallel surfaces: a- pressure profile; b- velocity profiles. by imposing aufay=O on the lower surface. The point of inversion y* is easily found by stating u(y*)=O, i.e.:
The Reynolds equation (Eqn 4.15), for h=const., is simply reduced to:
which, integrated twice with the boundary conditions p = p 1 for x=O and p=p2 for x=a gives the linear relationship: X
P = P 1 - @ 1 -P2),
(4.42)
The flow rate per unit length is, from the first of Eqns 4.32 and from Eqn 4.42:
For a length L, the flow rate is:
The "hydraulic resistance" of the clearance may be defined as the ratio of the pressure drop to the flow rate in the Poiseuille flow. So, from Eqn 4.43, for U=O, we obtain:
71
BASIC EQUATIONS
(4.44)
The tangential force on the moving surface is given by Eqn 4.35:
It should be stressed that, in order that flow is laminar as presumed, Reynolds number ReZPuh P
(4.45)
u
should be lower than the critical value Re=2300.In Eqn 4.45 is the mean velocity
4.7.2
Infinite-length rectangular pad.
Let us consider a segment L of a hydrostatic pad of infinite length, as in Fig. 4.8. The recess is bounded by two clearances. If we now use p to indicate the relative pressure in the clearances, and p r to indicate the relative recess pressure (which can be considered to be constant, since the recess depth is much greater than the thickness of the film), from Eqn 4.42 we get: (4.46)
The load capacity of the pad is found by integrating the pressure on the two land areas and adding the contribution prLb of the recess: ~ r 2
(4.47)
The total flow of the pad is found from Eqn 4.43: 11
L
&=-3 P h3 ~ -Prb The hydraulic resistance may therefore be defined as
(4.48)
72
HYDROSTATIC LUBRICATION
Fig. 4.8 Hydrostatic pad of infinite length.
Note that, in Eqn 4.48, the term that i s proportional to the velocity (shear flow) no longer exists. Indeed, it is easy to establish that the flow rates crossing each clearance are Ql2ULUl2, and hence their s u m does not depend on U.
Similar considerations hold good for the tangential force. In this case, the terms due to the pressure flow are opposite in sign for the two clearances, and the drag on the whole pad is null when U=O. The friction force on the lands is hence proportional to the velocity:
(the contribution of the recess area is often negligible because the recess depth h, is much greater than h). In the equation above we have released the minus sign, since it is implicit that the friction force on the moving member is opposite in direction to the velocity.
73
BASIC EQUATIONS
4.7.3
Flow recirculation Inside recess
Examining the simple pad in Fig. 4.8 it should be easily understood that velocity U induces a continuous recirculation of the fluid in the recess, and hence a velocity field exists even when Q=O (see Fig. 4.9.a). The simplest approach for studying the flow recirculation is to consider that huhrnb, hence disregarding the end effects at the recess edge. Assuming laminar flow,the velocity profile is given by Eqn 4.12
$[u - $8 (1 - $)] 2
u=
At any given section, flow rate must be null; hence, imposing hr
Judy=O 0
it follows
(the same result could have been directly obtained from Eqn 4.32). - b-
-a-
h
X
b
m
-
10
lo2
3
3
lo3
Re
)
lo4
lo5
lo6
Fig. 4.9 Recess flow recirculation: a- velocity profile; b- friction factor.
One consequence is that pressure is not constant, and a pressure drop exists across the recess. Furthermore, a critical speed exists a t which pressure near the leading edge becomes atmospheric (4.50)
74
HYDROSTATIC LUBRICATION
The friction force over the moving member is (Eqn 4.35)
F C , . = 4 U A IL 'hr For a sliding hydrostatic pad this force should be added to the friction force on the lands, obtaining a total friction force
where A1 and A,. are the land and recess area, respectively, and factor fr takes on a value f,.=4-
h hr
under the simplifying assumptions made above. More detailed and thorough studies of recess flow recirculation can be found in ref. 4.5. and ref. 4.6. Better approximations for critical velocity and recess friction factor are to be found in ref. 4.6: (4.52)
h h hr hr
f,. = (4 - 3 - ) -
(4.53)
All the foregoing equations are based on the assumption that recess flow is laminar, i.e. that the Reynolds number
is smaller than 1000. Actually, the problem is more complex because of the pressure head build-up a t the inlet of the clearance (ref. 4.7). For turbulent flow it may be stated that: f
r
=--f Re
h 2h,P
(4.54)
and f, may be read in Fig. 4.9.b (proposed in ref. 4.5 on the basis of experiments on journal bearings) or calculated (ref. 4.8) from the equation (4.55)
BASIC EQUATIONS
75
Annular clearance.
4.7.4
Let us examine the flow between two parallel ring-shaped walls (Fig. 4.101,rotating around the z-axis with a relative velocity a, which, however, is not so great as to require the effects of the inertia of the lubricant to be considered. Since the thickness of the film is small, compared to width r2-r-1,Eqn 4.23 (the Reynolds equation) is valid, which, due to the radial symmetry, is now reduced to (4.56)
Integrating it twice, with the boundary conditions at r = r l ,
p=p1
p = p 2 at r = r 2 ,
the pressure field is obtained: P 'P2
In rlr2 +
(P1 - P 2 ) & &
i.e. p decreases on the land area with a logarithmic trend. The flow rate per unit length is, from the first of Eqns 4.33 and Eqn 4.57,
Fig. 4.10 Annular clearance.
(4.57)
76
HYDROSTATICLUBRICATION
Consequently, the total flow across any circle of radius r1esr-2 is (4.58)
Hence, the “hydraulic resistance“ of the clearance may be defined as:
=P 1 - p 2 = 6 Q n
In r 2 h p 7
(4.59)
The friction moment, from Eqn 4.36, is:
The Reynolds number is again given by Eqn 4.45. Since the mean velocity at mean radius 7=(rl+r2)/2is
--8 U. 2xFh
’
it follows that: 1 P Re=--Q ”
4.7.5
1
(4.61)
r2(1+3
Circular pad
The simple hydrostatic bearing, shown in outline in Fig. 4.6.b, is made up of a circular recess bounded by a n annular clearance, like the one examined in the previous section. Assuming that p2=0 (ambient pressure), pressure p r in the recess is related to the rate of flow by Eqn 4.58, which can be rewritten as (4.62)
The pressure in the clearance decreases with a logarithmic trend, given by Eqn 4.57, in which pl=pr and p2=0. The hydraulic resistance is still given by Eqn 4.59. The load capacity of the pad is found by integrating the pressure on the land area and adding the term m$pr due to the recess pressure: (4.63)
BASIC EQUATIONS
77
If Eqn 4.62 is introduced in Eqn 4.63, we obtain
$(
W=3-Q
i)
1--
(4.64)
The friction moment is given by Eqn 4.60, since the friction in the recess is negligible.
4.7.6
Pipes
In the case of laminar flow in a straight circular pipe, whose length is much greater than the diameter, certain simplifying hypotheses, similar to those stated in section 4.3, may be applied to the Navier-Stokes equations. In particular, the pressure can be considered to vary with the z coordinate alone. The third of Eqns 4.7 is now reduced to the following:
Integrating it with the boundary conditions w=O for r=d/2 and aw/&=o for r=O (due to matters of symmetry), the velocity field is obtained as a function of the pressure gradient: (4.65)
which is the equation of a paraboloid. The maximum velocity of the fluid is a t r=O:
while its mean velocity is:
Since the continuity equation (Eqn 4.8) is reduced to dpldz=const., the pressure drop along the pipe is linear. If a length 1 is considered, the flow rate is related to the pressure drop by the law: (4.66)
Equation 4.66 is used to evaluate the hydraulic resistance: (4.67)
78
HYDROSTATIC LUBRICATION
of a capillary pipe used as a compensating device for hydrostatic bearings. In general, in this kind of restrictor, the flow proves to be laminar, due to the smallness of the diameter. However, a check should be made to ensure that the Reynolds number
wpd 411 Re = p - l-t p d-Q
(4.68)
is lower than the critical value Re=2300. Equations similar to Eqn 4.66 are obtained in ref. 2.7 for straight pipes with differing sections (Fig. 4.11). Namely, for a rectangular section (Fig. 4.11.b) with a>h (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
-a-
-b-
-C-
-d -
-e-
A3 fJ Fig. 4.1 1 Cross sections of pipes.
79
BASIC EQUATIONS
4.8
FLOW INTO THE INLET LENGTH
inlet lengthIn section 4.7, a number of cases of laminar flow have been examined; the equations which have been obtained expressing the flow a s a function of the pressure step are not strictly accurate, since certain distributed energy losses have been disregarded. As a n example, let us consider a circular pipe. When the fluid begins to flow into the pipe, the velocity w is a constant for the entire section (Fig. 4.12); the velocity of the fluid on the wall of the pipe, however, is null, and this layer exerts a great shearing stress on the inner layers, whose velocity must become greater than the mean value W i n order to satisfy the continuity requirement. The thickness of the "boundary layer" grows until, a t a distance li from the inlet, the whole section of the pipe is involved; a t this point, the fluid velocity reaches the parabolic profile already seen in sect.4.7.6, and remains so thereafter.
\
'. ,-
I I
I 1
2
Fig. 4.12 Velocity profiles in the inlet length of pipes. The length Zi is named "inlet length" (ref. 4.9) and its value is found to be li=0.0575 d Re. When the length of the pipe is greater than li, which is generally true for the capillaries used as compensating devices for hydrostatic bearings, Eqn 4.65 should be substituted by the following:
Q = - -xd 41 128
'
1 d 1 1+ 0.0356 i Re
(4.69)
In other words, the value of the flow rate found in sect. 4.7.6 should be multiplied for a correction coefficient, which accounts for the pressure loss required to accelerate the inner fluid layers in the inlet length.
80
HYDROSTATIC LU5RlCATlON
Similar considerations hoId good for pipes of different shapes, and for the clearances of hydrostatic pads (see, for example, sect. 6.2.14~)).
INLET LOSSES
4.9
In the preceding section, the pressure loss in the inlet length of a circular pipe was examined; other pressure losses occur in the tank near the pipe inlet and a t the inlet itself. The loss in the tank may be considered to be the sum of two terms: the first is used to accelerate the fluid in the tank (in the conoid bounded by dotted lines in Fig. 4.12) to the inlet velocity, the other is due to the viscous dissipation in the same area. The loss at the inlet, on the other hand, is due to the "uena contructu", which is generally present, especially when the inlet edge is sharp. The overall pressure loss is (ref 2.7): (4.70)
The values of resistance coefficient k are given in Fig. 4.13 for the cases of (a) a well-rounded entrance, (b) a slightly rounded entrance, (c) a sharp-edged entrance. The pressure loss a t the inlet, given by Eqn 4.70, should be added to the pressure loss in the pipe, given by Eqn 4.69. -a-
-b-
-C-
K = 0.05
Fig. 4.13 Resistance coefficients due to the geometry of pipe entrances.
4.10
TURBULENT FLOW
The flow of the lubricant in the gap between two close surfaces is, in most cases, of a laminar kind, i.e. the Reynolds number is lower than 1000. Nevertheless, in certain circumstances, it may be not so. For instance, with reference to a circular pad and to Eqn 4.61, if high values are selected for p r and r h (due to the need to sustain a heavy load) and if the viscosity of the lubricant is low (e.g. when water or
81
BASIC EQUATIONS
liquid metals such a s Sodium have to be used), and, above all, if the gap is higher than the values commonly adopted, Re may be greater than 1000. When Re is greater than 1000, a transition regime sets in; a t Rez2300 the flow is prevalently turbulent, and it is totally turbulent for higher Reynolds numbers (see ref. 4.10). When the fluid gets into the clearance, the boundary layer is at first laminar but becomes turbulent (except for a thin laminar sublayer) a t a distance from the inlet of 25+40 times the clearance. The velocity profile is similar to the one plotted in Fig. 4.14.b and the maximum fluid velocity is about 1.2 times its mean value.
-a
-
-b-
Fig. 4.14 Velocity distribution in a pipe: a- laminar flow; b- turbulent flow. In turbulent flow, the velocity components and the pressure in the clearance a t a given time may be written as a sum of two terms: u=u*+u1; u=u*+u1; w=w*+wl; p=p*+pl: the first terms are the average values, the others are the fluctuations due to turbulence. These expressions should be introduced into Eqns 4.7 and Eqn 4.8 (i.e. the Navier-Stokes and continuity equations). Certain assumptions can now be made: the viscosity and density of the lubricant are considered to be constants; the average flow is stationary and merely radial (i.e. u*=u*(F,z);u*=w*=O); the turbulent components are considered to depend solely on r and z , due to the radial symmetry. The first of Eqns 4.7 becomes (ref. 4.2,4.4) (4.71) in which z,=p(au*/az) and zt=-p(ulwl)* are the viscous and turbulent components of the shear stress. It may be assumed that z,<
where l=zfor 02&/2,1=X(h-z) for h/2Q
x should be determined by
Equation 4.71 may be further simplified (ref. 4.11) and then integrated with the following boundary conditions
82
HYDROSTATIC LUBRICATION
where zo is the thickness of the laminar sublayer on the surfaces. We thus obtain (4.72)
where z'=zlh, zb=zdh, and
The blunt profile of u* (Fig. 4.14.b) is rather different from the parabolic profile (Fig. 4.14.a) pertaining to laminar flow. Equation 4.8 becomes u* -+-=oau* r ar
(4.73)
Introducing Eqn 4.72 into Eqn 4.73, and integrating with the following boundary conditions
we obtain
P* -PI =PI[--
1 - rz/r
11
(4.74)
For circular pipes, whose Reynolds number is given by Eqn 4.68, the drop in pressure is given by the well-known equation (ref. 2.7)
where the friction coefficient f can be evaluated, for ReelO5, by means of the experimental relation:
For pipes of different sections, the pressure drop can be approximately evaluated using the same Eqn. 4.75, in which the "hydraulic diameter" d=4AIS is introduced: A and S are the area and the perimeter of the cross section of the pipe, respectively.
83
BASlC EQUATlONS
4.11
THE FLOW IN ORIFICES
In a sharp-edge orifice (Fig. 4.15.a), the Reynolds number often has a very high value and the flow becomes "turbulent". The turbulence, however, is of a particular type. The fluid particles crossing the orifice are accelerated until the velocity of sound is reached, a t section 2 of the duct. The flow between sections 1 and 2 is of a potential kind (or, better, the energy dissipation is small). Area A2 of the critical section of the vein is lower than area A0 of the orifice (since the velocity has increased), and the ratio CC=A2IAois defined a s the "coefficient of contraction". Between sections 2 and 3, a rapid mixing of the jet occurs with the fluid downstream; "turbulence"sets up and the kinetic energy of the fluid is transformed into internal energy, with a consequent increase in its temperature. - b-
-a-
'
'
O
7
Fig. 4.15 Sharp-edge orifice: a- flow; b- contraction coefficient. Applying the Bernoulli equation between sections 1 and 2 (i.e. imposing the balance of the kinetic and potential energies of the fluid), one may write:
On the other hand, the continuity of the flow rate implies:
Eliminating the inlet velocity w1, the velocity w2 may be written as:
04
HYDROSTATIC LUBRICATION
Due to the viscosity of the fluid, the actual velocity w i a t section 2 is slightly lower than the above value, the velocity coefficient C,=w;/w2 being about 0.98. The flow rate of lubricant through the orifice is:
So, if we write
C" c,
we get
(4.76) The "discharge coefficient" Cd is approximately equal to C,, since Cu=l and, in general, Ao<
I"'
turbulent flow
0 0
I
10
I
20
I
VRe
I
30
Fig. 4.16 Discharge coefficient versus Reynolds number for an orifice.
In the case of a low pressure drop and if viscosity is high, the Reynolds number may become sufficiently low to permit laminar flow. Although the analysis above is not valid at low Re, Q may still be calculated by means of Eqn 4.76, in which the
85
BASIC EQUATlONS
discharge coefficient Cd is now given in Fig. 4.16. The initial value of Re has to be calculated for Cd=0.6. The sharp-edge orifices are sometimes substituted by less costly orifices (Fig. 4.17) whose length 1 is no longer negligible. In such cases, the discharge coefficient can be calculated by means of the following equations (ref. 2.7):
[
Cd = 1.5 + 13.74
1 m -m (a) ]
(4.77)
for ?<50
;
the relevant values of Cd also being plotted in Fig. 4.16. Re has to be initially obtained from Eqn 4.68, with Q being taken from Eqn 4.76 for Cd=0.6. Since Cd is now a function of the Reynolds number, the flow rate for any certain pressure step is no longer completely independent from variations in temperature and viscosity.
0s
1
o,61 0.4
cd
a2
I
0.14
10
I
lo* DRe / I
I
lo3
1 4
Fig. 4.17 Discharge coefficient for a short-tube orifice.
REFERENCES 4.1 43
45 4.4
Goldstein S. D.; Modern Deuelopments in Fluid Dynamics; Claredon Press, Oxford, 1952; 702 pp. Pinkus O., Sternlicht B.; Theory of Hydrodynamic Lubrication; McGraw-Hill Book Co., N.Y., 1961; 465 pp. Tipei N.; Theory of Lubrication; Standford Univ. Press, 1962; 690 pp. Cameron A,; The Principle of Lubricatioq; Longmans, Green and Co. Ltd., 1966; 591 pp.
86
HYDROSTATIC LUBRICATION
Shinkle J. N., Hornung V . G.; Frictional Characteristics of Liquid Hydrostatic Bearings; ASME Trans., J . of Basic Eng., 87 (19651,163-169. 4.6 Ettles C. H. M., O'Donoghue J . P.; Laminar Recess Flow in Liquid Hydrostatic Bearings; Instn. Mech. Engrs., C27 (19711,215-227. 4.7 Tipei N.; Flow Characteristics and Pressure Head Build-up at the Inlet of Narrow Passages; ASME Trans., J . Lubr. Tech., 100 (1978), 47-55. 4.8 El-Sherbiny M., Salem F., El-Efnawy N.; Optimum Design of Hydrostatic Journal Bearings; Tribology International, 17 (19841, 155-166. 4.9 Langhaar H. L.; Steady Flow in the Transition Length of a straight Tube; J . of Applied Mech., 9,2 (19421, A55-A58. 4.10 Streeter V.;Handbook of Fluid Dynamics; McGraw-Hill, 1961. 4.11 Bassani R.; Sul Regime Turbolento nei Cuscinetti Zdrostatici di Spinta Molto Caricati; Atti 1st. Mecc Appl. e Costr. Macch., 1968-69, N 15; 36 pp. 4.6
Chapter 5
PAD COEFFICIENTS
5.1
INTRODUCTION
In this chapter, generalized equations for load capacity W, stiffness K,flow rate Q, pumping power Hp, friction power H f and total power Ht are proposed, which are valid for every type of pad. In order to simplify the representation of the behavior of bearings and make their design easier, use is made of characteristic parameters, such as the effective area A, and the hydraulic resistance R, as well as of the power ratio ll and the pressure ratio J/.’ For the same purpose, certain p a d coefficients (A*,,R*, HF) are introduced, which are characteristics of the actual pad shape. They are also considered the effects on the bearing behavior of misalignment and of certain phenomena, such as turbulence, inlet pressure losses and lubricant inertia, which are not typical of plain hydrostatic lubrication.
5.2
GENERAL STATEMENTS
Whatever the shape of the bearing pads and the supply system, certain general expressions may be stated. The load capacity of a pad (the resultant of the lubricant pressure in the bearing) may be put in the following form:
W = Pr A,
(5.1)
p r is the recess pressure; A, is the “effective“pad area. This last is a fraction of the projected area and, in general, depends on the operating conditions; in the most common cases, however, A, may be considered as a constant, related solely t o the shape and size of the pad.
88
HYDROSTATICLUBRlCATlON
The lubricant flow rate may be related to the recess pressure by the "hydraulic resistance" R of the bearing lands:
W
Q=pr / R = At?R
(5.2)
If the land surfaces of the two bearing members are parallel to each other, as occurs in the majority of pad types, R is proportional to the inverse of the third power of the film thickness h: e.g. see Eqn 4.59. The 'pumping power" is the power needed to push lubricant into the bearing and dissipated, because of viscous friction, mainly in the compensating restrictor (when it exists) and in the bearing clearance. The total pumping power is
Hp=psQ
(5.3)
where p s is the pressure of the lubricant obtained by means of the pumping device. The pumping power dissipated in the bearing is:
Hi = P r Q
(5.4)
For a constant pressure system, p s may be much greater than the recess pressure, while for a constant flow rate system p s is only slightly greater than pr (the pressure losses are due, in this case, solely to the pipes linking the pump to the recess). Friction force Ff and friction moment M f have been defined in section 4.5.3. The lyriction power" is the power needed to keep a relative motion between the members of the bearing; it is due to viscous friction and it is given by the following equation:
Hf= Ff U + MfR
(5.5)
where U is the sliding velocity, and R is the turning velocity of the bearing. The total power dissipated is clearly the sum of the pumping power and the friction power:
Ht = Hp + Hf
(5.6)
Power H t is transformed into heat, part of which is lost through the walls bounding the oil path; the remainder, however, remains in the lubricant, whose temperature increases as it passes through the compensating devices and the bearing clearances. This increase in temperature must be evaluated, a t least approximately, to allow the supply devices to be correctly designed. In every case, AT should be kept a t a few degrees; indeed, if a large increase in temperature is allowed, the viscosity cannot be considered as a constant, and the performance of the bearing may be considerably affected.
PAD COEFFlClENTS
89
For a fast (and safe) evaluation of AT it is often assumed that all the dissipated power H , remains in the lubricant (adiabatic flow). As a consequence, we have simply (5.7)
where c is the specific heat of the lubricant, and p is its density (for further discussion on thermal running see chapter 12). The term H f I H , is commonly called the "power ratio" and has also been used (ref. 1.9) for the following grading of hydrostatic bearings: Hf/Hp=O, zero-speed bearings (purely hydrostatic bearings); H f I H p < l , lowlmoderate speed bearings; and H f / H p > l , moderate/high speed 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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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,<7.2 MPa, it follows that Ae>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 Mf<0.05 Nm, it follows from Eqn 5.26 hat h 9 5 . 5 pm. If a clearance h=100 pm is selected, it follows that R=3.62.109 NslmS (Eqn 5.21) and Q=1.95.10-3 m31s. Since Q is high, it is advisable to verify the effect of the pressure losses at the inlet. From Eqn 4.61, the Reynolds number is Re=319 and, from Fig. 4.6, Cp=1.26. Hence, the new value of the flow rate is calculated: Q=1.54.10-3 m3/s. This is still a high value; it should be borne in mind, however, that a very low friction was required, the friction coefficient being of the order of 2.10-5.
PAD COEFFICIENTS
5.3.5
101
Turbulent flow
Although the flow in the clearances of hydrostatic bearings is, generally, of a laminar type, in certain circumstances (as already noted in section 4.10) the Reynolds number (Eqn 4.61) may become greater than 2300, at which value the flow is prevalently turbulent. Equations giving pressure and lubricant velocity in the clearance of a circular pad in turbulent regime have been obtained in section 4.10. Integrating the mean pressure (Eqn 4.74) on both the recess and land area, the load capacity proves to be: K
W = 4 D2 r' p ,
(5.40)
while, if we integrate the radial velocity field (Eqn 4.72), the following flow rate is obtained:
where (ref. 4.11): (5.42) and (5.43) The nondimensional quantities x and A (which depends on zb and accounts for the rate of flow in the laminar boundary sublayer along the land surfaces) are evaluated in ref. 4.11, for Re>2300:
Comparing Eqn 5.40 and Eqn 5.41 with those obtained for the laminar flow, two correction factors are easily obtained for the effective area as well as the hydraulic resistance of the pad. After stating that: (5.44)
(5.45)
102
HYOROSTATIC LUBRICATION
it follows that:
(5.47)
When r’ is greater than 0.80, we have 0 . 9 9 < C ~ ~ 0.5, we have 6 % ~ ) l n ( l / r 7 , thus 1
‘RT
(5.48)
O X 5 ( l + A ) zb F(zb)
and, hence, C R T may be considered to depend on zb alone. Both CAT and CRTare plotted in Fig. 5.7.
-a-
1.oo
-b-
10
CAT CRT
0.95
5
0.90 0.50
0.75 f
1.oo
0
0.00
0.10
0.05 zb
Fig. 5.7 Turbulent flow: a- correction factor CATfor effective area; b- correction factor C,, for hydraulic resistance.
Since CRTproves to be greater than 1, we can see that, at high Reynolds numbers, turbulence brings about a considerable decrease in flow rate or, if flow rate remains constant, a considerable increase in load capacity. On the other hand, turbulence may cause a large increase of foam in the lubricant; the persistence of the foam
PAD COEFFICIENTS
103
depends on the physico-chemical properties of the oils, and may be counteracted by means of suitable anti-foam additives (see chapter 3). EXAMPLE 5.3 Let us re-examine the hydrostatic pad in Exumple 5.2, in order to evaluate how the flow rate would vary if the clearance and the viscosity were changed to h=200 pm and p=0.005 Ns I m2 ( I S 0 VG5),respectively. From Eqn 5.21 it follows that R=O.126.1@ NslmS, and hence Q=56.1.10-3m31s. Since the Reynolds number proves to be greater than 2300 (Eqn 4.61), a correction factor should clearly be evaluated. From Eqn 5.42 we obtain z&0.0112, and hence C~,=7.78, R=0.979.109 Nslm5,Q=7.23.103 m3/s (in spite of the considerable reduction due to turbulence, this is a very high value). The Reynolds number proves to be Re54260. Bearing in mind the above results, it may be concluded that, for values of R e lower than 10, which is the most common case for these hydrostatic thrust bearings, the whole pressure loss in the clearance is due to viscous forces; when Re is greater than 100, the pressure losses examined in section (v) become considerable; when Re is greater than 1000, the effect of turbulence should be taken into account. In both these cases, for any given load capacity and film thickness, the flow rate proves to be lower than that expected on the basis of the laminar theory.
5.3.6
Effects of the inertia forces
When the angular speed of the bearing in Fig. 5.1.a is high, the forces of inertia in the fluid can no longer be disregarded. Supposing h=const, but retaining the inertia term, Eqn 4.23 becomes: (5.49)
where the “inertia parameter”
(5.50) has been introduced.
This kind of problem has been solved in ref. 5.6. When r o o l , the depth h,. of the recess should be at least 5 times greater than the film thickness h, while hr=20h is considered to be a n optimal value.
104
HYDROSTATIC LUBRICATION
Fig. 5.8 Pressure distribution for certain values of inertia parameter (ref. 5.6).
In Fig. 5.8 the pressure field is plotted for a number of values of the inertia parameter Si;it is assumed that r1=0.5r2 and that the radius of the supply duct is ro=0.05r2.A pressure peak can be seen a t the inner edge of the bearing land. When no recess exists (i.e. ro=rl),negative pressures, and hence cavitation phenomena, may occur. Note that in Fig. 5.8 the depth of the recess is h,=5h. For higher values (i.e. h,.220h, as is common practice) the depth of the recess does not affect pressure distribution. When h,/h>>5, 0 . 0 5 ~ , / r 2 < 015, . 0.534'10.9 (i.e. in the large majority of cases), simplified equations are available for the pressure in the recess fro
Integrating p on the entire pad surface, we obtain the load capacity, and hence the effective pad area. Equation 4.20 gives the tangential velocity of the lubricant:
PAD COEFFICIENTS
105
Since u a n d p are now known functions, the radial flow velocity, and hence the volume flow rate, may now be easily obtained from Eqn 4.21. The effective area and the hydraulic resistance may thus be expressed as (5.53)
R=:$R*
cRI
(5.54)
where: c,=l+si
CRI=
[( $)+ ( 1--
1 1 + si(1-
);":I.
1-- -
3)
(5.55)
(5.56)
It is clear, however, that
c,=1+si CRI=
1
(
1--):2 :
(5.57)
(5.58)
are very good approximations in the large majority of cases. In Fig. 5.9 CAI,is given a s a function of Si for certain values of F'. In the same figure CRI is also given as a function of Si. The increase in flow rate (the inverse of CRI)may be appreciable even for moderate values of the inertia parameter. It is moderately compensated by the increase C increases with Si,load capacity W is increased in load capacity (i.e. C,). Since , by inertia effects if inlet pressure is held constant; thus these bearings are often named hybrid. However, as CRI simultaneously decreases more quickly, flow rate Q increases more greatly than W, so an efficiency loss of the bearing may occur. The Reynolds number, as given by Eqn 4.61, is usually very low. Nevertheless, turbulence could set in, due to the tangential flow velocity. In ref. 5.10, in which the circular pad is compared to the case of a couple of faced rotating disks (ref. 5.71, the Reynolds number Re,=ar;/ v is introduced, and the flow is proved to be laminar for
106
HYDROSTATIC LUBRICATION
Re,
L?Fh
Re, = 7
(5.59)
is smaller than lo3. The latter result agrees closely with the former, since
Re,
=:
Re,
r2
and r2/h>102. Reference 5.8 also shows that, when the flow is turbulent, for any given flow rate, the load capacity and the stiffness of the bearing increase notably with Re,; it should however be noted that values of Re, greater than 1000 are rarely reached.
i
f=0.9
0.8 CAI
0.7
CRI
0.6
0.5 1
0 0
1
2
Si
Fig. 5.9 Correction factor CA,versus inertia parameter Si for certain values of ratio r‘. Correction factor CRIversus inertia parameter Sk
EXAMPLE 5.4 A pad bearing, with D=O.l m, r’=0.9, sustains a load W=lO Mv. It is fed by a constant flow source at a rate &=0.3.1O3m 3 / s with a lubricant with the following characteristics: p=0.05 Nslm2, p=880 Kglm3, c=1900 JIK&jC Let us evaluate how bearing performance is affected as speed becomes G30Oz rad Is (9000 rpm). Proceeding as usual, we find Az=0.902, R*=0.105, Hf*=0.344. Thus, at Q=O, Ae=7.08.10-3m2, and p,=1.41 MPa, the hydraulic resistance of the pad must be R=p,l&=4.70.1O9 Nslm5 and hence h=129 pm. At the highest speed, inserting the value of pr calculated above into Eqn 5.50, we find Si=O.208, and (from Eqn 5.57’ C ~ = l . 0 9From . Eqn 5.53 it follows that Ae=7.73.10-3m2 and hencep,=1.29 MPa. For
PAD COEFFICIENTS
107
a more accurate calculation it is advisable to evaluate Si again. We now obtain: Si=0.227, CN=l.lO, C~1=0.816;consequently, A,=7.79.10-3 m2, p,=1.28 MPa. The hydraulic resistance of the pad becomes R=4.27,1O9 Nslm5 and, from Eqn 5.54, h=124 pm. The pumping power is Hi=384 W a n d the friction power is Hf=1209 W (disregarding the lubricant inertia, Hi=423 W and Hf=1167 W would be obtained, respectively). It is easy to verifv that the temperature step is small and the flow is laminar.
5.3.7
Thermal effects
When velocity L?of the pad is high, the temperature increase, due to the viscous friction in the lubricant, should also be taken into account, as well as the inertia forces. This kind of problem has been studied in ref. 5.9 (see also ref. 5.10). The energy equation must also be associated to the customary Navier-Stokes and continuity equations, as well as a viscosity-temperature relationship; this last may be Eqn 3.4, where it can be stated that p=0.04 for most mineral oils.
For a n axial-symmetric configuration, applying the simplifying hypotheses stated in section 4.3 to Eqns 4.7 (except that p is no longer considered a constant), the Navier-Stokes equations may be written as:
+$)=o a
(5.60)
For the same reasons, Eqn 4.8 is reduced to: (5.61) The energy equation has already been obtained in section 4.6(Eqn 4.40). Suitable boundary conditions should be associated to this set of equations €or pressure, velocities and temperature. In particular, as far a s temperature is concerned, adiabatic flow is often considered (ref. 5.11,5.12). In ref. 5.13 thermal exchange is considered between the lubricant and the pad surfaces held at constant temperature. In ref. 5.14,instead, a n approximate evaluation of the boundary temperatures is considered, obtained by means of experiments (ref. 5.15). The above equations can now be integrated by means of an iterative procedure, based on the finite difference method, to give p , T , u , and v , from which the load capacity, the flow rate, and the friction moment can be calculated. Let us now examine a few results.
108
HYDROSTATIC LUBRICATION
The temperature field plotted in Fig. 5.10 refers to a lubricant, with a n inlet temperature T=35'C, in the clearance of a bearing with the following characteristics: r2=0.0657 m, r1=0.0348m (r'=0.53),r0=0.006m, h=100 pm, &419 rads, pr=0.294 MPa; the data concerning the lubricant, at 2OoC, are: p=0.0645 Ns/m2, p=857 Kg/m3, c=1890 J/Kg"K, ~ = 0 . 1 4 W/m°K. 4 The maximum temperature, at r=r2 and z=0.54h, is T=5OoCwith an increase AT=15OC. In comparison with the case of isothermal flow, taking into account inertia effects, pressure in the clearance proves to be slightly lower. Moreover we have: W=2190 N, Q =13.8.10-6m3/s, Mf=2.22 Nm. So, compared to the case of isothermal flow, in which we should have W=2054 N, Q =18.3.10-6 m3/s, Mf=3.2 Nm, the load capacity proves to be 6.2% lower, the flow rate 24.5% greater, and the friction moment 30.6% lower. Moreover, in the case of isothermal flow, from Eqn 5.7, we should have AT=45OC.
34
35
37
39
43
41
45
47
49
50
T (OC)
Fig. 5.10 Temperature distribution in the clearance of a circular pad: rl= 0.0348 m, r2=0.0657 m, h=lO-4 m, -19 r d s , pr= 0.294 MPa.
As parameter Si,given by Eqn 5.50, increases, the deviation from the isothermal case becomes prominent. For example, for the pad considered above, let us now suppose f2~628.3rads, pr=98.1 KPa, and hence Si=2.196. The maximum temperature increase now proves to be AT=45"C a t r=r2 and z/h=0.8; along the upper pad surface we have AT=31°C and AT=25.5"C along the lower surface. The calculated pressure field is shown in Fig. 5.11, where a negative (relative) pressure (confirmed experimentally, ref. 5.15) indicates the possibility of cavitation phenomena. Moreover, we have W=642 N, Q =14.6.10-6 m3/s, Mf=2.25 Nm, and in comparison with the isothermal case, the load capacity proves to be 33% lower, the flow rate 68% greater and the friction moment 55% lower.
PAD COEFFlClENTS
109
Fig.5.1 1 Pressure distribution in two circular pads. To avoid the problems related to the relevant increase in temperature and to the onset of cavitation, a proper choice of pad geometry is recommended. Let us now consider a pad with r2= 0.0435 m, r1=0.0348 m (r'=0.8) and ro=0.006 m, while all the other characteristics remain as before; we now have Si=0.963. The
110
HYDROSTATIC LUBRICATION
maximum A T proves to be lower than 8°C;the pressure pattern (also shown in Fig. 5.11) is very close to the isothermal case, and so obviously occurs for the load capacity W=642 N, while the flow rate Q=17.7.10-6 m3/s is greater by 17.4%. In comparison with the previous pad, the load capacity is similar, but the flow rate is even greater. A possible countermeasure may be a reduction in film thickness. If we take h=50 pm, the maximum temperature increase proves to be AT=23'C, but the pressure pattern is still close to the isothermal case (Fig. 5.11) and we have W=618 N, Q=2.43.10-6 m3/s, Mf=0.821 Nm. So, due to the thinner clearance, the actual flow rate proves to be about one 7th of what i t was before.
We must point out i t is best to use pads with a small radius and a large recess, when Q is large. The convenience of large recesses will be demonstrated in section 6.4 and in chapter 11. All the above results agree closely with those obtained in ref. 5.11 and 5.12 in the simplifying hypothesis of adiabatic flow. Reference 5.11 suggests that the possibility of adopting the isothermal theory (ref. 5.6), instead of the adiabatic theory, depends on the "viscosity parameter"
where g is the acceleration of gravity. By way of example, for water-lubricated bear-
K'
Fig. 5.12 Correction parameters kw. k~ and k~ for thermal regime.
PAD COEFFICIENTS
111
(due to the low viscosity and high specific heat, ings, V takes on values of about compared to the mineral oils) and the isothermal theory is fairly valid. Although the iterative procedure mentioned above can be quite accurate, i t is still very complicated and does not lead to any direct expression of pressure, load capacity, flow rate, and friction moment as functions of the parameters of the pad. A simpler method is described in ref. 5.16,which is based on using the results contained in ref. 5.14,to obtain an approximate expression of viscosity a s a function of r, in the thermo-fluid-dynamic regime. The Navier-Stokes and the continuity equations are then solved to give the pressure field; finally we obtain the following:
(5.62)
(5.63)
(5.64) 5.16 as functions of
(5.65) and are plotted in Fig. 5.12. In Eqn 5.65,p l = p ( r l ) ,Q , and H f are the values that these parameters take on in the isothermal case. The method has been tested for a number of bearings, with k* in the 0.211.3 range, and good agreement was found with the numerical results referred to above (deviations were smaller than 5% for W and M ,and smaller than 10% for Q 1. It should be pointed out that, for k*<0.2, the isothermal theory may be applied, while k*=1.3is quite a high value.
112
HYDROSTATIC LUBRICATION
4
li
Fig. 5.13 Annular recess pad.
5.4
ANNULAR
5.4.1
RECESS PADS
Basic equations
Figure 5.13 shows an annular recess pad; the inner hole is, in most cases, crossed by a shaft: hence, in general, it is large. From Eqn 4.57, the pressure in the pad is easily expressed:
In rlrq p=pr P=Pr
113
PAD COEFFICIENTS
In rlr2
P=Pr
(l-w)
(rl I r I rz)
Integrating the pressure field on the whole bearing area, we obtain load capacity W. Bearing Eqn 5.1 in mind, we may express the effective pad area as follows: (5.66) where (5.67)
A*, is plotted in Fig. 5.14.a as a function of u'=(r4-r3)/(r4-rl), for certain values of r'=rllr4 and for the most common case r4-r3=r2-r1. The hydraulic resistances of the outer and inner clearances are (Eqn 4.59):
Since the flow rate is the s u m of Q3-4 and Q2-l, bearing Eqn 5.2 in mind, the total hydraulic resistance of the pad is: (5.68) where:
R* =
1 1
+- 1 In r4fr3 In r2/rl
(5.69)
R* is also plotted in Fig. 5.14.b. The pumping power lost in the pad (Eqn 5.4), written a s a function of the load, is still given by Eqn 5.24. Given the inner and outer radii of the pad, it is now possible to assess its "optimal" geometry, looking for the minimum of r*=l/(A*,'R*).If we state that r4-r3=r2-r1,we find that the optimal value of a' is very close to 113 (see Fig. 5.14.d). However, this is not a critical value, and a larger recess (perhaps a'=0.2) is often preferred in order to decrease supply pressure and friction power.
114
HYDROSTATIC LUBRICAT/ON
0.
R'
0
0.0 0.00
'
'
0
0.25
0.50 a'
0.50
a'
-c-
1.01
0.25
a'
Fig. 5.14 Coefficients of annular recess pad.
-d-
115
PAD COEFFlClENTS
It should be pointed out that to configure the pad with equal length of the lands is by far the most common practice, since it is simple and give very good results, but it is not, strictly speaking, the best choice from the point of view of pumping power. Indeed, looking for the absolute minimum of r* it is possible to find the values of ratios r3Jr4and rl lr2 that ensure the least loss of pumping power for any value of r'; the results contained in Table 5.1 show that the equal-length configuration practically gives the same results as the true optimal configuration. Other authors (ref. 1.8) prefer pads with equal radius ratios, i.e. r3/r4=r1/r2, in such a way that flow rates crossing the inner and outer clearances are equal; Table 5.1 shows that this practice also gives good results, especially a t the highest values of r'. The friction moment is, from Eqn 4.60, (5.70) where (5.71) which is plotted in Fig. 5.14.c. The friction power is, hence, (5.72) The friction coefficient, referred to the mean radius F=(r4+r1)/2of the pad, is therefore: (5.73)
- r' 0.4 0.5 0.6 0.7 0.8 0.9
General case rllr2 r3/r4 r* 0.680 0.758 0.823 0.877 0.924 0.965
0.791 22.1 0.827 35.8 0.863 65.7 0.898 146.8 0.933 468.5 0.967 3552
a'
0.337 0.335 0.334 0.334 0.333 0.333
r2-rI=r4-r3 rl/r2 r3/r4 0.664 0.749 0.818 0.875 0.923 0.964
r*
0.798 22.1 0.833 35.8 0.866 65.8 0.900 146.9 0.933 468.5 0.967 3552
rl11-2 0:746 0.798 0.845 0.889 0.928 0.966
r1/r2=r3/r4 r3/r4 r* 0.746 23.5 0.798 37.1 0.845 67.0 0.889 148.3 0.928 470.3 0.966 3555
116
HYDROSTATICLUBRICATION
Hydrostatic thrust bearings with annular recesses are frequently used, very often coupled to hydrostatic journal bearings. When the turning velocity is not high, pumping power is much greater than friction power; on the basis of the above results, it is easy to make certain elementary suggestions:
r' should be low. In practice, due to design constraints, we almost always find that r50.5. Values greater than r'=0.8 should be avoided;
a' should be in the 0.2+0.4 range;
D, i.e. the bearing area, should be as large as is allowed by the other design constraints; The film thickness h should be low; The lubricant viscosity p should be high.
EXAMPLE 5.5 A n annular recess bearing must sustain a load W=20 KN, rotating at n = 2 n radls (60 rpm); its outer diameter is D=O.l m, the inner one is 2rI=0.04 m (i.e. r'=0.4). For clearance h=50 pm is chosen, and for a' the near-optimal value a'=l/3. 0 The lubricant data at T=40OC are: p=O.l Nslmz, p=9ooKglm3, ~ 1 9 0 JlKgSC. From Eqn 5.66 we get A,=4.40.10-3 m2 and hence a recess pressure pr=4.55 MPa is needed to sustain the load. The hydraulic resistance of the pad (Eqn 5.68) is R=22O,lO9 Nslm5and the flow rate proves to be Q=20.7.10-6m31s. The pumping power is Hi=94 W. For friction we find: MfzO.0857 Nm, Hf=0.54 W, f=O.l22.lO-3, respectively. The temperature step of the lubricant, from the inlet to the outlet of the pad, is (Eqn 5.7) AT=2.7%, which is practically negligible. Comparing these results to those already obtained for a circular recess pad with the same diameter (Example 5.1), it may be noted that the friction of the annular recess pad is lower; slightly greater pressure is required, but the flow rate and power consumption are more than 4.5 times greater. EXAMPLE 5.6 Let us change the inner radius of the bearing in the former example to rI=0.03 m (i.e. r'=0.6), and the angular speed to R=4n radls (120 rpm). We obtain p,=5.97 MPa, Q=46.8.10-6m3/s, Hi=279 W, Mf=0.147 Nm, Hf=1.85 W, f=0.18.10-3,AT=3.5oC. In comparison with the former example, a small increase i n pressure may again be noted, while for flow rate and pumping power the increase is much larger. EXAMPLE 5.7 I f the inner radius is again increased to rl=0.04 m, r' becomes 0.8;let the angular speed now be Q=6n radls (180 rpm). We find: pr=10.6 MPa, Q=187.10-6m 3 / s , Hi=1989 W, Mf=0.146 Nm, Hf=2.76 W, f=0.16.10-3 and AT=6.2"C. Temperature in-
PAD COEFFlClENTS
117
crease may no longer be negligible: hence, it will be best to put a heat exchanger in the oil reservoir. When this bearing is compared to the one in Example 5.5, one may note that pressure is more than doubled, while a greater increase occurs for the flow rate and pumping power (the latter increases more than 10 times). It may be useful to increase the viscosity of the lubricant: if p is doubled, Q and Hi are halved; Mf and H f are doubled, too, but they still remain quite low. Another way to reduce Q is to use a lower value for a' (i.e. to reduce the width of the recess); on the contrary, a greater a' causes a decrease in pressure. It should be pointed out, however, that, for the highest values of r', a slight displacement of a' from its optimal value can lead to a notable increase in Hi (see Fig. 5.14.d).
5.4.2
Effects of errors in parallelism
The bearing pad in Fig. 5.15 is affected by a lack of parallelism of the surfaces of the bearing components. We may state:
ah;=%&* r4
Fig. 5.15 Misaligned annular recess pad.
(5.74)
118
HYDROSTATIC LUBRICATION
1.oo
0.95
CR
0.90 r'=0.4 0.6
'
0.85 0.0
0.8
1.o
0.5 Ah'
Fig. 5.16 Misaligned annular recess pad: correction factor C, versus parallelism error Ah' for two values of the land width ratio a'.
where Ah' is given by Eqn 5.31. Bearing the results of section 5.3.3 in mind, we find that, for a certain recess pressure p r and mean film thickness E, the only conspicuous effect of misalignment is an increase in flow rate; such an increase may be expressed as l/CR(Ah';r 3 / r 4 )for the external gap, and 1/cR(Ahi;r l / r z )for the internal one. Thus, while the effective area A, may be considered to be practically unaffected by a misalignment Ah'<1, the hydraulic resistance R may be expressed in the form of Eqn 5.34; the correction factor CR, calculated as above, is plotted in Fig. 5.16 as a function of Ah', for certain values of r' and a'.
5.4.3
Effects of pressure losses at the inlet
When the Reynolds number is greater than 100, the pressure losses at the inlet should be taken into account. Applying the results of section 5.3.4 to both the gaps in the annular recess pad, the whole hydraulic resistance of the pad should be corrected as in Eqn 5.36, where Cp is given approximately by Eqn 5.39 (or by Fig. 5.61, but the Reynolds number is (5.75) and r', r and A r must be substituted by the following:
PAD COEFFICIENTS
119
- rl + r2 + r3 + r4 r=
Ar =
4
r2 - rl + r4 - r3 2
EXAMPLE 5.8 A pad bearing with an annular recess has an outer diameter D=O.l m; the inner one is 2rI=0.08 m (r'=0.8). It has to sustain a load W=15 KN, with a recess pressure pr lower than 8 MPa. The angular speed is Q=6.28 ra d ls (60 rpm); the main characteristics of the lubricant are: p=0.03 Nslm2,p885 e l m 3 . Let us select a'=0.3; h=100 pm. We immediately find A,=1.98.10-3 m2 and, hence, pr=7.58 MPa, which is safely smaller than the maximum pressure allowed. I f the pressure losses at the inlet are disregarded, we also find R=1.91.109 Nslm5, and Q=3.97.1 0-3 m3 I s. From the above equations, we find Re=207, F'=0.935, r=0.045 m, Ar=3 mm. A correction factor Cp=1.28 can be obtained from Fig. 5.6. Since the flow rate is inversely proportional to the hydraulic resistance, a new value is found: Q=3.10.10-3 mats, which is still very high. On the other hand, the friction power is very small, compared to the pumping power (Hf=O.O4 W,His23.5 Kw). I f such a low friction is not a design constraint, it is clearly advisable to reduce the selected clearance and to increase lubricant viscosity, in order to greatly reduce flow rate and pumping power.
5.4.4
Turbulent flow
When the Reynolds number is high, the flow becomes turbulent. The results already obtained in section 5.3.5 can be extended to the annular recess pad. In particular, since in most cases both r l l r 2 and r31r4 are greater than 0.8, the influence on the effective bearing area can be disregarded. The hydraulic resistance of the pad, on the other hand, can now be written as: (5.76)
Correction coefficient C R is~still given by Eqn 5.48 (also plotted in Fig. 5.7.b)in which parameter zb is now defined as follows:
120
HYDROSTATIC LUBRICATION
The flow rate may prove to be much lower than expected on the basis of the laminar theory, a s in the case of the central recess pads, but the problems connected with turbulence also remain the same.
5.4.5
Effects of the inertia forces
When the turning velocity of the pad is high, the forces of inertia in the lubricant should be taken into account. The Reynolds equation (Eqn 5.49) can be solved on the land surfaces to give the relevant pressure field, while the recess pressure is hypothesized to be a constant (see ref. 5.17). Integrating the pressure field, we find the load capacity of the pad, while the flow rate is obtained by integrating qr, as obtained from Eqns 4.33, along the recess boundaries.
Fig. 5.17 Annular recess pad: correction factor CR,versus inertia parameter Si for certain values of ratio r' and of land width ratio a'.
As usual, Eqn 5.66 and Eqn 5.68 have to be substituted by the following equations: (5.77)
(5.78)
where
121
PAD COEFFICIENTS -a-
-b-
I.€
11
1.1
I?
1.2
1.: 2 1 -
SI'O
1.0
1.c
-P
P Pro
Pro
a8
0.8 ~
0.6
0.6
0.4
0.4
0.2
0.2 r; =0.95
0
0 0.6
0.8
1
0.6
0.8
r
-
-r
4'
r4
1
Fig. 5.18 Annular recess pad: pressure distribution for one value or ratio r'=r1/r4and certain values of inertia parameter S,.[Note that pm=(3/7r)pQ( l/h3)ln(r2/r,)].
(5.80)
122
HYDROSTATIC LUBRICATION
For the correction factor CAI, however, we always have 0.99
123
PAD COEFFICIENTS
5.4.6
Thermal effects
It has been shown for the central recess pads that, at very high turning speeds, temperature, and hence lubricant viscosity, vary considerably in the bearing clearances. In the annular recess pads, however, the ratio of the inner to the outer radius of each clearance is, generally speaking, higher, and this leads to lower increases in temperature. This is confirmed by the examples above.
5.5
5.5.1
TAPERED PADS Basic equations
Figure 5.19 contains a sketch of two tapered pads fitted with central recess (a) and annular recess (b).
* -a-
-b-
Fig. 5.19 Tapered pads: a- central recess; b- annular recess.
The Reynolds equation for a conical clearance was obtained in section 4.3.2; assuming steady state operation] and uniform clearance h, on all the land surface, Eqn 4.26 is reduced to:
(5.81)
124
HYDROSTATIC LUBRICATION
The term on the right-hand side accounts for the inertia forces due to angular velocity 0, in the simplest case, it is negligible, and hence Eqn 5.81 is reduced to Eqn 4.56, which does not depend on the aperture angle a; thus the pattern of the pressure field is exactly the same as for a plane clearance (a=d2) of identical radius ratio r'. Stating pr=l, and integrating p.sina on the recess and land surfaces, we obtain the effective area of the pad. Again it turns out to be the same as for the plane bearings, i.e. Eqn 5.20, where A: is given by Eqn 5.22, or Fig. 5.l.b, for the central recess pad, and by Eqn 5.67, or Fig. 5.14, for the annular recess pad. The flow rate a t radius r is obtained by integration of the radial velocity: (5.82)
Let us disregard velocity R for the time being. The hydraulic resistance of the tapered pads may be written as follows:
(5.83) where R* is given by Eqn 5.23 or Fig. 5.l.b, and by Eqn 5.69 or Fig. 5.14, respectively. It follows that the flow rate through a conical clearance will turn out to be proportional to sina. The moment of the friction forces for an axial-symmetric conical land of length dr is dMf = 2 x- CrQ hn
r3 sin3a
dr
which may be easily integrated to give the friction moment: (5.84)
and the friction power:
(5.85) HF is exactly the same as that already calculated for the plane bearings: i.e. Eqn 5.28 for the central recess pad, and Eqn 5.71 for the other. When a tapered pad is compared to a plane bearing of the same diameter, with equal values of the normal film thickness and the same radius ratios, it is easy to see that:
125
PAD COEFFICIENTS
the load capacity for any given recess pressure is the same (or, for an equal flow rate, is greater for the conical pads: actually, it is proportional t o llsina); the flow rate, a t any given load, is proportional to sinq i.e. it is lower for the conical pads. The same Consequently, the same occurs for the pumping power; on the other hand the axial stiffness, which proves to be inversely proportional to h=h,lsina, for any given load capacity, is lower (proportional to sina) for the conical pads; the friction power (proportional to llsina) is greater for the conical pads. Furthermore, tapered pads have a greater axial size than the plane pads and may prove to be more sensitive to assembling misalignments.
EXAMPLE 5.11 An annular recess tapered bearing has to sustain a load W=20 KN, rotating at Q=4a radls (120 rpm). The main geometrical parameters are: D=O.l m; L=0.02 m, a=45'(i.e. rt=l-(2Ltga)lD=0.6).The viscosity of the lubricant is p=O.l N s l m z at operating temperature. I f we select a'=113 and hn=50 pm (or h=70.7 pm), we easily obtain A50.427, A,=3.35.103 m2, and hence p,=5.97 MPa; moreover, R*=O.0835, R=180.109 Nslm5, and hence &=33.10-6 m 3 / s . The pumping power is thus Hi=197 W. The friction moment is easily calculated as Mf=0.208 Nm, while Hf=2.6 W. Compare these results with those in Example 5.6.
5.5.2
Effect of the inertia forces
As has been pointed out before, when Q is large, the inertia term should be retained in Eqn 5.81. This kind of problem is examined in ref. 5.17 for the annular recess pad, and in ref. 5.19 for the central recess pad. In the latter, thermal effects are also considered, as in ref. 5.13. If we solve Eqn 5.81, and integrate the pressure field, we finally find that the effective area A, is again given by Eqn 5.53 and Eqn 5.77 for the circular-recess and the annular-recess pads, respectively. In the same way, the hydraulic resistance will be: (5.86)
126
HYDROSTATIC LUBRICATION
where correction coefficient CRIis given by Eqn 5.58 or by Eqn 5.80 (or by the relevant plots).
5.5.3
Effect of misalignment
Two possible types of misalignment are shown in Fig. 5.20. In Fig. 5.20.a, the shaft is tilted (by a n angle y) with respect to the axis of the fixed member; this case is similar to the error in parallelism for the plane bearings, seen in sections 5.3.3 and 5.4.2. The coaxiality error (case b), on the contrary, is meaningless for the plane pads. Of course, all these errors may occur together and the problem will prove to be quite complicated. -b-
-a -
aw I
-+I
Fig. 5.20 Tapered pads: a- misaligned; b- non coaxial.
Since the radial symmetry is lost, and hence the film thickness h is no longer a constant, but a function of the coordinates, Eqn 5.81 should be substituted by the twodimensional Reynolds equation:
(5.87) obtained from Eqn 4.26 in the steady state case. Equation 5.87 should be solved, in general cases, by means of suitable numerical methods, as seen in the case of the error in parallelism of the plane bearing. Some rough approximation will, however, be enough in most cases. For instance, in the case of Fig. 5.20.b, one may assume that, for small eccentricities, the flow remains
127
PAD COEFFlClENTS
essentially radial. In this assumption, the effective area does not vary; the flow rate on a sector dfi of the conical surface is (see Fig. 4.5):
The clearance may be written in the following form:
[ " cos (19- @]
h, = C cosa 1-
where C&tga is the mean radial play. The flow rate is, hence: 21r
x x3
Q = J d Q = G z p r sin4a(l+t$) 0
Thus correction factor C R x for the hydraulic resistance can be evaluated as: (5.89)
Friction is also affected by a radial displacement x . Proceeding as above, we can assess a correction factor Cfx for M f and Hf, which may be written as:
1.21 1.1
-
1.o
0.9 -
0.8 * 0.7 -..
0.0
0.1
0.2
0.4
0.3
0.5
X/C Fig. 5.21 Non coaxial tapered pad: correction factors C R for ~ the hydraulic resistance and Cfx for friction versus radial displacement xlC.
HYDROSTATIC LUBRICATION
128 1
Cfr =
(5.90)
$ <
It is clear that a radial displacement, comparable to the design value of the film thickness, greatly affects the hydraulic resistance of the pad, and consequently the performance of the bearing. This point should be carefully considered when designing. For example, if a conical thrust bearing sustains a shaft which is also loaded in a radial direction, the stiffness of the radial bearings will interact with the axial load capacity. The case of Fig. 5.20.a (tilting error) is examined in ref. 5.20 by means of numerical integration (finite differences) of Eqn 5.87. The effect of a relative tilt 1y'=1yDl(2h,)10.6 on the hydraulic resistance (at low speed) is found to be always smaller than for a flat bearing (a=90"). In particular, it is negligible for a=60"+75", while a=45" is a n intermediate case. When the speed is high, the hydrodynamic effect, due to the uneven clearance, may in part counterbalance the effect of the tilting error, and a tilted bearing may even prove to perform better than a centered one!
SPHERICAL PADS
5.6
To overcome the problems connected with the tilting errors, a spherically-shaped pad (Fig. 5.22) could be used, instead of the flat or tapered types.
As usual, to calculate the performance of the pad, the relevant Reynolds equation must be solved to find the pressure in the clearances. Such a n equation has been obtained in section 4.3.3. For an axial-symmetric spherical clearance, Eqn 4.30 is reduced to:
The above equation is easily solved when E=O (concentric configuration), giving:
tan (~112
Pr
(5.91)
In for the central recess pad, with p=pr for p q l andp=O for p q 2 . A general solution may however be obtained (ref. 5.21, 5.22), although the relevant equations are quite cumbersome.
129
PAD COEFFICIENTS
-a-
-b-
Fig. 5.22 Spherical pads: a- central recess; b- annular recess.
The load capacity is calculated by integrating the pressure on the recess and land area. For E=O, we obtain: (5.92)
From the first of Eqns 4.28; the velocity of the fluid in the tangential direction is: u=-
*(@-h,@ DPaP
and hence the rate of flow, for E=O, is: (5.93)
Load and flow rate can be written in the usual form (i.e. Eqn 5.1 and Eqn 5.21, where the effective area is: A, = x4 0
2
sin2q2A*,
and A*,, calculated from Eqn 5.92, is plotted in Fig. 5.23.a.
(5.94)
HYDROSTATIC LUBRICATlON
130 2
A', Hi
0.8
0.8 0.6
1
0.4
0.4
t I"
60"
90"
0.0 30'
90"
60" (92
(92
-c-
-d0
0.05
.
1
5
-
7
1
0.00' 30"
'
'
.
'
60"
.
'
'
90" (92
Fig. 5.23 Spherical pads. Pad coefficients A*,, R* and H,? (in the case of E=O) versus included angle (p2 for certain values of ratio (p1/(p2:a, b- central recess pad; c, d- annular recess pad with (pa=O.25'((p2-(pl).
PAD COEFFlClENTS
131
The hydraulic resistance may be written as: (5.95) Again R* is plotted in Fig. 5.23.b. The friction moment is:
in which the shear stress
should be substituted. As usual, we may write the friction moment and the friction power in the following form:
Mfo=
2$ f2
0 4 sin4q2 H,F
(5.96)
where HT (which is also plotted in Fig. 5.23.a) is: (5.98) Unlike for the plane pads, A*, cannot be considered as a constant] but depends on E especially when cp2 is great and q1/cp2 is small. Also Eqn 5.12 and Eqn 5.13 cannot be considered valid for large included angle q2. For instance] Fig. 5.24 (ref. 5.22) shows how load capacity and flow rate are subjected to change with E for a pad with cp2=85". It is easily seen that for negative displacements the effective area may experience a severe reduction] while the increase of hydraulic resistance is smaller than the cubic trend predicted on the basis of Eqn 5.12. Depending on the supply system, even negative stiffness may occur, and the bearing may prove to be unstable (ref. 5.23). However, this kind of problem may be prevented by using q1/q2 ratios in the order of 0.5 and values of q2S75".
A peculiar type of central-recess spherical pad is the "fitted' type, characterized by having a null clearance in the concentric configuration. It is obvious that most of the foregoing equations now become meaningless, and it is necessary to write load
132
HYDROSTATIC LUBRICATION
-a-
-
b-
1.0
0.8
w
20 0.6 2
n
a
5
0.4
0.2
0
0.2
0.4
0.6
to
a8
? 1%
0
0.2
0.4
0.6
0.8
4 ’%
Fig. 5.24 Spherical pad with cenh-a1 recess: a- load factor W.4/(zDzprsin&p) and b- flow factor Q-pl[(l+~)%?p,]versus ratio q1/(p2for a number of values of eccentricity E (ref. 5.22).
capacity and flow rate a s functions of the displacement h=h,(cp=O). We obtain (ref. 5.24):
tan2q2 - tan2cpl
x
W = 4 D2 pr
tan 472
tan2472 - tan2ql + 2 In tan 471
& = -xh3 - p
1
3 P
tan472 tan2472 - tan2cpl + 2 In tan471
Torque is found to be (ref. 5.21)
M r=
gf RD4
-
( ~ 0 ~ 2 4 7cos2ql ~ -2
ln-
Spherical bearings have also been studied from the point of view of the effects of fluid inertia (ref. 5.21) and of change in viscosity due to adiabatic flow (ref. 5.25). It
PAD CO€FF/CENTS
133
has been shown that high values of the inertia parameter Si defined in Eqn 5.50 (perhaps Sp0.25) are detrimental for divergent film shapes, i.e. for negative eccentricities: load capacity may be notably reduced, and even cavitation is likely to occur. On the contrary, for converging film shapes (e.g. for the fitted type of bearing) load capacity is improved and, due to the large pressures developed near the edge, bearings with a wide included angle (p2 also feature a notable radial load capacity. The annular recess pads (Fig. 5.22.b) feature an effective area W l p , that does not vary much with eccentricity, especially when cpz is small. Proceeding as above, we can calculate the relevant values of parameters A:, R* and HF to be inserted in the equations from Eqn 5.94 to Eqn 5.97. They are plotted in Fig. 5.23.c and Fig. 5.23.d. For A*,it may be taken approximately (see also ref. 5.26): (5.99)
at least when (pa is smaller than 0.25((p2-cpl).
5.7
RECTANGULAR PADS
For a rectangular pad, such as in Fig. 5.25, the two-dimensional Reynolds equation Eqn 4.15 has to be solved, which, if h=const or U=O, is simply reduced to Eqn 4.31 (Laplace equation). The boundary conditions are, as usual, p=pr at the inner boundary of the clearance, and p=O at the outer one. Once the pressure on the land area has been evaluated, the load capacity is obtained by integration. The rate of flow, on the other hand, is given by the following: (5.100)
where S is the inner boundary (or any closed contour including it), and n is the outer normal direction of S (see also section 4.5.2). In spite of its simplicity, Eqn 4.31 may be solved, in the general case, only by numerical methods. A first rough approximation can be very easily obtained by partitioning the land area as in Fig. 5.26 (ref. 1.8).The pressure is assumed to vary with a linear trend (see section 4.7.1) along the z. coordinate, for parts [ll and 121, and along z for parts 131 and [41. In the corners, a logarithmic trend is assumed (see section 4.7.4); if we have ri=O, the corners may be neglected altogether. Calculations for load and flow rate are now straightforward: on the analogy with the other pad types, the effective area A, and the hydraulic resistance R may be written as:
134
HYDROSTATIC LUBRICATION
4
J
Fig. 5.25 Rectangular pad.
A, = L B A:
(5.101)
R=$R*
(5.102)
with
- 4 ri 6' (a' + rib' B') + Za'
a'lB'+ 2 rib' a' In l+rib B )
(5.103)
(
R* =
6
(5.104)
n:
(
In l+- rl E ' B ) where a' - a
-L;
b
b'=B;
B ' =B -
and
ri q'=Z
PAD COEFFlClENTS
135
Fig. 5.26 The assumed shape of a rectangular pad.
A more refined approach is explained in ref. 5.27, in which a n approximate solution of the Laplace equation is searched for as a linear combination p=Xcjqj of n harmonic functions qJx, z ) ; coefficients cj are determined by imposing that the mean square error in certain points (xk, Zk) of the boundaries (on which pressure is known) is the least possible. In other words, the following system of n equations must be solved
If a certain skill is used in selecting the harmonic functions, only a few terms are needed (perhaps n=4) to obtain a good approximation of the pressure field: the relevant error can be easily checked, since it can be shown that it is maximum on the boundary. Once a closed form solution has been obtained for p , load capacity is calculated by integration, while flow rate is simply proportional to coefficient c1 (this is another consequence of the appropriate selection of functions 9). A t present, the solution of Eqn 4.31 is based on numerical methods, such as finite-difference or finite-element methods (for instance see ref. 5.28, 5.29, 5.30): in particular, the charts in Fig. 5.27, 5.28 and 5.29 (taken from ref. 5.31 and ref. 5.32) are calculated by means of a finite-difference method.
The pumping power dissipated in the bearing is still given by Eqn 5.4. As for the circular pad, it is useful to write Hi as a function of the load capacity: (5.105)
r* also is plotted in Fig. 5.29; it appears that an “optimal”value exists for a‘,in connection with the minimum of the r* curves. Concerning the effect of the inner
136
HYDROSTATIC LUBRICATlON
-b-
-a0.
0.
0.3'
0.0
0.2
a'
0.4
.
'
0.0
'
' 0.1
.
.
'
0.2
a'
Fig. 5.27 Rectangular pads: pad coefficient keversus land width a'.
3
F]
.
R'
2-
1.o
1
0.5 0.5
0.5 0.25
0
-b-
1.5
0.0
0.1
a'
0.2
Fig. 5.28 Rectangular pads: pad coefficient R* versus land width a'. fillet radius ri, although this yields a lower load capacity, it appears to be effective from the point of view of power. On the contrary, the outer radius r,, which is not considered in the charts above, has negative effects on both A, and H i , and should be avoided (however, when r,+r,lBs0.25, it has no practical influence on the performance of the pad). The friction force on the pad, due to a sliding velocity U,is (from Eqn 4.35):
137
PAD COEFFICIENTS
-b8
r
0.25
r 6-
0.5
54 .J
0.o
0.1
a'
0.2
Fig. 5.29 Rectangular pads: pad coefficient r*versus land width a'.
(5.106)
and, consequently, the friction power is:
The effective friction area Af coincides, as a rough approximation, with land area Al; as a matter of fact, the contribution of recesses to friction is often small, since recess depth h, is much greater than h. However, this is not always true; the flow recirculation in the recess can make recess friction comparable to land friction (and even greater, for turbulent recirculation). Moreover, h, should not be too high for reasons connected with the dynamic behavior (see chapter 10) and, when the speed is high, the lands should be narrow and, hence, the recess area greater than the land area. A better approximation for the effective friction area may be written in the following form:
A t = Al -I-f, A, = =L€?{l-(l-f,)(l-2~')
+ B ' ( 4 - d [ ( l - f r ) ( 1 - 2 8a' ;) 2 ri2-ri2]]
Coefficient f, can be calculated as indicated in section 4.5.3.
(5.108)
138
HYDROSTATIC LUBRICATlON
EXAMPLE 5.12 A rectangular pad has to be designed, able to carry a 45 KN load. The width of the pad should be B=O.I m; the recess pressure should be lower than 4 MPa and the friction coefficient lower than 10-4 at a speed U=0.2 m l s . The viscosity of the lubricant to be used is p=0.03 Nslm2. Due to the constraint on pr, the effective area (Eqn 5.1) must be greater than 11.3.10-3 m2. Figure 5.27 shows that, for values of a’ near to the “optimal”,A: takes on values greater than 0.55; consequently, we may take LB=0.02 m2, i.e. L=0.2 m and B’=0.5. If we select a’=0.125 (i.e. b’=0.5) and ri=0.5, in Fig. 5.27 and Fig. 5.28 we read Az=0.6, R*=0.77; hence A,=0.012 m2, pr=3.75 MPa. The land area is easily calculated as A1=0.013 m2; from Eqn 5.106 it follows that, for f d O - 4 , it must be h>17.3 pm: a very small value. A more suitable selection may be h=30 pm; now Eqn 5.104 gives R=0.86.1012 Nslm5 and hence &=4.4.10-6mats; the pumping power dissipated in the bearing clearaAce is Hi=16.5 W. A more accurate calculation of the effective friction area (Eqn 5,108) gives, for hr=0.5 mm, Af=0.015 ma; the friction power is, hence, Hf=O.6 W (Eqn 5.107) and the friction coefficient is f=0.67.lfY4. It should be stressed that in this case a relatively small recess has been selected since the speed was quite low: indeed, the friction power turns out to be much smaller than the pumping power. For higher speeds, a larger recess would have been a more suitable selection. This kind of problem, extensively dealt with in chapter 11, will also be considered in section 6.4.
5.8
CYLINDRICAL PADS
Cylindrical pads may be used to support a shaft in a radial direction. They may be thought of as extensions of any flat pad shape (e.g. the pad with a circular projected area in Fig. 5.30.a), although the rectangular shape is by far the most common (Fig. 5.30.b). It must be pointed out that a single-recess cylindrical pad is practically unable to sustain side loads; this greatly reduces the usefulness of such pads, unless they are a part of a multipad bearing system (see section 7.5). The study of these pads is made, as usual, by finding a solution for the relevant Reynolds equation, i.e. of Eqn 4.18.In that equation, the right-hand side accounts for the effects of the turning velocity of the journal and for the squeeze effect under dynamic loading. Film thickness h depends on the displacement of the journal center:
h = C [I- E C O S ( ~ - ~ ) ]
(5.109)
in which C is the film thickness in the concentric configuration E=O (see also Fig. 4.4).Once the pressure field on the pad lands is known, both the load capacity and the rate of flow are easily evaluated, as in the preceding sections.
139
PAD COEFFICIENTS
-b-
-a-
IW
IQ
lQ
.i Fig. 5.30 Cylindrical pads: a- circular recess; b- rectangular recess.
Even in the simplest case (E=O, &=O, h=C), in which the right-hand side of Eqn 4.18 vanishes, a numerical solution is needed. In Fig. 5.31, 5.32 and 5.33,the pad coefficients are plotted, for certain rectangular shapes, obtained by means of a finite element approximation. Other data are available in the literature: e.g. in ref. 5.33, where they are obtained by means of the electric analogue technique (ref. 5.34).
As in the case of the flat pads, one may write the effective bearing area and the hydraulic resistance of the pad (for E=O) in a form: a A, = L D sinTA*,
(5.110)
R~=&R*
(5.111)
Certain notable differences should, however, be stressed; first, the reference config uration is no longer arbitrary: a natural reference point exists, in which the centers of curvature of the two pad surfaces coincide, and h=ho=C.Coefficient A*, is not independent from the journal displacement (however, this dependence is in most cases negligible). The hydraulic resistance is no longer exactly related to the dis-
140
HYDROSTATIC LUBRICATION
placement by Eqn 5.12, which, however, remains approximately valid for small included angles a and small values of displacement E = e / h o in the normal direction rp=O.
As for other types of pad, it is convenient to write the pumping power Hi=p,Q as a function of the load, in order to try to obtain a n optimization of the pad design:
-a-
1.o
A',' o.8#
0.6
\ 60"
0.4
-b-
75"
a'
0.4' 0.0
.
'
'
' 0.2
.
.
a'
'
0.4
Fig. 5.31 Cylindrical pads. Pad coefficient Heversus land width a'.
-a-
R'
0.0
0.1
0.2
a'
0.3
0.0
Fig. 5.32 Cylindrical pads. Pad coefficient R* versus land width a'.
0.2
a'
0.4
141
PAD COEFFICIENTS
In particular, given L, D, and a,the relevant "optimal" value of a' may be readily evaluated by means of the plots of T*=lfR*A*, in Fig. 5.33. The friction moment, in the concentric configuration E=O,is: (5.112)
The friction power is, consequently, (5.113)
The effective area Af may be calculated as for rectangular pads; namely, as a rough approximation, Af coincides with the developed land area; if instead the recess friction is also taken into account, Afmay be written as: (5.114)
The friction factor f , is evaluated as in section 4.7.3;however, in most cases one may simply take f,=4C/h,.
-a-
-b-
6-
r* 4
u=90"
t
2
2'
0.0
0.1
0.2
a'
0.3
0.0
0.2
a'
0.4
Fig. 5.33 Cylindrical pads. Pad coefficient r*versus land width a'. 5.9
HYDROSTATIC LIFTS
A particular application (ref. 1.5) of externally pressurized lubrication consists in relieving large hydrodynamic bearings in certain critical operations, such as starting or stopping, in which, due to insufficient velocity, the lubricant film would be ruptured, leading to high friction and fast wear. In certain applications, in
142
HYDROSTATIC LUBRICATlON
which the torque available is not much greater than the normal regime torque, starting can even prove to be impossible without the aid of a hydrostatic lift. This kind of bearings (Fig. 5.34) are often referred to a s "hybrid" journal bearings; it seems, however, more appropriate to define them as hydrodynamic bearings with a hydrostatic lift, since external pressurization is generally confined to low speed running, while it is removed during normal running.
)Wa-
I
IQ
IQ
lw
-b-
Li II Fig. 5.34 Hydrostatic lift: a- axial recess; b- circumferential recess.
Behavior of such a "hybrid" bearing (studied for instance in ref. 5.35-5.37) is beyond the scope of a work on hydrostatic lubrication; we shall therefore present results relevant to the hydrostatic lift alone. Due to the large included angle (we may assume a=180°) and to the smallness of the recess, Eqn 5.12 is clearly far from representing the actual increase of hydraulic resistance with eccentricity, and even the effective area can no longer be considered as a constant. In Table 5.2, values are reported of the nondimensional load capacity for eccentricities from E=O &=0.8,assuming constant-flow direct supply, obtained by means of a finite element program (ref. 5.38). Several sizes of recesses have been
143
PAD COEFFICIENTS
considered, developed in the axial direction (small a,,Fig. 5.34.a), as well as in the circumferential direction (small I', Fig. 5.34.b). Table 5.3 also contains an evaluation of the ratio of recess pressure p , to mean pressure WILD. It should be noted, however, that the actual recess pressure can be notably higher; in particular in the case of axial recess (Fig. 5.30.b), a t starting the recess may be shut off as a result of elastic deformation and hence the pressure can be very high until the journal rises: values up to five times the normal running pressure are reported (ref. 1.5). It is good practice to use a pressure relief valve in order to protect the pump, limiting the maximum pressure. Concerning the dimensions of the hydrostatic pocket, it is generally suggested that its area LDl'sina, is 2.54% of the projected area LD. Indeed, from Table 5.3 it is clear that larger recesses have no practical advantage from the point of view of ratio W / Q , while the advantage in terms of pocket pressure is small; furthermore, the interference of recess with the hydrodynamic pressure pattern should be borne in mind.
T A B L E 5.2 Iydro: tic lift:
-
on-dimensional load capacity. W DLpQ/C3
a 1' VD L 1
0.1
1
0.1 0.1
E
=O
E = 0.3
E
= 0.6
E
= 0.8
E
=0.9
~~
2.19
7.88 6.67
29.6
48O
1.03 0.99
2.36
20.3
44.6
72"
0.93
1.95
5.17
2.31 2.14
7.73 6.50
12.3 29.1
20.6 87.3
19.8
43.5
1.73 1.60 1.44 1.68
35.2 24.1 14.3 34.3 23.3
130.8 64.7 28.4 127.6 62.4
24O
88.2
1 1
0.2
24"
1.01
1
0.2
48"
0.5
0.1 0.1 0.1 0.2 0.2
48" 72" 24"
48"
0.96 0.66 0.64 0.61 0.65 0.62
1.56
7.18 6.07 4.74 6.98 5.90
6" 12O
0.88 0.87
2.08 2.05
7.34 7.22
31.0 30.0
115.5
6' 12"
0.65 0.61
1.57
5.82
1.54
5.67
26.1 25.1
102.3 94.3
0.5 0.5 0.5
0.5 1 1 1
0.5 0.5 0.75 0.75
24"
1 --
106.6
144
HYDROSTATIC LUBRICATION
TABLE 5.3 [ydrol itic lift: o-f
non-dimensional recess pressure. PI
W I D L 1' UD a, 1 1 1
1 1 0.5 0.5 0.5 0.5 0.5 1 1 1 I
0.1 0.1 0.1 0.2 0.2 0.1 0.1 0.1 0.2 0.2 0.5 0.5 0.75 0.75
24" 48" 720 24"
48" 24O
48" 720 Z"
48" 6" 12" 6" 12"
- --
E=O
3.38 2.73 2.36 2.97 2.42 3.97 2.95 2.41 3.55 2.65 2.76 2.56 2.40 2.23
= 0.8
E = 0.3
E = 0.6
E
3.75 2.95 2.49 3.26 2.58 4.15 3.06 2.47 3.72 2.74 3.05 2.81 2.63 2.42
4.53 3.38 2.73 3.85 2.90 4.55 3.25 2.58 4.05 2.90 3.66 3.31 3.12 2.82
5.85 4.02 3.06 4.81 3.34 5.21 3.52 2.7 1 4.55 3.11 4.72 4.13 3.92 3.43
E
= 0.9
7.35 4.67 3.38 5.85 3.77 5.88 3.80 2.82 5.08 3.30 5.99 5.03 4.83 4.05
EXAMPLE 5.13 The following are the data of a hydrodynamic journal bearing: D=0.5 m, L=0.4 m, C=0.3 mm, W=300KN. The lubricant is a SAE 30 oil (p=O.l Nslm2 at operating temperature). Let us try to design a suitable hydrostatic lift. First, one may establish the dimensions of the recess. For instance, we may select 1=60 m m (i.e. l'=0.15) and a circumferential width b=100 m m (ar=23?. The flow rate and pressure of the pump can be calculated once the working eccentricity has been selected. If we choose &=0.8(corresponding to a minimum film thickness h=60 pm), by interpolating data i n the relevant column of Table 5.2, the flow rate m31s. From Table 5.3 we may obtain a turns out to be Q=WC31(32~DLp)=0.013~103 recess pressure pr=Wl(0.19.LD)=7.9 MPa; as noted above, the lifting pressure will be much greater, and the value above should be considered as only a rough indication.
5.10
SCREW AND NUT ASSEMBLY
This particular kind of bearings (ref. 5.39),shown in Fig. 5.35, is dealt with in the usual way, i.e. by writing down the Reynolds equation in a suitable coordinate system, and solving it for the pressure field. Then, to find the load capacity and the rate of flow is straightforward: e.g. see ref. 5.40 and ref. 5.41.
145
PAD COEFFlClENTS
From the point of view of load capacity, every turn of a screw flank may be regarded, with a very good approximation, a s a tapered annular-recess pad (section 5.51, i.e. the effective bearing area is: x
A, = n 4 0 2 A*,
(5.115)
Fig. 5.35 Hydrostatic screw and nut.
where n is the number of active turns and A*,is given by Eqn 5.67, or Fig. 5.14. In evaluating flow rate, on the other hand, the effect of the helix angle, which actually increases the length of the recess boundaries in relation to the analogous tapered pad, should be taken into account as follows: (5.116) where
is the mean value of the helix angle of the flank
2. = atan(p*/2xr)
and R* is given by Eqn 5.69, or Fig. 5.14. Note that the actual film thickness is h,=h cose c o d , which, strictly speaking, varies with r; however, Eqn 5.116 retains a very good approximation for all the usual values of ratio rl/r4 and of the screwpitch (i.e. r1/r4<0.6,AclO"). Another approximation made in obtaining Eqn 5.116 is that no lubricant leaks out through the clearance a t the ends of the recess: this amount is generally negligible (ref. 5.421, except in certain special cases, e.g. partial-arc hydrostatic nuts. In addition to the misalignments already mentioned in the case of tapered pads, an error of pitch between screw and nut may considerably affect the performance of
146
HYDROSTATIC LUBRlCATlON
the assembly (ref. 5.43).Film thickness is no longer a constant: if 6 is the mean film thickness in the axial direction, the hydraulic resistance becomes:
(5.117)
where Sp is the pitch error, i.e. the pitch difference between screw and nut, which is assumed to be a constant. The friction power is:
(5.118) where HHf*may be taken from Fig. 5.14,or obtained from Eqn 5.71.
REFERENCES
Bassani R.; Cuscinetti d i Spinta a Facce Piane a Lubrificazione Forzata Rotanti a Piccola Velocita; Ingegneria Meccanica, 16,12(1966),45-57. Bassani R.; Conseguenze dell%rrore di Parallelismo tra le Facce Piane dei 52 Cuscinetti di Spinta a Lubrificazione Forzata (Part 1 and 2); Ingegneria Meccanica, 17,8(1968),17-28; 17,9(19681,21-28. 53 Safar Z. S., Mote C. D.; Analysis of Hydrostatic Bearings under Nonaxisymmetric Operations; Wear, 61 (19801,9-20. Bassani R.; Sulla Portata Effettiva di Lubrificante nei Cuscinetti Zdrostatici di 5.4 Spinta; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno Acc. 1971-72, N. 35;39 pp. Bassani R.; Perdite Totali di Pressione e Valori Effettivi della Portata di Lu5.5 brificante e Della Capacitb di Carico nei Cuscinetti Zdrostatici d i Spinta; Fluid - Apparecchiature Idrauliche e Pneumatiche, 113 (19731,43-49. 5.6 Dowson D.; Inertia Effects in Hydrostatic Thrust Bearings; ASME Trans., J. of Basic Engineering, 83 (19611,227-234. 5.7 So0 L. S.;Laminar Flow over an Enclosed Rotating Disk; ASME Trans., 80 (1958))287-296. 5.8 Shen F. A.; Optimum Stiffness of Externally Pressurized Thrust Bearings in Turbulent Regime; ASME Trans., J. of Lubrication Technology, 92 (1970))457465. 5.9 Bassani R.; Le Equazioni del Regime Termofhidodinamico nei Cuscinetti Idrostatici di Spinta; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno ACC.1968-69,N. 7;50 pp. 5.10 Bassani R.; The Thermodynamic Flow Equations for Hydrostatic Thrust Bearings; Meccanica, 5 (1970))262-269.
5.1
PAD COEFFICIENTS
147
6.11 Ting L. L., Mayer J. E.; The Effect of Temperature and Inertia on Hydrostatic Thrust Bearings Performance; ASME Trans., J. of Lubrication Technology, 93(19711,307-312. 6.12 Kapur V. K., Verma K.; The Simultaneous Effects of Inertia and Temperature on the Performance of a Hydrostatic Thrust Bearing; Wear, 54 (19791, 113-122. 6.13 Kennedy J. S., Sinha P., Radkiewicz Cz. M.; Thermal Effects i n Externally Pressurized Conical Bearings With Variable Viscosity; ASME Trans., J. of Tribology, 110 (1988),201-211. 6.14 Bassani R.; I Cuscinetti Idrostatici di Spinta in Regime Termofluidodinamico; 1st AIMETA Congr., Udine, 1971; p. 25-68. 6.16 Bassani R.; Ricerca Sperimentale sul Regime Termofluidodinamico nei Cuscinetti Idrostatici di Spinta; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno Acc. 1968-69, N. 9; 40 pp. 6.16 Bassani R.; Proporzionamento dei Cuscinetti Idrostatici di Spinta Rotanti a Velocitd Anche Elevate; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno ACC.1968-69, N. 12; 56 pp. 6.17 Prabhu T. J., Ganesan N.; Characteristics of Conical Hydrostatic Thrust Bearings under Rotation; Wear, 73 (1981), 95-122. 6.18 Bassani R.; Cuscinetti Zdrostatici di Spinta a Recess0 Anulare; Atti Dip. Costr. Mecc. Nucl., Univ. di Pisa, DCMN 003(87), 1987; 37 pp. 6.19 Salem E., Khalil F.; Thermal and Inertia Effects i n Externally Pressurized Conical Oil Bearings; Wear, 66 (1979), 251-264. 6.20 Prabhu T. J., Ganesan N.; Non Parallel Operation of Conical Hydrostatic Thrust Bearings; Wear, 86 (1983),29-41. 6.21 Dowson D., Taylor C. M.; Fluid Inertia Effects in Spherical Hydrostatic Thrust Bearings; ASLE Trans., 10 (1967), 316-324. 5 s Ragab H.; Pe4ormance q f Spherical Thrust Bearings; Wear, 29 (1974)) 11-20. 6.23 O'Donoghue J. P., Lewis G. K.; Single Recess Spherical Hydrostatic Bearings; Tribology, 3 (19701,232-234. 624 Sasaki T., Mori H., Hirai A.; Theoretical Study of Hydrostatic Thrust Bearings; Bull. JSME, 2,5(19591, 75-79. 5.26 Salem E., Khalil F.; Variable-Viscosity Effects in Externally Pressurized Spherical Oil Bearings; Wear, 50 (19781,221-235. 526 O'Donoghue J. P.; Design of Spherical Hydrostatic Bearings; Mach. Prod. Engineering, Oct. 21, 1970; p. 660-665. 527 Massa E.; Sulla Determinazione della Forza di Sostentamento e della Portata nella Lubrificazione Idrostatica; Apparecchiature Idrauliche e Pneumatiche, 3,16 (1963),31-39. 6.28 Castelli V., Shapiro W . ; Improved method for Numerical Solutions of the General Incompressible Fluid Film Lubrication Problem; ASME Trans., J. of Lubrication Technology, 89 (19671,211-218. 5.29 Reddi M. M.; Finite Element Solution of the Incompressible Lubrication Problem; ASME Trans, J . of Lubrication Technology, 91 (19691,524-533. 6.30 Szeri A. Z.; Hydrostatic Bearing Pads: a Matrix Iterative Solution; ASLE Trans., 19 (1975),72-78. 6.31 Bassani R.; Rectangular Hydrostatic Bearings; Ann. CIRP, 19 (1971), 53-59.
148
HYDROSTATIC LUBRICATION
532
Bassani R., Culla C.; Coefficienti Caratteristici di Pattini Zdrostatici di Forme Diverse; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno Acc. 197273, N. 45; 29 pp. Rippel H. C.; Hydrostatic Bearings. Part 8: Cylindrical-pad Performance; Machine Design, Nov. 7, 1963; pp. 189-194. Loeb A. M.; Determination of the Characteristics of Hydrostatic Bearings Through the Use of the Electric Analog Field Plotter; ASLE Trans., 19 (19751, 72-78. Heller S., Shapiro W.; A Numerical Solution for the Incompressible Hybrid Journal Bearing with Cavitation; ASME Trans., J . of Lubrication TechnolO ~ Y 91 , (19671, 508-515. So H., Chen C. R.; Characteristics of a Hybrid Journal Bearing with one recess. Part 1: Dynamic Considerations; Tribology Int., 18 (19851, 331-339. S o H., Chang T. S.; Characteristics of a Hybrid Journal Bearing with one recess. Part 2: Thermal Analysis; Tribology Int., 19 (19861, 11-18. Straccia P. F.; Dimensionamento dei Cuscinetti Zdrostatici Conici; Doct. Thesis, 1985; 263 pp. Rumbarger J. H.;Wertwijn G.; Hydrostatic Lead Screws; Machine Design, April 11, 1968; pp. 218-224. Lombard J., Moisan A.; Caractkristiques Statiques et Dynamiques d'un Syst&meVis-Ecrou Hydrostatique; Ann. CIRP, 18 (19701, 521-525. El Sayed H. R., Khataan H. A.; The Exact Performance of Externally Pressurized Power Screws; Wear, 30 (19741,237-247. Bassani R., Piccigallo B.; Perdite per Trafilamento dalle Tenute di VitiMadreviti Zdrostatiche; Ingegneria, 1981; pp. 213-224. Bassani R., Piccigallo B.; Effetti delle Tolleranze d i Fabbricazione sulle Prestazioni di Coppie Zdrostatiche Autoregolate; I1 Progettista Industriale, 1,7 (19811, 62-72.
5.33
5.34
5%
5.36
537 5.38 539
5.40 5.41
SA2 5.43
Chapter 6
SINGLE PAD BEARINGS
6.1
INTRODUCTION
Pad coefficients, obtained in chapter 5 for several types of hydrostatic pad bearings, are not enough to completely define the static behaviour of a bearing, since it also depends on the behaviour of the supply system. In the simplest case, the pad is directly fed by a pump at a constant flow rate Q and supply pressure practically coincides with recess pressure (that is ps=pr); the performance of the bearing is then easily assessed by means of equations from Eqn 5.1 to Eqn 5.8. On the other hand, in the case of a compensated supply, pr is smaller than p s and a further relationship between these pressures is needed, which depends on the characteristics of the compensating restrictor. In this chapter the steady-state performance of thrust bearings is studied, when these are directly fed by a pump as well as by means of several types of compensating devices. Finally, a number of remarks on the optimum design of these bearings are made and design procedures are proposed.
6.2 6.2.1
DIRECT SUPPLY Bearing performance
When a pad bearing is directly fed by a volumetric pump, flow rate Q may be said to be constant; moreover, if the hydraulic resistance of the lubricant supply ducts is low compared to the hydraulic resistance R of the clearances of the pad, the recess pressure is nearly equal to the pressure a t the pump: pr=ps. The equations examined in section 5.2 can now be rearranged as follows:
150
HYDROSTATICLUBRICATlON
(6.1)
w(t”
H p = p s Q = p,. Q = Q2 R =A; R~R’ H f = Hfo H i
(6.3) (5.11 rep.)
In the above equations, the subscript “0“ refers to the “design configuration“ in which the film thickness is h=ho for all the land surfaces. R’ and H i are non-dimensional functions of the non-dimensional displacement and are given by Eqn 5.12 and Eqn 5.13. These equations are, strictly speaking, only valid for the plane pads, in which the lands bounding the lubricant film always remain parallel to each other; however, they c a n also be used as a n approximation for the other pads when small eccentricities are involved. In the rest of this chapter we shall assume that Eqn 5.12 holds good. Combining the foregoing equations, it is possible to study the behaviour of the clearance and of the power consumption when the load varies in relation to the reference value Wo=A,QRo. Figure 6.1 shows a plot on non-dimensional film thickness, pumping power and friction power as functions of the non-dimensional load. The bearing stiffness is easily obtained from Eqn 5.9. Since Q and A , do not depend on the clearance, while R is proportional to h-3, it follows that: (6.5)
The stiffness KO in the reference configuration is, of course, (6.6)
and, hence, (6.7)
151
SINGLE PAD BEARINGS
0
1
W
2
3
w,
Fig. 6.1 Direct supply. Film thickness, pumping power, friction power and stiffness versus load.
As has already been observed above, Eqn 6.5 (and consequently Eqn 6.6 and Eqn 6.7) is only valid as an approximation for the pads whose film thickness is not uniform (e.g. cylindrical and spherical pads).
6.2.2
Temperature and viscosity
The temperature step of the lubricant (assuming adiabatic flow) is given by Eqn 5.7; in the reference configuration we have:
When the load increases, the temperature step also increases: AT Ht -=-=
ATo Hto
R'+nHj-
l+n
(6.9)
In Fig. 6.2 it is plotted as a fhction of the load. Careful consideration of AT is of great importance when a constant-flow supply system is selected, since a change in lubricant temperature involves a variation in its viscosity, which, in turn, directly affects the bearing performance. Let us consider Eqn 6.2, which gives the load capacity: the hydraulic resistance of the pad (evaluated in chapter 5 for several types of pad) proves to be proportional to viscosity
152
HYDROSTATIC L UBRlCATlON
consequently, if the lubricant, say, warms up, Ro must be substituted in Eqn 6.2 by the lower value ROpIpO and a higher value of R’ is needed to sustain the same load. In other words, this will result in a smaller clearance. The influence of the viscosity on the clearance-load relationship is shown in Fig. 6.3.a. ,LL;
3
2
AT AT0 1
-0
n
1
2
3
Fig. 6.2 Direct supply. Temperature step versus load, for certain values of reference power ratio.
-a-
-b-
2
6
K KO
-tl h0
4 1
2
0
Fig. 6.3 Direct supply. Clearance (a) and stiffness (b) versus load for certain values of viscosity.
SINGLE PAD BEARINGS
153
On the other hand, it is easy to verify that p has no effect on pumping power (indeed, Q is fixed and the recess pressure depends on the load alone, since A, is not affected by p); friction power, for any given load, decreases if the temperature goes up:
(Hf is proportional to p, for a certain clearance). Because of the reduced film thickness, stiffness is increased if viscosity is reduced; on the contrary, a cooler lubricant (e.g. during the starting phase) may cause a considerable reduction in stiffness:
The effect of p on bearing stiffness is shown in Fig. 6.3.b.
6.3
COMPENSATED SUPPLY
Let us now consider a pad bearing, supplied through one of the compensation devices seen in section 2.3,from a lubricant source, which is maintained a t a constant pressure p s . Whatever the kind of compensation system we have, Eqn 5.1 and Eqn 5.2 remain valid for any given pad. The recess pressure is then a fraction, proportional to the load, of the supply pressure:
The relationship between the hydraulic resistance of the pad (which in its turn is closely connected to clearance h ) and the load is easily evaluated by equating the flow rate crossing the restrictor Q =-Ps Pr
Rr
to the flow rate crossing the clearances of the pad
We obtain: (6.10)
154
HYDROSTATIC LUBRICATION
The reference pressure ratio is commonly named 8: (6.11) it also represents the ratio of the design load to the limiting value A g , .
From Eqn 6.10 and Eqn 6.11 we obtain:
R, B WIWO R'=R&pW/Wo
(6.12)
The hydraulic resistance of the compensation device is, in general, dependent on flow rate and on recess pressure. Once such a relationship is known, as well as the relationship between R' and h, Eqn 6.8 allows us to obtain the clearance as a function of the load. For the plane pads, Eqn 5.12 and Eqn 6.12 give: (6.13) The required flow rate, at any given value of the load, is
PPW 1 W Ro Wo R' - Q0
Q="----
h 3
(G)
(6.14)
Consequently, the relevant pumping power is: (6.15) The friction power is given by Eqn 5.11:
(values of Iffo for several types of pad are calculated in chapter 5). Hence, the reference power ratio is (6.16) The temperature step, in the design configuration, is still given by Eqn 6.8, while, for different values of the load,
(6.17)
155
SlNGLE PAD BEARINGS
The bearing stiffness is, on the basis of Eqn 5.8,
W
1
K = 3 r
(6.18)
w a,
1
l - p w / w o + KdW
It is worth noting that in order to obtain explicit equations connecting the bearing parameters (h, K, etc.) to the load, knowledge is needed of the actual value of R,.. In other words, the foregoing equations need to be specialized for the various compensating devices.
Laminar flow restrlctors (capillaries)
6.3.1
The recess is supplied by a source of lubricant, kept at a constant pressure p s , by means of a capillary tube or some other laminar-flow fixed restrictor. The hydraulic resistance R, of such devices proves to be a constant for any given lubricant viscosity (see section 4.7.6). It may be expressed in terms of Ro and 8:
(6.19)
- b-
3
1.0 -
h
2
K KO
.
ho 0.5 -
1
t
0
1
2
3
w
OAe PS
4
5
0
0
1
2
3
W PS
Fig. 6.4 Capillary compensation. Bearing clearance (a) and stiffness (b) versus load.
4
5
156
HYDROSTATIC LUBRICATION
The equations describing the static behaviour of the bearing may be easily obtained from Eqn 6.10 and the following, up to Eqn 6.18. The reference values of flow rate and pumping power are: (6.20)
The main non-dimensional bearing parameters are also plotted in Fig. 6.4 and Fig. 6.5 against the non-dimensional load W/ Wo=W/(jlpsAe).In particular, it follows that: n
(6.21)
Bearing stiffness comes from Eqn 6.18 and Eqn 6.19. In the reference position, it is: (6.22)
thus (6.23) KO is plotted in Fig. 6.6 as a hnction of 8.
-b-
0
1
3
2 W BA, Ps
4
0
1
3
2
4
W 0 4 Ps
Fig. 6.5 Capillary compensation. Flow rate, pumping power and friction power (a), and temperature step (b) versus load.
157
SINGLE PAD BEARINGS
0.0
1.o
0.5
f3 Fig. 6.6 Reference values of stiffness versus design pressure ratio for capillary and orifice compensation.
When the temperature of the lubricant is different from its design value, both the hydraulic resistances of the restrictor and of the bearing gaps vary as does lubricant viscosity p. The effect of such a viscosity change is easily evaluated by substituting RopI po and €€fop/p o for Ro and H f o , respectively, in the foregoing equations. It is easy to see that the clearance, as well as the stiffness, do not depend on the actual value of p . Flow rate and pumping power are proportional to l / p . Friction power is proportional to p and the power ratio is therefore proportional to p2.
6.3.2
Orifices
When orifices are used instead of laminar restrictors, it should be borne in mind that the hydraulic resistance R, of such devices is no longer independent from the recess pressure. The flow rate through these devices is proportional to the square root of the pressure drop ps-pr (see section 4.11). Since in the reference configuration we must have
the hydraulic resistance of a sharp-edge orifice may be written in the following form: (6.24)
158
HYDROSTATIC LUBRICATlON
By introducing Eqn 6.24 into Eqn 6.13, the relationship between clearance and load may be explicitly rewritten as: (6.25)
The other bearing parameters can now be obtained straightforwardly (Eqn 6.14 to Eqn 6.18) and are also plotted in Fig. 6.6, Fig 6.7 and Fig. 6.8. In particular, stiffness becomes: (6.26)
where (6.27)
is its reference value. The orifices are clearly characterized by greater stiffness than the laminar-flow restrictors (see Fig. 6.6). On the other hand, a more careful control of the lubricant temperature is now required.
-a-
-b-
3
L
1.0 -
h ho
2 K KO
. .
0.5 -
0
1
1
2
3 W
OA, Ps
4
5
a
0
1
2
3
W OA, Ps
Fig. 6.7 Orifice compensation. Bearing clearance (a) and stiffness (b) versus load.
4
5
159
SINGLE PAD BEARINGS
-a-
4
I
0.7
I
1
/ 0.5
/'"=0.3
9 QO
1
L!f HfO
c
2
1
3
4
1
0
W fl Ae PS
3
2
4
W fl Ae PS
Fig. 6.8 Orifice compensation. Flow rate, pumping power and friction power (a), and temperature step (b) versus load.
0
1
2
JL
3
4
PO Fig. 6.9 Orifice compensation. Effect of viscosity (temperature) on clearance and stiffness. Unlike laminar restrictors, the temperature of the lubricant affects the load capacity of the bearing. The hydraulic resistance of the pad is proportional to the viscosity of the lubricant, which in turn depends on temperature; this last, on the contrary, has no practical effect on R,. Consequently, if a viscosity p occurs, differe n t from the design value po, each value W of the load will be sustained with a clearance which does not tally with the value calculated by means of the preceding
160
HYDROSTATIC LUBRICATlON
equations. It is easy to see that the actual film thickness is proportional to the cubic root of the viscosity; the opposite applies for stiffness (see Fig. 6.91, whereas the flow rate does not vary. Actually, we have Q
=-Ps Pr
Rr but R , depends on pressure drop ps-pr alone; this last depends on the applied load and is not related to viscosity. In brief, the consequences of a change in lubricant temperature are the following: a cooler lubricant causes lower stiffness; a warmer lubricant causes smaller clearance and, consequently, higher friction and higher temperature step.
6.3.3
Constant flow valves
A simple constant flow device is shown in Fig. 2.11. The spool compares force F, exerted by the spring with the action of the differential pressure pv-pr on spool area A,. If restrictor RV1varies much more rapidly with spool displacement than spring force F,, differential pressure pv-pr=F,/ A , is practically kept constant; the same occurs for flow rate Q , which depends on pressure drop pv-pr across reference restrictor R,. If R, is a true sharp-edged orifice, and its bore diameter is small enough to ensure that the discharge coefficient Cd is not dependent on the Reynolds number (see F'ig 4.161, Q does not actually depend on any change in lubricant viscosity. Indeed, we have (Eqn 4.76):
Q can be adjusted by setting up the spring force. However, since the orifice section A, cannot be too small (to avoid an excessive drop in pressure in the device), and true sharp-edged orifices are difficult to build (and to maintain), a certain dependence on p often exists (see Fig. 4.16 and Fig. 4.17), at least near the lower end of the flow rate range. If, on the other hand, R, is a laminar-flow restrictor, its hydraulic resistance becomes a constant, and thus, for a capillary restrictor, Eqn 4.66 gives:
i.e. Q becomes inversely proportional to viscosity (in the last equation, 1 and d are the length and equivalent diameter of the restrictor). It should be noted that, when a
I61
SINGLE PAD EEA RINGS
hydrostatic pad has to be supplied, the proportionality of Q and p ensures that the bearing stiffness is independent from lubricant temperature.
As long as recess pressure p r is lower than p s - F v / A , , the bearing simply behaves like a constant flow system: (6.28)
(6.29)
Ps B
(6.30)
=&-P2B
(6.31)
Q=&o=R, H~ = H~~
(5.11 rep.)
(6.32)
(6.6 rep.)
(6.7 rep.)
Once Fv is fixed, pressure drop Ap=pV-pr=Fv/ A v is a characteristic parameter of the device (in most cases, a few bars). Of course, it must always be pv
the device ceases to produce a constant flow rate, with a sharp drop in stiffness. In Fig. 6.10 clearance and stiffness are plotted versus load. Dashed lines represent indicatively the limit behaviour of the valve for a number of values of @, for the sample case Ap=O.lps.
162
HYDROSTATIC LUBRICAT/ON
-a-
1.5
61.o
K KO
h -
’
4 -
h0
0.4
0.5
2-
0.0 0
1
2
3
W 0 % Ps
4
5
0
1
2
3
4
5
W OA, Ps
Fig. 6.10 Constant flow valves. Bearing clearance (a) and stiffness (b) versus load.
The response of the system to a change in lubricant temperature is closely connected to the behaviour of the reference restrictor R,. As noted above, if R, is a laminar-flow device, Q is inversely proportional to p . Since the hydraulic resistance of the pad is proportional to p, h and K are not affected by temperature change, while friction power proves to be proportional to p . If R , is a true orifice, the system simply reacts like the “direct-supply” system examined in section 6.2; i.e. actual clearance is proportional to the cubic root of p , whereas stiffness is inversely proportional to the same value; in general, however, orifice-type valves also show a certain dependence on lubricant viscosity.
6.3.4
Spool valves
The controlled restrictors in Fig. 2.8 are able to decrease their own hydraulic resistance a s recess pressure (i.e. the sustained load) increases; consequently, greater bearing stiffness occurs than for a fixed restrictor featuring the same pressure ratio 8. Furthermore, a lower rate of flow is obtained for lower loads. The hydraulic resistance R, of the device depends on the displacement of the spool by means of a linear relationship:
SINGLE PAD BEARINGS
R,.=C,x
163 (6.34)
On the other hand, for the equilibrium of the spool, the force exerted on it by the spring is:
where FvS is the spring force for complete spool displacement (x=O) and K, is the stiffness of the spring. Under the design load, we have Pr=/3ps and Rro=Ro(1-a)/a; thus F0s,/J, and Ro are connected together by the relationship: (6.35)
If the device is such as to allow us to adjust the spring force, it is possible to achieve any design value of Ro (i.e. of film thickness ho)under the design load psA$. Note that, since R,. is a laminar-flow restrictor, both Ro and C, are proportional to viscosity; F,s does not then depend on lubricant temperature. The displacement of the spool under the design load is
Combining the foregoing equations, the hydraulic resistance of the controlled restrictor may be written as follows: (6.36)
where
For any given value of p, flow rate Q, is a characteristic parameter of the device. Introducing Eqn 6.36 into Eqn 6.13, it is possible to obtain clearance as a function of the load:
"--;I
-a
1 w/wo PWIW,
(
1 W ) & + U
l-woPo
(6.37)
a
where (6.38)
164
HYDROSTATIC LUBRlCATiON
-b-
-a1.5
1.0 -
h
.
ho
.
-
0.5
0
2
1
0.0 0
W fi Ae PS
-d-
-c-
2
4
2
W fi Ae PS
10
Op0.25
0.3 0.4
K Koc c
c .
.
,
, 0 . 2 ; g j (
.c
0
2
1
W fi Ae PS
2
1
W fi Ae PS
Fig. 6.11 Cylindrical spool valve. Bearing clearance (a, b), flow rate (c) and stiffness (d) versus load for certain values of valve parameter.
SINGLE PAD BEARINGS
165
A plot of hlho as a function of W /Wo is given in Fig. 6.11.a,b for certain values of 8 and 8,. It should be pointed out that, when we have Pu=/3, the system produces a flow rate which does not depend on load. The spool displacement will generally be limited to a certain range X,+XM: outside the relevant load range W,+W,, Eqn 6.37 is no longer valid and the bearing would merely perform like a capillary compensated one; furthermore, for very small values of x,, the flow is no longer laminar. The other bearing parameters (flow rate, power, etc.) are still given by Eqn 6.14 and the following equations; the flow rate is also plotted in Fig. 6.1l.c. The bearing stiffness is obtained by combining Eqn 6.18 and Eqn 6.36. In particular, for the reference load, it is (6.39) where KO, is the stiffness of the same pad supplied at the same pressure ratio through a fixed restrictor (Eqn 6.22). For a generic load we have: 1- Pl8v
1-8 W
1-8- 1 + WO
El??
WOPV 1- WIWO
p
1-8
Pv
8
(6.40)
Nondimensional stiffness K/Kocis plotted in Fig. 6.11.d as a function of Wl W O(it has been chosen to show K / K o c instead of K in order to display the gain in stiffness of the system, compared to a capillary-compensated one). By tuning the elastic constant of the spring, a very great, and even negative, stiffness could clearly be obtained. On the other hand, the stiffness would be much smaller than KO for loading conditions different from the reference one, and the flow rate would increase sharply for loads greater than Wo. Furthermore, the behaviour of the bearing under dynamic loading must be carefully considered. Concerning the effects of a change in lubricant temperature, i t has already been pointed out that, provided R , is a laminar-flow restrictor, the ratio C, / Ro does not depend on p; thus, only flow rate and power are affected, Q and H p being proportional to Up and H f being proportional to p. Other types of devices have been proposed, with better performances than the cylindrical spool. For example, let us consider the tapered spool valve (ref. 2.5) in Fig. 2.9. Its hydraulic resistance may no longer be expressed by a linear relationship such a s Eqn 6.34. Provided aperture angle a: is very small, we have:
(6.41)
HYDROSTATIC LUBRlCATlON
166
-b-
-a-
0.0'
0
'
' . . '
1
.
'
.
'
0
2
2
-d-
-c-
2
4
W DA.9 Ps
W BAe Ps
10
9
K
00
Koc
5 1
0
1
2
-5 0
2
1
W BAe PS Fig. 6.12 Tapered spool valve. Bearing clearance (a, b), flow rate (c) and stiffness (d) versus load for certain values of valve parameterg, and R,=Ro.
SINGLE PAD BEARINGS
167
where &xal c is the non-dimensional spool displacement. Proceeding as in the former case, it is easy to verify that F,s (spring force for {=O), to(spool displacement in the design configuration), p and Ro are connected by the following equations:
(6.42)
The hydraulic resistance of the device may thus be rewritten as:
(6.43)
where 1 c pu = ---(i2aA,
Ro Ku RlJ Ps
(6.44)
The clearance for any load W now proves to be (from Eqn 6.13):
h
-
q-
3
11 - p WlWo Ro
1
(6.45)
The rate of flow is still given by Eqn 6.14.The stiffness for the reference load is again given by Eqn 6.39,whereas, for different loads, the relevant stiffness may be evaluated by introducing Eqn 6.43into Eqn 6.18.Film thickness, the flow rate and stiffness are plotted in Fig. 6.12as functions of the load. Again no effect follows a change in viscosity, except for different values of flow rate and pumping power.
6.3.5
Diephragm-controlled restrictors
The device shown in Fig. 2.10 (ref. 2.6)works like the spool valves examined in the previous section, but its hydraulic resistance varies according to a cubic law:
(6.46)
168
HYDROSTATIC LUBRICATION
where <=x / lo is the non-dimensional displacement from an arbitrary configuration (say, lo may be assumed a s the restrictor clearance when no force is exerted on the diaphragm, Fig. 6.13).
Fig. 6.13 Diaphragm-controlled restrictor.
For the equilibrium of the membrane, the elastic force has to be:
A , is the effective area of the membrane, Fvs is the elastic force exerted by the spring, for x=O, from which the constant effort exerted on the membrane by the supply pressure p s has been subtracted. To have a hydraulic resistance Ro of the pad when the design load Wo=ppsA,is applied, F,s must be:
Fvs = P P s A , -k 5OlOKu
(6.47)
where
Note that we should have Fus cKvloto prevent a complete shut-off of the restrictor for small loads. By rewriting R, in the form
1-8
1
(6.48) 1-P
SINGLE PAD BEARINGS
169
where
(6.49) and by inserting it into Eqn 6.13,one can obtain the relationship between load and clearance:
(6.50)
A plot of the film thickness , as well as of the flow rate (obtained as usual from Eqn 6.14))as a function of the load, is given in Fig. 6.14. The stiffness in the reference configuration is again given by Eqn 6.391,where Eqn 6.49 has now to be used for Po. For different loads, the non-dimensional stiffness is:
K
W
1 -PIP,
h0
h
WP 1-p- wo P u 1-8-w 1--11-w/wop
wo
3
1-p
(6.51)
P"
Comparing Fig. 6.14 to Fig. 6.11 and Fig. 6.12,we see that the performance of the diaphragm-controlled restrictor is better than that of the spool-controlled devices; in particular, the load range is larger. Furthermore, due to the small mass of the membrane, the dynamic behaviour should also be better.
As happens in the case of the spool valves, any change in the temperature of the lubricant does not affect the set-up of the device, but only flow rate and pumping power.
6.3.6
Infinite-stiffness devices
Certain controlled restrictors, such as the one shown in Fig. 2.12 (ref. 2.81,are able to keep bearing film thickness constant for a wide range of loading conditions. For the equilibrium of the spool we must have:
(6.52) When the load, and therefore p r , vary, the spool is displaced, changing the inlet restrictor of the device, until Eqn 6.52 is again fulfilled. The flow rate is:
170
HYDROSTATIC LUBRICA TlON
-b-
-a-
1.5
1.o
h h0
0.5
OS 0
2
2
1
-c-
2
Qo
-d-
1C
\
9
K Koc
1
(
0.5 1
pzq
n . 0.25
0
4
W OAe PS
W DAe PS
2
1
W fl Ae PS
-11
2
1
W fl A, Ps
Fig. 6.14 Diaphragm-controlled restrictor. Bearing clearance (a, b), flow rate (c) and stiffness (d) versus load for certain values of valve parameter.
171
SINGLE PAD BEARINGS
-b-
*
1.o
.
\ yp0.25
\
\
\
0.5\
0.75\
\
\
\
" \
' \ ' \\
0.5 -
\
\ \'
\\ \ 0.0 0.0
0.5
-!&
1.o
Ae PS
0.5
W -
1.o
Ae PS
Fig. 6.15 "Infinite-stifSness"valve. Bearing clearance (a) and flow rate (b) versus load, for certain values of the valve parameter.
Q
=-Pv - Pr
R,
This means that the hydraulic resistance of the pad clearances is pegged to the constant value
R=Ro=R,
Yv l-~,
(6.53)
and thus the film thickness is kept constant as well. ho can be adjusted by setting restrictor R,. It is noteworthy that, if R , is laminar in kind, the system is not sensitive to any change in lubricant viscosity, except as far as flow rate and pumping power are concerned. The flow rate proves to be proportional to load W=p,.Ae:
W Q= A x
(6.54)
We obviously must have pu yvps, the whole hydraulic
172
HYDROSTATC LVBRlCATlON
resistance of the device will be reduced to the fixed value R,, and the system will merely work like a capillary-compensated bearing with a pressure ratio b=y, (see Fig. 6.15).
6.3.7
Inherently compensated bearings
The simplest of the inherently compensated bearings described in Chapter 2 is represented in Fig. 2.17.a (see also ref. 2.13). From Eqn 4.58,flow rate is:
then pressure at radius rl is:
Ps lnrz/rl
1
+E& (1+ hp/h)3 Load capacity is:
Inherently compensated bearings are generally used with compressible Iubricants.
6.4
DESIGN OF SINGLE-PAD THRUST BEARINGS
The specifications of a pad bearing consist, in most cases, in a range of loads W,+WM to be sustained with a certain stiffness at any given sliding or turning velocity. A further set of constraints is generally specified, involving important parameters such as: size and shape of the pad film thickness overload capacity lubricant viscosity flow rate supply pressure friction. The first step in design often consists in establishing suitable pad dimensions, in such a way that the effective area of the pad be greater than the ratio of the maximum expected load t o the maximum available pressure. The shape of the pad
SINGLE PAD BEARINGS
173
is often constrained. However, a few remarks on the most convenient land width to choose may be useful. In the foregoing sections it has been shown that, for any pad shape, the land width ratio (e.g. parameter a' for annular-recess pads) may be chosen in such a way a s to minimize pumping power. It is clear that when pad velocity is null, or very low, total power consumption practically coincides with pumping power, and then such a choice of the land width may be considered to be the "optimum"one. However, this is not a critical point: for instance, it may be convenient to select narrower lands to increase load capacity without affecting the external size of the pad. On the other hand, when the speed is high enough, it is generally more convenient to select narrow lands to minimize the total power Ht=Hp+Hp The selection of the thickness of the film depends on certain contrasting factors: it must be small enough to ensure sufficient stiffness a t any load to meet the specifications; it must be small enough to ensure small flow rate and pumping power (of course, this point, as well a s the next, is also connected with the choice of lubricant viscosity); it must be large enough to avoid excessive friction; it must be much larger than the geometric errors of the mating surfaces (roughness, planarity, parallelism).
As far as the problem of a suitable choice of h and p is concerned, i t is possible to demonstrate that, for any given pad and any given viscosity, an "optimum" clearance may be calculated (from the point of view of total power) on the condition that Hf=3Hp. Conversely, if h is given, "optimum" viscosity is the one which gives Hf-&lp (ref. 2.3). Let us now briefly examine this problem applied to rotating pads. Total power in the reference configuration may be written in the form:
valid for constant-pressure supply, as well as for direct constant-flow supply (in this case it must be stated that /j'=l).For axisymmetric pads, such as plane or tapered circular pads, the effective area can be written a s in Eqn 5.20, and the hydraulic resistance as in Eqn 5.83: of course, the relevant equations must be used for A*, and R*, and a=d2 for plane pads. For these pads total power may hence be rewritten as:
(6.55)
174
HYDROSTATIC LUBRICATION
Once all the other parameters are given, the power ratio will depend on turning velocity; from Eqn 6.16 and Eqn 5.85:
Conversely, the last equation may be used to evaluate the viscosity needed to obtain a given power ratio: (6.56)
Introducing the last equation into Eqn 6.55, we obtain: (6.57)
where
(6.58)
The dimensionless value H,*, plotted in Fig 6.16 and in Fig. 6.17, only depends on the geometric ratios of the pad. It is clear that a similar equation for Ht can be obtained for any pad shape o r any supply system; however, for sliding bearings H , will no longer be independent from the pad size (e.g. B.L or D2) or supply pressure. Equation 6.57 allows us to draw some important conclusions concerning "optimum" bearing design. First, it clearly seems convenient to get the greatest values for H t : that is, to reduce land width, obtaining the further advantage of lower supply pressure and temperature step. However, in many cases, the variation of H,* with the land width is small (see Fig. 6.16). There is no doubt, on the other hand, that film thickness should be as small as possible, since the power expense for any given value of the power ratio is proportional to ho (for the sake of completeness, however, i t should be noted that the increase in stiffness which follows the choice of a smaller clearance is not always beneficial, since it may lead to smaller margins of stability). Finally, the reader may easily verify that, once ho has been fixed, the minimum power is obtained when n=1, although Ht varies little for values of the power ratio in the range 113SIlS3. To obtain the desired value of IT, viscosity must satisfy Eqn 6.56, which for the sake of practicalness may be rewritten as: (6.59)
SlNGLE PAD BEARlNGS
175
where (6.60)
or alternatively as: (6.61)
where (6.62)
These equations are valid for any supply system (for direct supply p=1 and ps=pro). Both p$ and p; are plotted in Fig. 6.16 for central-recess circular pads and Fig. 6.17 €or annular-recess pads.
A problem now arises, since in practice it is not always possible to select a lubricant with the theoric "optimum" value of viscosity given by Eqn 6.59 or Eqn 6.61 for n=1.In particular, a t low speed, this optimum viscosity may prove to be impracticably high. Although one may try to increase land width or pad size, sometimes a value of l 7 smaller than 1 must be accepted: indeed, for Q=O, a n infinite viscosity would be the optimum! On the other hand, a t high speed, the optimum viscosity
Hi
Fig. 6.16 Circular recess pad. Total power H: and optimum viscosities p$ and p; versus radius ratio r'.
176
HYDROSTATIC LUBRICATION
-b-
-a40
4
400
\
r'=O.8
Hi
GI
2
0 0.0
20
0.4"
0.2 a'
200
O
0.0
0.2
a'
0.4
Fig. 6.17 Annular recess pad. Total power H: and optimum viscosities p$ and p; versus land width ratio a'.
would be the optimum! On the other hand, a t high speed, the optimum viscosity might prove to be very low, even after having reduced the land length to a minimum and having increased recess pressure (reducing the pad diameter) as much as possible. In these cases i t is recommended t o select a suitably low viscosity (sometimes, however, p is explicitly prescribed) increasing ho in such a way as to have n13.Actually, it can be shown that, once p is fixed, the "optimum" clearance is the one which leads to n=3,but the problem is complicated by the fact that h and p may not be considered to be constant values, since they are subject to change according to load and temperature. The following is a design criterion which generally produces good results: take a s reference load Wo the one (in the normal load range) which gives the minimum value of power ratio H f / H p :namely, Wo=WMfor constant-flow supply and W,=W, for constant pressure supply; select h,, p, and the land width ratio in such a way a s to obtain n=1.The value of p selected in this way should be the one which tallies with the maximum lubricant temperature allowed by the supply system. However, when velocity is too low or too high, it is necessary to accept values of ll that are smaller than 1 or greater than 3, respectively.
SINGLE PAD BEARINGS
177
For a more general and complete discussion of the optimization problem for hydrostatic thrust bearings, the reader is referred to chapter 11. Let us now examine detailed design procedures for constant-flow or eonstantpressure systems.
6.4.1 Direct supply (constant flow)
As stated above, it is convenient, in this case, to take the greatest load of the normal load range a s reference ( W o = W ~ The ) . design may then be carried out as follows: i)
choose a trial set of geometric parameters. If pad velocity is very low, select a land width ratio close to the one which gives minimum pumping power, otherwise, select narrower lands. For instance, for a central-recess circular pad, one may select r’=0.53 in the former case, and a greater value (perhaps r’=0.9)in the latter;
ii)
choose a suitable effective area A,. This parameter is closely related to the supply pressure and the load, since ps=pr=WIAe. If a certain overload capacity is required, it must be A e > W ~ l pwhere ~ , WEand pE, are, respectively, the extreme load and the greatest allowable pressure (i.e. the the pressure tallying with the set-up of the relief valve); evaluate the size of the pad, and check that it is compatible with the specifications;
iii)
select the minimum film thickness ho: bear in mind that the thinner ho is, the smaller the power loss will be (provided a sufficiently low viscosity is available) and the greater the stiffness; on the other hand, greater manufacturing accuracy will be required; calculate KO(Eqn 6.6); (by means of Eqn 6.2) calculate the film thickness h~ which tallies with the minimum load W m ; calculate the minimum stiffness Km=3Wml hM; if Km does not meet the specifications, decrease ho. However, when a wide load range is expected, there will be great variations in stiffness as the load changes (see Eqn 6.5); in this case, the use of an opposed-pad bearing may be more suitable (Chapter 7);
iv)
calculate the optimum viscosity, that is the value b p t that makes Z7=1 (Eqn 6.4). For circular bearings, this may be done quickly with the aid of Fig. 6.16 o r Fig. 6.17; check that b p t is a plausible value (sometimes p may be directly imposed by the specifications). If it is not, modify the geometrical parameters (namely
178
HYDUOSTATIC L UBUlCATlON
the land width ratio) or ho and start again from point (ii) o r (iii). If popt cannot be brought into the admissible range, because the velocity is too small, select the highest available viscosity. On the other hand, a very high speed would require too low viscosities to be optimized with n=1,and consequently a greater value of the power ratio must be selected; values of n greater than 3 should, however, be avoided, or else excessive temperature steps could follow (Eqn 6.8); if the supply system is not able to hold the temperature of the lubricant within narrow limits, the designer should take a certain viscosity range into account, instead of a single value. A simple method consists in considering popt as the lower limit of the viscosity range, which tallies with the highest lubricant temperature. For a cooler lubricant viscosity will be greater, as well as film thickness and, above all, bearing stiffness will be lower. Consequently, it is necessary to check K for the whole temperature range; v)
calculate the required flow rate; calculate pumping power H p o and friction power H f o (for circular bearings Eqn 6.57 and Fig. 6.16 o r Fig 6.17 give total power Hto: now Hpo=HtoI(l+17) and Hfo=Hto17).For loads smaller than Wo, both the powers will be lower; check the maximum temperature step in the lubricant (Eqn 6.9); if a range of viscosities is to be allowed in the plant, check friction power and temperature step for the maximum viscosity too;
vi)
check the Reynolds number; for bearings rotating a t high speed also check the effect of inertia forces (parameter Si) and the circumferential Reynolds number;
vii)
check the dynamic behaviour of the system (see chapter 10).
In certain cases it may be necessary to have a flow rate which differs from the value calculated a t point (v) above (for instance, because the pump which has to be used does not feature an adjustable flow rate); clearly, if this happens it is necessary to repeat the calculations (in particular, as far as film thickness and stiffness are concerned) using the actual value of the flow rate. If the difference is large, it may be considered convenient to modify viscosity o r geometrical parameters in order to approach the optimum value of the power ratio again.
EXAMPLE 6.1 Design a circular-recess pad to sustain a thrust varying in the W,=30 KN t W ~ = 4M 0 v range at a speed G=lOradfs, with a friction moment lower than 1 Nm. Further constraints are the following: the outer diameter must be DSlOO mm; the lubricant viscosity can be chosen in a range of 0 . 0 2 4 1 Nslm2; the displacement as
SINGLE PAD BEARINGS
179
the load varies (slowly) from the minimum to the maximum value must be smaller than 10 pm; the recess pressure must always be smaller than 7 MPa. i) Since the turning velocity is low, a first trial value for r' could be r'=0.5, and hence A2=0.541. In order to sustain the maximum load WO=wM=40KN, the effective area must be greater than w 0 / 7 MPa, i.e. A , ~ 5 . 7 . 1 0 -m2; ~ consequently, due to the limited diameter, it must be (Eqn 5.20) A*,>0.727, and a considerably larger value is needed for r' (Fig. 5.1. b). ii) Let us select D=lOO mm, r'=O.75; it is easily obtained (from Fig. 5.1.6 or by means of the relevant equations) A50.760, R*=0.288, Hf*=0.684;the effective area is then A,=5.97.10-3 m2 and the maximum recess pressure is pro=WolA,=6.70 MPa. iii) A suitable value for minimum film thickness may be ho=30pm. The maximum film thickness, which tallies with the minimum load Wm=30KN, is (Fig. 6.1) h~=1.10.ho=33pm. Note that the displacement h ~ - h o = pm 3 is much smaller than required. iv) In Fig. 6.16 we may read &=8.72, i.e. bpt=0.31Nslm2. This last value is too high, hence the maximum allowable ualue p=O.1 Nslm2 should be selected, accepting a power ratio smaller than 1 (the friction power will prove to be much smaller than the pumping power). v) The reference hydraulic resistance is easily calculated as Ro=2.03.1012Nslm5 (Eqn 5.21), and hence Q=prolRo=3.29.10-6m3/s, Hpo=proQ=22W. A s far as friction is concerned, we have Mf0=0.22Nm, Hf0=2.2 W. Obuiously, for loads smaller than Wo, both Hp and H f will be smaller. The temperature step is maximum in correspondence to the maximum load: from Eqn 6.8, assuming c.p=l.7.10-3 J l m 2 " C , dTod.3 "c. EXAMPLE 6.2 Design an annular-recess bearing to sustain a load W=75 KN with stiffness K0>5.109N l m , at a speed l2=628 radls (6000 rpm). The outer diameter of the pad must be smaller than 200 mm, and the inner one greater than 120 mm. The maximum recess pressure can be as high as 12 MPa, but the bearing should be able to KN. the data of the lubricant sustain, exceptionally, a maximum overload d W ~ = 7 5 to be used are: p=0.02+0.03 N s lm2 (depending on the actual operating temperature), p =go0 e l m 9 c=1890 JlQOC. i) Since the speed is high, let us start with a small value of a', say a'=0.1. I f the greatest allowable pad area is chosen, it follows that r'=120/200=0.6. ii) It follows immediately that: A*,=0.576 and A,=0.0181 m2. The maximum load WE=150KtV hence tallies with a recess pressure p ~ = 8 . 2 9MPa; for the normal load Wo we have pr0=4.14 MPa. iii) Since Ko=3W0Ih, to obtain the required stiffness we must have hoc45 pm. However, since a wide viscosity range is expected, to obtain hc45 p for the highest
180
HYDROSTATIC LUBRICATION
viscosity value (0.03 Nslm2) we must have ho<39 pm for the lowest viscosity p=0.02 N s I m2 (see Fig. 6.3). iv) From Fig. 6.1 7, p:=15.2: to get Il=l with p=0.02 Ns lm2 a clearance ho=60 pm is needed (Eqn 6.60).Since ho must be much lower to ensure adequate stiffness, friction power would be much greater than pumping power. To reduce total power it is possible to reduce the land width. For instance, a=2.5 mm (a'=0.0625) may be selected. From equations of section 5.4 it follows that A:=0.60, R*=0.0156, Hf*=0.119; hence Ae=0.019 m2, pro=WolAe=3.98MPa. In Fig 6.17 we may read pz=25.2, thus for l7=3 and p=0.02 Nslm2 a clearance ho=36.9 pm is needed, whereas we may select ho=39pm in order to have Il=2.4 for p=0.02 Ns I m2 and n=3for p=0.03 N s I m2. v) Flow rate is &=0.395.10-3mats. The power consumptions are Hp0=l.6 KW, Hf0=3.8 KW and Ht0=5.4KW. When the lubricant is cool (p=0.03 Nslm2), clearance increases to 44.7 pm (Fig. 6.3.a) and friction power increases to 5.0 KW. Note that if viscosity is further increased, stiffness may fall under the designed minimum value. The maximum temperature step is (Eqn 6.8)AT=6.61(0.395.0.9.1.890)=9.7"C. ui) The Reynolds number is given by Eqn 5.75: Re=17.7; the circumferential Reynolds number in the outer clearance is (from Eqn 5.591 Re,=109; consequently, no turbulence is expected. Also, the inertia forces i n the lubricant not need be taken into account since we have Si=O.13 (Eqn 5.50).
6.4.2
Compensated supply (constant pressure)
In this case we suggest taking the smallest load in the normal load range as reference. The design procedure is then similar, in its main aspects, to the one outlined in the previous section. Although the procedure below refers to bearings compensated by means of capillary restrictors, it may be easily extended to other compensating devices, including controlled valves: i)
choose a trial set of geometric parameters. If the pad velocity is very low, select a land width ratio close to the one which gives the minimum pumping power; else select narrower lands. For instance, for a central-recess circular pad, one may select r'=0.53 in the former case, and a greater value (perhaps r'=0.9) in the latter; choose pressure ratio dependent on load ratio WMIWo (for instance, with the aid of Fig. 6.4): i t is advisable for to be considerably smaller than Wo1 WM(e.g. 8 ~ 0 . 8W . o /W M )to ensure enough overload capacity and to avoid the stiffness corresponding to W M being too small. In particular, as far as overload is concerned, if a certain overload capacity AWE is given, we must obviously have PCWOIWE, where WE=WM+AWE.I t should, however, be stressed that the total power will turn out to be proportional to (Eqn
l/fi
SINGLE PAD BEARINGS
181
6.57), but, on the other hand, stiffness KO is proportional to (1-p).When the load range is wide (i.e. when Wo is much smaller than W,) a very small value is needed for p; in this case, an opposed-pad bearing may be the most suitable choice (see chapter 7); ii)
choose suitable values for effective area A , and supply pressure p s , in such a way as to obtain Wo=AJ?p,; evaluate the size of the pad, and check it is compatible with the specifications;
iii)
select the minimum clearance h,, tallying with the maximum load W M ; evaluate the reference clearance ho (e.g. for capillary compensation from Eqn 6.21 or Fig. 6.4.a). The thinner ho is, the smaller the power losses (provided a sufficiently low viscosity is available) and the greater the stiffness will be; on the other hand, greater manufacturing accuracy will be required; calculate KO (e.g. from Eqn 6.22) and stiffness K=KoK' tallying with load W, (e.g. from Eqn 6.23 or Fig. 6.4.b). For controlled restrictors, the relevant parameters (e.g. 8,) must also be selected; check that stiffness is great enough for the whole load range: if i t is not, one may try to decrease p, starting again from point (i), or to decrease ho, if possible, starting again fkom point (iii);
iv)
calculate the optimum viscosity, that is the value popt that makes I7=1(Eqn 6.4). For circular bearings, this may be done quickly with the aid of Fig. 6.16 or Fig. 6.17; check that k p t is a plausible value (sometimes p may be directly imposed by the specifications). If it is not, modify the geometrical parameters (namely the land width ratio) or ho and start again from point (ii) o r (iii). If popt cannot be brought into the admissible range, because the velocity is too small, select the highest available viscosity. On the other hand, a very high speed would require too low viscosities to be optimized with n=1,and consequently a greater value of the power ratio must be selected; values of Il greater than 3 should, however, be avoided: otherwise excessive temperature steps could follow; if the supply system is not able to hold the temperature of the lubricant within narrow limits, the designer should take a certain viscosity range into account, instead of a single value. A simple method consists in considering popt a s the lower limit of the viscosity range, which tallies with the highest lubricant temperature. For a cooler lubricant viscosity will be greater: for many compensating devices this will have no effect on clearance and stiff-
182
HYDROSTATIC LUBRICATION
ness; instead, the power ratio will be greater (proportional to p2 for laminarflow devices); v)
calculate the flow rate; calculate pumping power Hpo and friction power Hfo (for circular bearings Eqn 6.57 and Fig. 6.16 o r Fig 6.17 give total power Hto: now H p o = H t o / ( l + ~ ) and Hfo=Hto17).For loads greater than WO,the pumping power is smaller, but Hf increases (Eqn 5.11); check the maximum temperature step in the lubricant (Eqn 6.8 and Eqn 6.17); if a range of viscosities is to be allowed in the plant, check friction power and the temperature step for the maximum viscosity, too;
vi)
design the restrictor: its hydraulic resistance RrO must be given by Eqn 6.19; if capillary pipes are used, select length and diameter by means of Eqn 4.67; check the Reynolds number in the restrictor to ensure laminar flow (Eqn 4.68);
vii)
check the Reynolds number; for bearings rotating a t high speed, also check the effect of inertia forces (parameter Si) and the circumferential Reynolds number;
viii)
check the dynamic behaviour of the system (see chapter 10).
EXAMPLE 6.3 A large slide is sustained by a set of rectangular pads (supplied through capillary restrictors) whose dimensions are B=0.3 m and L=0.4 m. Each pad must sustain a load W=60 KN with a stiffness K>2.5.109 N l m and a f i l m thickness not smaller than 40 pm. The friction force should not be greater than 6 N at a speed U=0.05 m I s. i) Since velocity is low, it is convenient to have large lands; i f we select a'=0.15 and rL: =0.25, from Fig. 5.27 we obtain Az=0.645. Since no great load variations are expected, we may try to select a relatively high value, such as 8=0.6, for the pressure ratio. ii) The pad dimensions are given: the effective area is A,=LBA:=0.077 m2. The supply pressure should hence be p,=1.29 Pa. iii) I f we select ho=40 pm, Eqn 6.22 immediately gives K0=1.8.109 N l m , which is not enough. Hence it is necessary to select a smaller value for p, such as p=0.4. Repeating the above calculations, we now get: ps=1.95 MPa and Ko=2.7.1@ N l m . iv) From Fig. 5.28 we obtain R*=0.75, and from Eqn 5.108 (stating h,=l mm) Af=0.079 m2. Since the friction force must be smaller than 6 N,Eqn 5.106 gives an upper boundary for viscosity: p=0.0607 Nslm2. For such a viscosity Eqn 6.16 gives n=0.14 (of course, the pumping power will prove to be much greater than the fric-
SINGLE PAD BEARINGS
183
tion power). A lubricant is therefore selected, whose viscosity is p=0.06 Nslm2 at the lowest operating temperature. v) Calculations of flow rate and power are now straightforward, leading to: Qo=1.1.10-6 m3/s; Hp=2.1 W, H ~ 0 . 3W. Temperature step proves to be slightly greater than 1 "C(assuming c.p=1.7.106 Jlm3"C). For lower viscosities H f decreases while Q and Hp prove to be proportional to 1 I p. vi) The hydraulic resistance of the restrictor can be calculated by means of Eqn 5.102 and Eqn 6.19, which lead to R,/p=17.6.1012 m-3. From Eqn 4.67 it follows that such a hydraulic resistance may be obtained with a pipe 27 mm i n length and with a 0.5 mm bore. The Reynolds number, calculated by assuming p=0.06 N s l m 2 and p 9 0 0 Kglm3, is Re=42, considerably less than its critical value. EXAMPLE 6.4 Design a capillary-compensated annular-recess bearing to sustain a load which varies i n the 2 0 4 5 KN range, with a displacement Ah smaller than 20 pm. The outer pad diameter must be smaller than 0.2 m, and the inner diameter must be equal to 0.1 mm. A further requirement is that the friction moment needed to rotate the bearing at e 6 2 8 radls (6000rpm) must be smaller than 5 Nm. The supply system to be used delivers lubricant at pressure ps=4 MPa; viscosity is expected to vary i n the p=0.0154.020 Ns 1m2 range, depending on actual operating conditions; density and specific heat are p=920KgIm3andc=1850 JIKgOC. i) Since the speed is high, choose narrow lands, say a=2 mm. The load ratio is WMIWo=1.75,hence p=0.45 should be a proper selection (see Fig. 6.4). ii) W i t h the given supply pressure, the effective area should be A e = 2 ~ / ( O . 4 5 . 4 . 1 O 6=)0.0111 m2. For the sake of simplicity, let us choose D=0.16 m (i.e. r'=O.625; a'=0.067), obtaining Az=0.569, R*=0.0154, Hf*=0.122;hence, A,=0.0114 m2 and p=0.437. iii) From Eqn 6.21 (or Fig. 6.4.a), for any certain design clearance ho the minimum clearance (which tallies with the maximum load) is hm=0.62.ho. To ensure Ah=ho-hm<20 pm we must therefore have ho152.7 pm. On the other hand, to have Mf<5 N m for p=0.02 Nslm2, the minimum film thickness cannot be smaller than 19.8 pm, i.e ho must be not smaller than 31.8 pm (see Eqn 5.70). iv) From Fig. 6.17, pJ=119; to have ll=1 for p=0.015 Nslm2 we should have (Eqn 6.59) ho=41.4 pm. Selecting ho=40 pm, the reference power ratio will vary in the 1.1411712.04 range, depending on the actual value of the viscosity. Flow rate, powers and temperature step depend on load and viscosity and are summarized i n the following table. v) The hydraulic resistance of the restrictor must be R,=1.29.Ro (Eqn 6.19), i.e. R,.=8.87.109 Nslm5 for p=0.015 Nslm2. Note that, due to the high value of the flow rate, such a restrictor cannot be made with a small pipe without accepting high
184
HYDROSTATIC LUBRICATION
Reynolds numbers. A more suitable approach may consist in using a narrow annular clearance as a restrictor (Fig. 4.11.e). A further solution may consist in substituting the laminar-flow restrictor with a n orifice, which works well at a high Reynolds number (section 4.11).
W
p
h
(KN) (Ns/m2)
(m3/s) (KW)
0.015 20
Q.103 H p
Hf
AT
(KW)
("(2
0.25
1.02
1.16
5.0
0.19
0.76
1.55
7.1
0.11
0.42
1.87
12.8
0.08
0.32
2.49
20.7
40. 0.020 0.015 24.8
35 0.020
vi) It is easy to verifv that the radial Reynolds number of the bearing (Eqn 5.75) is suitably low, as well as the tangential Reynolds number (Eqn 5.59) and the inertia parameter (Eqn 5.50).
EXAMPLE 6.5 The stiffness of the bearing designed in Example 6.4 needs to be improved, without changing the dimensions of the pad, nor the supply system (except for the compensating device). Namely, Ah needs to be reduced to a value of 10 p m or smaller. i) First, one may try to reduce the design value of clearance. In the previous example it is shown that ho cannot be smaller than 32 pm, in order to avo& excessive friction; consequently, Ah cannot be smaller than 32.(1-0.62)=12 pm. It is clear that another type of restrictor must be used to emure adequate stiffness. W One test may consist in substituting the laminar flow restrictor with a sharpedge orifice. In this case, Eqn 6.25 gives a minimum film thickness hm=O.717.h0; assuming ho=30 pm, it follows that h,=21.5 pm and Ah=8.5 pm. For orifice compensation, however, clearance depends on the actual value of viscosity, as shown in Fig. 6.9: i f p is increased from 0.015 NsIm2 to 0.02 Nslm2, clearance experiences a 10% increase, and Ah becomes greater than 9 pm. By repeating the calculations, the flow rate, powers and temperature step are obtained, as shown in the following table. The stiffness constraint has clearly forced us to select values of the power ratio greater than 3, leading to large temperature steps in the lubricant. The diameter of the orifice can be calculated by means of Eqn 4.76, in which Q=Qo=O.ll m3/s, Ap=(l-p)p,=2.25 MPa. The discharge coefficient depends on practical considera-
185
SlNGLE PAD BEARlNGS
tions (see Fig. 4.16 and Fig. 4.17), a typical value being Cd=0.65. d=1.72 m m is readily obtained. W
p
h
(KN) (Ns/m2) (pm)
Q.103 H p (m3/s) (KW)
Hf
AT
(KW)
(T)
0.015
30.
0.11
0.43
1.55
10.9
0.020
33.
0.11
0.43
1.88
12.7
0.015
21.5
0.07
0.28
2.16
20.8
0.020
23.7
0.07
0.28
2.61
24.7
20
35
iii) In the last case examined, it is clear that power, flow rate and temperature step are subject to large variations when the load and viscosity change. A great improvement from this point of view may be obtained by means of a constant flow valve. Figure. 6.10 shows that now h,=0.830.ho; it is therefore possible to select a greater value for clearance, such as ho=40 pm. Assuming that a temperature compensated valve is used, calculations now give:
It may be verified that by reducing the flow rate, say to 0.1 .lO-3 m3/s, clearance hoproues to be close to 30 pm, with a higher friction power, but the total power expense will not be notably affected (it will be slightly reduced).
Chapter 7
OPPOSED-PAD AND MULTIPAD BEARINGS
7.1
INTRODUCTION
A multipad bearing system is made up of a certain number of pads, with a common moving member. The pads may be, in general, quite different, and they, too, might be fed by means of differing devices.
All these bearings may be studied by applying to each pad the equations already developed for the single pads in chapter 5, and by summing together the effects of all the pads, after having expressed all the pad clearances as functions of the displacement of the moving member. The case of the opposed-pad bearings is of particular importance.
7.2
OPPOSED-PAD BEARINGS
For any opposed-pad bearing, such as the one in Fig. 7.1, the total load capacity is, obviously,
The flow rate is, from Eqn 5.2 and Eqn 5.10, (7.2)
OPPOSED-PAD AND MULTIPAD BEARINGS
187
It is useful to take the origin of the displacements in such a way as to split the axial play into two equal parts; thus, if e is the axial displacement (see Fig. 7.11,ho is the axial half-play, and &=-
e h0
(7.3)
is the non-dimensional eccentricity; the non-dimensional clearances are:
(7.4)
Fig. 7.1 Opposed-pad bearing.
The bearing stiffness may now be defined as:
(7.5) It is obvious how the expressions for H p and H f may be obtained. In the most common case of symmetrical bearings ke. that in which the two pads are equal), we have Ae,=Ae2=Ae and R10=R20=R0, and the above equations turn out to be simplified even further.
188
HYDROSTATIC LUBRICATlON
7.2.1
Direct supply
(i)Basic equations. When the two opposite recesses are fed by separate pumps,
providing the constant volume flows Q1 and Q2, respectively, the recess pressures are:
The load capacity is given by Eqn 7.1. For a symmetrical bearing, provided that Q1=Q2=Q/2(see Fig 7.21, we have: 1
W=ijRoA, Q W ( E )
(7.7)
where
A, and Ro are the relevant values, for each type of pad, calculated in chapter 5. When Eqn 5.12 is valid, i.e. for plane pads, and, approximately for many other pads, Eqn 7.8 becomes
Fig. 7.2 Opposed-pad bearing: direct supply.
OPPOSED-PAD AND MULTIPAD BEARINGS
189
In the rest of this chapter, Eqn 5.12 will be considered to be valid. The bearing stiffness, calculated by means of Eqn 7.5, may be put, as usual, in the following form:
K = KO K ( E )
(7.10)
where (7.11)
is the stiffness in the unloaded configuration (E=O), and (7.12)
The pumping power is now expressed by the equation: (7.13)
where (7.14)
For the friction power, we have:
H f = 2 Hfo H i
(E)
(7.15)
where
Hi. '
1
1
1
Z ( E +F&)
(7.16)
Again, H f o is the relevant value of the single pads (see chapter 5). The power ratio in the reference configuration is, quite obviously,
n=4Hfo Ro Q2 and the relevant temperature step is
(7.17)
190
HYDROSTATIC LUBRICATION
When the bearing is loaded, the lubricant experiences different temperature steps in the two branches of the circuit; since, however, the reservoir is generally the same for both pumps, a mean temperature step AT may be evaluated
H'
AT
1++7
a=%l + l 7 P
(7.19)
The main bearing-parameters are plotted against the non-dimensional load in Fig 7.3, from which certain conclusions may be easily drawn: any given bearing could, in theory, support any load however high, but considerable eccentricities and high pressures should be allowed; the stiffness and the power always increase with the load; indeed, K and H p are practically proportional to W for ~ 0 . 2 ; for ~ > 0 . 5 the , loading performance of the opposed-pad assembly practically coincides with the performance of the more loaded pad alone; when a constant load has to be supported, the single pad clearly proves to be more convenient, while the opposed-pad is more suitable for loads varying over a wide range.
-a-
-b-
20
K
4 Q Ro/2 10
v.v
0
0 5
10 W &Q Ro/2
0
5
10
15
W &QRo/2
Fig. 7.3 Direct supply: a- eccentricity, pressure, and stiffness versus load; b- pumping power, friction power and temperature step versus load.
191
OPPOSED-PAD AND MULTIPAD BEARINGS
(ii) Working tolerances. Once film thickness ho has been chosen on the basis of the required stiffness of the bearing, of the available rate of flow and pumping power, and of the moment of friction developed, it has to be borne in mind that, due to the working tolerances, the actual axial play might be quite different from the designed value. In order to study the consequences of this fact, we shall assume that the reference value 2ho is also the maximum allowable value of the axial play. Let the actual axial play be g12ho; clearance error may be then defined as: (7.20)
If the flow rate is assumed to be equal to design value Qo, we can now plot the lesser film thickness, the higher recess pressure and the stiffness as functions of the load, as in Fig. 7.4.
-a-
0.4'
0
'
'
'
'
'
5
'
.
'
W
-b-
.
10
&QW
0
5 W &QRo/2
10 -0
5 W A,Q Fb/2
10
Fig. 7.4 Direct supply; working tolerances. Effects of clearance error q. The bearing stiffness may clearly be much greater when gc2ho, whatever the load. On the other hand, the clearance of the more loaded pad is smaller, and the relevant recess pressure (and therefore the power required, too) are higher. Both the pumping power and the friction power are greater when the axial play is reduced, but the pumping power (which depends on h-3) is more notably affected. The power ratio (for W=O)proves to be proportional to (g/2ho)2.
192
HYDROSTATIC LUBRICATION
In order to avoid excessive pressures, one could adjust the flow rate on field to a value Q0(l-9)3, depending on the actual axial play. In this case, the eccentricity and the recess pressures are equal to the design values, whatever the load. As a result of this, however, the minimum film thickness for any given load proves to be reduced even further. The pumping power also proves to be notably reduced, thanks to the smaller flow rate, and the power ratio rises to I7( l-q)?
(iii) Lubricant temperature. When a constant-flow supply system is adopted, adequate control of lubricant viscosity is very important, i.e. control of its temperature. Indeed, since the hydraulic resistance of the pads proves to be inversely proportional to p, i t is obvious from Eqn 7.7 that any given load will require a greater bearing displacement if the viscosity is lower than the design value po. On the contrary, higher viscosity will lead to smaller displacements, but also to a greater waste of power. In Fig. 7.5 eccentricity, pressure and stiffness are plotted as functions of the load, for several values of the actual viscosity p.
-b-
-C-
20
10
0
0
5
W
10 -0
&,Q5/2
5
10 -0
W &gab/:!
5
W &Q5/2
10
Fig. 7.5 Direct supply. Effects of viscosity on bearing performance. 7.2.2
Capillary compensation
(i) Basic equations. When laminar-flow restrictors are used as compensating devices, the recess pressures are (see Eqn 6.10 and Eqn 5.12):
193
where P1 and P2 are the relevant pressure ratios for h1=h2=h0. The load capacity is given by Eqn 7.1. For a symmetrical bearing (Fig. 7.61, it may be written as follows: (7.21)
W = ps A, Wc(1;~) where p=P1=p2and 1
1
W' =
1-131 +
R'(4
l+uL P RYE)
Provided Eqn 5.12 is valid, the last equation may be rewritten as (7.22)
The total flow rate is:
Fig. 7.6 Opposed-pad bearing: capillary compensation.
194
HYDROSTATIC LUBRlCATlON
-a-
0.6
w
1.o
0.8
A, Ps
-b-
0.6 0.5
0.0I . . . . , . . . . 0.0 0.5 1.o
0.0
1.o
0.5
& Ae PS
Ae PS
Fig. 7.7 Capillary compensation: a- eccentricity, b- flow rate and c- stiffness versus load, for certain values of the pressure ratio.
OPPOSED-PAD AND MULTIPAD BEARINGS
195 (7.23)
(7.24)
The bearing stiffness may be expressed in the usual form:
K = KO Fa&)
(7.25)
where KO(stiffness in the centered configuration) and K are: At!
KO= 6 ~ p s (B1 - 8 )
(7.26)
(7.27)
Eccentricity, flow rate and stiffness are plotted in Fig. 7.7 against load. The pumping power is: (7.28)
and the friction power is still given by Eqn 7.15, as in the case of the constant-flow supply system. Hence, the reference power ratio is
n=-lROH BP,2 f0
(7.29)
The temperature step (for E=O) is
AT^=$ P (i+m
(7.30)
and, in general cases,
(7.31)
However, the variation of HflHp and AT with the load is small enough to be disregarded in the actual design. From the above equations, it is clear that maximum stiffness in the centered position is achieved for p=0.5; this is the reason why p=0.5 is often indicated as the
196
HYDROSTATIC LUBRICATION
"optimal" pressure ratio. However, lower values of p lead to greater stiffness for medium and high loads and, furthermore, to less waste of flow rate and power (both are halved for p=0.25). On the contrary, no benefit ensues for higher pressure ratios, which, consequently, should be avoided.
(ii) Working tolerances. Let us now examine, as in section 7.2.1.W, the consequences of clearance error q , defined in Eqn 7.20. If the restrictors are fixed, the bearing simply behaves as an assembly with an actual pressure ratio Bq =
1
1+
(7.32)
(1- q)3
The actual power ratio becomes and an actual hydraulic resistance Roq=R0/(l-q)3. (7.33)
-a-
-b-
-0 . 1
8 0.4
0.3
6-
n
0.2
4 -
6.00
0.25
0.50
rl Fig. 7.8 Capillary compensation: a- actual pressure ratio, and b- actual power ratio versus clearance error.
In Fig. 7.8, pq and l7,, are plotted as functions of q for certain values of p. Figure 7.9 shows how the tolerances may affect lesser film thickness and bearing stiffness, for certain values of load. It is clearly advisable to carry out the design in such a way a s
197
OPPOSED-PAD AND MULTIPAD BEARlNGS
to have small values of p (perhaps j?=0.3) when the axial play is the maximum allowed by the tolerances. As an alternative, one could use adjustable restrictors, to restore the designed pressure ratio 8, whatever the actual axial play g is. In this case, one simply has to do the calculations again, using the actual value of ho.
-a-
o .a
h2 h0
0.4
1.3 1.4
1.5
---
W=O.S&,P,
\
0.4\ 0.5\ \
0.0 0 1
0.25
'
\ \
0.50
6.00
0.50
0.25
11
11
Fig. 7.9 Capillary compensation; working tolerances. Effects of clearance error. (iii) Lubricant temperature. Concerning the effects of the temperature of the lubricant, it should be noted that, since laminar-flow restrictors are used, the pressure ratio B is not affected by changing viscosity; thus, no close control of the temperature is required to ensure constant loading performance. Q and H p , on the other hand, are inversely proportional to p and H f is proportional to p; hence, the actual power ratio HflHp is proportional to p2. Thus, for high-speed bearings, it may be advisable to avoid full operation while the lubricant is much cooler than planned in the design, since the friction torque might prove to be too high; the higher value of the power ratio should, however, help the lubricant to warm up quickly (see Eqn 7.30).
7.2.3
Orifices
(i) Basic equations. If true sharp-edge orifices are used as compensating devices, the pressure drop across the restrictors becomes proportional to the square of
198
HYDROSTATIC LUBRICATION
the flow rate (Eqn 4.76). With the help of the results obtained in sect. 6.3.2, it is possible to obtain a set of equations describing the static behaviour of such bearing systems. For a symmetrical bearing, the load capacity may be written in the form of Eqn 7.21, but W now stands for the more complicated equation 1
w’=2[l+
-1+
+4
1+4
p2
(1 + &)6
]
(7.34)
In the same way, the total flow rate, the pumping power, the friction power, the power ratio and the temperature step are given by Eqn 7.23, Eqn 7.28, Eqn 7.12, Eqn 7.29 and Eqn 7.30, respectively, in which Q’ stands for the following equation:
&’=
(1 - &)3
(1 + &)3
+
l++q&z-
(7.35)
1+4+
Bearing stiffness may still be put in the form of Eqn 7.25, where
(7.36)
+
4-
(1 + &)5
[ ++ q G z 1
1”
(7-37)
The main parameters are plotted against the load in Fig. 7.10. It is easy to see that the orifices yield greater stiffness than the capillaries (the maximum stiffness is now obtained for p=0.586).
(ii) Lubricant temperature and working tolerances. One drawback to orifices is that the system now proves to be highly sensitive to the viscosity of the lubricant, which directly affects stiffness and load capacity.
If 3/ is the design value of the pressure ratio for design axial play 2ho and viscosity po, the actual pressure ratio p,, proves to depend on the clearance error q and on the actual value of viscosity p:
199
OPPOSED-PAD AND MULTIPAD BEARINGS
-a-
0.0
0.6
0.4
0.2
1.o
0.8
w Ae PS
-c-
-b-
0.5 0.4
0.0
0.5
w Ae PS
1.o
0.0'
0.0
. .
'
.
' 0.5
. . .
Jv-
'
1.o
Ae PS
Fig. 7.10 Orifice compensation: a- eccentricity, b- flow rate and c- stiffness versus load, for certain values of the pressure ratio.
200
HYDROSTATIC LUBRlCATlON
-a-
-b15
1.C
! ? l
n
h
10
0.5 5
0.C C
1
0.25 1- (1
0.50
0.25
0.00
0.50
-v)m
11
Fig. 7.11 Orifice compensation. Effects of clearance error and lubricant viscosity on: a- pressure ratio; b- power ratio.
1.o
-b-
-ar -
- W=0.3&,P, 2 K ..
1
0 1 . ’ ” 0.00
’
. 0.25
.
‘
.
I 0.50
11 Fig. 7.12 Orifice compensation. Effects of clearance error and lubricant viscosity on: a- lesser film thickness; b- stiffness.
OPPOSED-PAD AND MULTIPAD BEARINGS a m
2
=
20 1 (7.38)
In Fig. 7.11; Pq is plotted as well as the actual value of the power ratio: (7.39)
Fig. 7.12 shows certain effects of 7 and p / p 0 on the lesser film thickness and on stiffness. Clearly, the wider the temperature range allowed for the lubricant, the smaller the tolerances on the axial play should be.
7.2.4
Constant flow valves
(i)Basic equations. An opposed-pad bearing can be fed by delivering a flow Q/2 to both recesses by means of constant-flow devices, of the type examined in section 6.3.3. Obviously, the assembly behaves like a constant-flow system until the maximum pressure p s -Ap is reached in the more loaded recess, i.e. until 3
I & l I & & f =$3/(l-.4p/ps) l-
(7.40)
(Ap is a minimum pressure drop, characteristic of the device). For larger displace-
ments, the more loaded pad is, in practice, fed through a fixed restrictor, and the stiffness collapses. In the normal working range, the equations describing the performance of the system are written straightforwardly; for the main bearing parameters we may retain Eqn 7.21, Eqn 7.23, Eqn 7.25, Eqn 7.28, Eqn 7.15, Eqn 7.21, and Eqn 7.30, in which we have: 1
I-
w ' = p [m
1
(7.41)
(7.42)
(7.43)
In Fig. 7.13, E, Q, and K are plotted against the load for certain values of j3.
202
HYDROSTATIC LUBRICATlON
-a-
0.6
W A,
0.0' 0.0
'
'
'
.
' . 0.5
'
'
W A, Ps
'
1.o
0.0
1.o
0.8
Ps
0.5
W -
1.o
A, Ps
Fig. 7.13 Constant-flow values: a- eccentricity, b- flow rate and c- stiffness of the bearing versus load, for certain values of the pressure ratio and Ap/p,=O.I.
203
OPPOSED-PAD AND MULTIPAD BEARINGS
(ii) Working tolerances. Let us now consider the effect of the working tolerances. If the axial play g is a fraction (1-9) of design value 2h0, the actual hydraulic resistance of the pad clearances becomes Roq=Ro/(l-q)3and hence the actual pressure ratio rises to p,=p/( 1-9)s. The main consequences are: stiffness increases: e.g. K(W=O)proves to be proportional to V(1-9); on the other hand, the load range +W(EM),in which both the feeding devices act correctly, is reduced (substitute pq forp in Eqn 7.40) as shown in Fig. 7.14; friction power and power ratio are proportional to 1/(1-9).
-a-
v.v
n
0.00
0.25
n
0.50
Fig. 7.14 Constant-flow valves: maximum displacement and relevant load versus clearance error, for certain values of the pressure ratio and Apip,=O.l. Clearly, narrow tolerances and small pressure ratios are required to ensure high load-capacity; thus, the designer may not make the most of the potentiality of such a supply system; furthermore, the stiffness of the actual system would be much higher than necessary. This kind of problem may be overcome by planning for a n on-field adjustment of the flow rates delivered by the valves, in such a way as to restore the planned pressure ratio whatever the actual value of the play is: i.e. the total flow rate must be Q(l-9I3instead of Q.In this way, a n error concerning the play no longer affects load capacity. Stiffness in the unloaded configuration is now proportional to l/(l-q). The friction power is also proportional to l/(l-q), whereas the power ratio becomes nq=n/(l-q)4.
204
HYDROSTATIC LUBRICATION
(iii) Lubricant temperature. The effect of a change in lubricant viscosity is, of course, connected with the behaviour of the flow-control valve. It has been already noted that the "ideal" device (for this kind of application) should be able to provide a flow rate that is independent from recess pressure, but inversely proportional t o lubricant viscosity. In this case, the loading performance of the system would not be affected, while the pumping and friction powers would be proportional to l / p and to p, respectively. A "true"constant flow valve, on the other hand, provides the same flow rate, whatever lubricant viscosity is. The main consequences are very similar to those already seen in the case of the bearings directly fed by positive-displacement pumps; in particular, the actual pressure ratio is proportional to p, thus a warmer lubricant leads to less stiffness, while a cooler lubricant leads to greater stiffness, counterbalanced by a narrower load-range (it is easy to see, from the equations above, that if viscosity is too high, the valves cannot deliver the flow rate planned, even at W=O!).
7.2.5
Flow dividers
(i)Basic equations. Flow dividers may be used to improve the performance of the opposed-pad bearings, as compared to the fixed-restrictor supply systems. Some of these devices are introduced in section 2.3.2.
To start with, let us consider the cylindrical-spool valve in Fig. 2.13.The hydraulic resistance of the two branches of the device depends directly on the displacement x of the spool:
(7.44)
The displacement in its turn depends on the recess pressures, i.e. on the load. In non-dimensional form it may be written as follows (7.45)
Bearing in mind Eqn 7.2 and Eqn 7.21; displacement may be rewritten as follows: <=-'-A
ioKu p s
w'
Since we have
(7.46)
OPPOSED-PAD AND MULTIPAD BEARINGS
205
it follows that:
As usual, the reference pressure ratio p has been introduced above; in the present case we have (7.47)
In the same way p 2 is also obtained, and, finally, the non-dimensional load W' (to be introduced in Eqn 7.21): (7.48)
Equation 7.48 may be solved together with Eqn 7.46 in order to obtain the nondimensional bearing displacement E as a function of the non-dimensional load W' and of parameters p and
, a, =-loAK ,
5
pS=
w'
(7.49)
The flow rate is obtained by adding Q1 and Q2 together; bearing Eqn 7.23 in mind, the non-dimensional flow rate is: (7.50)
In Fig. 7.15, E and Q' are plotted against the non-dimensional load, for certain values of a,, in the typical case of p=O.5. It should be stressed that high values of a,, may lead to a negative stiffness, coupled with a relatively high flow-rate. In Fig. 7.16 E is plotted against p, for W=0.6 and W=0.3. Pumping power, friction power, the power ratio, and the temperature step are still given by equations Eqn 7.28, Eqn 7.15, Eqn 7.29, and Eqn 7.30, respectively. The above equations are clearly no longer valid after the maximum displacement of the spool has been reached. Since we have {=1 for W'=lla,, it follows that
206
HYDROSTATIC LUBRICA TlON
a,
wM -- aAe, p s
(7.51)
For greater loads, the spool cannot displace further and the stiffness strongly decreases (in practice, the spool-travel allowed should be even smaller).
-a1.01
I
-b-
1.o
Q 2k'%
0.5
0.0
0.5
W Ae PS
1.o
0.0 0.0
0.5
-
1.o
Ae PS
Fig. 7.15 Cylindrical-spool flow divider: a- eccentricity and b- flow rate of the bearing versus load, for certain values of the valve parameter.
The stiffness of the assembly may be calculated by considering that, after substituting Eqn 7.46 for 5, Eqn 7.48 gives W' as an implicit function of E; thus, i t may be differentiated locally. In particular, in the centered position, i t is easy to find that (7.52)
where Koc is the reference stifmess of the same bearing fed a t the same value of the pressure ratio p through fixed restrictors, i.e. Eqn 7.26.It would seem possible to obtain any stiffness whatsoever by adjusting the parameters of the device; very great stiffness would, however, require values of a,,that exceed the limit aU=1 quoted above.
207
OPPOSED-PAD AND MULTlPAD BEARINGS
Better performances may be obtained with the tapered-spool value in Fig. 2.14. The hydraulic resistances of the two branches of the device may be expressed in the following form:
(7.53)
-a-
-b-
0.d
E
0.
0.1 0.0
1.o
0.5
0.0
1.o
0.5 B
0
Fig. 7.16 Cylindrical-spool flow divider: eccentricity of the bearing versus pressure ratio, for certain values of the valve parameter.
The position of the spool depends on the load, according to Eqn 7.46. Proceeding as before, we again obtain Eqn 7.21 and Eqn 7.23, in which:
w'=
1
1
(7.54)
208
HYDROSTATIC LUBRlCATfON
-a1.01
-b-
1.a
Q 24’%
0.5
0.0
0.5
A,
1.o
0.c
I
0.5
w
1.o
A, PS
PS
Fig. 7.17 Tapered-spool flow divider: a- eccentricity and b- flow rate of the bearing versus load, for certain values of the valve parameter.
-b-
-a0.E
E
0.4
0.C I
I3
1.o
0.5 I3
Fig. 7.18 Tapered-spool flow divider: eccentricity of the bearing versus pressure ratio, for certain values of the valve parameter.
209
OPPOSED-PAD AND MUlTlPAD BEARINGS
It is now possible to plot the displacement of the bearing and the flow rate as functions of the load and of parameter a,, as in Fig. 7.17, or E as a function of 8 , a s in Fig. 7.18. The reference stiffness now becomes:
and a very high stiffness may now be obtained, even for a,
Displacement x depends on the differential pressure p2-pl, according to Eqn 7.45, in which A, is understood as a n "effective"area of the diaphragm. Coefficient RvOfor diaphragms may be calculated from Eqn 4.59 (substituting h with lo), while the effective area is (7.58) Stiffness K, may be calculated as
r2
I
Fig. 7.19 Diaphragm flow-divider.
!
I
-b-
21 0
HYDROSTATIC LUBRICA TlON
(7.59)
for diaphragms as in Fig. 7.19.b,whereas for diaphragms as in Fig. 7.19.a the following may be taken:
(7.60)
Proceeding as above, the non-dimensional load and flow rate are easily found (see also Fig. 7.20 and Fig. 7.21):
1
1
(7.61)
(7.62)
-a1.0r
0.0
-bI
0.5
*e
PS
1 .o
0.0
0.5
w
1 .o
PS
Fig. 7.20 Diaphragm flow-divider: a- eccentricity and b- flow rate of the bearing versus load, for certain values of the valve parameter.
21 1
OPPOSED-PAD AND MULTIPAD BEARINGS
-a-
-b0.8I
0.4
I
E
E
0.4
0.:
0.1
1.o
0.5
0.0 0.0
D
1.o
0.5 D
Fig. 7.21 Diaphragm flow-divider: eccentricity of the bearing versus pressure ratio, for certain values of the valve parameter.
The bearing stiffness, in the unloaded configuration, is: (7.63) Great stiffnesses may be obtained for smaller values of a, than with the preceding devices. Moreover, thanks to the small mass of the diaphragm, the dynamic behaviour is generally better.
(ii) Working tolerances. Once a device has been selected, with a parameter a,, to obtain a given performance with a given opposed-pad bearing, i t has to be borne in mind that, due to working tolerances, the actual axial play g may differ from the design value 2 4 . Consequently, the actual hydraulic resistance of the pads becomes Roq=RoI(l-q)3 in the unloaded configuration, and the actual pressure ratio is now given by Eqn 7.32. As in the preceding sections, q is given by Eqn 7.20.The actual performance of the assembly may thus be evaluated again with these new data. It is also necessary to take into account the working tolerances of the valve (see Example 7.1).
21 2
HYDROSTATIC LUBRICATlON
(iii) Lubricant viscosity. All the devices considered in this section are controlled restrictors of a laminar kind; consequently, any change in lubricant viscosity has the same effect upon the hydraulic resistances of both the pads and the dividers. Therefore, stiffness and load capacity are not affected, while the flow rate and the pumping power prove to be proportional to lip. Friction power, on the other hand, is proportional to p, and thus the power ratio Il is proportional to p2.
EXllMPLE 7.1 A n opposed-pad thrust bearing has been designed to sustain loads of up to 35 KN; it is fed at ps=4.5 MPa, through capillary restrictors; lubricant viscosity at the maximum operating temperature is p=0.05 Nslm2. The bearing is made up of two annular-recess pads with D=200 mm, r'=O.72, a=5 mm; the measured axial float is g=70 pm, hence ho=35 pm. It is required to improve bearing stiffness by substituting the fixed restrictors with a diaphragm-controlled flow-divider; namely we wish to obtain a situation in which, for any load W smaller than 20 KN, displacements are smaller than Wl(5.loS Nlm) For the given pad shape, it is easy to calculate the effective area (Eqn 5.66): Ae=0.0124 m2; at W=20KN the load parameter is hence W=WIAeps=0.358.Parameters b and a , should now be selected in such a way to have, at W=20 Mv, an eccentricity ~IWlho(5-109Nlm)=0.114. With the aid of Eqn 7.49 and Eqn 7.61 it is possible to draw a plot of eccentricity at W=0.358 (as a function of 3/ and a") like the one in Fig 7.22. It is clear that a wide range of possibilities exists for a selection of both parameters. For instance, since flow rate Qo is proportional to B, it seems to be convenient to choose p=0.3 and aU=0.55.
a, Fig. 7.22 Example 7.1; eccentricity against the valve parameter.
OPPOSED-PAD AND MU1 TlPAD BEARINGS
21 3
Of course, it is necessary to also verify the behaviour of the system for different loads. For small loads, Eqn 7.63 gives K0=3.26.K,=6.5S.1@ N l m (bear in mind that KO,is the stiffness tallying to simple capillary restrictors, Eqn 7.26), which is much higher than required. At the maximum load W=35 KN (i.e. W'=0.626), eccentricity may be calculated as ~=0.22,a good result compared with capillary compensation. Flow rate at W=O can be easily calculated: for p=0.05 Nslm2, &0=41.7.10-6m3/s. The relevant pumping power is, hence, Hp0=188W. Equation 7.62 shows that Q and H p are smaller when the bearing is loaded. Concerning the design of the controlled restrictor, the reader should verify that the selected value of the pressure ratio may be obtained (for instance) with r2=6 mm, r,=3 mm, and 10=76 p m (see Fig. 7.19) and that the relevant Reynolds number is quite low. Parameter A, is easily calculated: A,=0.061~10-3m2. From Eqn 7.49 it follows that the stiffness of the diaphragm should be K,=6.59.106 N l m ; for instance, using a steel diaphragm (El(l-v2)=226GPa) with a thickness s=l mm, the required value of K, may be obtained selecting rv=12.6mm, for the diaphragm i n Fig. 7.19.a. It is clear that the designed values of a, and jl can only be obtained with a certain approximation, but they are not critical, in this case, since limited variations can be allowed, as may be seen by examining Fig. 7.22. In particular, it may seem difficult to obtain exactly the calculated value of clearance lo; nevertheless a greater clearance would clearly lead to quite small changes in stiffness, For instance, to increase lo by 20% would give av=0.458 and /3=0.425, and hence the displacement at W=20 KN remains practically unchanged. Of course, flow rate would increase as the pressure ratio increases.
7.2.6
Design of opposed-pad bearings
In most cases the specifications of thrust bearings consist in a certain value W , of the load to be sustained a t a displacement smaller than a certain value eM. Certain obvious constraints generally exist for the size of the bearing, as well as for the supply system (max. pressure and flow rate; viscosity of the lubricant) and for the friction (max. torque). Since too many parameters and constraints are involved, a general design procedure cannot be given, except for certain main aspects: let us briefly examine how the choices of the designer may affect the final result, and try t o assess "optimal" values of certain parameters, on the usual basis of the least power loss. For the sake of brevity, the following considerations will refer mainly to the case of annular-recess pads fed through laminar-flow restrictors, but they could easily be extended to other pad shapes and supply systems.
As a first design step, one may try to select a maximum value EM of the eccentricity, commonly & ~ = 0 . 5however, ; it is not advisable to allow eccentricities greater
21 4
HYDROSTATIC LUBRICATlON
than 0.6. When a maximum value of the displacement is given, this immediately leads to a n upper constraint gk=2 eM/&M for the axial play g=2ho. When high stiffness is required, g),,may prove to be too small; in this case i t may be useful to select a smaller value for the eccentricity.
A set of geometrical parameters can now be stated: as for the single-pad bearings, it is generally convenient to select large lands for low-speed bearings, the contrary for high-speed bearings. The pressure ratio should remain in the 0.3+0.6 range; i t is advisable to start fixing a low value for the design pressure ratio j? in order to allow greater margins for the working tolerances: this point is clarified below. The load parameter W ' ( E M $ ) = A can ~ , now be calculated, or read in the appropriate chart. Supply pressure is often prescribed, and then the effective pad area (i.e. the pad size) is readily obtained; otherwise, both A, andp, have to be suitably chosen. Caution is necessary in selecting high values for the supply pressure, since an excessive temperature step in the lubricant may follow: see Eqn 7.30. The problem of a suitable selection of the axial play and of the lubricant viscosity may be tackled on the usual basis of minimization of the total power consumption:
Proceeding as in the case of the single-pad bearings (sect. 6.21, it may be found that, for plane or tapered circular pads, rotating a t a speed a, the total power may be written as (7.64)
the relevant value of the viscosity being (7.65)
Equations for H; and p i are given in Chapter 6 (namely, Eqn 6.58 and Eqn 6.62; see also Fig. 6.16 and Fig. 6.17). Equations 7.64 and 7.65 are valid for all constant-pressure supply systems, provided the relevant values for W' are selected. Examining Eqn 7.64, it should be noted that, as for the single-pad bearings, it is convenient to select large recesses and small clearances, provided that a lubricant with sufficiently low viscosity is available, in order to get n=1.If, instead, j l is fixed, the optimal value for axial play is the one which gives n=3.
OPPOSED-PADAND MULTlPAD BEARINGS
215
Further observations are, however, necessary. In particular, it must be stressed that the choice of axial play is meaningless if the relevant working tolerances are not specified. As a matter of fact, a whole range (2h0-6~&<2ho should be selected, instead of a single value, for the axial play g. To reduce manufacturing costs, it is advisable for tolerance Sg to be as large as possible, but it has been shown that large tolerances can notably affect the performances of the bearing (e.g. see Fig. 7.8 and , and Sg=h0, it may be seen that the Fig. 7 . 9 ) .For instance, selecting e ~ = 0 . 5/3=0.5, actual pressure ratio may prove to be as great as 0.89 (take q=0.5 in Eqn 7.32);the main consequence is that load capacity is strongly reduced and the load designed to be sustained at ~ ~ = 0cannot . 5 be reached even at e = l (this may be verified by means of Eqn 7.22). It should therefore be clear that smaller tolerances must be selected. The actual value of the power ratio will also be dependent on the actual clearance (see Eqn 7.33);for instance, forfi=0.3, 6g=ho/1.5and I7=1, the actual power ratio may prove t o be as great as 2.6.
A convenient design practice consists in first selecting a suitable value for toleraxial play may then be chosen (e.g. with the aid of Eqn 7.32)in such a way ance as to keep the actual pressure ratio within a reasonable range. For instance, we may take ho=l.6.6g iffi=O.3, or ho=2.2.6gif/3=0.4. Smaller values for ratio ho/Sg can ) expected. be selected, when small eccentricities (e.g. ~ 0 . 3are
4;
In order to free the loading performances from the adverse effects of manufacturing tolerances, it is possible to select adjustable restrictors, that allow us to restore the design value of the pressure ratio on field, whatever the actual value of axial play. Nevertheless, for high-speed bearings, ,6 should not be greater than ho/2 to avoid excessive values of the power ratio. Instead, for low-speed bearings, tolerances may be larger. Concerning viscosity of the lubricant, it should be borne in mind that, in general, it may not be considered a constant, since it largely depends on lubricant temperature, and hence on the working conditions and on the effectiveness of the cooling of the reservoir. Unless a temperature control system is provided, it is advisable to verify the performances of the bearing for all the expected range of viscosity. In particular, when laminar-flow restrictors are used, fi is not affected by lubricant temperature, nor load capacity. For low-speed bearings it is convenient at the design stage t o consider p as the lower end of the expected viscosity range (in this way flow rate and power loss should always turn out to be smaller than calculated). Otherwise, optimization may be looked for, for a center-of-range value of viscosity: it is then necessary to check flow rate and power for all the viscosity range (it is convenient, however, for the supply system to be able to keep lubricant temperature within a narrow range).
21 6
HYDROSTATIC LUBRICATION
As a result of the above remarks, a simple procedure may be proposed that should lead to a quick and "optimized"design of the opposed-pad bearings. Let us assume a certain load W M is given, to be sustained with a displacement smaller than eM;further constraints may be required (e.g. viscosity or supply pressure may be imposed): the device may then be designed following the steps listed below. i)
Choose a trial set of geometric parameters: select large recesses unless low speeds are expected. Choose a trial value of the maximum eccentricity, such as ~ ~ = 0 . 5 . Choose a trial value of the design pressure ratio: perhaps p=0.3.
ii)
Evaluate coefficient W' (e.g. from Eqn 7.22, for capillary compensation). Decide supply pressure p s , when it is not prescribed, and effective area A,, in such a way to satisfy Eqn 7.21. Calculate the pad size, in order to get the above value of A,.
iii)
Select a suitable value for the manufacturing tolerance on axial play: bear in mind that the narrower Sg is, the lower the power losses can be. Select the design film thickness ho (the actual value g of the axial play must consequently lie in the 2hot2ho-Sg range); for capillary compensation, i t should be ho>1.5.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
OPPOSED-PAD AND MULTIPAD BEARINGS
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
HYDROSTATIC LUBRICATION
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).
OPPOSED-PAD AND MULTIPAD BEARINGS
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'<0.4 we must have hO>% pm (Eqn 7.67). On the other hand, Fig. 7.24 shows that, assuming @=0.3,at the maximum load WM=15KN, we have 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.
OPPOSED-PAD AND MULTIPAD BEARINGS
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
HYDROSTATIC LUBRICATION
(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
OPPOSED-PAD AND MULTIPAD BEARINGS
-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
HYDROSTATIC LUBRICATION
-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)
OPPOSED-PAD AND MULTIPAD BEARINGS
227
Let us now compare the SRB to the annular-recess opposed-pad bearing, with the same radii, fed by two pumps (see section 7.2.1 and section 5.4 for the relevant equations). It is easy to see that the 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
229
OPPOSED-PAD AND MULTIPAD BEARINGS
1-Y R,.=-Ro Y
and
thus, bearing in mind Eqn 7.75: (7.99)
The main bearing parameters now become: (7.100)
W = p s p ' A e W'
n R= H U P? Y
(7.101)
R' H'
P AT,, = S ( i+ m P C
l + & p
AT =
AT^ --&+
(7.102)
The main effects of the inlet restrictor are also shown in Fig. 7.28.b. What is stated a t the end of the preceding section, concerning the effects of the working tolerances, also holds good here (see also ref. 7.7).self-regulating bearing
7.5
HYDROSTATIC SLIDEWAYS
One of the most common uses of hydrostatic bearings is to ensure that the heavy carriages of large machine tools move smoothly. In this case, four, or more, pads are interposed between the carriage and the guides; for instance, in Fig. 1.16, all the loads and moments acting on the carriage, except the axial component of the load, are supported by a n assembly of 12 pads, which are so arranged a s to form 6 opposed-pad bearings. The axial load may, in its turn, be supported by a hydrostatic lead screw, which ensures a smooth and frictionless drive of the carriage, with a high degree of stifiess. Such assemblies are studied by examining how the displacement of the carriage affects the mean film thickness of each pad (the lack of parallelism of the pad surfaces, deriving from tilting displacements, can, in general, be disregarded). By way of example, let us consider the very simple four-pad assembly in Fig. 7.29. When no external force is acting on the carriage, we will have, for each pad, hi=ho,pi=pro=Wd4Ae,Qi=prdRo.Wo is the weight of the carriage itselc the coeffi-
230
HYDROSTATIC LUBRICATlON
cients A, and Ro depend on the type of pad: for rectangular pads, for instance, see Eqn 5.101and Eqn 5.102. Under a n external load, the carriage is displaced, until a new equilibrium is reached:
(7.103)
My = a Ae (P1 + P4 - P2 - P 3 )
Fig. 7.29 Four-pad assembly.
If each pad is directly fed by a separate pump, the flow rates Qi do not vary; hence, from Eqn 5.2,Eqn 5.10and Eqn 5.12,we obtain
(7.104) By introducing Eqn 7.104 into Eqns 7.103,a set of equations is obtained, which relates the external force to the mean film thickness of the pads. Each hi can be expressed in terms of the angular displacements (Ox, 0,) of the carriage and of the vertical displacement of its center Ahc.
231
OPPOSED-PADAND MUiTlPAD BEAMNGS
In Fig. 7.30 the displacements are plotted against the component W of the load along axis z , for a few values of moments M, and My. If the pad is supplied a t a constant pressure p s , through fixed laminar restrictors, Eqns 7.104must be substituted by
(7.105)
where
A plot of the relevant displacements, for certain loading conditions, is given in Fig. 7.31. The total flow rate is the s u m of the pad flows
-a-
1.25
k
ex
h0
1.oo
0.75
-b-
0.4
b ha
0.2
0
1
rc wl
2
0.0 0
1
JY
2
M6
Fig. 7.30 Load-displacement performance of a four-pad slideway (constant flow supply). (1): M,=M =O; ( 2 ) :MX=O.25.bWo,M,=O; (3): M,=O.S.bWo, My=O; (4): MX=O.25*bWo, My=0.25*ado.
HYDROSTATIC L UBRlCATlON
232
-b-
-aI.2
ex- b
G.!
’
h0
h0
0.2 -
1.o(
0.7! 1
M
2
L
0.0 0
M6
1
M
2
M6
Fig. 7.3 1 Load-displacement performance of a four-pad slideway (capillary compensation; J = 0 . 2 5 ) . (1): M , = M 4; ( 2 ) : M X = 0 . 2 5 . b W o ,M,=O; (3): M x = 0 . 5 . b W o , M y = ( ) ; (4): MX=0.25,bWo,k f y = d 2 5 . 0 W o . Of course, the pads can be fed as well through controlled restrictors of one of the types seen in sections 6.3.3 to 6.3.6. Use can even be made of the flow dividers shown in section 7.2.5;in this case, two devices are needed: one to feed pads 1 and 3, and the other for pads 2 and 4. Such an arrangement leads to greater stiffness, particularly as far a s the moments are concerned (see, for instance, ref. 7.8).
When a high degree of stiffness is required, especially in the case of loads of variable direction or dynamic loads, opposed-pad bearings may be used instead of single pads, as in Fig. 1.16. Using pil and pi2 t o indicate the recess pressures of the upper and lower pads of the i-th couple, and Ael and Ae2 to denominate the relevant effective areas (in general it is advisable to have different values of the effective areas for the upper and lower pads in order to compensate weight Wo of the carriage), it is clear that:
W=Wo
+ AW=A,1
C ~ i l - A e 2Cpi2
(7.106)
Similar equations may be obtained for moments M, and M y . As before, the recess pressures can be related to the eccentricity of the relevant couple by means of a relationship which depends on the type of supply system: for instance, for a constant-flow supply, we have
OPPOSED-PAD AND MULTIPAD BEARINGS QiRoi
Pil=(l-Ei)s
;
Pi2 =
Q2
Ro2
233 (7.107)
The eccentricities may be written a s functions of the average value E = ( E ~ + E ~ ) / ~ = =(%+e4)/2 and of the angular displacements 6, and Sy of the carriage:
e1=E+Syu/hO-6xb/h0, and so on. It is then possible to evaluate the performance of the assembly. Greater details concerning an analytical treatment of complex pad assemblies may be found in ref. 7.9; ref 7.10 contains results concerning a vertical boring machine.
7.6
MULTIPAD JOURNAL BEARINGS
The multipad journal bearings may be regarded a s a set of cylindrical pads acting on the same journal (Fig. 1.12.a). Consequently, the load capacity of such bearings can be evaluated, for any given displacement of the axis of the shaft from the axis of the bearing, by calculating the load sustained by each pad, as in section 5.8, and then summing all of them together vectorial. Although the working of the multipad bearings is simpler to understand and to calculate (ref. 7.111, compared to multirecess ones, they are less effective and (due to the drainage grooves separating the pads) more expensive to build. Besides, a t high speed, the grooves may lead to the inlet of air in the clearances. For these reasons, multirecess bearings are generally preferred. Here we shall confine ourselves to showing the load-displacement performance of a typical four-pad bearing, fed through laminar restrictors (see Fig. 7.32). Of course, all the other supply systems could, in principle, be used. The reference hydraulic resistance Ro of each pad, which is necessary to design the supply restrictors, may be calculated from Eqn 5.111 and Fig. 5.32. Hence, the flow rate in the concentric configuration is :
Ps Q=nPRO
(7.108)
( n is the number of pads). When the bearing is loaded, Q proves to be slightly smaller. The pumping power and the friction power are (for E=O) (7.109)
(7.110)
234
HYDROSTATIC LUBRICATION
-a-
0.4
-b-
-
.
0.0
0.8
0.4 E
Fig. 7.32 Multipad journal bearing: load versus eccentricity (LID=O.5,llL=O.5, a=85", ai=70.7O).
Af is the equivalent friction area of each pad, i.e. the sum of the land area Al and of a fraction of the recess area A,. (Eqn 5.114).The reference power ratio is: (7.111) and the temperature step (from Eqn 5.7):
For remarks on the design, the reader may refer to the multirecess bearings (section 8.2.3).In particular, we wish to stress the advisability of planning for low values of p and l7 when the value of clearance C is at the maximum allowed by the tolerances. Finally, it is worthwhile to mention that the complete and optimized design of multipad journal bearings is the subject of a German industry standard (see ref. 7.12 and 7.13).
OPPOSED-PAD AND MlJlTlPAD BEARINGS
235
REFERENCES 7.1 7.2
7.3
7.4 7.6
7.6 7.7
7.8 7.9
7.10
7.11 7.12
7.13
Mueller-Gerbes H., Ernst P.; Maschinenelemente z u m Umformen Drehender in Geradlininge Bewegung; Werkstatt und Betrieb, 11,2 (1978),65-77. Mizumoto H., Matsubara T., Makimoto Y.; A Hydrostatically Controlled Restriction System for a Hydrostatic Lead Screw; Bull. Japan SOC. of Precision Eng.,20 (19861, 195-196. Mizumoto H., Okasaki S., Matsubara T., Usuki M.; An Active Restriction System for a Hydrostatic Lead Screw; 7th Word Congr. IFToMM, Seville, 1987; 4 P. Mizumoto H., Matsubara T., Kubo M.; Effective Improvements in the Design of Hydrostatic Lead Screw; Proc. 24th Int. MTDR Conf., 1983; p. 369-374. Bassani R.; Z Pattini Zdrostatici Contrapposti Autoregolatori della Portata d i Lubrificante; Ingegneria Meccanica, 24,12 (1975)) 11-18. Bassani R.; The Flow Self-Regulating Hydrostatic Screw and N u t ; ASME Trans., J. Lubr. Tech., 101 (19791,364-375. Bassani R., Piccigallo B.; The Dynamic Performance of the Self-Regulated Hydrostatic Opposed Pad Bearing; AGARD Conf. Proc. 323, 1982, paper 21; 12 p. Bassani R., Recchia L.; Slitta Zdrostatica Alimentata Tramite Divisori d i Flusso a Spola Conica; Ingegneria, 1979, p. 129-137. Decker O., Shapiro W.; Computer-Aided Design of Hydrostatic Bearings for Machine Tool Applications. Part 1: Analytical Foundation; Proc. 9th Int. MTDR Cod., 1968; p. 797-818. Decker O., Shapiro W.; Computer-Aided Design of Hydrostatic Bearings for Machine Tool Applications. Part 2: Applications; Proc. 9th Int. MTDR Conf., 1968; p. 819-834. Rippel H. C.; Hydrostatic Bearings. Part 9: Single and Multiple-Pad Journal Bearings; Machine Design, Nov. 12, 1963; p. 199-206. Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Berechnung von Olgeschmierten Gleitlagern mit Zwischennuten); DIN 31656, Teil 1, Nov. 1984; 29 p. Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Kenngroben f u r die Berechnung von Olgeschmierten Gleitlagern mit Zwischennuten); DIN 31656, Teil2, Nov. 1984; 9 p.
Chapter 8
MULTIRECESS BEARINGS
8.1
INTRODUCTION
A multirecess journal bearing may be thought of as being derived from a multipad bearing, in which the drainage grooves separating the pads disappear (Fig. 1.12.b). Even though these bearings are simpler to build, more effective and therefore much more widely used than the multipads, they are rather more difficult to calculate, since a certain amount of flow is now exchanged between the adjacent recesses.
In Chapter 1it has already been noted that multirecess bearings may be classified according to the direction of the load that they are able to bear; namely, we have axial bearings (Fig. 1.3.c), radial bearings (Fig. 1.12.b), conical and spherical bearings (Fig. 1.13).A typical application of combined radial and axial bearings, as well as of conical bearings, is in the spindles of machine tools.
8.2
ANALYSIS
The problem consists in solving the relevant Reynolds equation on the developed land surface. For example, in the case of cylindrical journal bearings (Fig. 8.11, Eqn 4.18 should be used, that, in static loading conditions, is reduced to the following:
which may be solved by means of numerical methods, such as finite differences or finite element methods.
237
MULTIRECESS BEARINGS
Fig. 8.1 Multirecess journal bearing: a- cross section; b- developed surface of sleeve.
A problem now arises, since the boundary conditions are not explicitly known. Indeed we have p=O on So andp=pi on Si, but the recess pressures pi depend on the supply system and are related to the (unknown) flow rates Qi. For instance, in the common case of compensation by means of fixed laminar-flow restrictors, we have:
A solution can be found (ref. B.l), thanks to the linearity of the %operator, splitting the differential problem a8 follows: Q&o)=Gp~CL?sin
p(O)=O on So ;
p(0)= 0
on Sj
(j=l ...n)
These n + l differential problems can be solved t o find the pressure fields p(i)(note that, after discretization, this may be accomplished by means of only one matrix inversion; see, for instance, ref. 8.2). One can now evaluate the rates of flow across the boundaries Si:
238
HYDROSTATIC LUBRlCATlON
I t is now clear that p=L?p(O)+Cpip(i),and hence Qi=QQY’+ZpjQY’. Since the recess flow rates Qi are also given by Eqns 8.2, the actual values of the recess pressures pi can be obtained, and, consequently, the pressure field p and the actual flow rates Qi. For different compensation systems, one should simply replace Eqns 8.2 by the appropriate set of equations; for instance, in the case of constant-flow supply, these are simply Qi=const. The components of the load can be calculated by integrating the pressure field on the recesses and on the land surfaces. The superposition technique is clearly very effective, but cannot be directly used when linearity is lost, as in the case of cavitation. Other workers prefer to solve Eqn 8.1 and Eqns 8.2 together (ref. 8.3) or to start with estimated values of recess pressures and then to correct them by means of an iterative procedure (ref. 8.4). To account for cavitation, since the boundaries of the cavitated region are not known a priori, an iterative procedure must generally be used (for each step, it is common practice to assume p=O in the region where negative pressure was found in the previous section). The foregoing procedures entail the use of computer programmes requiring lengthy calculations. Obviously, then, a number of simplified procedures have been devised to allow easier, but fairly accurate, evaluation of the bearing performance (ref. 8.5 is one of the earliest works on this matter). We shall now briefly explain the widely-used method of lumped resistances (see also ref. 8.6), also known as thinland method. The lubricant flowing out from the i-th recess can be divided into three parts: q f ) going into recess i-1, qj+) flowing into i + l (circumferential flows), and the axial
flow qjo). Of course, Qi=qi-)+q!+)+qj0)and qi-)=-q{:), qp)=-qiii),Fig. 8.2.a. Each qi depends on the geometrical configuration of the bearing ( E , #) and on the recess pressures. It may be stated as follows:
The hydraulic resistances Rf” and Riij are calculated approximately by solving the Reynolds equation on the rectangular lands, with the further simplifying hypothesis that the flow is directed only in the circumferential or axial direction, respectively. So, for low turning velocities and #=O, we find:
239
The pi's are now easily obtained, by schematizing the bearing as a network of hydraulic resistances (Fig. 8.2.b). Now, from the Reynolds equation, the pressure field is known approximately over the entire land surface of the bearing; consequently, its load capacity can be calculated.
-a-
Fig. 8.2 Multirecess journal bearing: a- axial and circumferential flow paths; b- equivalent hydraulic circuit.
The method can be extended to account for the effect of the velocity R (ref. 8.7) and for different compensation systems. Good results are obtained, except for large displacements ( ~ > 0 . 6 o) r large lands, that, in any case, should be avoided in practice. Most of the numerical results reported in the following sections were obtained by means of finite-element computer codes (ref. 8.8, 5.38).
8.3
MULTIRECESS JOURNAL BEARINGS
In the following section we shall introduce plots of the non-dimensional load and of the non-dimensional flow rate, a s functions of the displacement and of the concentric pressure ratio /J, for a few typical geometries. Only the use of laminar
240
HYDROSTATIC LUBRICATlON
restrictors a s compensating devices will be considered in detail (plots for orifice compensation may be found, for instance, in ref. 8.9 and for constant-flow valves in ref. 8.10). Parameter p is defined, as usual, a s the ratio of recess pressures pi, in the concentric configuration, to the constant supply pressure p s . A few approximate expressions will also be given for load capacity, flow rate and stiffness. The influence of the main geometrical parameters will then be explained. Finally, a number of remarks will be made about the design of such journal bearings.
8.3.1
Bearing performance
In Fig. 8.3, the non-dimensional load capacity
w'=- W PSLD
(8.3)
of a four-recess bearing is plotted as a function of eccentricity E for certain values of ratio L I D and of pressure ratio p. It is assumed that laminar restrictors are used as compensating devices. In the next section, we shall show how load capacity may be affected by the varioue geometrical parameters.
In ref. 8.11 a systematic algorithm is used to obtain the load capacity, at small
-a-
-b0.4
0.4
W -
W -
LDPS
LDPS
0.2
0.1
0.L
0.0
0.8
0.4 E
0.0 0.0
0.4
0.8 E
Fig. 8.3 Multkecess journal bearings: load versus eccentricity, for certain values of pressure ratio3 and of ratio LID ( n 4 , a'4.25, 8=30', $4').
241
MULTIRECESS BEARINGS
eccentricities, €or bearings with any number of recesses and different supply systems. The result, when expressed in the nomenclature of the present book, is: (8.4)
W = 6 p ~ ( - a1’ ) A
where the non-dimensional parameter A depends on the geometry of the bearing and on the characteristics of the supply system. Unfortunately, the equations that allow us to calculate A are rather complicated. In the particular case of n=4, however, we have simply n
1
where 0=0, 1, 2, for constant flow, orifice compensation and capillary compensation, respectively. For n=6, we have plotted A in Fig. 8.4.
-.-
a=const
---
Orniies
- Capillaries
0.0 0
2
1
rn Fig. 8.4 Multirecess journal bearings: bearing parameter A versus geometric parameter rn=2d(1-2a’)lBD2.
In ref. 8.12 another approximate equation is given, valid for any number of recesses; using the same notation we have:
242
HYDROSTATIC LUBRICATION
It is easy to see that it does not coincide, sometimes significantly, with the results of ref. 8.11. Although the load capacity increases with E, even when ~ - 0 . 8 ,it is advisable in designing not to consider values of E greater than 0.5. Indeed, a t high eccentricities, bearing stiffness rapidly falls (see Fig. 8.51, and then there is not guarantee of a good safety margin against overloads or against geometric errors of the mating surfaces. -a-
-b0.6
0.6
0.4 -
0.4
K L Dp,/C
K
LDPX 0.2
-
0
0.2
I
0
0.2
I
I
0.4
0.6
E
0
0.8
0.6
0. 8
Fig. 8.5 Stiffness versus: a- eccentricity, and b- pressure ratio, for a journal bearing with n=4, LID=l, a'=0.25, 0=30°.
The effect of the pressure ratio p and of the attitude angle $ on load capacity is easier to see in Fig. 8.6.a. For low values of and high eccentricities, the load capacity is clearly considerably affected by $. This may lead to small oscillations of the journal when the load rotates in relation to the recesses. In the most common cases, the variation of W with $ creates no serious problems; however, i t may be effectively counterbalanced by using a greater number of recesses n (see section 8.3.2). The choice of p has a great influence on load capacity (and hence on stiffness: see Fig. 8.5) as well as on flow rate. Load capacity and stiffness have a maximum which depends on E and $. /3=0.5 is often referred to as an "optimal"value, while it is advisable to avoid values outside the 0.3c~c0.7range. It should be borne in mind that the actual value of will depend on the actual value of the radial clearance C (unless adjustable restrictors are used) and then the working tolerances have a direct influence on the behaviour of the bearing; this point will be examined more closely later. Stiffness may be deduced from the slope of the (W-E)characteristics of the bearing. In Fig. 8.5 the stiffness of a typical four-pad bearing is plotted as a function of
MULTIRECESS BEARINGS
243
the displacement and of the pressure ratio. It may be seen that, in the most useful range of 8 , stiffness does not vary much with E, thus the reference stiffness Ko=K(&=O) may be used with sufficient approximation up to ~ = 0 . 5 An . approximate value for KOimmediately follows from Eqn 8.4:
LD (1 - u ' ) A KO 6 B p , 7
(8.7)
-a0.4
-9
'
I
/
/-
= 0
\ \
Fig. 8.6 a- load and b- flow rate, versus pressure ratio for certain values of eccentricity, for a journal bearing with n=4, L / D = l , ~ ' 4 . 2 50=30°. ,
Up to now we have assumed that the turning velocity of the journal is small enough to have no practical effect on load capacity. Actually, if both E and R are not null, a hydrodynamic pressure field (whose resultant is perpendicular t o the direction of eccentricity) is superimposed on the hydrostatic one. Consequently, 4 no longer coincides with the loading angle +L (strictly speaking, even when R=O, we have exactly +=$L only when +L=O or when it is a multiple of d2n). The hydrodynamic effect is particularly marked at high eccentricities, but, in the most common cases ( ~ < 0 . 5and moderate velocities), it may be disregarded in designing, especially a s the bearing stiffness turns out to be increased. On the other hand, at high velocities the increase in static stiffness may become important. The influence of the velocity parameter
244
HYDROSTATIC LUBRICATION
-a-
2.0 W
'
0.0
0.4
0.2
0.0
Sh
0.4
0.2 Sh
Fig. 8.7 a- load and b- attitude angle, versus speed parameter S (Sommerfeld hybrid number) for two journal bearings with different ratio LID and n=4, a'd.25, b=3Oo, #=Oo,J?=0.5.
(often called the "Sommerfeld hybrid number") on W and on the angle #L-#,is shown in Fig. 8.7 for a few typical cases. Hydrodynamic load capacity is calculated in ref. 8.11 as
Total load capacity is the vectorial sum of the hydrostatic and hydrodynamic terms; therefore, a "speed enhancement factor" may be calculated: (8.10) which is not dependent on the number of recesses. For small values of E, the actual load capacity may be approximately evaluated by multiplying W(a=O)by the same factor K , . Other workers suggest different equations. For instance, in ref. 8.12 similar results have been obtained, except for (1- 2 a ' ) in Eqn 8.9 and Eqn 8.10 is substituted with (1- a'). In consequence of the above, it seems profitable to design in such a way as to have high values of Sh, in order to take advantage of the great increase in load capacity. However, it must be stressed that the increase in the speed parameter creates certain undesirable consequences. In particular, in certain recesses, the pres-
MULTIRECESS BEARINGS
245
sure becomes greater than the supply pressure, causing a reverse flow in the restrictor; in other recesses, the pressure tends to become lower than the ambient pressure, causing cavitation and entrainment of air in the lubricant. These and other factors, such as non-linear behaviour of restrictors and thermal effects, may notably affect bearing performance; indeed, experimental evidence (ref. 8.13 and ref. 8.14) has shown that, in certain cases, load capacity may even be worsened by rotational speed. There is also great friction a t high speed, especially when a turbulent flow recirculation starts up in the recesses. Moreover, if a bearing is designed for high-speed operation, the alternative choice of a hybrid or purely hydrodynamic bearing should be considered. Finally, for high speeds, the damping progressively decreases, and the dynamic behaviour of the system gets worse, until, a t a critical speed, instability sets in (see Chapter 10). When the speed is high i t is necessary to check that the Reynolds number
R e = -1 -PC D R 2P
is smaller than 1000: above this value the effects of turbulence in the lubricating film can no longer be disregarded. Turbulent flow is unusual in oil-lubricated hydrostatic bearings: on the contrary, it is often present in certain particular applications, in which low-viscosity fluid (such as water, liquid metals or cryogenic fluids) are used. A considerable amount of work has been done on this matter and complete numerical solutions have been obtained in which turbulence a s well a s the effects of inertia, cavitation and shaft misalignment are taken into account: see, for instance, ref. 8.15. The flow rate of the lubricant is not greatly affected by eccentricity (at least when &<0.6,as shown in Fig. 8.6.b), nor by the attitude angle 9, whereas in practice it is proportional to p. The non-dimensional flow rate
-&§!
Q'-ps
c3
(8.11)
may be read in the plots given in Fig. 8.6.b and in the next section, or evaluated by means of the approximate equation (8.12)
Since Eqn 8.12 is a limiting value, calculated a t &=O, disregarding the effect of the axial lands, it proves to be slightly overestimated. The pumping power is clearly
246
HYDRWTAT1C LUBRlCATlON
(8.13)
and hence the variations of H p with E and
I$
are also small.
The friction power may be considered to be the s u m of two terms:
which represent the contributions of the lands and of the recesses. Land friction Hfi should be calculated from the integral
extended to the whole land surface, whose area is: (8.14)
The shear stress z is given by the Newton formula (see sections 3.2 and 4.5.3). The velocity field is made up of the superposition of a "Couette" flow (due to the turning velocity 52) and of a "Poiseuille" flow (due to the pressure gradient). In the centered configuration, however, the shearing stress due to the pressure-induced flow constitutes a symmetric field, whose integral, therefore, vanishes. The velocity gradient may be substituted by its average value LlDJ2C, and, hence: (8.15)
Even when the bearing is displaced, the contribution of the Poiseuille-type flows to the friction is, in general, small, and the equation (8.16)
gives a good approximation, while more accurate equations may be found in ref. 8.16 or in ref. 8.17. The power loss in the recess may be comparable to Hfl, and it may even become dominant if the fluid recirculation in the recess is turbulent. The friction power lost in the recess may be expressed in the following form: (8.17)
MULTIRECESS BEARINGS
247
where A,. is the total recess area:
(
"7
A,. = R D L (1 - 2a') 1 -%
(8.18)
The coefficient f, depends on the ratio Clh, and on the Reynolds number 1P Re, = -D 52 h,
2P
(8.19)
If we have Rer
(8.20)
In the case of turbulent flow, f, may be calculated from Eqn 4.54. Lastly, the friction moment and the friction power may be written as follows: D2 1 M f =;I p R A f
(8.21)
Hf=RMf
(8.22)
c
where the equivalent friction area Af may be calculated from the following equation: (8.23)
Plots of A; are given in Fig. 8.8.
A t the highest speeds the torque required to accelerate the fluid entering the bearing (momentum torque) may become notable. It may be calculated (ref. 8.19) as: 1 D MQ = ;IP Q Re
c
where the Reynolds number is R e = -1 -PC D R 2P
The relevant power loss is obviously H Q = N Q .
It is interesting to compare HQ with the pumping power:
(8.24)
248
HYDROSTATIC LUBRICATION
- b-
-aE-0.5
0.3 0
% 20000
10000
0.5'
0.1
'
'
'
'
'
0.2
.
.
a'
'
.
'
0.3
0.8 0.1
0.2
0.3 a'
Fig. 8.8 Multirecess journal bearings: equivalent friction area A? versus ratio a'; a- for certain values of eccentricity' E and of ratio h,JC (ReS1000); b- for certain values of the Reynolds number (~4 h,./C=50). ,
If we assume, for instance, that supply pressure is ps= 4 MPa, it is easy to see that a velocity of the journal as high as 10 m/s is needed for H g to be 1% of Hp. Since in common design practice we generally have Hpmf, i t follows that momentum torque may be disregarded in the vast majority of cases. The power ratio can be evaluated, from the foregoing results, as follows:
(8.25) Finally, it is easy to evaluate the temperature step in the lubricant. If we assume, as usual, that flow is adiabatic, for E=O we have: (8.26)
Since both Hfand H p undergo only small variations when the bearing is displaced, AT may also be considered, roughly speaking, not to be dependent on the loading conditions.
249
MULTIRECESS BEARINGS
Effect of geometrical parameters
8.3.2
The geometrical ratios which may affect both load capacity and flow rate are LID, a'=alL, 8 (see Fig. 8.1). Besides, the number of the recesses should also be considered. The value of n could, in theory, be anything, provided it is greater than 3, but values greater than 6 are seldom encountered; indeed, the benefit of further increasing n diminishes and hardly counterbalances the greater manufacturing costs and the greater complexity of the supply system. In brief, the main consequences of increasing the number of recesses are that load capacity and stiffness are higher and, moreover, are less dependent on the attitude angle $. The rate of flow a t any given eccentricity is practically unaffected. The load capacity of a 6-recess bearing is plotted in Fig. 8.9, which should be compared with the analogous Fig. 8.6. -a-
-b4
3.2
Q21A P,C"P 1.6
0.8
0
I
I
I
Q2
0.4
0.6
B
0 0.8
1
0
0.2
0.4
0.6
0.8
1
B
Fig. 8.9 a- load and b- flow rate, versus pressure ratio for certain values of eccentricity, for a , journal bearing with n=6, L/D=l, ~ ' 4 . 2 5 8=30°.
The effect of a l L and L I D is shown in Fig. 8.10. To increase the parameter a l l means to increase the land area to the detriment of the recess area; consequently, the load capacity decreases with a quasi-linear trend. On the other hand, the flowrate is approximately proportional to L 1a;since the pumping power is proportional to Q ' I W'2 (once load W, eccentricity E, and the other geometrical parameters are fixed), a value of a' clearly exists a t which the required pumping power is a minimum. This occurs, roughly speaking, when a'=0.3; however, this value is not critical and, in general, it proves to be advisable to design a larger recess to step up load capacity and reduce friction, which, indeed, mainly depends on the developed land
250
HYDROSTATIC LUBRICATION
-a-
0.0
0.1
0.2
a'
I 0.3
-b-
"
0.1
0.2
a'
0.3
Fig. 8.10 Multirecess journal bearings: a- load and b- flow rate, versus a', for certain values of ratio LID ( n 4 , 8=30",$=45O,J=O.4).
area Al. Most authors propose "optimal" values for a' in the 0.2510.1 range, or even smaller (see ref. 8.17 for a review); some maintain that a recess area A,.=O.hDL is a good compromise between high load capacity and low power loss (see ref. 8.20). Actually, a more detailed analysis (ref. 8.21) would show that even a t moderate i t is convenient to select small values of a' (a'=0.1)in order to reduce Ht speeds (n3), the power loss is the smallest for vanishing recesses: this explains the peculiar design of hybrid bearings (see chapter 9). Concerning LID, since W' decreases as LID grows, it is clearly more convenient to increase D instead of L in order to enlarge the projected area and hence increase load capacity. On the other hand, flow rate and pumping power are approximately proportional to D J L ,and do not depend on the projected area. Furthermore, Giction power is proportional to D4L (Eqn 8.22). Thus, to reduce power loss, i t seem advisable to have high L 1D ratios. L /D=1 is often proposed as a good compromise.
251
MULTIRECESS BEARINGS
The effect of the land angle 8 is shown in Fig. 8.11. As can be seen, the flow-rate does not vary so much with 0, whereas the load capacity improves, increasing 8,to a certain extent, which depends on E, 4, a' and p. This is not surprising, because the separation of the recesses is improved even though the recess area is reduced. However, values of 8 greater than 2 d 3 n (i.e. 30 degrees for n=4) provide no great benefit, while friction power always grows with 8. Also H , and AT are influenced by the recess angle 8.In ref. 8.21 it is shown that the temperature step decreases when 8 is increased up to a certain value, which depends on ll and a', and then remains practically constant. For instance, for L I D = l , n=4,8=0.5, a'=0.1 and I7=1, 8 should not be smaller than 18". Greater values may be suggested for higher values of I7 and a'. From the point of view of total power an optimal value of 8 may be determined: for n=4 and moderate speed, this vanes in the 18"+30"range, when a' is between 0.1 and 0.2. In ref. 8.23 i t is suggested that8 =0.5 rad be selected for n=4 and L I D = l and smaller values when n is greater and L I D is smaller (e.g. 8 =0.5 for n=12 and LID4.3).
-a-
0.4
W -
LDP,
0.2
-b-
a'=0.1
@ 0.2 0.25
0.0 10"
"
20"
30"
40"
10"
20"
30"
40"
Fig. 8.1 1 Multirecess journal bearings: a- load and b- concentric flow rate, versus land angle 8, for certain values of ratio a'; ( n 4 ,L/D=l,J=O.4).
8.3.3
Design of multirecess journal bearings
Design criteria for journal bearings are not so different from those proposed in section 7.2.6, concerning opposed-pad bearings. As usual i t will be assumed that the main objective consists in obtaining a certain load capacity W, with a smaller dis-
252
HYDROSTATIC LUBRlCATlON
placement than eM=EMC.Of course, several further constraints may be added, such as given supply pressure o r viscosity. Optimization may be tried by means of the usual criterion of minimum power consumption (see also chapter 5). Since total power H , does not generally experience great variations for usual eccentricities, analysis may be limited to the centered configuration (E=O). Total power is the sum of pumping power and friction power:
Equation 8.25 gives (8.27)
and, hence, (8.28)
Shaft diameter and supply pressure cannot be regarded as independent variables, since their selection is connected with the load capacity of the bearing (Eqn 8.3)
D
WM
Ps D2 = L W'(EM;rO)
(8.29)
Finally, total power and the relevant value of viscosity may be written in the form (8.30)
(8.31)
where (8.32)
(8.33)
The non-dimensional quantities H: and 1 ; can be calculated approximately (e.g. by means of Eqn 8.4, Eqn 8.12 and Eqn 8.23) or read directly in Fig. 8.12.
253
MULTIRECESS BEARINGS
Examining Eqn 8.30, one may easily draw the same conclusions as those reached in the case of Eqn 6.57 or Eqn 7.64, concerning thrust bearings. For given values of clearance and speed, the "optimal" value of viscosity is the one which ensures a power ratio n=1.However this value is not critical, since Ht experiences only small variations for values of the power ratio in the range 1/3t3. Concerning clearance, i t is clearly convenient to reduce i t a s far a s possible, in order to obtain small power and high stiffness. Limits to the selection of very small clearances follow from two types of considerations. Firstly, Eqn 8.31 shows that a very small clearance may require an impracticably low viscosity to obtain n=1,even after a small land width has been selected, in order to reduce pp*. In certain cases the value of viscosity is a design constraint, and i t may be shown that when p is given, the lowest value of Hto is obtained by selecting C in such a way as to have n=3.
-b-
1.5
- &=0.5 --- €10.3
0.0'
I
0.1
0.2
a'
0.3
0.1
.
I
0.2
.
.
a'
.
.
0.3
Fig. 8.12 Multirecess journal bearings: a- total power H: and b- relevant viscosity p; versus ratio a' for certain values of ratio LID and of working eccentricity E; (n=4, 8=3#",$=45",J=#.4, f,=#.2) 1
Another limit to the selection of C is connected with manufacturing tolerances (ref. 8.241. Indeed, the designer must often select a whole range CmlCIC, instead of a single value and, if the tolerance 6g=2(CM-Cm)is large, the actual performance of the bearing may differ notably from the one calculated. Let us assume that the maximum radial clearance CM.has been selected in such a way a s to satisfy all the design constraints with a certain value B of the concentric pressure ratio. When
254
HYDROSTATIC LUBRICATION
E=O, the flow rate is proportional to will be
1
fill=
1-p c 3 l + P (G)
C-3: it
is now clear that the actual pressure ratio
(8.34)
Since it is convenient to impose an upper limit /lM on the pressure ratio, to avoid a sharp decrease in load capacity (see Fig. 8.6.a)) we readily obtain
(8.35)
For example, if p=0.4 and 8 ~ = 0 . 7we , must have c , / c ~ > 0 . 6 6 , i.e. 6g<2/3'cMM. Consequently, CM cannot be too small, otherwise an excessively small tolerance would be required. The actual value of the power ratio also depends on the actual value of clearance. Namely, if n is the designed power ratio tallying with the greatest clearance , C for any other clearance we get
which, strictly speaking, is not valid when recess friction makes a predominant contribution to the total power loss (e.g. in the case of turbulent flow in the recess). It is worth noting that if p=0.4, n = 1 and c,=2/3'cM, the greatest relevant value of the concentric power ratio proves to be close to 3. We wish to stress that the foregoing considerations concerning the optimal design of such bearings are based on a simplified analysis, which nevertheless gives good results. In particular, the increase in load capacity due to the hydrodynamic effect has not been taken into account, and total power has been evaluated for E=O instead of a t a working eccentricity. It is also worth noting that there is a turbulent recess flow in the case of high-speed bearings. Consequently, coefficient f,is much greater than the value assumed in Fig. 8.12 and, above all, H t and p; cannot be considered to be virtually independent from p, D and C . Several authors express different points of view, obtaining somewhat different results (see, for example, ref. 8.17 and 8.18 for a review). In particular, it must be remembered that the optimized design of multirecess journal bearings is the subject of a German industrial standard (ref. 8.22 and 8.23). Direct optimization methods have also been proposed, based on well-known optimization algorithms such as the complex method or the flexible polyhedron method, in which the object function (e.g. the H , J W ratio) is found by solving the Reynolds equation by means of finite-difference or finiteelement methods (ref. 8.25).
255
MULTIRECESS BEARINGS
Besides the effects of the error on the clearance value, to be controlled by means of size tolerancing as indicated above, other kinds of manufacturing errors may affect the bearing performance. For instance, ovality may cause a loss of stiffness and greater directionality. In ref. 8.26 it is reported that a 10% ovality (with respect to the mean clearance) causes a loss of stiffness in the order of 20% in a typical bearing. The effects of journal misalignment on load capacity have been studied in ref. 8.27, by means of thin-land analysis, and the results have been verified by means of experiments. It has been concluded that misalignment reduces load capacity and causes directionality of stiffness. The effect is most marked near the optimum pressure ratio @=0.5);it is rather small for values of slope 6 lower than 0.2, but becomes quickly appreciable for higher values (a loss of stiffness smaller than 10% may be assumed when 6=0.4). Slope 6 used in ref. 8.27 may be practically defined as iy being the angle a t which the shaft is tilted. 6=(L/2C)tg~,
It may be concluded that clearance should be carefully selected, also bearing in mind that a very small clearance enhances the effects of journal misalignment; moreover, i t is recommended to design for moderate eccentricities (e.g. ~ ~ = 0 . 5 ) , thus allowing considerable safety margins to compensate for the effects of manufacturing errors. Finally, for the sake of completeness, a number of remarks may be made on the roughness of land surfaces and on the shape of recesses. Roughness is not generally a problem. Indeed, it is widely accepted that the roughness of the land surfaces should be smaller than 0.1 times the lowest operating film thickness, which often leads to values of R, greater than 1 pm, that are quite easy to obtain by grinding. The roughness of pockets may even be much greater, except in the case of the high speed bearings, where it may cause turbulence. For recesses, a shape of type (a) in Fig. 8.13 requires a dismountable construction of the bearing sleeve, so a shape of type (b) is often selected, because i t may be obtained by milling; of course, the convergent recess outlet may involve some differ-
-a-
/----\. i
Fig. 8.13 Shapes of recesses in journal bearings.
\
-b-
256
HYDROSTATIC LUBRICATION
ences in the behaviour of the bearing. Particular recesses have also been proposed in order to reduce recess friction in high-speed bearings (ref. 8.28).
8.3.4
Design procedure
Following the remarks made above, a simple procedure may be proposed, that should lead to a n "optimized" design of opposed-pad bearings. As before it will be assumed that a certain load WM is given, to be sustained with a displacement smaller than eM at a speed iz; some further constraints may be required: the design may then be carried out following the steps below. i)
Choose a trial set of geometric parameters; perhaps L I D = l , alL=0.15, 0=30°, n=4; i t is convenient to select large recesses (i.e. small values of a ) unless low speeds are expected. Choose a trial value of the maximum eccentricity, such a s ~~=0.5. Take p=0.4 as the design pressure ratio: a smaller value can be accepted if n>4 or &&0.5.
ii)
Evaluate coefficient W' by means of figures given in the preceding sections. Select a suitable supply pressure ps, if it is not given. Calculate a first value for diameter D, by means of Eqn 8.29. Check the temperature step (Eqn 8.26); since for many mineral oils we have ~ ~ - 1 . 6 . J/m3"C, 1 0 ~ and i t should be n<3, the maximum temperature step is expected to be AT
iii)
Select a suitable value for the manufacturing tolerance Sg=2(C~-Cm): see also Fig. 8.14; bear in mind that the narrower Sg is, the lower the power losses can be. Take Crn+ and cMM=cm+6g/2. Check if stiffness is great enough. Otherwise reduce Sg or select a smaller value for EM, restarting from point (ii).
iv)
Calculate the "optimal"value of viscosity, i.e. the value k p t that leads to n=1 when C=C, (Eqn 8.31). Check that kptis 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), the clearance or the supply pressure. However, if R is very low it will not be possible to get n=1(indeed, for 8=0,a n "infinite" viscosity would be required!); in this case select the highest allowable viscosity, and a larger land width, such as a'=0.25, On the contrary, a very high speed would lead to values too small for h p t ; in this case it will be necessary to accept values of the maximum power ratio (tallying with the least clearance C,) greater than 3: the ensuing hydrodynamic effect will enhance
257
MULTIRECESS BEARINGS
iool IT=7
"1 0
18
30
50
80
120
180 250
D [mml Fig. 8.14 Values of parameter 6g=2(C,+,-C,),
for certain international tolerance (IT) grades
(ISOIR286). bearing stiffness, but any of the problems already noted might arise (cavitation, excessive warming of lubricant, and even dynamic instability). v)
Calculate flow rate, pumping power, friction torque and friction power, with reference to the design configuration (C=CM,E=O). Check the same parameters for different values of C and E. If flow rate seems too great, and 6 , cannot be further reduced, i t will be necessary to accept a power ratio n>l (e.g. increasing p or reducing p s ) consequently increasing friction. Conversely, friction may be easily reduced with a larger expense of flow rate and pumping power. If the supply system is not able to maintain lubricant viscosity within a narrow range, check friction and the temperature step for the maximum expected viscosity, and check the flow rate for the minimum viscosity.
vi)
Design the compensating restrictors.
vii)
Check the Reynolds number in the bearing clearances and in the recess.
viii)
Check the dynamic behaviour of the system (see Chapter 10).
EXAMPLE 8.1 Design a multirecess bearing to sustain a journal, whose diameter is D=80 mm, rotating at a speed i2110.5 radls (-1000rpm); the operating load WltO KN must be sustained with a displacement smaller than e ~ = 2 pm; 0 manufacturing tolerances
258
HYDROSTATIC LUBRICATION
should not be smaller than IT5 grade for both journal and sleeve; supply pressure may not be greater than 5 MPa. i) Let us select a trial set of parameters: LID=l, a'=0.15, n=4, 9=30°, ~ ~ = 0 . 5 , 8=0.4. ii) From Fig, 8.11 we may read W ( ~ ~ ) = 0 . 2 6it9is ; clear (Eqn 8.29) that, due to the constraints on pressure and shaft diameter, the maximum operating load cannot be sustained with a n eccentricity ~ ~ = 0(actually .5 it should be W p 8 . 6 KN). I n order to increase load capacity, one may try to increase the axial length o f the sleeve and to reduce the width of the circumferential lands (to increase the number o f recesses would also have a notable effect on load capacity). Selecting LID=1.25 and a'=O.1, a simple interpolation of available data gives W'(&=0.274, and hence wM=ll.o kW with a supply pressure ps=5 MPa. iii) Since the tolerance grade is given, the width of the tolerance range for diametral clearance must be Gg=2(C~-C,,J=26 pm; hence, a suitable choice for radial clearance may be C=40+27 pm. Note that, assuming h o = C ~ = 4 p0 m as the reference clearance, a t the eccentricity EM =0.5 we have a n actual displacement e=CMEM=20pm; for smaller clearances, of course, stiffness is greater. iv) From Fig. 8.12.b one may obtain pi=1.12, thus to have Il=1 a viscosity p 4 0 2 0 N s 1m2 is required (Eqn 8.31). v) Non-dimensional flow rate may be evaluated by means of Fig. 8.10.6 as Q'(O)=1.57; for the "optimal" viscosity calculated above, it follows that Q=25.10-s m 3 / s , when C=CM,and Hp0=126 W , Stating hr=l mm,it follows that frZO.155 (Eqn 8.20); Eqn 8.23gives A;=0.549 and hence Hf0=122 W, and Mf=1.16 N m ; at the eccenit is easy to calculate: Aj-=O.622,Hf=138 W, and Mf=1.31 N m . IL intricity ~ = 0 . 5 stead, the actual clearance is the lowest value allowed C,=27 pm, the actual pressure ratio becomes 8,,=0.684; nevertheless flow rate and pumping power will be nearly halved; on the contrary friction will be notably increased, giving Aj=0.523, Hf=172 W, Mf=1.64 N m at E=O; Aj=0.595, Hf=195 W, Mf=1.86 N m at ~=0.5. All these results are subject to change when actual viscosity does not tally with the value assumed above; in particular during starting-up operations, with cool lubricant, p may be much greater, and hence friction will also be proportionally higher; i n this condition, however, the temperature step is also greater, contributing to bringing the lubricant to the normal operating temperature. vi) Since flow rate is low, it is possible to use small bore pipes as compensating restrictors; its hydraulic resistance in the design conditions (i.e. C=cM=40 pm, 8=0.4, p=0.02 N s l m z ) should be Rr=(l-8)p,l(Q/n)=480.109 Nslm5. From Eqn 4.67 it follows that, using a pipe with a 1 m m internal bore, a length 1=589 mm is necessary for each restrictor; the relevant Reynolds number proves to be low enough. It is obviously possible to select a smaller bore to reduce the pipe length; it is, however, necessary to check the relevant Reynolds number.
259
MULTIRECESS BEARINGS
vii) It is easy to verify by means of Eqn 8.19 that the Reynolds number in the recesses is quite low (Rer=189 assuming p=900 @In3). The Reynolds number in the clearance is obviously much smaller. A concluding remark concerning the velocity parameter Sh: this should be in the 0 . 0 6 7 4 1 4 7 range, depending on the actual value of C,and hence a n enhancement of load capacity of between 13% and 21% may be expected (Eqn 8.10). EXAMPLE 8.2 Design a journal bearing able to sustain a load W ~ = 2 5 K 0 N rotating at 0=157 r a d l s (1500 rpm). Total power consumption should be smaller than 100 KW, supply pressure should not be greater than 8 MPa. Let us select L / D = l , alL=O.l, &SO0, ~ ~ = 0 p=0.4. . 5 , It follows that W’=0.306, Q’=l.86, At this point, diameter and pressure ratio may be selected in such a way as to satisfy Eqn 8.29. Since this is clearly a high speed bearing, Eqn 8.31 suggests that supply pressure should not be low, to avoid too small a value o f the optimal viscosity. Let us select D=0.32 m and, hence, p,=7.97 MPa. The next step consists in selecting the value of radial clearance. If we assume 6=50 p m (tallying with IT5 quality grade), we should select the greatest radial clearance as C ~ = h o = 7 5pm: i n this case, however, this does not prove to be a suitable selection. Stating h,=5 mm (and hence fr=0.06), we find Air0.5 (Eqn 8.23) and pt=1.34 (Eqn 8.33); consequently, in order to get n=1 the optimal viscosity should be p=0.006 Nslm2, a rather low value. Furthermore, it is easy to see from Eqn 8.19 that the high velocity of the journal would cause a turbulent recess flow (Rer=19000), and a much higher recess friction factor f , than the value considered above (namely, Eqn 4.54 gives fr=0.78). This leads to a higher value for power ratio n.Of course, if the actual clearance is smaller than cM’75 pm, the power ratio proves to be even greater (e.g. see table below for C=50 pm).
-
---
Mf 0
(Ns/m2
(Nm)
Hf 0 (KW)
0.006 0.010 0.006 0.010 0.013 0.010 0.013 0.016 0.0 13 0.016
115 168 91 128 155 108 129 149 113 130
18.1 26.5 14.3 20.1 24.3 17.0 20.2 23.4 17.8 20.4
P
100 100 100 125 125
0.63 0.48
-
n
AT0
(“0 7.36 17.90 1.73 4.04 6.34 1.44 2.22 3.17 1.oo 1.41
41.6 94.0 13.6 25.1 36.5 12.1 16.0 20.7 9.9 12.0
---
260
HYDROSTATIC LUBRICATION
In order to avoid excessive friction power and too great a temperature step, one may use very narrow tolerances or a lower viscosity (the advisability of selecting viscosity on the basis of the actual value of clearance may also be considered); another suitable approach may consist in increasing clearance, which also makes it possible to select a lubricant with a greater viscosity. Calculations for certain values of C and p are summarized in the above table.
8.4
ANNULAR MULTIRECESS THRUST BEARINGS
The thrust bearing shown in Fig. 8.15 works in a very similar way to the annular-recess pads, seen in section 5.4, from the point of view of axial load capacity. Furthermore, since each recess is supplied through an independent compensating
Fig. 8.15 Annular rnultirecess plane bearing.
261
MULTIRECESS BEARINGS
device, it is able t o react t o tilting moments. In the present work, we shall confine ourselves to considering constant pressure supply systems which feed the recesses through laminar-flow restrictors (capillaries). Calculation of the performance of such bearings is based on the approximate solution of Eqn 4.23 (the Reynolds equation in polar coordinates), for example, by means of the finite-element method (ref. 8.29).
For y=O, the bearing simply behaves like an annular-recess pad, and hence an "effective" area A, may be defined (Eqn 5.1) as the ratio of the load to the recess pressure. A, turns out to be independent from the film thickness, and it may be calculated approximately using Eqn 5.66, which proves to be accurate especially when the angle 8 is small (however, it is advisable for the lands separating the recess to be a t least twice as wide as the annular lands, to improve tilting stiffness). The values plotted in Fig. 8.16.a, on the other hand, were calculated by means of a finite-element computer code (ref. 5.38).
-a-
0.: 0.5
-b-
0.00
0.6
0.7 r'
0.8
'
0.5
0.6
0.7
0.8
r'
Fig. 8.16 Annular multirecess thrust bearings: a- Effective area #,and b- hydraulic resistance R* versus ratio r1/r4for certain values of land width ratio cd(r4-t-I). [ n d ,6=(1-r?.25"]. The flow rate and the film thickness, for any given axial load, as well as the axial stiffness and the pumping power, can be calculated as in section 6.3.1, taking into account that R, is now the equivalent resistance of n restrictors in parallel. The hydraulic resistance of the pad may be written as in Eqn 5.68, in which R* may be read from Fig. 8.16.b or roughly calculated from Eqn 5.69.
262
HYDROSTATIC LUBRICATION
In calculating the friction power, the contribution of the recess, mainly caused by the damming effect of the radial lands, should be taken into account. Evaluating i t as in section 4.7.3, friction moment and friction power may still be expressed by Eqn 5.70 and Eqn 5.72, where: (8.36) Assuming h,.>>h and laminar flow, i.e. a Reynolds number
Re,. =$:
O h , D ( 1 + r')
(8.37)
less than 1000, it may be stated that fr=4h/h,.
At high speed the effects of lubricant inertia become appreciable. Certain numerical calculations are reported in ref. 8.30; although the data are presented in a completely different form, a detailed analysis would show close agreement with the results presented in section 5.4.5 for the continuous-recess pad (i.e. Eqn 5.77 to Eqn 5.80). Centrifugal forces cause an increase in flow rate, for a given recess pressure, and, conversely, a decrease in recess pressure and load capacity when supply pressure or flow rate are given. As has already been noted, the main distinguishing characteristic of these bearings is their ability to sustain tilting moments. Tilting stiffness, defined as the ratio of the tilting moment Mt to angle ry, may be written as: (8.38) The non-dimensional factor K ; is plotted in Fig. 8.17 (taken from ref. 8.31) a s a function of the non-dimensional axial load WIA,ps and of the radius ratio r4/rl.A pressure ratio /3=0.5 is clearly advisable, since it brings maximum stiffness both for axial and tilting loads. Axial load capacity proves, in practice, to be unaffected by a limited tilting angle, a t low speed; the problem may be quite different when tilt is coupled t o high speed. This kind of problem is dealt with in ref. 8.32, in which Eqn 4.23 is solved, for certain bearings. It transpires that load capacity slightly increases with tilt angle v (thanks to the hydrodynamic effect), until cavitation occurs and load capacity breaks down. For the bearings considered in the reference cited above [a'=0.25; 8=30"; /3=0.5; il=(3/8)pLUI2/@,h2)=50]this occurs a t y'=vD/2hz0.4 when r'=0.4.Higher values of y' could be allowed for r50.5. However, the bearings considered i n ref. 8.32 are characterized by high values of power ratio n, thus, it may be concluded that, if the bearing is designed on the usual basis of a power ratio Z7=Hf/Ht smaller than 3,
263
MULTIRECESS BEARINGS
025
0.2
0.15
K; 0.1
\
0.05
0
0.2
0.3
0.4
0.6
0.5
0.7
0.8
4 A,P,
Fig. 8.17 Annular multirecess thrust bearings: tilting stiffness Ki versus load WIA,p, for certain values of ratio r&,.[n=6; a'=0.25].
and large tilting angles are not allowed, no problem should arise. If, on the other hand, the speed is very high, more careful numerical calculations should be carried out. Tilting stiffness also seems to be enhanced by the hydrodynamic effect (ref. 8.291, although the direction of tilt no longer coincides with the direction of the tilting moment. Concerning the design of a multirecess thrust bearing, all the remarks made about the annular-recess pad could be repeated, adding the obvious constraint of the minimum tilting stiffness that may be required by the specifications.
8.5
TAPERED MULTIRECESS BEARINGS
The conical bearing in Fig. 8.18 may be considered as a generalization of the cylindrical journal bearings examined in section 8.3. The main advantage of such bearings is their ability to support loads in the axial a s well as in the radial direction. For this reason, they may be used to substitute assemblies made up of a journal plus a thrust bearing when the thrust has a constant direction.
264
HYDROSTATIC LUBRICATION
Fig. 8.18 Tapered multirecess bearing. Unlike cylindrical bearings, the radial play C is not fixed but depends on axial loading, and hence the single-cone bearings may not be suitable when great variations of the axial thrust are planned. The opposed-cone assemblies, on the other hand, are able to sustain a wide range of loads in every direction (Fig. 8.191, with greater stiffness, thanks to the effect of the preload. In order to assess the performance of tapered multirecess bearings, it is necessary, as outlined in section 8.2, to solve, a t least approximately, the relevant Reynolds equation on the conical surface. From Eqn 4.26, if the inertia of the lubricant and the squeezing components of the surface velocity are disregarded, we obtain:
In the following pages will be found results (load capacity, flow rate) obtained by means of a finite-element technique (ref. 5.38). Other results can be found elsewhere, for instance in ref, 8.33.
Most of the remarks made in section 8.3 could be repeated here, but with certain important differences. First of all, the concentric radial clearance C is no longer a
265
MUL TIRECESS BEARINGS
constant, being related to the axial film thickness, i.e. t o the axial load. Consequently, the actual value of the concentric pressure ratio p also depends on the axial load.
As for the other multirecess bearings, we shall concentrate on the most common constant-pressure supply systems, with capillary compensation.
-a-
_-
-b-
-c--
Fig. 8.19 Arrangements of tapered multirecess bearings. 8.5.1
Slngle-cone journal bearings
Let us begin by examining the performance of the single-cone bearings, fed by means of laminar-flow restrictors. If no radial load is present (i.e. it is E=O), axial load capacity may be written in the following form:
Wz =A, p r = A, P Ps
(8.40)
where, as usual, A, is the effective bearing area and p is the concentric pressure ratio. A, may be considered to be a constant. At least when no radial load is applied,
266
HYDROSTATIC LUBRlCATloN
the conical journal bearing merely behaves like the single-pad thrust bearings examined in chapter 6. Furthermore, even after radial displacement, axial load capacity experiences variations of a few percent (at least for ~<0.6). In other words, radial loads have practically no effect on axial stiffness (whereas the opposite is not true!). In what follows, for the sake of simplicity, we shall consider axial load W, to be independent from radial eccentricity E. Axial stiffness is clearly the same as for single-pad thrust bearings; hence it is given by Eqn 6.22, for laminar-flow restrictor compensation. Nor does the flow rate greatly vary with E (at least for the usual values of E and B); consequently, it may be written in the following form:
P B
w,
Q=*=m
(8.41)
and (8.42)
may be considered a constant. On the other hand, Q clearly depends greatly on the axial load, just as in the case of single-pad bearings (see section 6.3 and Fig. 6.5.a). An initial approximation for (8.43)
and R* can be obtained from the results given in section 5.5, i.e. by considering the multirecess cone as an annular-recess pad with geometrical parameters of the same value, while values obtained by numerical calculation are given in Fig. 8.20, as functions of the semi-cone angle, for certain values of ratio LID. The pumping power is directly calculated from:
(8.44) The friction power developed on the land surface is calculated, when no radial load is applied, by integrating the elementary power
and we easily find: (8.45)
267
MULTIRECESS BEARINGS
-a-
0.e
0.15
-b-
I I I /
UD=1
R'
0.75
10.5
0.6 0.10
A', 0.4
0.05 0.2
0.00
0.0
15"
a
30"
45"
a
Fig. 8.20 Tapered multirecess bearings: a- Effective area A*, and b- hydraulic resistance R* versus semi-cone angle a for certain values of ratio LID; (n=4,8=30°). Friction power, of course, varies with the eccentricity, but it is often enough for the designer to consider the value above alone. When the recesses are large, their contribution to friction should also be taken into account. Proceeding as in the case of the journal bearings (sect. 8.3.11,we find: (8.46)
where it may be stated that fr=4h,lh, if h,>>h, and the Reynolds number in the recess I P L Re = - - h, D i2 (1- tga) 2P
is smaller than 1000. The friction power may now be written as follows: (8.47)
268
HYDROSTATIC LUBRICATlON
(8.48)
Hf*is plotted for several geometries in Fig. 8.21. The power ratio Zl=Hf,,lHpo and the step in temperature can now be obtained from Eqn 6.16 and Eqn 6.17.
0.8 UD=1
'
/ /
0.01.". '0 15'
4 . 7 5 /0.5
0.25,
/
.
.
45'
30"
a Fig. 8.21 Tapered multirecess bearings: friction power H; versus semi-cone angle a for certain values of ratio LID; (n=4,8=30°, E=O, f,=0.2).
The effects of turning speed (in the case of E=O) on axial load capacity and flow rate have been studied in ref. 8.30. Results obtained there agree closely with those presented in section 5.5.2 for the continuous-recess pads; therefore, Eqn 5.77 and Eqn 5.86 may be used to evaluate A, and R accounting for the effect of inertia force, as long as cavitation does not occur. The response to a static radial load of a number of typical bearings is shown in Fig. 8.22. In particular, in Fig. 8.22.a, the non-dimensional radial component of the load
is plotted as a function of the radial eccentricity E; in Fig. 8.22.b i t is plotted as a function of the concentric pressure ratio P, i.e. of the axial load. As for cylindrical
269
MULTIRECESSBEARINGS
journal bearings, the pressure ratio, which now depends on the axial load, should not fall outside the 0.4t0.7 range. Beyond these limits, the bearing stiffness rapidly falls, both in the axial and radial directions. Moreover, at low pressure ratios (small axial loads) the performance is largely dependent on the load angle, whereas, a t the higher values of p, the limit load capacity (i.e. the load which corresponds to the higher eccentricities, ~>0.8)is greatly reduced, as in the case of cylindrical bearings (see Fig. 8.3). A greater range could be allowed for p by using 6-recess bearings.
-a-
-b-
0.47 0.3 W L D Ps
0.2
0.1
0.0
0.0
0.4
0.8 E
0.0
0.5
0-
J!h-
1.o
A, Ps
Fig. 8.22 Tapered multirecess bearings: load W': a- versus eccentricity E and b- versus pressure ratio4 for four bearings with n=4, 8=30°, a'=0.2. (1)LID=l. a=lOO;(2) LID=0.75, a=20°; (3) LID=O.5, a=3Oo;(4) LID=O.25, a=40°.
In Fig. 8.23 the non-dimensional radial load a t ~ = 0 . 3and ~ = 0 . 5is plotted for certain values of the geometric parameters. Since the radial load, for small eccentricities, is virtually proportional to E, Fig. 8.23.a may also be used to evaluate the radial stiffness
D L p S W ( E ) D L p S WYE) KO = =C E htga E
(8.50)
The design of a single-cone bearing could be based on the following procedure: state the minimum axial load Wzm=Wzoand the relevant maximum radial load W M .State a value for the maximum eccentricity EM and the minimum pressure ratio /3 (perhaps eM=0.5 and p=0.4). Note that the maximum axial load should
270
HYDROSTATIC LUBRlCAT/ON
-aI
-b
0.4 I
0.2 -
0.3
w
w
0.2 0.1
\ 0.1
\0.75
0.0 0"
'
\ 1
\
- a'=0.1 --- a'=0.3 V."
15"
30" a
45"
0"
15"
30"
45"
a
Fig. 8.23 Tapered multirecess bearings: load W versus semi-cone angle a for: a- eccentricity ~ = 0 . 3 and , b- ~ 4 . 5 (n=4, ; 9=30°,J=0.4, 4 4 5 " ) . not exceed 1.75.W,,, otherwise the pressure ratio under the highest load may prove to be too high to give good radial stiffness; assume a/L=O.l (a greater value is, however, suitable for slowly rotating bearings). Select (for example with the help of Fig. 8.241 the aperture angle a and the length to diameter ratio LID; the bearing can now be calculated just like a single-pad thrust bearing (section 6.4.21, except that, in selecting the film thickness, the required radial stiffness must also be considered. In other words, it must be
C = h t g a s -eM EM
where eM is the maximum allowable value of the radial displacement. "Optimal" values of the viscosity and pumping power are given by Eqn 6.56 and Eqn 6.57. Figure 8.25 contains plots of HF and p; for certain bearing shapes.
8.5.2
Opposed-cone assemblies
Let us now consider an opposed-cone assembly, such as those in Fig. 8.19. The axial play 2ho is established during manufacturing (although plans can be made to
271
MUL TIRECESS BEARlNGS
adjust it during assemblage); hence, the radial play at W,=O is Co=hotgafor both cones. The concentric pressure ratio is 1
(8.51)
R, p 1+n R* h: sin3a
where R, is the hydraulic resistance of each restrictor and n the number of recesses of each cone: the two cones are assumed to be identical.
-a-
1 .I
-b-
W wz
02
0.1 0"
15"
30"
a
45"
0.c
0"
15"
30"
45"
a
Fig. 8.24 Tapered multirecess bearings: ratio of radial load W to axial load W, versus semi-cone angle a for: a- eccentricity ~=0.3,and b- ~=0.5;(n=4, 0=30",&0.4, 4=45").
In this case, the response to an axial load is the same as that of the opposed-pad bearings, that is:
wz = A, Ps w;v;4
(8.52)
K = KO K&3; &,I
(8.54)
where:
272
HYDROSTATIC LUBRICATION
1
w; = I+?
1 ( 1 - ~ 3I+?
(8.55)
(1+.~3
(8.58) The radial stiffness of each half bearing is greatly influenced by the axial load: when the shaft is displaced in the z direction, the more loaded cone actually works with a radial play
c, = c,
(1 - 4)
and a concentric pressure ratio
-a-
5
-t
3
4
UD=1
2
\,
0.5,
0.25
3
6
"; 2
1 1
0
0.75
1
0"
- a'=0.1 _ _ -a ' 4 . 3
15"
30"
a
45"
0
15"
30'
45"
a
Fig. 8.25 Tapered multirecess bearings: a- total power H:. and b- viscosity p; versus semi-cone angle a for certain values of ratios alL and Y D ;( n 4 , 9=30°,&0.4, $ 4 5 ' , fr=0.2).
273
MULTIRECESS BEARINGS
82 =
1 I+$-@ ( 1 - ~ 3
Whereas for the other cone we shall have:
c1=c, ( 1 + &,I
Figure 8.26 contains a plot of G,81, and &, as functions of the axial Ioad.
-a-
I
0=0.6
0.0
0.4
w,
0.8
0.0"'". 0.0
'
0.4
w,
'
0.8
A, Ps A, Ps Fig. 8.26 Opposed cone bearings: a- axial eccentricity & and b-pressure ratiosJ1 and&, versus axial load W;for certain values of reference pressure ratioj.
Bearing Fig. 8.22 in mind, it should be clear that the radial load capacity and the stiffness of each cone are greatly affected by the axial load. An example is given in Fig. 8.27, in which the non-dimensional radial load capacities W i and W; are plotted, for a typical bearing, against the axial load Wl. When large axial displacements are allowed to sustain a certain load W,,arrangements such as in Fig. 8.19.a and Fig. 8.19.b are clearly not able to sustain high radial load components on both the half bearings a t the same time.
274
HYDROSTATIC LUBRICATION
The opposed-cone arrangement shown in Fig. 8.19.c may be regarded as a single bearing, able to react to external forces in every direction. The main difference as compared to the arrangements examined above is that the eccentricities and ~2 of the two cones may no longer be considered independently, since we have:
0.0
0.2
w,
0.4
0.6
Ae PS
Fig. 8.27 Radial loads W ; and W ; versus axial load W; for certain values of pressure ratio$, for a bearing with LID4.75, a=2Oo, alL=0.2, n=4, 0=30°.
0.4
I
- .-
0.0
0.2
w,
0.4
0.6
Ae PS
Fig. 8.28 Radial load versus axial load. for certain values of pressure ratioj, for an opposed-cone bearing with LID4.75. a=20°, a/L=0.2,n=4. 8=30°.
MULTIRECESS BEARINGS
275
The radial load capacity is, of course, W=W,+W2. When a n axial load is applied, becomes smaller than E,, and hence the radial load capacity is mainly due to W,. Figure 8.28 shows, by way of example, how the radial load capacity of a given bearing is affected by a radial component W,.
8.6
SPHERICAL JOURNAL BEARINGS
Spherical multirecess bearings (Fig. 8.29) have the advantage, as compared to similar conical types, of being intrinsically insensitive to the tilting misalignment of the journal. On the other hand, they are more difficult and more expensive to build.
Fig. 8.29 Spherical journal bearing.
276
HYDROSTATIC L UBRICATlON
To evaluate the performance of such bearings, the relevant Reynolds equation (Eqn 4.30) should be solved numerically. Due to the large number of geometrical parameters, general charts are not given, but in Fig. 8.30 (obtained from ref. 8.34, as are the following figures in this section) the main performance of a typical 6 recess bearing is shown. In more detail, in Fig. 8.30.a, the axial and radial components of the load, i.e. W, and W, are plotted as functions of the eccentricity E and the relevant angle whereas the effect of the attitude angle is small. In Fig. 8.30.b the flow rate for W=O: (8.59) is given as a function of E. The pressure ratio in the concentric position is assumed to be p=0.5. As usual, this value is considered to be the optimal one, when fixed laminar restrictors are used as compensating devices, from the point of view of both axial and radial stiffness. Furthermore, for different values of p, higher interaction occurs between the loads in the axial and radial directions, whereas from Fig. 8.30 it is clear that the axial load has little influence on radial stiffness, a t least when yis between 60"and 90". In Fig. 8.31.a the effect ofa on load capacity a t ~ = 0 . 5and on the concentric flow rate is shown for the same bearing as in Fig. 8.30. Concerning the geometrical parameters, ref. 8.34 shows that to decrease rp, to below 30" has a detrimental effect on radial load capacity. It may be advisable to use
- b-
-a12
Q'
8
4
0.04
-
0.12
0
Ps D'
Fig. 8.30 a- Load-displacement characteristics; b- flow rate versus eccentricity, for a spherical bearing with 'pl=5Oo,q2=850, qa=8=3S0,n=6.J=0.5 (ref. 8.34).
277
MULTIRECESS BEARINGS
larger lands than for the bearing considered above; radial load capacity is virtually unaffected by an increase to 0.25-((p4-(pl)of the width of the axial and radial lands, while the reduction in axial load capacity is largely compensated by the decrease in flow rate. With reference to Fig. 8.31.b, the non-dimensional axial load capacity is clearly reduced by a factor 0.9 when passing from a land angle of 3.5" to a land angle of 7". This may be compensated by a reciprocal variation in supply pressure. On the other hand, the non-dimensional flow rate is halved; consequently, taking the pressure increase into account, the actual flow rate will be reduced by almost 44%, and pumping power by 38%. Furthermore, thanks to the small increase in supply pressure, the radial stiffness will also actually be better.
-b-
0.2 I
I
W (0.5,90°)
20
10
W D2Ps
0.0 0.10
0.15
0.20
0.25
'Pa 92-91
Fig. 8.31 Axial load, radial load and flow rate versus a- pressure ratio and b- land width ratio, for a spherical bearing with (pl=500,(p2=850, n=6 (ref. 8.34).
Friction power may be roughly calculated, in the concentric position E=O, by the equation:
This last equation was obtained on the assumption that the recess depth hr is much greater than the radial clearance C. For the friction coefficient f, it may be
278
HYDROSTATIC 1UBRICATION
assumed, by way of approximation, that fr=4Clhr,provided that the flow in the recess is laminar. From Eqn 8.60 and Hp=psQ
the concentric power ratio 17=HfdHpo can be easily assessed. As usual, it is advisable for II to be in the li-3 range. The performance of the opposed spherical bearings is easily obtained by summing together the effects of the two halves. In Fig. 8.32 the axial and radial load capacities are plotted for a typical symmetrical opposed bearing.
03
0.18
0 06
0.03
WZ Ps
0.09
015
D2
Fig. 8.32 Load-displacement characteristics for an opposed spherical bearing arrangement with 'pl=5O0, 'p2=85", (~,=8=3.5~, n=6,J=0.5 (ref. 8.34).
For the design of spherical bearings, similar remarks to those concerning the other multirecess types can be made. As usual, the designer should select bearing size and supply pressure in such a way that, in the worst loading conditions, the eccentricity is smaller than a prefixed maximum value EM; it is advisable to choose a relatively small value for EM (e.g. ~ ~ = 0 .to 5 )allow for sufficient stiffness and overload capacity. A pressure ratio lower than the optimal value (e.g. 8=0.5) should be selected for these initial calculations. The minimum and maximum values of the radial clearance can now be selected, in such a way that p remains within a suitable range. As for cylindrical
MULTIRECESS BEARINGS
279
bearings, the smaller the working tolerances, the smaller the minimum value of the clearance, and, hence, the smaller the power expenditure and the higher the bearing stiffness. A lubricant should now be selected whose viscosity leads to a power ratio n = 1 for the maximum value of the radial clearance. Of course, if the turning velocity is low or even null, this last condition will be impossible to satisfy, in which case the maximum allowable viscosity must be selected.
8.7
YATES BEARINGS
Figure 8.33.a contains a sketch of the so-called "Yates" bearing, which in practice consists of a multirecess journal bearing with, in series, two opposite thrust pads, fed by the lubricant flowing out of the circumferential lands of the journal bearing. For the sake of precision, it must be said that in the original proposal (ref. 2.21) there was no side recess, i.e. Di=D. The same operating principle may be applied to various other types of multidirectional bearing arrangements (ref. 8.35).
8.7.1
Axial load
When no load is applied (fig. 8.33.b), in all the recesses of the journal bearing pressure is pjo=plp,, and in the side recesses pz0=P2ps.When a n axial thrust is applied (Fig. 8.33.c), the side pads perform just like an opposed-pad bearing, supplied a t a pressure pjo through the circumferential lands of the journal bearing, that consequently act as compensating restrictors. Their hydraulic resistance is (8.61) Values of Q'(O)/jl may be found in section 8.3 (e.g. Fig. 8.10.b); as a preliminary approximation we may take
The axial load capacity may now be calculated as
wz = A , Pjo w;(a,;%)
(8.62)
In the last equation, W;is given by Eqn 7.22, ~~=(h~-hO)lho is the axial eccentricity, and the effective pressure ratio 8, is Pz = Pzo lPj0 = 82 18 1
(8.63)
HYDROSTA,TIC LUBRICATION
280
t+zi
m
-a-
-b-
-c-
Fig 8.33. a- Yates bearing, b- pressure profile in the no-load condition, c- pressure profile in the case of axial load, d- pressure profile in the case of radial load.
The effective area of the thrust pads is equal to the effective area of analogous circular pads, from which the cross-section area of the shaft must be subtracted:
A, =
2 [DO"A*,(r')-
0 2 1
(8.64)
In Eqn 8.64 it is r'=Di/D,; the non-dimensional value A*,isgiven by Eqn 5.22. Pressure pjo in Eqn 8.62 is not constant, due to the existence of the restrictors R, in series with the recesses of the journal bearing. However, Fig. 7.7.b shows that the
MULTIRECESS BEARINGS
281
total hydraulic resistance of an opposed-pad bearing slightly increases when a load is applied; consequently, pjo should actually be greater than @lps, and therefore it may safely be taken that (8.65) From the point of view of the axial load capacity, it is clearly suitable to take means to select ~z=P1/p2=0.3+0.5 and p1 close to 1; unfortunately, to take a large compensating restrictors R , with low hydraulic resistance, hence causing poor radial stiffness.
8.7.2
Radial load
Let us now examine how this bearing works when subjected to a purely radial load (Fig. 8.33.d). Due to symmetry, in the side recesses there is the same pressure pzo; consequently, the performance under a radial load is that of a journal bearing supplied a t a relative pressure ps-pzo, i.e.
W = L D (ps - pzo) W'coJ.,E)
(8.66)
The non-dimensional function W' may be drawn from section 8.3 (e.g. Fig. 8.6.a or Fig. 8.10.a); the effective value of the pressure ratio is pj
81 - 8 2
=1-p~
(8.67)
Besides a radial displacement E, pressure pzo clearly varies; but, again, since the total hydraulic resistance of a journal bearing is not very sensitive to radial displacements (see Fig. 8.6.b), it may be taken that (8.68) For good radial stiffness P j should clearly be close to 0.5 and p2 should be small. This last requirement contrasts with the advisability of having a high p1to increase the axial load capacity, as pointed out above. A compromise is clearly necessary; for this purpose, Fig. 8.34 shows the interrelations between PI,8 2 , /3j and Bz.
8.7.3
Combined axial and radial loads
For a combined load it is possible, as a preliminary approximation, to consider the bearing responses to the axial and radial load components independently; actually, a certain degree of interaction exists, although it is not so great as in the case
282
HYDROSTATIC LUBRICATlON
of the opposed-cone assemblies seen in section 8.5.2.An example is given in Fig. 8.35,taken from ref. 8.36.Loss of axial load capacity due to radial displacement and (to a lesser degree) loss of radial load capacity due to axial displacement are considerable at high eccentricities; on the other hand, such losses are fairly small when E and E, are lower than 0.5.
Fig. 8.34 Pressure ratiosfl, andJ2 versus pressure ratioJz, for two values of pressure ratioJj,
0.3 -
0.2-
0
0.1
w .. -
0.2
0.25
L DP,
Fig. 8.35 Load-displacement characteristics for a typical Yates bearing (ref. 8.36). 8.7.4
Other bearing parameters
Let us now examine briefly the other bearing parameters (see also ref. 8.37 for further details).
MULTIRECESS BEARINGS
283
Flow rate in the reference configuration E=G=O is easily calculated from Eqn 8.11 or Eqn 7.23
(8.69) where R* is given by Eqn 5.23.When any load is applied, flow rate should be slightly smaller, if both P j and P, are lower than 0.5. Pumping power is Hpo = ~s QO
(8.70)
Friction may be evaluated by summing the effects due to the journal and to the side pads. In the unloaded configuration we find with no difficulty that
(8.71)
+
Factors and H; may be calculated by means of Eqn 8.23 and Eqn 5.27, respectively. It should be noted that C and ho cannot be chosen in a completely independent fashion, since in order to obtain the selected values of the pressure ratios we must have
(8.72) The power loss due to friction is easily calculated by multiplying the friction moment by the angular speed
HfO = MfO a
8.7.5
(8.73)
Design procedure
The design of a Yates bearing is not very different from the design of journal and opposed-pad bearings. Once the maximum values of the load components are given, the first step consists in selecting the values of the pressure ratios (see Fig. 8.34);it should be taken into account that PjeO.5 is the optimum for the radial stiffness (however, values smaller than 0.4 should be avoided). a s far as p, is concerned, selecting too small a value would give poor axial stiffness; on the contrary, too high values lead to poor radial load capacity (since i t is proportional to 1-PZ)and even the axial stiffness breaks down when is close to unit. A typical selection might be &=0.2, and p1=0.6; possibly, both may be increased in order to increase the axial load capacity; the opposite may be done when the radial load is more critical.
284
HYDROSTATIC LUBRICA TlON
The next step may consist in dimensioning the journal bearing (see also section 8.3.3). After suitable trial values for a / L and LID (say a/L=0.1 and L I D = l ) have been stated, diameter D and supply pressure p s may be calculated by means of Eqn 8.68. In designing it may be best to slightly increase loads (perhaps by 10%)to account for the losses in load capacity due to interactions between thrust and radial load. The radial clearance C may then be selected; as usual, the narrower C is the higher radial stiffness will prove to be and the lower the total power consumption (provided that it is possible to select a lubricant with a viscosity close to the relevant "optimal" value). The thrust bearing may now be designed (see also section 7.2.6). First, one may choose the radius ratio r' (say r'=0.9);in general, thin lands are more convenient, unless R is very slow. The axial gap ho may not be chosen at will but must be calculated by means of Eqn 8.72; should it prove t o be too narrow, it might be necessary to reduce r'; on the contrary, if ho seems too thick, causing poor axial stiffness, it may be necessary to reconsider the journal bearing, increasing a I L . The effective area needed to sustain the axial load may be calculated by means of Eqn 8.65, and then the outer diameter Do by means of Eqn 8.64. Should the axial load be small, it could turn out that Di=r'D@: in this case a smaller value can be chosen for E ~ as , well as for 81and 82. The selection of the "optimal" value of viscosity will be based, as usual, on the power ratio 17=HfdHpo.From Eqn 8.70 and Eqn 8.73 one may obtain (8.74)
in which one should state l7=1to obtain the optimal viscosity. As for the other bearing types considered in previous sections, it may happen than this calculated viscosity is too low or too high; the remedies are, of course, similar: increase film thickness and I l in the former case (avoid, however, values of Il greater than 3); increase the land width and decrease I7 in the latter case. When the viscosity value is a constraint, the same Eqn 8.74 allows us to select clearance C in such a way as to get a power ratio in the l t 3 range. Once the main bearing parameters have been established, it is possible to calculate flow rate, friction torque, and the relevant power losses. It is also advisable to check the Reynolds numbers and the temperature steps, as in the case of the other bearings. Finally, the hydraulic resistance of each of the n restrictors R, which compensates recess pressures in the journal bearing has to be
MULTIRECESS BEARINGS
285
where Rj may be calculated by means of Eqn 8.61. One final remark: it should be noted that the actual values of C and ho have a considerable influence of the loading performance of the assembly. For instance, let us assume that radial clearance proves to be greater than the selected value C , while all the other remain the same; consequently, the resistance of the lands of the journal bearing (proportional to C-3) will be lower, and we actually get a smaller value of PI, and a greater value of Icj2, i.e. both axial and radial load capacities have been lowered. Of course, if clearance is narrower than designed, and p 2 change in the opposite way; nevertheless, stiffness may become very poor, since Bj and 0, differ from their optimal values.
REFERENCES
O'Donoghue J. P., Hooke C. J., Rowe W. B.; A Solution Using Superposition Technique for Externally Pressurized Multirecess Journal Bearings Including Hydrodynamic Effects; Proc. Instn. Mech. Engrs.; 185,5 (1970-71), 57-61. 82 Colsher R., Anwar I., Katsumata S.; An Advanced Method for Predicting Hybrid Bearing Performance; AGARD Conf. Proc. 323, 1982, paper 28; 13 pp. 8.3 Ghai R. C., Singh D. V., Sinhasan R.; Load Capacity and Flow Characteristics of a Hydrostatically Lubricated Four-Pocket Journal Bearing by Finite Element Method; Int. J . Mach. Tool Des Res, 16 (19761, 233-240. 8.4 Shapiro W.; Computer-Aided Design of Externally Pressurized Bearings; Instn. Mech. Engrs., C 10/71 (1971); 22 pp. 8.5 Raimondi A. A., Boyd J.; An Analysis of Orifice and Capillary Compensated Hydrostatic Journal Bearings; Lubr. Eng., 13 (1957), 28-37. 8.6 Kher A. K., Cowley A,; The Design and Performance Characteristics of a Capillary Compensated Hydrostatic Journal Bearing; Proc. 8th Int. MTDR Cod. (1967), pt. 1,397-418. 8.7 Davies P. B.; A General Analysis of Multirecess Hydrostatic Journal Bearings; Proc. Instn. Mech. Engrs.; 184,l (1969-701,827-836. 8.8 Bettini B.; Calcolo di Cuscinetti Radiali Zdrostatici; Progetto d i una Attrezzatura d i Prova; Doct. Thesis, 1985; 269 pp. 8.9 Singh D. V., Sinhasan R., Ghai R. C.; Finite Element Analysis of Orifice Compensated Hydrostatic Journal Bearings; Tribology Int., 9 (1976), 281-284. 8.10 Sinhasan R., Sharma S. C., Jain S. C.; Performance Characteristics o f a Constant Flow Value Compensated Multirecess Flexible Hydrostatic Journal Bearing; Wear, 134 (19891,335-356. 8.11 Davies P. B.; The Dynamic Behaviour of Passively Compensated, Hydrostatic Journal Bearings with Various Number o f Recesses; J. of Mech. Eng. Science, 18 (19761,292-302. 8.12 Rowe W. B.; Dynamic and Static Properties of Recessed Hydrostatic Journal Bearings by Small Displacement Analysis; ASME Trans, J . of Lubrication Technology, 102 (1980),71-79. 8.1
286
HYDROSTATIC LUBRICATION
8.13 Ho Y. S., Chen N. N. S.; Performance Characteristics of a Capillary-Conpensated Hydrostatic Journal Bearing; Wear, 52 (19791,285-295. 8.14 Lingard S.,Chen N. N. S., Kong Y. C.; Aspects of the Performance of Externally Pressurized Journal Bearings; Wear, 78 (19821,343-353. 8.15 Artiles A., Walowit J., Shapiro W.; Analysis of Hybrid, Fluid-Film Journal Bearings with Turbulence and Znertia Effects; Advances i n Computer-Aided Bearing Design, ASME-ASLE Lubrication conf., Washington D.C., 1982; p. 25-51. 8.16 Manea G.; Machine Elements; Technical Publishing House, Bucharest; 1970. 8.17 Dumbrawa M. A.; Review of Principles and Methods Applied to the Optimum Calculation and Design of Externally-Pressurized Bearings. Part 1: Low /Moderate Speed Bearings; Tribology Int., 18 (19851, 149-156. 8.18 Dumbrawa M. A.; Review of Principles and Methods Applied to the Optimum Calculation and Design of Externally-Pressurized Bearings. Part 2: ModeratelHigh-Speed Bearings; Tribology Int., 18 (1985), 223-228. 8.19 Rowe W. B., Koshal D., Stout K. J.; Investigation of Recessed Hydrostatic and Slot-Entry Journal Bearings for Hybrid, Hydrodynamic and Hydrostatic Operation;Wear, 43 (19771,55-69. 820 El Sherbiny M., Salem F., El Hefnawy N.; Optimum Design of Hydrostatic Journal Bearings. Part 2: Minimum Power; Tribology Int., 17 (19841, 162-166. 8.21 Vermeulen M.; Optimisation of Hydrostatic Journal Bearings Zncluding Hydrodynamic Effects; Proc. Eurotrib 81, Varsaw, 1981; V. 2, pp. 371-388. a22 Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Berechnung von Olgeschmierten Gleitlagern ohne Zwischennuten); DIN 31655, Teil 1, Nov. 1984; 29 p. Hydrostatische Radial-Gleitlager im Stationaren Betrieb (Kenngropen fur die Berechnung von Olgeschmierten Gleitlagern ohne Zwischennuten); DIN 31655, Teil2, Nov. 1984; 11 p. 824 Rowe W. B., Stout K. J.; The Design of Externally Pressurized Bearings for Reliability with Particular Reference to Manufacturing Errors; Instn. Mech. Engrs.; C310 (19731,421-429. 8.25 Xu S., Chen B.; Optimum Design and Automatic Drawing of Recessed HydrostalCc Bearings; Tribological Design o f Machine Elements, proc. 1 5 t h Leeds-Lyon Symp. on Tribology, Leeds, 1988; p. 411-418. 8.26 O'Donoghue J. P., Rowe W. B., Hooke C. J.; Some Tolerancing Effects in Hydrostatic Bearings; Proc. 11th Int. MTDR Cod. (1970); pp. 317-322. 827 Sato Y., Ogiso S.; Load Capacity and Stiffness of Misaligned Hydrostatic Recessed Journal Bearings; Wear, 92 (1983), 231-241. 82% Moshin M. E.; A Hydrostatic Bearing for High Speed Applications; Tribology Int., 14 (1981),47-54. 829 Aoyama T., Inasaki I., Yonetsu S.; Friction and Tilting Characteristics of Hydrostatic Thrust Bearings; Bull. Japan SOC. of Precision Eng., 10 (19761, 6870. 8.30 Prabhu T. J., Ganesan N.; Analysis of Multirecess Conical Hydrostatic Thrust Bearings under Rotation; Wear, 89 (19831, 29-40. 8.31 O'Donoghue J. P.; Design of Annular Multi-Recess Hydrostatic Thrust Bearings; Mach. and Prod. Eng., Nov. 18, 1970; p. 830-834.
MULTIRECESS BEARINGS
8.32
8.33
8.34
8.36 8.36 8.37
287
Prabhu T. J., Ganesan N.; Behaviour of Multirecess Plane Hydrostatic Thrust Bearings under Conditions of Tilt and Rotation; Wear, 92 (1983), 243251. Hessey M. F., O'Donoghue J. P.; The Performance of a Four-Pocket Conical Hydrostatic Bearing; Externally Pressurized Bearings / Joint Conf. Instn. Mech. Engrs. - Instn. Prod. Engrs., 1971; p. 133-145. Rowe W. B., Stout K. J.; Design data and Manufacturing Technique for Spherical Hydrostatic Bearings in Machine Tool Applications; Int. J. Mach. Tool Des Res, 11 (1971), 293-307. O'Donoghue J. P., Wearing R. S., Rowe W . B.; Multirecess Externally Pressurized Bearings Using the Yates Principle; Proc. Instn. Mech. Engrs, C44 (19711,337-351. Wearing R. S., O'Donoghue J. P., Rowe W . B.; Design of Combined Journal Thrust Hydrostatic Bearings (the Yates Bearing); Mach. and Prod. Eng., Aug. 19, 1970; p. 301-308. Lund J. W.; Static Stiffness and Dynamic Angular Stiffness of the Combined Hydrostatic Journal-Thrust bearing; Mechanical Technology Inc., Rept. No. MTI-63TR45, 1963.
Chapter 9
HYBRID PLAIN JOURNAL BEARINGS
9.1
INTRODUCTION
When a comparison is made between hydrostatic and hydrodynamic (selfacting) bearings, the performance of the former a t low speed is clearly unique, while the latter may show a better load-to-power ratio a t high speed. Although the turning velocity increases the load-carrying capacity of the recessed hydrostatic journal bearings, too, the hydrodynamic effect can only develop on a fraction of the total area (i.e. on the circumferential lands). Furthermore, the "hydrostatic" contribution to the load capacity of a recessed bearing can be negatively affected, a t high velocity, by a number of factors such as cavitation in the inter-recess lands and even in the recesses, lubricant recirculation in the recesses, reversal of the flow in certain restrictors, local antagonistic superposition of the Couette and Poiseuille flows (ref. 8.18). The hybrid plain (i.e. non-recessed) bearings may be seen as special hydrostatic bearings whose shape has been optimized to take the maximum advantage from the hydrodynamic effect on the load-carrying capacity; in particular, the recesses are reduced to a minimum, since they are made up of one o r two rows of 8t16 entry ports (holes or slots). Consequently, these bearings are well suited to support journals that may rotate a t different velocities, especially when the load grows with the velocity (due, for example, to the existence of some unbalanced mass).
HYBRID PLAIN JOURNAL BEARINGS
9.2
289
PERFORMANCE OF THE HYBRID PLAIN JOURNAL BEARINGS
To evaluate the performance of the hybrid journal bearings, shown in Fig. 9.1, the relevant Reynolds equation (Eqn 4.18) should be solved, which for static loading may be rewritten as follows: (9.1)
where Sh is the so-called Sommerfeld hybrid number (Eqn 8.8). An approximate solution to Eqn 9.1 is easily obtained in the trivial case E=O, in which the film thickness is a constant h&. Now, in the whole portion of the bearing surface included between the two rows of entry holes (i.e. for l z I I(L/2-a)),the relative pressure takes on a constant value p0=Dps which depends on the hydraulic resistance of the inlet restrictors and of the outer clearances. This last resistance is easily calculated by considering the clearances in the zone l z I >(L/2-a)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
HYBRID PLAIN JOURNAL BEARINGS
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
HYBRID PLAIN JOURNAL BEARINGS
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
HYDROSTATIC LUBRICATION
-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
HYDROSTATIC LUBRlCATlON
(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 3
Select a set of geometrical parameters, with such typical values as L / D = l , a /L=O.l, n=12 (this last parameter has little influence in hybrid operation, provided n>8).
HYBRID PLAIN JOURNAL BEARINGS
297
ii)
Select a suitable range for 8, and a maximum value for eccentricity, perhaps 0.4
iii)
Select a tolerance range 6, for the clearance (see Fig. 8.14). The maximum and minimum values C M and C, are then calculated in such a way as to ensure that p always remain in the proper range: e.g. if it has been stated that 0.4 1 . 5 . 6on ~ ; the other hand, radial clearance should be small, since i t is more suitable from the point of view of the power expense. The choice of the tolerance range clearly follows from a reasonable compromise between manufacturing and running costs. Furthermore, a n upper constraint e M may be required for the displacement e=&Cof the journal. In this case, if it turns out that E&M>eM and Sg cannot be reduced, a n attempt can be made to select a smaller value for EM after which one must start again from point (iii).
iv)
From Eqn 9.8 calculate the optimal value of the viscosity:
If this value is impracticably great, and it is not possible to further reduce C M , one may try to reduce p s , increasing the bearing dimensions. If, on the other hand, p is too small, one may choose to increase the clearance, but it might be more advisable to select a higher value for l7,and restart the calculations from point (iii). v)
Calculate flow rate, pumping power and friction power. Check the rise in temperature, using Eqn 9.13.
vi)
Design the compensating restrictors.
vii)
Check the Reynolds number in the bearing clearance.
viii)
Check the dynamic behavior of the system, especially as far as whirl instability is concerned (see Chapter 10).
If different loads have to be sustained a t different velocities, the performance of the bearing should be checked in all the situations foreseen. In particular, at low velocities (hydrostatic operation) stiffness is strongly reduced and, moreover, great eccentricities (&>0.6)should not be allowed.
298
9.4
HYDROSTATIC LUBRICATlON
CONCLUDING REMARKS
Certain problems connected with hybrid bearings do not seem t o have been dealt with yet in any depth. In particular, the isothermal flow model may be far from representing the actual temperature profile in the clearance. Due to the great values of the power ratio needed to get a high hydrodynamic load capacity, the temperature step (Eqn 9.13) may prove to be considerable; furthermore, this is only a n average value, obtained by dividing the total power by the total flow rate. It is easy to see that, actually, most of the lubricant flows directly outward (the entry ports are close to the bearing edge) while the heat due to friction mainly develops in the lubricant enclosed between the rows of supply ports, in the area of minimum clearance. The temperature in this area is clearly much higher than the average value. Furthermore, this hot lubricant, instead of being readily flushed out, may be recirculated round the bearing, due to the presence of the rows of entry ports (especially of the slot type, that occupy practically the entire circumference with no interecess land) and of large cavitated regions (ref. 9.5). Although other circumstances (heat transport through the bearing sleeve, smaller friction power due to cavitation, back flow through the most heavily-loaded restrictors) may attenuate the temperature peaks, i t is clear that bearing performance may be considerably affected. Other problems involving the hybrid bearings may be (ref. 9.5) the existence of large cavitated regions (which may cause severe starvation problems), back flow through the restrictors in the high pressure area (which occurs a t high speed and eccentricity, and thwarts the build up of the hydrodynamic pressure profile) and the possibility of whirl instability. Instability is connected with the existence of a hydrodynamic side thrust on the journal, revealed by the attitude angle (Fig. 9.4). In this sense, cavitation may be considered beneficial and, hence, eccentricities larger than &=0.6should be selected for hybrid operation. Lastly, a peculiar type of hybrid bearing exists, characterized by the asymmetrical configuration of its entry ports (ref. 9.5). Namely, each row is made up of a few slots clustered symmetrically a t the bottom dead center, while a t the top dead center there is a n axial groove supplied a t low pressure (Fig. 9.7.b). This bearing presents certain advantages, as compared with a usual slot-entry hybrid bearing (Fig. 9.7.a), such a s higher load capacity a t small eccentricity, reduced cavitated regions, smaller flow rate and easily flushed-out hot fluid. On the other hand, a t high eccentricity, load capacity is smaller, and (at low speed) great negative stiffness occurs.
A similar bearing (Fig. 9.8) is described in ref. 9.6 as an alternative to conventional generator bearings. The latter usually have small high-pressure pockets to provide hydrostatic lift during start-up and shut-down operations, whereas in the case of the hybrid bearing two rows of slot-type entry ports are also pressurized
299
HYBRID PLAIN JOURNAL BEARINGS
-a-
- b-
3
n=9
*.
2
W LDPS 1
a =I
0 a2
04
&
0.6
0
aa
I
I
I
I
Fig. 9.7 Load versus eccentricity for a hybrid journal bearing with: a- twelve symmetrical slots per row; b- five slots per row and an axial groove. [a=a,dRD is the slot width ratio].
during normal running. The load support arc is 120 degrees wide and is bounded by large low-pressure inlet recesses. This type of hybrid bearing is claimed to be more efficient than conventional generator bearings.
---
I O
I Fig. 9.8 Slot-entry generator bearing.
I
---
I
l o
I
1
300
HYDROSTATIC LUBRICATION
REFERENCES
9.1 9.2 9.3
9.4
9.5 9.6
Ichikawa A.; A study of High Speed Hydrostatic Bearings (Part 1, Theoretical Analysis of Static and Dynamic Characteristics of Hybrid Plain Journal Bearings; Bull. JSME, 20 (19771,652-660. Rowe W. B., Xu S. X., Chong F. S., Weston W.; Hybrid Journal Bearings; Tribology Int., 15 (19821,339-348. El Kayar A., Salem E. A., Khalil M. F., Hegazy A. A.; Two-Dimensional Finite Difference Solution for Externally Pressurized Journal Bearings of Finite Length; Wear, 84 (1983),1-13. Rowe W. B., Koshal D.; A New Basis for the Optimization of Hybrid Journal Bearings; Wear, 64 (1980), 115-131. Ives D., Rowe W. B.; The Effect of Multiple Supply Sources on the Performance of Heauily Loaded Pressurized High-speed Journal Bearings; Proc. Inst. Mech. Engrs., C199 (19871, 121-127. Ives D., Weston W., Morton P. G., Rowe W. B.; A Theoretical Znuestigation of Hybrid Journal Bearings Applied to High-speed Heauily Loaded Conditions Requiring Jacking Capabilities; Tribological Design o f Machine Elements, proc. 15th Leeds-LyonSymp. on Tribology, Leeds, 1988;p. 425-433.
Chapter
10
DYNAMICS
10.1
INTRODUCTION
In previous chapters we have examined the behaviour of hydrostatic bearings loaded by static forces. We now intend to evaluate the bearing response to timedependent loads; this means: studying bearing stability; i.e. is the bearing able to return to its previous equilibrium when excited by a small perturbation? assessing bearing behaviour under given loads (impulsive and periodical, in particular 1.
As a first step, the lubricant film may be assimilated to a mechanical system formed by a (nonlinear) spring and a viscous damper, whose coefficient depends (in a nonlinear way) on film thickness (Fig. 1O.l.a). Stiffness and damping can easily be evaluated, without taking inertia and the compressibility of the lubricant into account. On such assumptions, hydrostatic bearings always prove to be stable, but in practice instability can sometimes arise. Actually, the compressibility of the oil volume contained in bearing recesses and supply pipes and the compliance of the tubing itself can play an important part in lowering the dynamic stiffness of bearings and in causing instability. Moreover, it has to be borne in mind that, in practical cases, lubricant compressibility may be greatly increased by aeration phenomena. Finally, the influence of the dynamic behaviour of the supply system (e.g. controlled valves) must also be considered. For a better approach to bearing dynamics, more elaborate mechanical models are often used (ref. 10.1, 10.21, like the ones in Fig. 10.l.b and Fig. lO.l.c, which
302
HYDROSTATIC LUBRICATlON
allow compressibility to be accounted for. Also electrical analogies have been proposed. A more effective and more general approach may be based on the results of control theory (ref. 10.3, 10.4).
Fig. 10.1 Equivalent mechanical systems of hydrostatic thrust bearings. 10.2
EQUATION OF MOTION
As will be shown in the following sections, in a thrust bearing the lubricant fluid exerts a force W on the moving member; this force may be written in the form:
where A, is the effective area, p r is the fluid pressure in the recess and B is a squeeze coefficient, which depends on clearance h. For the pads whose clearance is not the same for the whole land surface (e.g. spherical pads), h is understood as being the clearance in a reference point of the surface. In every case, h does not indicate the fiam thickness along the normal to the surface, but the clearance measured along the direction of the displacement. Let us now consider a reference configuration in which h=ho, and let us take E=e 1ho to define the non-dimensional displacement of the "moving member", whose mass is M ,from this reference position. As noted in chapter 5, the reference configuration is completely arbitrary for plane and tapered pads, whereas it is convenient to assume as reference the centred configuration for cylindrical and spherical pads. In certain circumstances, e.g. when studying the performance of the bearing as a vibration attenuator, it is also necessary to consider the displacement <=z / ho of the "foundation" (Fig. 10.2). Hence in genera1,we have:
h - ho = e - z = ho ( E -
0
(10.2)
DYNAMICS
303
Prs
1
Fig. 10.2 Dynamic pressure profile for a pad bearing.
Thereafter, it will be generally assumed that z=O, i.e.:
Recess pressure will be related to the position of the moving member by a differential equation f ( p r ;p, ; h; h)=O,which, in general terms, is nonlinear and depends on the supply system. Hence, the law of motion is found by solving the differential system: (10.4)
In Eqns 10.4, M is the moving mass and F the external force acting on the bearing. F can be put in the form
F = F,
+ 6F(t)
where F, is a constant force and 6F a time-dependent perturbation, which is assumed not to be dependent on the configuration of the bearing. In studying system stability, the bearing can be assumed to make small vibrations ~ E = E - E , about the static equilibrium point E,, a t which F,=-W,=-A,~,(E,);it is thus possible to linearize Eqns 10.4 and then apply the Laplace transformation. Equations 10.4 are transformed as follows:
(10.5)
304
HYDROSTATIC LUBRICATION
where s is the Laplace operator and B,=B(hL,).Hereafter subscript "S" will mean "in the static case". Since the reference configuration is often arbitrary, it could be assumed hs=hO and, hence, E,=O and ~ E = E . By examining Eqns 10.5 it may be stressed (ref. 10.1) that the damping capacity of the system (that is its ability to change the pressure distribution to react to the squeeze velocity) is associated with two factors. The first one is the change in recess pressure 6p, caused by pressing away the lubricant from the pad through bearing gaps and inlet restrictors (see Fig. 10.2). The second is the change of the shape of pressure distribution across the lands (squeeze film effect represented by coefficient B ) . This factor is often much less important than the former, but because it is practically unaffected by lubricant compressibility, it may even become dominant when lubricant stiffness is poor (e.g. due to a low bulk modulus caused by air entrainment: see section 3.2.4). Introducing the second of Eqns 10.5 into the first, we obtain:
W -Ae jlp 6E = ho s (B, + M
S) 8~
(10.6)
The vibrating systems described in Eqn 10.6 may be represented by the block diagram in Fig. 10.3. Block &(s) depends on the lubricant feeding system, whereas coefficient B depends on the pad shape: in the following sections will be shown how they can be obtained in several cases.
Fig. 10.3 Block diagram for hydrostatic pad bearings.
The transfer function of the system is easily obtained from Eqn 10.6: 1
8E -=
W h, M
~2
+ ho B, s +Ae ;lp
(10.7)
By means of Eqn 10.7 one may study bearing stability and assess the linearized frequency response for small vibrations.
DYNAMICS
10.3
305
PAD COEFFICIENTS
The aim of this section is ta show how Eqn 10.1 may be written for a pad bearing. In particular, the value of squeeze coefficient B must be calculated.
10.3.1
Circular-recess pads
Let us start with the plane circular pad (Fig. 5.1.a) whose static behaviour has already been examined in section 4.7.5 and section 5.3. As usual, we can start from the Reynolds equation, in this case Eqn 4.23 which, assuming that film thickness is uniform, becomes:
The above equation may be integrated twice to obtain the pressure pattern as a function of the boundary pressures, as well as of the angular and squeeze velocities L 2 and h:
The pressure at a certain radius proves to depend on the boundary pressures, on the square of the rotation speed and on the squeeze rate in a linear fashion; this will be true for every shape of pad, since it is a consequence of the linearity of the Reynolds equation. The load capacity of the pad may therefore be calculated (as in Eqn 10.1) by adding the term A,p,, already known from chapter 5, to the term Wd=-Bh obtained by integrating the following squeeze overpressure on the land surface:
The following is easily found (10.8) Coefficient B is plotted in Fig. 10.4.a, in which it must obviously be taken that a=d2. Another parameter related to the squeeze effect, that will prove to be useful in the following section, may be introduced at this point; it is defined by the following equation:
(10.9)
306
HYDROSTATIC LUBRICATION
It can be noted that, for any given pad shape, B is proportional to the square of the pad area, to viscosity, and to h-3 (see for example Eqn 10.8); R is proportional to p and to h-3,thus ay* depends on the pad shape alone, and not on its actual size, nor on any other parameter like p and h. For central-recess pads, parameter ay* has also been plotted in Fig. 10.4.a.
-a-
r'
-b-
a'
Fig. 10.4 Squeeze coefficient B*=B .32h3sin4a/(3qfD4)and squeeze parameter ay* for plane (a=x/2) and tapered circular pads: a- central recess: b- annular recess.
It should be pointed out that A,pr is not actually the "static" load capacity, because recess pressure p r is also affected by squeeze velocity h. Indeed, p r is related to the flow rate by a law which depends on the supply system; the flow rate, in its turn, depends on the squeeze velocity and may be written as the sum of the usual term p r I R and of the squeeze term (obtained, like the former, by the integration of Eqns 4.33): (10.10)
Note that Qd(r2)-Qd(rl)=-&(r~ - r:); that is, the volume of lubricant squeezed gut from the land area because of a reduction in clearance (or, conversely, flowing back when h increases) is partly added to the pressure-induced flow p r I R leaving the
DYNAMICS
307
outer boundary of the land, and partly subtracted from the same flow p , l R entering a t the inner boundary. In particular, it may be easy to see that the flow rate crossing the recess boundary (i.e. a t radius r=rl) is:
where A, is the projected area of the recess (in this case, simply Ar=zr;) and A, is the effective bearing area of the pad, viz. the ratio of the static load capacity to the recess pressure. Equation 10.11 may be considered to be of general validity, whatever the shape of the pad.
10.3.2
Annular-recess pads
Proceeding as above, the squeeze coefficient is easily obtained for the annularrecess pads of the type shown in Fig. 5.13:
I (10.12)
A plot of B and w* (Eqn 10.9) is given in Fig. 10.4.b, both for tapered and flat (a=x/2) annular-recess bearings.
10.3.3
Tapered pads
For the pads in Fig. 5.19, we may assume that h=h,lsina, where h, is the film thickness over the land surface. The following is easily obtained (see ref. 10.5) for a central recess pad: (10.13) that is, it equals the coefficient B already obtained for the flat pad (Eqn 10.81, divided by sin4a. Equation 10.13 is plotted in Fig. 10.4.a. Coefficient B for the annular-recess pad may be obtained in the same fashion, namely by dividing Eqn 10.12 by sin44 and it may be found in Fig. 10.4.b.
308
HYDROSTATIC LUBRlCATlON
Screw and nut assemblies
10.3.4
Equation 10.13 and Fig. 10.4may also be used for the hydrostatic screw and nut assemblies (see Fig. 5.35), substituting sina with cose, i.e, with the cosine of the flank angle. If hydrostatic lubrication is extended over more than one turn of the screw, B must also be multiplied by n (the number of active turns).
Other pad shapes
10.3.5
In general, the Reynolds equation cannot be explicitly integrated, for a generic pad shape, and approximate solutions should be looked for. For the sake of clarity, let us consider a plane pad of any shape; the relevant Reynolds equation will be (from Eqn 4.15):
a z
a
(h3
Z P )+
a
a
(h3
(10.14)
Z P ) = 12 P h
to which the inner and outer boundary conditions p = p r on ri and p=O on rohave to be added. If the film thickness h is not uniform (e.g. tilted pads), another term should be added, depending on the bearing velocities in directions x and z. Since Eqn 10.14is linear, the differential problem can be split into two parts: $(ha
,PO)+ a
g(h3$ P O )
=0 ;
p O = p r on
; p O = O on To
(10.15)
(10.16) where P'Pa+Pd
is the solution being sought.
It is easy to see that p a is the static pressure distribution for the given boundary conditions, while Pd is the dynamic overpressure due to the squeeze effect. If tangential velocities have to be accounted for, another term p u could be evaluated in the same way. Similar considerations hold good for pads of any shape, cylindrical, spherical, and so on. The solution of the static problem (Eqns 10.15)has been dealt with in Chapter 5 , leading to the evaluation of the effective area A,, and Eqns 10.16 may be solved with the aid of the same numerical methods. Whatever the method (for instance finite differences, finite elements, as well as boundary elements), a pressure distribution pd, proportional to the squeeze rate h, will be obtained. If we integrate the pressure Pd on the whole land area, we may evaluate the contribution Sh of squeezing to the load capacity of the land itself, which has to be added to the static load capacity and, if necessary, to a hydrodynamic term depending on tangential velocity.
309
DYNAMICS
In many cases, however, coefficients B and ty* may be totally disregarded. Indeed, it will be shown in section 10.5 that, when the compressibility of the lubricant is negligible, the damping of the bearing is little affected assuming B=O. On the contrary, when the lubricant stiffness is poor, as compared to the bearing stiffness, the squeeze film effect may become important in assessing bearing stability. However, B may be evaluated roughly by dividing the land area into parts with simple shapes, whose contribution to squeeze may be easily obtained. For instance, the land of a rectangular pad may be split up in the way shown in Fig. 5.26;for parts 1 to 4 the expression of the squeeze load of an indefinite rectangle can be used:
while each corner can be approximated by an arc of a circular ring, for which
The flow rate can be found for a given pressure distribution by integrating the flow vector q,whose components are given in Eqns 4.32:
Q = I q v dT r where I- is a closed contour of which v is the normal external direction. Again, the flow rate will be the sum of the static and dynamic terms. If Tis the inner contour of land area, the result will be an equation such as Eqn 10.11.
10.4
SUPPLY SYSTEMS
In order to write the second of Eqns 10.4,we may state the continuity between the flow rate Q delivered by the supply system and the flow rate entering the bearing. With reference to Fig. 10.5,the difference between flow rate Q and flow rate Qi entering the bearing clearance is equal to the volume variation of the recess and relevant tubing, to which the effect of the lubricant density variation has to be added:
-
Vb - --
Qi Q = V
P
(10.17)
The total volume V is the sum of recess volume V,. and of tubing volume Vt; hence:
31 0
HYDROSTATIC LUBRICATION
Fig. 10.5 Flow rates in a pad bearing: Q flow rate delivered by the supply system; Qi flow rate entering the bearing clearance (in static conditions Q=Qi).
av, .
V = V,. + V, =A, h + -pr
apr
(10.18)
(here we assume that pressure p , is uniform in the recess and pipes up to the compensation devices); A, is the projected area of the recess. The term aV,lap, is due to tubing compliance and, in general, may be regarded as a constant. Density p may be considered to depend solely on pressure, hence
where Kla is the apparent bulk modulus of the lubricant (see section 3.2.4). Let us now define the lubricant stiffness Kd as: (10.19) Bearing in mind Eqn 10.11, Eqn 10.17 becomes: (10.20) From the point of view of flow continuity, the bearing may be represented by the hydraulic system in Fig. 10.6; Q is the volume flow rate delivered by the supply devices, R is the hydraulic resistance of the bearing clearances. In dynamic conditions, the flow rates of a spring accumulator (whose spring stiffness and section area are Kd and A,, respectively) and of a piston (section area A,) fastened to the moving member of the bearing have to be added to flow Q delivered by the supply system. It should be pointed out that a number of simplifications have been introduced in developing the foregoing equations. In particular, we have assumed that the pressure is uniform in the recess and in the ducts connecting the recess to the compensation device (or to the pump), that the compressibility of the lubricant in the
DYNAMICS
31 1
Fig. 10.6 Equivalent hydraulic system of a pad bearing.
clearance may be disregarded (since the volume of lubricant in the clearances is very small), and that the inertia of the lubricant is negligible. On the other hand, in certain circumstances (such as long supply pipes and low viscosity), other dynamic phenomena may occur, such as pressure waves in the supply pipes, that will not be considered here. In the following sections, several supply systems will be examined, for each of which a n expression of the flow rate Q will be found and introduced into Eqn 10.20. The second of Eqns 10.4 will thus be obtained and then linearized in order to write a n equation for block A, which appears in the block diagram (Fig. 10.3) and in the relevant transfer function (Eqn 10.7).In applying small perturbation linearization, one should remember that, as a general rule, the hydraulic resistance of the bearing is proportional to h-3 (at least in the case of small amplitude vibrations), hence
(10.21) is the perturbation of 1lR.
10.4.1
Direct supply (constant flow)
The supply system is constituted simply by a positive displacement pump, which delivers a flow rate Q which is not dependent on recess pressure pr. Therefore, the equivalent hydraulic system shown in Fig. 10.6 may be completed by substituting the "supply" block with a constant-flow pump. Equations 10.4 can now be written in the form:
(10.22)
31 2
HYDROSTATICLUBRICATION
W,=-F, is the static load capacity of the bearing in the position of equilibrium es=O, and R, the relevant hydraulic resistance of bearing clearances.
Equations 10.22 constitute a third order nonlinear system, which may be integrated by numerical means or linearized (see also appendix A.2). Applying the small perturbation method and the Laplace transformation, we obtain: (10.23)
Bearing Eqn 10.21 in mind, and comparing Eqns 10.23 with Eqns 10.5, it is easy to find the following expression of block &: (10.24) The characteristic frequencies w1 and w 2 take on the following values:
Ks
01=A2,R,
;
Kd 0 2 = A x
;
(10.25)
K, is the static bearing stiffness
(see Eqn 6.51, and Kd is the lubricant stiffness. In this connection, in evaluating Kd (by means of Eqn 10.191, it is important to take into account the volume of lubricant contained not only in the recess but also in the entire length of the feeding pipes back to the pump. We can now resort to Eqn 10.6 governing the dynamic behaviour of the bearing system in the case of small-amplitude vibrations; the relevant transfer function (Eqn 10.7) will be examined in section 10.5 in order to assess stability and frequency response.
10.4.2
Compensated supply (constant pressure)
The bearing is fed by a system which is able to maintain a constant pressure p s over a compensation device, whose hydraulic resistance may be constant (capillary restrictor or similar devices) or depend on the pressure step (in most case orifices, though elastic restrictors have also been proposed). The overall hydraulic scheme of the bearing is, then, the one to be seen in Fig. 10.7.
DYNAMICS
31 3
Fig. 10.7 Equivalent hydraulic system of a pad bearing supplied at constant pressure, compensated by: a- a fixed restrictor; b- a spool valve; c- an infinite-stiffness valve.
The volume rate of flow passing through a laminar-flow restrictor R , is easily written (see Eqn 6.19)as: (10.26)
whereas for an orifice we find (Eqn 6.24): P s c E z Ro
p
&=
(10.27)
As usual, p is the pressure ratio prIps in the reference configurationh=ho. Equating the right-hand members of Eqn 10.20 and Eqn 10.26, the desired differential relation between p , and E is obtained
31 4
HYDROSTATIC L UBRlCATlON
In order to study stability and low-amplitude frequency response, one may linearize and then apply Laplace transformation, thus obtaining:
from which it is easy to find the transfer function of block $ in Fig. 10.3. A similar equation may clearly be obtained in the case of orifice restrictors:
In both cases Ap may just be written as in Eqn 10.24, but the characteristic fiequencies are now: (10.28)
The value of parameter
0 is:
(capillary) (10.29)
(orifice)
As usual, Kd indicates lubricant stiffness (Eqn 10.19),while Ksis the static stiffness of the bearing (Eqn 6.23 and Eqn 6.26) that may also be written in the following form: (10.30)
10.4.3
Controlled restrictors (constant pressure)
The bearing is now fed by a constant pressure system through a compensation device whose hydraulic resistance will depend on the degree of freedom of a moving element (e.g. a membrane) which, in its turn, will depend on the pressure step and sometimes on the rate of flow being supplied, too. The transfer function of the overall system will obviously depend on a larger number of time constants than in previous cases, since the parameters describing the dynamics of the valve have to be accounted for. This section will show the way the dynamic study of such a device may be planned; in the following sections a few particular kinds of valve will be examined in a little more detail. In order to evaluate A, for passively compensated systems, Eqn 10.20, expressing inlet flow rate Q as a function of the recess pressure and the position of the bear-
DYNAMICS
315
ing, has been put together with another equation expressing the rate of flow from the supply system as a function ofp, (e.g. Eqn 10.26); the block ;lp was easily obtained by eliminating Q and performing linearization. In the case of controlled devices, the supply flow rate may be expressed as a function of x (the degree of freedom of the valve) and of its time derivative, as well as of p,:
Since another degree of freedom has been introduced in the system, another equation is required, which can be obtained by writing down the balance of forces acting on the moving member of the pressure-compensating device; in general, this relation can be written as:
Substituting Eqn 10.20 for Q in both Eqn 10.31 and Eqn 10.32, after performing small perturbation linearization and Laplace transformation, the two equations will take the following form: (10.33)
The relevant block diagram in is shown in Fig. 10.8. Note that Ap0 is the value that the transfer function ;lp would have for &=O (i.e. if the moving member of the controlled valve was "blocked" a t its static position); and && must be null functions when valve control is not sensitive to pressure or to flow rate (and hence to E ) , respectively. An expression for block
is easily obtained from Eqns 10.33:
(10.34)
The bearing transfer function will, formally speaking, remain Eqn 10.7, provided the above value is substituted for %. In the following sections, we shall examine a few examples of controlled devices.
10.4.4
Spool or diaphragm valves
The static behaviour of spool and diaphragm devices has already been dealt with in chapter 6: in both cases the moving member reaches a position of balance due to the opposing thrusts of a n elastic element and recess pressure pr. The hydraulic resistance of the valve therefore depends on the degree of freedom (namely "x") of
31 6
HYDROSTATIC LUBRICATION
Eo,
Fig. 10.8 Block diagram for a hydrostatic pad bearing compensated by means of a controlled valve.
the device, with a law R,=R,(x), which, in general cases, is a non-linear one. If the valve restrictor is not laminar, a more complicated law R,=R,(x, p r ) will emerge. As indicated above, the flow rate delivered by the valve in dynamic conditions must be written a s a function of x and p,.. From Fig. 10.7.b it follows that Q is the
sum of the flow rate passing through the variable hydraulic resistance R , and of the volume of lubricant displaced by the moving spool: Q
=-Ps - Pr
+
A, x
Rr
(10.35)
Note that, in the case of the diaphragm valve, x will be the displacement of the centre of the membrane and A , will be an "effective"area. Substituting Eqn 10.20 for Q, Eqn 10.35 becomes: (10.36) Equation 10.32 is, in this case:
M,X+K,X+A,~,=F,~
(10.37)
where Mu, K , and A , are the spool mass, the spring stiffness and the spool section area, respectively; is a constant (namely, the spring force for x=O). The set of differential equations made up by Eqn 10.36, Eqn 10.37 and the first of Eqns 10.4 describes the dynamic behaviour of the bearing system. As usual, in the case of small amplitude vibrations, linearization may be carried out. For the sake of simplicity, let us limit ourselves to examine the case of small vibrations around the
DYNAMICS
31 7
reference configuration h,=ho (hence, E,=O and BE=&). I n the neighbourhood of E=O, R, may be written as:
R, = R,, + m, =
P
+ C, 6e
+cup ~ p
,
(10.38)
The coefficients C, and Cupshould be evaluated by considering the constructive details of the device. If R, is a laminar-flow restrictor, i t is simply C,=O; whereas, if the restrictor is a true orifice, it is 1 Ro
Cup=-@
p,
For the devices considered in sections 6.3.4 and 6.3.5, coefficient C, may be written as
c,=--1 Ku Ro P” A, Ps The reader may refer to the relevant sections for the meaning of the symbols used above. It is easy to see that the complex operators which appear in Eqns 10.33, and hence in Eqn 10.34, are in this case the following:
(10.39)
In the equations above, KO,is the static stiffness of the bearing supplied through a fixed restrictor Rr=RrOwith the same pressure ratio P:
w,
1
ho
0
Kw=3--
Coefficient CT is given by (10.40)
31 8
HYDROSTATIC LUBRICATlON
(the reader may easily check that, when the restrictor is a pure laminar-flow device or an orifice, Eqn 10.40 give the same result as the first or, respectively, the second of Eqns 10.29). The characteristic frequencies o are given by the following equations:
(10.41)
Note that q,is the natural frequency of the valve spool, whose internal damping has been taken to be negligible; should this last assumption not be applicable, the term (l+s2/a$) could be substituted by a more comprehensive one in the form (1+2CVS/ %+S2/ o&. Equation 10.34 can now be rewritten as (10.42)
The overall transfer function will, as usual, be provided by Eqn 10.7. By the way, the static stiffness of the bearing compensated by a controlled device (i.e. Eqn 6.39) may again be easily obtained from Eqn 10.7; for a laminar-flow valve:
10.4.5
Infinite stiffness devices
Let us now try to obtain the transfer function of a bearing, supplied by the controlled valve in Fig. 2.12. The static behaviour of the valve has already been examined in section 6.3.6. As has been outlined in previous sections, equations expressing the flow and force balance for the valve should be written down; by examining the flow paths in Fig. 10.7.c, we can obtain the following equation: (10.43)
where Qi is the flow rate that crosses the inlet restrictor Ri and y,=(Av-A,)lAv. Since pU=pr+(Q-AvX)Rv, we have:
(10.44)
DYNAMICS
31 9
Force balance on the spool gives: M , X + A , P ~ - ~ ~ ~ - A ~ ~ ~ , = O
(10.45)
and hence
The laminar restrictor Rd, which has no effect on static behaviour, may be added to increase the damping of the movements of the spool. As usual, the compressibility of the lubricant in the device has been disregarded. The hydraulic resistance Ri is, in general, dependent on the position of the spool and on pressure p u . For small-amplitude vibrations it may be stated, that
Of course, it would be C= ,O limit to this last case.
for plain laminar flow: for the sake of simplicity, let us
Eliminating Q from Eqn 10.44 and Eqn 10.46 by means of Eqn 10.20, two differential equations in E, x and pr are obtained, which, added to the first of Eqns 10.4, allow the problem to be solved numerically. For small vibrations around the reference configuration E=O, by linearizing and Laplace-transforming, a set of equations like Eqns 10.33 is obtained. After a number of manipulations, the following equations may be written:
Kocho &
0
=
7
1+-
s
"1
s
1+-
"2 S
1 ;, =p2
1-PSCV "3 Ro 1 + - s "2
A? &P=-AX
1 Kocho l-pAvAeRo
;1 =V&
(10.47)
1 - 1 Rd 1 + -s l+YVRV @4 1 Rd
l . + -Yu Rv
1+-
s
01
(
i4)
s 1+-
320
HYDROSTATlC LUBRICATION
As before, Koc and A,, are, respectively, the static stiffness and the value of A, for Cv=O(that is, the values that would be obtained if the spool was blocked). We have stated
10.5 10.5.1
DYNAMICS OF SINGLE-PAD BEARINGS Transfer function
The block diagram representing the dynamics of a thrust bearing has been presented in section 10.2 (Fig. 10.3), as well as the relevant transfer function (Eqn 10.7). The block A, which appears in Eqn 10.7 stands for the feedback of the system: namely it shows how the system reacts, dynamically changing the recess pressure, to displacements induced by load variations. In section 10.4 it has been shown how $ depends on the lubricant supply system and on the lubricant itself (namely, on its compressibility). In particular, in the case of passive compensation (e.g. a constant flow pump or fixed restrictors) Ap is given by Eqn 10.24. The characteristic frequencies o1 and 0 2 ,on which mostly depends the dynamic behaviour of the system, turn out to be proportional to bearing stiffness K , and to lubricant stiffness Kd, respectively (of course l/w2=0 when compressibility is negligible); they may generally be written in the form of Eqn 10.28, in which the value of constant Q depends on the type of supply system (namely, we have to take o=l in the case of direct supply by a constant-flow pump, or a value calculated by means of Eqns 10.29 for capillary and orifice compensation). In the case of a n "active" supply system (e.g. constant flow valves or "infinite stiffness" devices) block $ is obviously more complicated (see also ref. 10.4 and 10.6): recess pressure p,., in fact, depends also on another variable x (the degree of freedom of the supply device), which, in turn, depends on recess pressure and sometimes on the hydraulic resistance of the bearing, and hence on E. This argument has been treated in sections 10.4.3 to 10.4.5. In the case of passive compensation it may be interesting to return to the equivalent mechanical system of Fig. 10.l.b, and search the values to be assigned to operators K 1 and B,. We can write the linear differential equations which describe the mechanical system and then obtain the relevant transfer function; having established that it must be identical to Eqn 10.7, it turns out that:
321
(10.49)
It is obvious that K, and B, turn out to be negative when &>K8),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
HYDROSTATIC LUBRICATION
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
HYDROSTATIC LUBRICATION
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 nt
nt = - nP
(10.59)
Since np20,the system clearly cannot be stable if the turns are clockwise, that is if (-1,O) is an internal point for the Nyquist diagram. By way of example, let us consider the typical case of a restrictor-compensated bearing, for which the block Ap takes on the simple form of Eqn 10.24, where 01 and 0 2 = 6 0 1 are proportional to the static bearing stiffness and to the lubricant stiffness, respectively. For the sake of simplicity, let us first consider that the squeeze coefficient B is negligibly small, which gives us:
As long a s {>1, 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 <.The limit of GH(iw) for o+O is now =-eid2 and there is only one infinite half circle. As pointed out above, if B and are large enough, the stability of the system is ensured, whatever the value of compressibility; in other words, all the diagrams of the family obtained by varying 5 do not cross the real axis, or cross it at a point on the right of GH=-l. Whereas, in the case shown in Fig. 10.10, the system proves to be unstable for the lowest values of 5. The crossing frequency onmay be
326
HYDROSTATIC LUBRICATION
Fig. 10.10 Nyquist diagrams for restrictor-compensatedbearings ( B S ) . For the sake of clarity, the drawing is out of scale. found by solving the real equation Im[GH(io,)]=O. Provided a finite real solution exists, the system will be unstable if IGH(iw,)l>l.
10.5.3
Frequency response
The frequency response of the system (i.e. the amplitude and phase shift of the steady vibration of the bearing when the force perturbation has a sinusoidal shape with unitary amplitude and frequency f=wl2x) can be found by substituting s=iw in the transfer function (Eqn 10.7). A complex number is obtained, whose modulus and argument represent the amplitude and phase shift of the vibration of the 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
HYDROSTATIC LUBRICATION
-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
HYDROSTATIC LUBRICATION
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 <,one could try to increase slightly the film thickness or the effective area of the bearing. I n the first case greater flow rate and pumping power would be required, while stiffness and damping factor would turn out to be lowered. In the latter case, on the contrary, a great increase in lubricant stiffness (namely, parameter 5 proves to be proportional to the fourth power of D) would match a consistent reduction in flow rate, supply pressure and total power (because turning speed is low); damping factor, too, would be greater. EXAMPLE 10.2 The pad bearing already considered in Example 6.4 bears a load W0=20KN, with a clearance ho=40 pm, when fed at a constant pressure ps=4 MPa through a compensating restrictor, with a pressure ratio p=0.437. Under a load W ~ = 3KN, 5 clearance is reduced to 24.8 ,um and pressure ratio rises up to 0.765. Assuming that moving mass is M=2000 h ' & that the equivalent bulk modulus of lubricant is Kla=500 MNI m2 and that the lubricant volume comprised between the restrictor and the pad clearances is V=20.10-6m 4 we have to check for stability and to assess the frequency response of the system. From Example 6.4 we get: D=0.16 m2, r'=O.625, a'=0.067, A,=O.0114 m2, R*=0.0154. Hydraulic resistance R may be calculated from Eqn 5.68, squeeze coefficient B from Eqn 10.12, parameter w*from Eqn 10.9 (or from Fig. 10.4.b), static stiffness K, from Eqn 10.30, while lubricant stiffness Kd is given by Eqn 10.19, i n which the contribution of the compliance of supply pipe may be disregarded. The main dynamic parameters can now be evaluated, as shown in table below.
F
P (Ns/m*) (KN)
20 35
0.015 0.015
h
urn) 40 24.8
prlps
0.437 0.765
Ks.10-9 (N/m) 1.78 0.845 4.25 0.995 d
<
w,,
C
Y
0.196 0.315
0.003 0.007
@-I)
3.87 3.29
Eqn 10.29 Eqn 10.30 Eqn 10.54
650 705
Eqns 10.51
Eqn 10.52
Stability is clearly out of question, since 5>1; the linearized frequency response has been plotted for both the greatest and the least loads in Fig. 10.15. Comparing this diagram with the frequency response in the case of incompressible lubricant (Fig. 10.11) it should be evident that compressibility produces a noticeable decrease
DYNAMICS
331
>4
Fig. 10.15 Example 10.2: frequency response.
in damping, that is the amplitude of oscillation in the neighborough of resonance is greater. In order to reduce the peaks of vibration one should increase 5 and, above all, This may be obtained reviewing the design and using, if possible, a slightly larger pad.
c.
10.6
OPPOSED-PAD BEARINGS
Opposed-pad thrust bearings (Fig. 7.1) may be regarded a s a set of two singleeffect pad bearings. For the sake of convenience, the position of the moving member which divides the axial p l a y g into two equal parts may be taken as reference; hence, hlo=h20=ho=g12. For the two pads, we have (see Eqns 7.4): hl -ho
& 1 = h = & 0
,
(10.62)
If each pad is supplied independently (e.g. by two pumps or through two restrictors), the relevant block diagram can be obtained by summing the effects of both components (Fig. 10.16). The equation of motion may be obtained from the balance of forces acting on the moving member:
By linearizing, Laplace-transforming and bearing in mind that (as follows from previous sections):
332
HYDROSTATIC LUBRlCATION
Fig. 10.16 Block diagram for opposed-pad bearings.
(10.64)
equation 10.63 becomes:
M ho S ~ S +E (B1, + B a ) ho s BE+ (Ae, &I +Ae2 &2) S E = 6F
(10.65)
Squeeze coefficients B1, and Bas can be calculated from the results in section 10.4, while blocks jLP have been the matter of section 10.5. In what follows, symmetrical bearings alone (i.e. Ael=Aez=Ae and R ~ ( E ) = R ~ ( - E ) ) will be studied in a greater detail. For such bearings the block scheme in Fig. 10.16 may be substituted by the one in Rg. 10.17; the latter is valid even when the two recesses are not supplied by independent devices (e.g. when a flow divider is used), after the relevant expression for & has been found. In the case of symmetrical bearings, Eqn 10.65 may be rewritten as: M ho s2&+ B, ho s &+Ae jLP 8 ~ 6F =
Fig. 10.17 Block diagram for symmetrical opposed-pad bearings.
(10.66)
DYNAMICS
333
and, thus, the relevant transfer function is formally identical to Eqn 10.7.Block & is clearly the sum of AP1 and APz. The squeeze coefficient may be calculated from the following equation:
where Bo has the same value a s each pad (for example, for circular bearings, see Fig. 10.41,while B' is given (for pads having uniform film thickness) by
(10.68) A plot of B' is also given in Fig. 10.18. 10 -
B
5-
0
1
Direct supply (constant flow)
10.6.1
When each recess is supplied independently by a pump delivering a constant flow Q/2, we have:
S
5P2 42'
-6E =-
where:
1 + (1 - &) -
Koho 1 2Ae - 1 +
w1
1 s (1-EYw2
(10.69)
334
HYDROSTATIC LUBRICATION
(10.70)
(hereafter we shall omit the subscript s, taking i t for known that in linearized equations all the parameters take a value corresponding to the steady part of the load). As usual, KOis the static stiffness of the bearing for E,=O (Eqn 7.11). The equations above have been obtained from Eqn 10.24 and Eqns 10.25, taking into account that static stiffness and hydraulic resistance of each pad depend on E , a s indicated in Eqn 6.7 and Eqn 5.12. The transfer function of block & is easily obtained adding &1 and h2together: (10.71)
In the particular case of small amplitude vibrations around cS=O, Eqn 10.71 takes the simpler form of Eqn 10.24.
10.6.2
Compensated supply, passive compensation (constant pressure)
We may obtain the feedback functions ;Ipl and Ap2 for a n opposed-pad bearing compensated by laminar restrictors (Fig. 7.6) from Eqn 10.24 and Eqns 10.28; proceeding as in the previous section, namely bearing in mind Eqn 6.22, 6.23 and 5.12, we find
(10.72)
where function 0 is given by
[A
= (1 -P)
+ (1 +
(10.73)
and (10.74)
KO is the static stiffness of the bearing in the centre (unloaded) position (Eqn 7.26). Block 4 2 is obtained changing the sign of E in Eqn 10.72.
As before, the total feedback ,$ will be the sum of jZpl and ;Lpa (i.e. Eqn 10.71). In the particular case of eS=O, we have 0=1; thus, an equation similar to Eqn 10.24 will again be obtained (the relevant values must of course be used for KO,~ 1 ~,2 ) .
335
DYNAMlcS
The reader may obtain similar equations in the case of orifice-type restrictors. E W P L E 10.3 Let us examine, from the point of view of dynamics, the hydrostatic lead screw considered in Example 7.3, assuming that the reduced mass of moving members is M=3000 I@. For what concerns lubricant, we must consider an effective bulk modulus Kla=109N l m z a n d a viscosity varying in the range p=0.05+0.2 Nslm2, depending on the actual temperature; the volume of recess and relevant tubing is V=10-5m3 for each side. Let us consider first the unloaded case (E~=O). We can calculate the static stiffness from Eqn 7.26 and the stiffness of lubricant (for each side) from Eqn 10.19, thus obtaining K0=1.29.1# N l m , Kd=1.92.1@ N l m ; from Fig. 10.4 weget Bol(np)=0.89.106 m. The other main parameters, which in general depend on viscosity, may be calculated as shown in the following table. B,-10-6
o1
w2
on
(NsIm5) (Ns/m) 0.093 0.18 0.185 0.35 0.277 0.53 0.370 0.71
(s-1)
(s-1) 1545 772 515 386
(s-1)
P,.10-'2
Eqn 5.116 Eqn 10.67
518 259 173 130
w
5 0.68
655
1.35 2.03
0.07 1
2.7 1 Eqn 10.52
Eqns 10.51
Eqns 10.74
-b-
1.51
1.o
0.5
'
0.0 0
I
1
I
100
200
300
Fig. 10.19 Example 10.3: frequency response for two values of eccentricity.
336
HYDROSTATIC LUBRlCATiON
In practice, in centred position, the system is the sum of two identically behaving pads and, therefore, the considerations made about stability of thrust bearings can be used again. Namely, stability depends on the value of parameter t=%/wl=2KdlK&since it is greater than unity, a good margin of stability is ensured. Frequency response is obtained substituting s=iw in the transfer function (Eqn 10.71,in which B, is calculated by means of Eqn 10.67,and ;\p is the sum of ;\pl and App2 (Eqns 10.72). I n Fig. 10.19 we have plotted the amplitude of frequency response for E,=O and for ~,=0.37(which is reached under a load W=l5KP& when pitch error is null).
10.6.3
Flow dividers
If the bearing recesses are fed by means of two independent valves, one may proceed as in the above cases by evaluating Ap for both recesses and then summing their effects. If, on the other hand, a flow divider is used (see sections 2.3.2 and 7.2.51, the block diagram in Fig. 10.16 is no longer valid, while the diagram in Fig. 10.17 is still useful (it is assumed that the pads are symmetrical). In the latter case, block may be substituted by the one in Fig. 10.20, if the valve is controlled by the recess pressures alone. Proceeding in the same way as in section 10.4.4,we can write two equations connecting the recess pressures, the displacement of the bearing and the degree of freedom of the controlled device:
I I I I I I
I I I I I I I
:ti€ I
I I I I I I I
I I I
DYNAMICS
337
(10.75)
to which we may add another equation expressing the balance of the forces acting on the moving member of the valve:
In the above equations K , is the stiffness of the elastic member of the valve (spring or membrane), M , and A , are, respectively, the reduced mass and the effective area of the spool or membrane (see section 7.2.5). The hydraulic resistances R,1 and Rv2 of both sides of the valve depend, very often in a nonlinear way, on the degree of freedom x . For the sake of clarity let us consider the case of a diaphragm valve (Fig.2.15): the hydraulic resistance of each side proves to be inversely proportional to the third power of the relevant gap, as can be seen in Eqns 7.57, in which we introduced the non-dimensional diaphragm displacement <=x / l o ; as usual, p is the ratio of static recess pressures a t E=O to supply pressure p s . After linearization and Laplacetransformation, Eqns 10.75 and Eqn 10.76 take on the following form:
(10.77)
In the case of ~~'0,the A blocks are given by the following equations:
1 --
S
"3
1+-
(10.78)
S
"2
The value of KW is given by Eqn 7.26;q,=d w u is clearly the natural frequency of the diaphragm; for the other characteristic frequencies we have:
338
HYDROSTATIC LUBRICATION
(10.79) Finally, the transfer function of block ,Ip can be easily obtained:
Once & has been evaluated, the dynamic behaviour of the relevant bearing can be examined as in section 10.5.
EXAMPLE 10.4 T h e opposed-pad thrust bearing already examined in Example 7.1 reaches a high stiffness because its supply pressure is compensated by means o f a diaphragm-controlled flow divider. Let us now examine it for what concerns stability and frequency response, assuming that moving mass is M=lOOO @. From data reported in Example 7.1 it is easy to find: A,=12.4.10-3 m2 (Eqn 5.66),
-a-
-b-
Re(GH)
\ 0'
0
I
200
400
600
$$ ( H a Fig. 10.21 Example 10.4: Nyquist diagrams (a) and frequency response curves (b) for certain values of {=w2/w,=2Kd/KOc (dotted line represents the frequency response of the same bearing with capillary compensation and <=S).
339
DYNAMICS
Ro=64.8.109Nslm5 (Eqn 5.68), Kk=2.01.109 N l m (Eqn 7.26),B0=0.158.106N s l m (Eqn 10.12) and B,=2Bo=0.315.106 Nslm. Also the parameters of the controlled restrictors were selected in Example 7.1 (in particular 8=0.3 and a,=0.55), therefore, from Eqns 10.79 we can obtain the relevant characteristic frequencies: W1=144 s-1 and 03=13.5.1@ s'. A first rough assessment of stability could be made, as indicated in section 10.5.2, disregarding the squeeze coefficient and the mass of the diaphragm; the Routh criterion applied to the relevant third-order characteristic equation would indicate that the system is stable when 5 = w 2 / o1> 1/[1-6@(1-P)a, (1 + w1 / 03)1 = 3.34 ,
that is when Kd>l.7~Koc=3.36~109 N l m . A more detailed analysis, however, shows that w, is large enough to have no practical effect, whereas squeeze parameter contributes to increase the margin of stability. In Fig. 10.21 Nyquist diagrams and frequency response curves have been plotted for a number of values of parameter 4. It may be seen that the controlled supply device may very easily enhance static stiffness at will, but this gain is rapidly lost as the frequency of exciting force increases.
10.7
SELF-REGULATING BEARINGS
The dynamic behaviour of SRBs (Fig. 7.25) can be studied in a similar way to usual bearings. For the sake of simplicity, the theoretical case alone will be considered here, in which all bearing clearances are equal to ho when the external load is F=O. A quantitative evaluation of the consequences of working tolerances may be found in ref. 7.7. Dynamic load capacity is, a s usual, the sum of a term proportional to recess pressure and of another one due to squeeze:
W = A ,p,. W ' ( E + ) B ( E ho ) &
(10.81)
where W and A, are given by Eqn 7.78 and Eqn 7.79, respectively, and (ref. 10.7): B = Bo B'(E)
(10.82)
Bo and B' are plotted in Fig. 10.22 for certain values of r', while rh is assumed to take on the relevant "optimal" value as in section 7.4 (see Fig. 7.26). Note that, in evaluating B, lubricant compressibility in gaps and in secondary recesses has not been taken into account. Comparing Fig 10.22 with Fig. 10.4.b, the intrinsic damping of the SRB proves to be much greater than for conventional opposed-pad bearings. This occurs because the SRB has a built-in pressure-compensating system, whose damping effects are
340
HYDROSTATIC LUBRICATION
-b-
2
B
B*
1
0
"."
0.5
0.9
0.7
I
I"
1.o
0.5 E
Fig. 10.22 Self regulating bearings: a- damping coefficient B*=B0.32h30/(3x/d)4)versus radius ratio r'; b- damping coefficient B'= BIB, versus eccentricity E ; (ri=ri,opt). reflected in the high value of coefficient B. Namely, for the opposed-pad bearings, B may even be disregarded, since the damping relies mainly on the external compensating system; the contrary happens in the case of the SRB. The balance of the forces applied to the moving member of the bearing is expressed by:
In studying small amplitude vibrations around position E = E ~ (static displacement under load Fs),we may apply the small perturbation method and Laplace transformation to Eqn 10.83, which becomes
The lubricant flow rate delivered by the supply system is:
(10.85) which may be transformed into:
D WAMlCS
341
(10.86)
(Equations for Roand R i are given in section 7.4).
10.7.1
Constant flow feeding
The behaviour of the system is represented by Eqn 10.83 and Eqn 10.85 in which Q is assumed to be a constant. In the case of small amplitude vibrations, Eqn 10.84 and Eqn 10.86 m a y be used instead, and SQ=O. Equation 10.86 leads to:
(10.87)
where 4 KO 3A,Ro
" 1 = - 2
''
"2==
Kd e
O
;
(10.88)
KO is the bearing stiffness in the unloaded configuration (Eqn 7.92), Kd is the stiffness of lubricant contained in the central recess and in the relevant supply pipes (Eqn 10.19); coefficients G1and G2 are non-dimensional functions of the displacement, given in Fig. 10.23. It should be noted that if the static part of the load is null, ;lp also vanishes. The transfer function of the whole system is (from Eqn 10.84): (10.89)
where
K is given by Eqn 7.93 (and Fig. 7.27), coefficients G2 and G3are plotted in Fig. 10.23 as functions of the static displacement. It is easy to see that, for static loads (s=O), Eqn 7.91 is again obtained.
HYDROSTATIC LUBRICATION
342
-a-
100
Gl
-b-
11
G2
G3
10
O! 1
0.' 0.o
1.o
0.5
0.1
1.o
0.5
E
E
Fig. 10.23 Self-regulating bearings: coefficients G , , G2 and G3 versus eccentricity. When the static load is null, we get G3=0and K'=l; hence, the bearing behaves like a second order system with undamped natural frequency 0 ~ 1 2and ~ a damping factor In general cases, it will be a three-pole system.
c.
For what concerns stability, i t is easy to see that no problem generally exists, even when { = w , / q is small, since the damping properties of the system rely mostly on the bearing itself (namely on coefficient B ) rather than on the supply system.
10.7.2
Constant pressure feeding
When the SRB is directly fed by a hydraulic network at constant pressure p s , we shall obviously have pr=ps and 6pr=0. Hence, the behaviour of the system is described by Eqn 10.83 or, in the case of the linearized model, by Eqn 10.84 which may be written as follows: (10.91)
where KOis the static stiffness in the centre position (Eqn 7.97) and K is given by Eqn 7.98. Note that, under the simplifying assumptions we have made, lubricant compressibility has no influence on the dynamic behaviour of the bearing, which is reduced to a second order system.
DMvAMlCS
343
When the SRB is fed through a laminar-flow restrictor R, (see section 7.4.21, the variation in flow rate due to a change in recess pressure is
which can be substituted in Eqn 10.86 to obtain the feedback function Ap=6p,./8&. Then, the transfer function may be obtained from Eqn 10.84. In this case, too, we get a second-order system when the static part of the load is null.
10.8
MULTIPAD BEARING SYSTEMS
This section deals with the hydrostatic bearing systems consisting of a certain number of independent pads, which are able to sustain loads in multiple directions. The multipad journal bearing in Fig. 1.12.a and the hydrostatic slideway in Fig. 1.16 are examples of such systems. In the first example the bearing is able to SUStain loads along any radial direction; hence, its displacement may be described by a set of two coordinates. In the second case, the 12 hydrostatic pads take the loads in every direction, except along the x: axis: five coordinates (and five equations) will therefore be required to describe the static and dynamic behaviour of the carriage. In every case the system may be studied along the following lines: i)
ii)
iii)
first, we must fix a suitable set of n generalized (Lagrangian) coordinates, able to describe any displacement of the system: obviously, n is the number of degrees of freedom constrained by the bearing system; we must obtain a n equation for each pad, giving its dynamic load capacity Wj as a function of the parameters which characterize the supply system, of the generalized coordinates and of their time-derivatives; we must write down a set of n independent differential equations, expressing the balance of the external forces, of the inertia forces and of the load capacities (D'Alembert o r Lagrange equations). In general, a complicated set of nonlinear equations will be obtained, requiring numerical simulation to trace the system response to large-amplitude loads. In most case, however, i t will be enough to linearize these equations and to examine them to judge the stability of the system and to obtain its response to dynamic loads.
A number of other considerations must be borne in mind in certain cases. For instance, when the tangential speed is high and the thickness of the film of lubricant is not uniform, as happens in journal bearings, the forces due to the hydrodynamic effect should be taken into account.
344
H YDRCSTATIC LUBRICATION
Hydrostatic slideways
10.8.1
Let us now see, with the aid of a practical example, how the dynamic study of a system of pads may be stated. The simple carriage sketched in Fig. 7.29 is supported by four hydrostatic plane pads. In any given steady configuration, the mean film thickness of the j-th pad is hjo and e&-hjo)/hj0 is the relative variation of the gap. Each pad exerts on the carriage a force Wj, that may be written in the following form: Wj = Aj pj - BjC&j)hjo Ej
Cj = 1 ... m)
(10.92)
The effective area Aj may be considered to be a constant; actually certain displacements of the carriage may make the bearing surfaces out of parallel, and consequently may affect the coeficient Aj: however, such changes are generally negligible. The squeeze coefficient Bj is greatly affected by the actual value of the film thickness; on the other hand, as already noted in section 10.5, initially it can be totally disregarded; alternatively, i t may be substituted by its reference value Bjo=Bj(0), when small displacements from the steady configuration are considered. The equations of motion of such systems can be obtained equating the external forces to the inertia forces; their general form is, therefore: (z = 1...n)
(10.93)
where m is the number of pads, n the number of generalized coordinates xi 6.e. the degrees of freedom constrained by the pads); the terms auWj are the generalized components of the pad forces, i.e. aii=hjo.(dej/dxi), and Fi are the generalized components of the external forces. In the case of the system in Fig. 7.29, it is clearly m=4, n=3; the xi are the axis z and the tilt angles 8, and 8,; therefore Eqns 10.93 become:
-M2 + -J, 8,
C, (Wj)+ Fz= 0 J
+ b (W, +Wz -W3 -W4) + M, = 0
I-J, 8, - a (w,-w,-w3+ w 4 )+ M, = o (it is assumed that x , y and z are principal axes). The Ji are the inertia moments, the Fi and Mi are the components of the resultant and of the resultant moment of the external forces. The forces Wj are given by Eqns 10.92, in which the recess pressures pj are still unknown. If each pad is fed by a n independent supply device, each pressure p, is related to the relevant displacement 5 by a n equation
DYNAMICS
345
like the ones already examined in section 10.4.Of course, if the supply devices are not independent (for instance when flow dividers are employed), more complicated relations are needed, the general form being:
(10.95) The pad displacements nates:
~j may
be written as functions of the Lagrangian coordi-
For instance, for the system in Fig. 7.29,we have: E~
=
Q=
1 (Z
+ b 0,
- a eY)
g1 (z + 6 0, + a OY)
and so on. If we introduce Eqns 10.92 and Eqns 10.96 into Eqns 10.93,these last, together with Eqns 10.94 (or, more generally, Eqns 10.95))constitute a set of non-linear differential equations that are clearly difficult to handle. As usual, a very great simplification is obtained by limiting ourselves to studying the system for the case of small vibrations. It is possible, therefore, to linearize and Laplace-transform the foregoing equations. ARer the transformation, Eqns 10.94may be written in the form: spj
= -&jW 6Ej
(10.97)
Functions Apj of the complex variable s may be written as in section 10.4,depending on the type of supply device. Typically, each &j can be written in the form of Eqn 10.24,in which w1 and w 2 are given by Eqns 10.25 when multiple pumps are used, or by Eqns 10.28 for restrictor compensation. A couple of opposite pads may be treated also a s a single opposed-pad bearing, especially when compensated by means of a controlled restrictor: in this case, h j will be obtained as indicated in section 10.6.Equations 10.92now become
SWj = - [Aj APj(s)+ Bj hjo S] &j
(10.98)
and, thanks to Eqns 10.96,may further be written in terms of the coordinates x i , instead of the ~ j .
346
HYDROSTATIC LUBRICATION
Finally, by linearizing Eqns 10.93, a set of n linear equations is obtained which constitutes a model of the dynamic behaviour of the carriage. For instance let us consider again the system in Fig. 7.29, with some further simplifications: all the pads are equal (the same values for the effective area A,,, and the same clearance ho in the steady configuration) and are fed through capillary restrictors with the same pressure ratio 8. Equations 10.98 now give (see section 10.4.2): where
A = -Bo s + l + s1/+osz/ w 1 KO
and w1 and 0 2 are given by Eqns 10.28. Proceeding as outlined above, the following set of equations is obtained:
M s ~ &+ 4 KO A & = SF, J, ~ 2 6 +0 4~b2Ko A 60, = W x J Y s 2 66,
+ 4a2 KOA6ey = SM,
These equations are completely uncoupled: obviously this is only a consequence of the simplicity of the system we have considered and will not be generally obtained. If the lubricant is sufficiently stiff (w2>>w1), the operator A becomes A=1+ (l/q+Bo/&)s. No stability problem should hence exists: indeed, this kind of systems often feature a great damping.
10.8.2
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: <=x/C and q=y/C. The general problem is quite difficult: the hydraulic resistance Ri and the load capacity Wi of each pad show a nonlinear dependence on the shaft displacement and on its velocity; Wi is not always directed toward the centre of the bearing, but a tangential component may exist; furthermore, the Reynolds equation may not be directly solved, and hence numerical computing should be used to obtain load and flow rate. However, if the displacement is not too great (EcO.~), if the arc taken by each pad is smaller than 90° and the turning speed of the journal is not so high as to give appreciable hydrodynamic effects, great simplifications can be introduced. First the tangential component of the load capacity may be disregarded: Wi is directed a s ~ i . Then it may be assumed that the load capacity and hydraulic resistance of the i-th pad depends only on the relevant components of the shaft displacement and velocity, namely on ~i and E i . The load capacity of each pad may be written in the usual form
The effective area A, may be considered a constant (see section 5.8). The coefficient Bi depends mainly on the clearance: a rough evaluation is often enough; i t may even be totally disregarded. The relevant perturbation is, therefore, SWi =A, Spi - Bi, C s 6 ~ i
(10.100)
The perturbation of each recess pressure may be written in the usual form (Eqn 10.97) and each operator $(s) obtained a s shown in the preceding sections. For instance, in the classical case of capillary compensation (see also section 10.6.2) we obtain:
where KO (reference stiffness of each pad) is given by Eqn 6.22, w1 and o2are given by Eqn 10.28 and O(E)by Eqn 10.73 (hereafter we shall omit the subscript "s",by which we mean that all the parameters are calculated in the point of static equilibrium). If n is an even number, it may be preferable to consider the multipad bearing as a set of n/2 opposed-pad bearings, obtaining the operators as in section 10.6. In any case, Eqn 10.100 becomes
348
HYDROSTATIC LUBRICATION
(10.101) The equations of motion for the journal may now be easily written:
(10.102)
where W t and 6Fq are the components of the external perturbation.
At this point the problem is completely defined, for we have a set of differential equations connecting the displacement of the journal to the external excitation. In facts, i t is easy to see that the pad displacements S E ~depend on the journal displacement: (10.103) Introducing Eqns 10.101 and Eqn 10.103 into Eqn 10.102, the equations of motion become:
(10.104)
Apparently, this is a complicated set of equations; in particular cases, however considerable simplifications can be introduced: for instance, if we assume that n=4, &=O, and that 5 is directed toward the centre of a pad, it is very easy to see that the above set splits into two independent equations:
Very simple equations are obtained in the case of small vibrations around the centred configuration E ~ O since , for all the pads we have:
Clearly no stability problem should occur when the lubricant is sufficiently stiff (w2>w1). Actually, when the turning speed is high, self-excited vibrations may set in
(ref. 10.9); these 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
HYDROSTATIC LUBRICATION
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 (~<0.5).For a better approximation we must return to the general form of Eqns 10.115 and calculate the four stiffness coefficients (that is the elements of matrix K)and the four damping coefficients (matrix B)for given values of static displacement kS, and turning velocity a.A simple approximate way is the thin land assumption, that is to assume that, when land width is small and eccentricity is not great, pressure variation over the lands is linear (ref. 10.13). Axial and circumferential lands may be treated separately and flow calculations may be performed using the simple Poiseuille and Couette flow equations. In practice, the lumped resistance method (see section 8.2) has been extended to account for turning velocity and for the additional flow rates caused by the shaft movements around its static equilibrium position (the squeeze effect of the lands is disregarded, since it is negligible when lands are narrow and fluid is incompressible). Once the force W due to the pressure of the lubricant has been obtained as a function of the displacement and of the shaft velocity, the dynamic coefficients can be calculated by differentiating the components W, and W ywith respect tox,y,x andy.
A more general approach is based on the numerical solution of the Reynolds equation, which also allows us to account for possible effects of cavitation (see, for instance, ref. 10.14, 10.15, 10.16). Equations 10.118 also allow us to foresee the dynamic response of the bearing to external actions. In particular, let us examine the classical case of a harmonic ex-
357
DYNAMICS
citing force of a given angular frequency o.Taking W,=O (since for small eccentricities the influence of the load angle is negligible, there is no loss of generality), Eqns 10.118 give:
We may now substitute iw for s, obtaining two complex numbers, whose moduli and arguments give the amplitude and phase of the oscillation in the 5 and q directions. The shaft axis is seen to cover a n elliptical orbit, the maximum displacements in the 5 and 17 directions being:
ISF I
I "I =
& [(l-w2/w;I4+ 16
(1-02/w,2 12 + 4 r" o"/O,"
(a2-w2/w;)2 + 8
c2(1-w2/o;
)2
(R'2+~2/o:)
(10.122)
' "'
=
16F I
4 pa2
[(l-02/c$I4+ 16 c4(Rr2-02/oi + 8 r" (1-w2/02 (R.2+q2/o$ )2
)2
)
in which R' is the ratio of the actual turning speed to the critical one:
Examination of Eqns 10.122immediately leads to a number of considerations. First, the vibration amplitude in the normal direction, with respect to the load, vanishes for non-rotating bearings. Actually, this is due to our simplified approach, since a certain cross-coupling generally exists (namely, B and K in Eqns 10.115 are not diagonal matrices even when a=O,although off-diagonal terms are often quite small). Then it is easy to see that the amplitude of vibration becomes infinite when eu,,and a=R*,which agrees with the above remarks concerning stability limits. If we now plot the dynamic flexibility as a function of the exciting frequency for a given value of the damping coefficient, and certain values of the turning speed (Fig. 10.29),we see that, at high values of R', the hydrodynamic effect makes a considerable (indeed dominant) contribution to the static load capacity. On the other hand, the damping progressively decreases: a t low speed we get a n overdamped behaviour, then a resonant peak occurs, whose amplitude increases, until i t becomes infinite a t S'=1. In conclusion, we see that whirl instability requires particular care during the design stage, since the actual behaviour of the bearing might prove to be vastly dif-
358
HYDROSTATIC LUBRICATION
-a-
0.0
0.5
w
1.o
Wn
1.5
0.0
05
Wn 0
1.o
1.5
Fig. 10.29 Multirecess journal bearings: typical frequency response for various turning speeds (
c=a.
ferent from what foreseen when the turning speed was not taken into consideration. Fortunately, however, in most cases, the turning speed in actual applications is far from dangerous limits.
10.9.4
Compressible lubricant
Let us consider a four-recess bearing, rotating a t low speed; if the axis of the journal is assumed to undergo small vibrations along the z axis, directed toward the centre of a recess, the pressure in the side recesses may be considered to remain practically constant. In this case, a simplified approach may be attempted and the bearing may be studied just like an opposed-pad bearing. Even, i t is easy to consider particular types of compensating devices, such as diaphragm-controlled flow dividers (ref. 10.4).
A more general approach is delineated in ref. 10.17. The main simplifications introduced in that work consist in assuming that eccentricity is not too high (~<0.5), in order to avoid cavitation and large attitude angles, and that, when a harmonic dynamic load is superimposed to the steady load, the journal centre executes plane small-amplitude harmonic vibration around its steady state position, namely
359
DYNAMICS
G&=QeiWt.The local film pressure may be expressed as the s u m of a static term plus a dynamic one, proportional to the vibration amplitude:
The dynamic film pressure can be obtained by numerical (finite differences) solution of the first-order perturbation of the Reynolds equation, with suitable boundary conditions. In particular, the boundary conditions a t the recess edges (that is the dynamic recess pressures) are obtained imposing continuity of recess flow and prove to be complex functions of the compressibility of the lubricant and of the frequency of vibration, besides of usual parameters, as eccentricity, pressure ratio, etc.
-a0.17
10’
K0.5
LDPslC
I-
A -2
lo3
u-
lo4
2x~04
-d-
-C4
-b-
7
I
J
*lo4 A =5 A =2
Fig. 10.30 Dynamic coefficients (for a multirecess journal bearing with r r 4 , L / D = l , a‘=0.25, 0=45q J 4.6, ~ ~ 4 . versus 5 ) frequency parameter a=3ptw’[pS(CID)z],for certain values of compressibility parameter y=psAri/CK, and of speed parameter .4=12xSh.
360
HYDROSTATIC LUBRICATION
By integration of the dynamic pressure field acting on the journal we can obtain the radial and tangential components of the dynamic load and, hence, the relevant stiffness and damping coefficients:
K,+iwB, K4&-+ i w Be Obviously, the four coefficients are functions of the frequency of vibration and also depend on the geometrical parameters, on eccentricity, on lubricant compressibility and on pressure ratio; moreover the cross coefficients K@& and B@ depend on angular speed, too, whereas the direct coefficients are practically not affected by 0. A lot of calculations are therefore required in order to give a full description of the dynamic characteristics of a bearing with given geometrical ratios: in ref. 10.17, for instance, the calculations of the coefficients for a typical bearing are condensed in several plots, some of which are shown in Fig. 10.30.
REFERENCES 10.1 Opitz H., Bottcher R., Effenberger W.; Znuestigation on the Dynamic Be-
hauiour of Hydrostatic Spindle-Bearing Systems; 10th Int. MTDR Conf., University of Manchester, 1969, pap. MS-21; 15 pp. 10.2 Masuko M., Nakahara T.; The Influences of the Fluid Capacitance in the Oil Feed Line System on the Transient Response of Hydrostatic Guideways; Int. J . Mach. Tool Des. Res., 14 (19741,233-244. 10.3 Wilcock D. F.; Externally Pressurized Bearings as Servomechanisms. - The Simple Thrust Bearing; ASME Trans., J . of Lubrication Technology, 89 (1967),418-4!24. 10.4 Chen K. N., Yang G. P., Wang X.,Yang H. H.; A System Approach to the Dynamic Characteristics of Hydrostatic Bearings Used on Machine Tools; Int. J. Mach. Tool Des. Res., 20 (19801, 287-297. 10.6 Prabhu T. J., Ganesan N.; Characteristics of Conical Hydrostatic Thrust Bearings under Rotation; Wear, 73 (1981),95-120. 10.6 Moshin M. E., Morsi S. A.; The Dynamic Stiffness of Controlled Hydrostatic Bearings; ASME Trans., J . of Lubrication Technology, 91 (19691, 597-608. 10.7 Wylie C. R., Barrett L. C.; Advanced Engineering Mathematics; MacGraw & Hill, 1985; 1103pp. 10.8 Ogata K.; Modern Control Engineering; Prentice-Hall, 1970; 836 pp. 10.9 Inasaki I.; Stability of Hydrostatic Journal Bearings; Eurotrib 81, proc. 3rd Int. Tribology Congr., Warszawa, 1981, vol. 2; pp. 116-122. 10.10 Ghosh M. K.; Dynamic Characteristics of Multirecess Externally Pressurized Oil Journal Bearing; ASME Trans., J. of Lubrication Technology, 100 (1978), 467-47 1.
DYNAMICS
361
10.11 Ghosh M. K., Majumdar B. C.; Stiffness and Damping Characteristics of Hydrostatic Multirecess Oil Journal Bearings; Int. J. Mach. Tool Des. Res., 18 (19781,139-151. 10.12 Leonard R., Rowe W. B.; Dynamic Force Coefficients and the Mechanism of Znstability in Hydrostatic Journal Bearings; Wear, 23 (19731,277-282. 10.13 Vermeulen M.,De Shepper M.; Theoretical and Experimental Study of the Dynamic Behaviour of Hydrostatic Radial Bearings; Eurotrib 89, proc. 5th Int. Congr. on Tribology,Helsinki, 1989, vol. 3; p. 180-185. 10.14 Chen Y.S., Wu H. Y., Xie P. L.; Stability of Multirecess Hybrid Operating Oil Journal Bearings; ASME Trans., J. of Tribology, 107 (19851,115-121. 10.16 Rowe W. B., Chong F. S.; Computation of Dynamic Force Coefficients for Hybrid (Hydrostatic JHydrodynamic) Journal Bearings by the Finite Disturbance and Perturbation Techniques; Tribology International, 19 (19861, 260271. 10.16 Lund J. W.; Review of the Concept of Dynamic Coefficients for Fluid Film Journal Bearings; ASME Trans., J. of Tribology, 109 (19871,37-41. 10.17 Ghosh M. K., Viswanath N. S.; Recess Volume Fluid Compressibility Effect on the Dynamic Characteristics of Multirecess Hydrostatic Journal Bearings with Journal Rotation; ASME Trans.,J. of Tribology, 109 (19871, 417-426.
Chapter
11
OPTIMIZATION
11.1
INTRODUCTION
In this chapter an important aspect of the study and design of hydrostatic bearings, already presented in previous chapters, especially in chapter 6, will be dealt with. The "optimum" conditions corresponding to the minimum power dissipated by the bearing will be identified and a general procedure for the solution of the problem will be described. Our investigation is carried out on an infinitely long pad directly supplied by a pump. The same procedure is then applied to real bearings. Afterwards, the investigation is extended to an infinitely long pad and real bearings supplied by means of compensators. The results will also show that a direct supply system, a s compared with a compensated supply system is more efficient (that is, less power is dissipated and stiffness is greater). This subject may be dealt with using only some of the elementary formulae of hydrostatic lubrication presented in chapters 4 and 5 .
11.2
GENERAL PROCEDURE
First of all we shall start with the study of the static behaviour of a n elementary hydrostatic bearing: the infinitely long hydrostatic pad, Fig. 11.1,of which we shall consider a finite part and afterwards we shall go on t o determine its optimum dimensions.
OPTlMIZ4TlON
363
The pad is first considered as being supplied directly (Fig. 1l.l.a) and then as being supplied by a compensating element (Fig. ll.l.b), and in both cases first when the pad is still and then when it is in motion. Therefore, we shall start with a pad that is supplied directly and still. Consider-
-a-
-b-
Fig. 11.1 Hydrostatic pad of infinite length. A finite length L of it is considered. a- Direct supply; b- Compensated supply (supply through compensating elements).
364
HYDROSTATIC LUBRICATION
ing that its behaviour is described by the fundamental flow rate-pressure relation, Eqn 4.48, we assume that: the lubricant flow rate Q is constant; we then verify the variation of recess pressure p r and of the other quantities: load capacity W, stiffness K and dissipated pumping power H p , as functions of the characteristic variables: length L, recess width b, film thickness h, and lubricant viscosity p; the pad width B is generally assigned. Afterwards we assume that: recess pressure p,. is constant; starting from Q we continue as above. Finally, we assume that: load W is constant and we continue as above. It should be noted that although W is a derived quantity it is fundamental for the determination of the dimensions of any hydrostatic system, as it is almost always assigned and often the most important required characteristic. In the investigation described above, particular care is taken in the determination of the constrained minima of H p , that is of the corresponding optimum values ofL, b, h and p, considered individually. Afterwards a moving pad is considered. Its performance is described by the fundamental relation Eqn 4.49, and, assuming that speed U is constant, the variations of friction F, of dissipated friction power H f and of friction coeficient f as functions of L, b, h, and p, are verified. Finally the variation of the total dissipated power H,, that is the s u m of H p and H f , is considered and its minima and the corresponding optimum values of L, b, h, and p in the three above-mentioned cases are determined: Q constant, p r constant, W constant and assuming in all cases that U is constant. Optimization is first carried out as regards one variable, then two, three and finally all four variables. The concept of pad emciency is also introduced. The optimization procedures obtained for the infinite pad, that are already indicative for any real pad, are transferred in the end to specific pads: the rectangular, the circular and the annular pads. Afterwards the pad supplied by means of a capillary tube is considered. In such a case the ratio /3 (Eqn 5.15) between recess pressure p r and supply pressure p s also plays a role. As done before, the pad is first studied when it is still and assuming that there is a constant supply pressure p s and then a constant load W. In both cases p is obviously taken into account. Afterwards the moving pad is studied.
OPTlMlZATlON
365
Finally the variation of total dissipated power Ht is studied, determining its minima and the corresponding optimum values of L, b, h, and p. The procedures of optimization obtained for the infinitely long pad supplied by a capillary tube are transferred in the end to other types of compensating elements: orifices and constant flow valves, and to the above-mentioned real pads. The single cases of optimization, with one or more variables, are all widely discussed to make their application easy and examples with three and four variables are given.
CONDITIONS OF MINIMUM
11.3
The evaluation of the condition corresponding to minimum pumping and friction power and, more generally, to the minimum total dissipated power of a bearing, requires, a s is well known, the solution of equations or systems of equations such a s
aH -=
axi
0
, i = 1, 2, ...n
(11.1)
where xi can be a dimension of the bearing, film thickness, lubricant viscosity, etc. Thus the "optimum" values of xi are obtained so as to make H a minimum. Sometimes, in addition to the above-mentioned condition, others are imposed: for example that film thickness, or load capacity or stiffness should not be lower than a n assigned value In such cases we have to deal with problems of "constrained optimization".
11.4
EFFICIENCY
Useful indications for optimization can be obtained from the ratios of total power and load and of total power and stiffness:
(11.2)
r w and r, will be defined afterwards as "efficiency losses". It should be noted that p s l W can be interpreted as a "pressure coefficient" on the analogy of the friction coefficient F l W which is interpreted as a n index of the emciency loss for friction of a kinematic couple.
366
HYDROSTATIC LUBRICATIOW
11.5
DIRECT SUPPLY
Let us consider the infinitely long pad in Fig. l l . l . c , of width B , directly supplied by a pump (Fig. 5.11.a), and let us study the performance of a portion of length L.
11.5.1
Steady pad
11.5.1.1Given flow rate. Flow rate Q is assumed to be constant. This is easily accomplished in the case of direct supply with a constant flow supply pump. Film pressure which is determined using Eqn 4.46, replacing z with x, considering Eqn 4.48 and disregarding the sign of absolute value, is
while recess pressure is ~r = 3~ Q
1 B-b
7
(11.3)
If the losses in the supply line are assumed to be equal to zero, p r equals the supply pressure p s . In this chapter the notation pr=RQ, where R is the hydraulic resistance of the bearing (section 4.7.21, will be rarely used, whereas, in previous chapters, i t was used for the sake of synthesis. The diagram of the recess and film pressures of the bearing is showing in Fig. 11.1.~. The dissipated pumping power H p , given by Eqn 5.3, with ps=pr, is 1 B-b Hp = 3p Q 2 h3 L
(11.4)
Load capacity W, obtained from Eqn 4.47 with Eqn 11.3, is 1 W =j3 p Q p (B2- b2)
(11.51
Therefore stiffness K, given by Eqn 5.8, is (11.6) Let us now consider the influence of the dimensions of the bearing, that is its length L,width B and recess width b, on the above-mentioned quantities and then on the performance of the bearing. Using B a s the “reference“ dimension we introduce
L L’=B
(11.7)
367
OPTIMIDITION
and so pressure p r may be expressed as a dimensionless function of L' (11.8)
where p,.o=pQ(B-b)/ Bh3 is the reference pressure. Pumping power Hp becomes (11.9)
where Hpo*=pQ2(B-b)/Bh3is the reference power. It should be noted that Hi=p;. Such equalities between dimensionless quantities will often be found from here on. In Fig. 11.2.a p ; and H i are presented as functions of L'. They are inversely proportional to L'. Introducing
b b'=B
(11.10)
the following dimensionless functions of b' are obtained:
p;=Hi=3(1-b') ,
W' =
= 3 (1- b'2)
(11.11)
In Fig. 11.2.b W', K , p;, H i are presented. They decrease as b' increases. Let us now consider the influence of film thickness expressed in the usual dimensionless form
h ' = -h h0
(11.12)
where ho is the reference film thickness. The following relationships are obtained: (11.13) In Fig. 11.2.c W', K ' , p ; and H i are given. They decrease rapidly as h' increases. These diagrams show the performance of the bearing working with varying loads. It should be noted that, as film thickness reaches zero, the bearing would bear infinite loads (with infinite stiffness). This is obviously not possible, since the pump should yield a n infinite pressure and the supply system should bear it: in practice the maximum pressure is limited by a relief-valve placed downstream from the pump (Fig. 1l.l.a).
360
HYDROSTATIC LUBRICATION
-b-
-a-
4
6
5
4, P; 4 H i , F' 3
f' 2
W' 1
x
K' 0
0
0
0.5
b'
L'opt
1/3
1 Ht
3
-HD
6
12
Hi
H;
10
Hb. ~
5
H;9 Pr
' r
8
4
H i , F'
Hi, F'
3
6
1'
W' 4
W'
2
K' 2
I
0
aI
K' 0
1
-c-
h'
0
-d-
Fig. 11.2 Load W , stiffness K , recess pressure p;, pumping power H', friction force F', friction power H j , friction coefficientf' and total power H ; (for speed factor k=4) versus: a- bearing length L'; b- recess width b'; c- film thickness h'; d- viscosity p', or c'=(l-b')p'. Finally let us consider the influence of the viscosity p of the lubricant, expressed in the dimensionless form = JL
PO
(11.14)
where again po is the reference viscosity. The following relationships are obtained
OPTiMiZ4 TION
369 (11.15)
In Fig. 11.2.d W', K',pi, Hi are presented. They vary linearly with p'. Such diagrams may be employed t o study the performance of the bearing in operating conditions, as p varies with temperature. So, if viscosity decreases because temperature increases, load also should decrease. Since that is generally impossible in operating conditions, film thickness must decrease.
11.5.1.2 Given pressure. Supply pressure p s is assumed to be constant; therefore if there are no friction losses in the supply line, recess pressure pr=ps is also constant. The following formulas are especially interesting for the study and design of a bearing operating at a given pressure which, for the sake of safety, is generally much lower than the maximum pressure the pump can supply or the supply circuit can bear. Load capacity is given by Eqn 4.47 which may be rewritten as 1
W = j p r L (B + b)
(4.47 rep.)
The product LB is the pad area. It is often defined as "projected pad area" to distinguish it from the product L[(B+b)/2] defined as the "effective pad area". Stiffness is 3P K=-'L(B 2h
+ b)
(11.16)
Flow-rate is (4.48 rep.)
Pumping power is (11.17) In dimensionless form the following expressions, functions of L', are obtained:
Figure 11.3.a shows the linear variations of W , K , Q' and H i with L'. Considering their dependence on b' they can be written as:
370
HYDROSTATIC LUBRICATlON
-a-
-b-
3
-7
3 Hi
Hi
lib , Q ' 2
2
, F'
Hi
Hb ,Q' 1.15H;,
H i , F'
~
Hi,
1
__-
I
f'
W ' , K'
W ' , K' 0
0
0
1
2
3
4
L'
5
u pr = const.
HP
3 Hi I
Q' 2
Hi , F' 1.15H;, Hi,
___ ~-
I f' 0
10
3
1
-Ht
l/s
1
3
hHP
-c-
"P -d-
Fig. 11.3 Load W', stiffness K', flow rate Q', pumping power H i , friction force F', friction power H i , friction coefficientf' and total power H;(for speed factor k=l) versus: a- bearing length 15';b- recess width 6'; c- film thickness h'; d- viscosity p', or c'=( 1-6')p'. 1 w=K'=2(1+b')
,
&'=H' P =-311 -16 '
(11.19)
W' and K' are often called "load and stiffness factors", respectively, while Q' and H i may be called "flow and pumping power factors", respectively. In Fig. 11.3.b W',
371
OPTlMlzATlON
K , Q' and H i are plotted against b'. While W' and K' vary linearly with b', Q' and H i vary exponentially. Considering the dependence on h' the following expressions are obtained: (11.20) In Fig. 11.3.c K ' , Q' and HL are plotted against h'. The first quantity is inversely proportional to h' while the others increase according to a cubic law. Finally, considering the dependence on p', the following expressions are obtained: (11.21) Figure 11.3.d shows the inversely proportional variations of Q' and H i with p'.
11.5.1.3 Given load. The load W carried by the pad is assumed to be constant and equal to an assigned value, as commonly occurs in practical applications and therefore in design. Recess pressure is (11.22) Stiffness is
W K=3h
(11.23)
Flow-rate is (11.24)
Pumping power is (11.25) In dimensionless form the following expressions, functions of L',are obtained: p;=L' 2
I
4 1 H;=jL'
(11.26)
In Fig. 11.4 p; and H b are plotted against L',with both quantities inversely proportional to L'. Considering the dependence on b ' the following expressions are obtained:
372
HYDROSTATIC L UBRlCATlON
-a
-
-b 1
0.8 0.6 b'opt
0.4 0.2
0
bopt
u 2
~
1/5
1
0
3
--
6
4
8k10
Hi HP
H;
3
H6 1.15H;,
2
Him -
Hi,F', f ' 1 Q' K' C
I h'opIt ' 1
h'
2
I
1/3
1
3
3HP -d-
-C-
Fig. 11.4 Stiffness K'. recess pressure p;, flow rate Q',pumping power H' friction force F', friction power H i , friction coefficient f' and total power H ; (for speed &;tor k = l ) versus: a- bearing length L', or g'=L'p'; b- recess width b'; c- film thickness h'; d- viscosity p'. In case b optimal recess width b& versus speed factor k is also represented.
Pi==
2
2
9
1
Q ' = 3- -1 - b" '
4
1
Hb = 3 (1 + b') (1 - b")
(11.27)
In Fig. 11.4.bp;, Q' and H i are plotted against b':p; decreases, Q' increases, H i has a minimum. Such minimum value is important for the optimization of a bearing
373
OPTlMlzATlOfU
and it can be determined by solving Eqn 11.1 for i = l and xl=b', thus yielding b'=1/3. It should be noted that around that value Hb is not critical so that greater values of 6' may also be used. The value of 0.5 is often suggested for b', with a n increment of H i lower than 6%, with respect to its minimum value. Considering the dependence on h' the following expressions are obtained: 1
K=r,
Q'=3h'3 2 ,
(11.28)
In Fig. 11.4.c K', Q' and Hi,are plotted against h'. The first quantity is inversely proportional to h' while the others increase with a cubic law. Finally, considering the dependence on p', the following expressions are obtained: (11.29)
Figure 11.3.d shows the inversely proportional variations of Q' and H i with p'
11.5.2
Moving pad
If the pad shown in Fig. 11.1 should move in the z direction, perpendicular to its length, the influence on its performance of the inertia forces acting on the lubricant in the recess as well as the effect of lubricant recirculation in the recess cannot be disregarded. On the other hand if the pad moves in the direction of its length, the influence is nil. In this case the expressions relevant to the motionless pad still hold good, as well as the following expression of the friction force in the recess.
11.5.2.1 Friction. The friction force in the film is given by Eqn 4.49
Ff=p U
L ( B - b)
(4.49 rep.)
The friction force in the recess is
Ffp= p U
1
LB
P
Friction power is given by Eqn 5.5, where &O. the film, this is
HfZpiFiL(B-6) while in the recess it is
As regards the power dissipated in
(11.30)
374
HYDROSTATIC LUBRICATION
If the recess is not too wide, a s often happens, andfor its height is much greater that the thickness of the film, Hfp is negligible compared to Hf, as will be assumed from here on. The power necessary to accelerate the lubricant in the film should also be added to I f f , that is:
However, even for high values of U , Hl is generally negligible compared to Hf, as will be assumed from here on.
Ff and H f may be expressed as dimensionless functions of L':
- 5 - Ff = L' Fi.-Ffo - p U K 1B ( B - b )
'
H H -f= Hi- - H f o p u 2 K 1 i ( B - b )= L'
(11.31)
where Ffo=pU(lfhfB(R-b) and Hfo=pU2(lfh)B(R-b) are reference values for the friction force and power. Figures 11.2.a, 11.3.a and 11.4.a show how F j and H i vary linearly with L'. Considering the dependence of Ff and Hf on b' they can be written as:
Fj = H i = 1-b'
(11.32)
F i may be called the "friction factor" while H i is sometimes called the "power factor". Figures 11.2.b, 11.3.b and 11.4.b show how Fj and H j decrease linearly a s b' increases. Considering the dependence on h' the following expressions are obtained:
Fj = H i
1 =r
(11.33)
Figures 11.2.c, 11.3.c and 11.4.c show the inversely proportional variations of Fj and H j with h'. Finally, considering the dependence on p' the following expressions are obtained:
F j = H j = p' Figures 11.2.d, 11.3.d and 11.4.d show how Fj and H j vary linearly with p'
(11.34)
OPTIMIZATION
375
11.5.2.2 Friction coefficient. If the flow rate is assigned and then load capacity is expressed by Eqn 11.5, the friction coefficientf=Ffl W becomes f
L 2 u h2 = -3Q B+b
(11.35)
If expressed in dimensionless form, as a function of L', 6' and h', i t becomes (11.36) In Fig. 11.2.a and Fig. 11.2.b, f is plotted against L' and b': its variations are similar to those of Fj and H i ; in Fig. 11.2.c, f ' is plotted against h': its variation differs from those of F j and H i as it increases with h'. If pressure is assigned and load capacity is given by Eqn 4.47, the friction coeficient is
f = 2 -Up - -1 B - b pr h B + b
(11.37)
When expressed in dimensionless form, as a function of b', h' and p', it becomes (11.38) In Fig. 11.3.b, 11.3.c and 11.3.d f' is plotted against b', h' and p': its variations are similar to those of F j and H i . Finally, if load is assigned, f ' may be expressed as
u 1k L (B - b)
f =p w
(11.39)
In dimensionless form, as a function of L', b', h' and p', it becomes
Figures 11.4.a, 11.4.b, 11.4.c and 11.4.d show how the variations off' are identical to those of F i and H i .
1125.3
Minimum dissipated power and efficiency losses
11.5.3.1 Given flow rate. In the case of direct supply and constant flow rate, pad total dissipated power, given by Eqn 5.6, is obtained by adding Eqn 11.4 to Eqn 11.30, that is
376
HYDROSTATIC LUBRICATION
(11.41) Introducing a reference pumping power, total power H t may be expressed in dimensionless form, as a function of L': (11.42)
where
k =-U B h
(11.43)
Q
k may be called a "speed parameter". Hi is minimum for
d3
(11.44)
LhPt = k and for such a value of L' i t is
(11.45) where H f I H , is the often mentioned "power ratio". Equation 11.45 and Eqn 11.60 below were given for the first time in ref. 2.3 and have been used many times in optimization problems in previous chapters. In Fig. 11.2.a H i is plotted for k = l . Within the range 1
where Hp0*=pQ2(B-b)lh3Band Hfo*=pU2B(B-b)Ih. Considering the dependence on b', HI can be expressed a s Hi=3(1-b')+kz(l-b')
,
with k
ULh
=
Qp
(11.46)
In Fig. 11.2.b H i is plotted for k = l . As b' approaches 1, H i as well a s W' and K approach 0.
H i can be expressed as the following function of h': 1 1 H;=37+k2j7 h
,
with k=-
ULho
Q
( 11.47)
OPTlMlZATION
377
In Fig. 11.2.c Hi is plotted for k = l . As h' increases, Hi decreases; W' also decreases but more rapidly and K even more so. Finally considering the dependence on p' the following expressions are obtained:
Hi = 3 p'
+ k2 p'
(11.48)
where K is given by the second of Eqns 11.46. In Fig. 11.2.d Hi is plotted for k = l . Hi decreases linearly with p' as do W and K'.
As regards the variation of H't as a function of B, considered a reference quantity until now, it is sufficient to put
B
(11.49)
B'=b and we immediately get the variation of H&B') from that of Hi(b').
The study of the variation of total dissipated power Ht becomes more complicated if it is considered as a function of two or more variables unless they are b (or B ) and p. Indeed, in such a case the following "compound variable can be introduced: C'=(l-b')p'
(11.50)
and Eqn 11.41 becomes
Hi = 3 C ' + k
2 ~ '
(11.51)
similar to Eqn 11.48 and Eqn 11.46 and with the same k. In Fig. 11.2.d Hi(c') is also plotted, for k=l.
10In the previous paragraphs it has been shown that if b, h, p (and B ) vary in such a way as to make H, decrease, W and K also decrease though in a different way; this must generally be taken into account in dimensioning a bearing. In this connection useful informations may be obtained from the "efficiency loss" rw, or from its inverse (ref. 9.41, and from r K given by Eqns 11.2. Considering the dependence on b', dividing Eqn 11.46 by the second of Eqns 11.11gives us: (11.52)
Both rb and r k decrease as b' approaches 1 suggesting the choice of a wide recess. In Fig. 11.5.a rw and rk are plotted for k=O, 1,2. Considering the dependence on h', dividing Eqn 11.47 by the second and the third of Eqns 11.13,respectively, gives
378
HYDROSTATIC LUBRICATlON
In Fig. 11.5.b Tb(h') and rk(h')are plotted for h=O, 1, 2. riy decreases as h' approaches 0 except for k=O, while decreases as h' approaches 0 in every case. This suggests the choice of a small film thickness. -b-
-a 5
E l Q =const.
4
3
rh 2
1
0
0.5
Fig. 11.5 Efficiency losses tain values of speed factor k.
bl
1
1
h'
2
rb and rKversus: a- recess width b', b- film thickness h', for cer-
Finally from Eqn 11.48 and from the second of Eqns 11.15 it transpires that the efficiency loss does not depend on lubricant viscosity. The choice of the values of the variables B, L, 6, h, and p also depends on various other conditions which may be encountered in the design of a bearing, regarding its operation and construction. For example, in actual pads, Fig. 6.25, it is convenient to choose BSLSZB.
11.5.3.2 Given pressure. Total dissipated power is
H t , expressed in dimensionless form, as a function of L', is given by:
OPTIMIZATION
379 (11.55)
where k is similar to the Sommerfeld number of plane hydrodynamic pads. In Fig. 11.3.aHi is plotted for k=l. Hi decreases linearly with L' as well as W and K'. Considering the dependence on 6')Hi can be expressed as H i =1 3 11 +6'2(1-6')
,
with k -_l!.L!.E P r h2
(11.56)
Hi is minimum for 6ipt = 1-
11 3
(11.57)
to which Eqn 11.45 still corresponds. It should be noted that Eqn 11.57 leads to 6Apt<0 for k
H i can be expressed as the following function of h': (11.58)
Hi is minimum for hApt =
6
(11.59)
to which the following relation corresponds:
It should be noted that from Eqn 5.7 high values of the ratio HfIHp can lead to high temperature increments. In Fig. 11.3.c Hi is plotted for k=l. Finally considering the dependence on p' the following expressions are obtained: (11.61)
HI is minimum for 11
Pbpt
=3i
to which Eqn 11.45 again corresponds. In Fig. 11.3.d Hi is plotted for k=l.
(11.62)
380
HYDROSTATIC LUBRICATION
It should be noted that, within the range 1O, too. For example, with reference to Fig. 11.3.b for k = l , putting B'=-, the value of H i would not exceed its corresponding value for B&l/b~pt=2.37 by more than 15%. Considering Eqn 11.50, Eqn 11.54 becomes (11.63)
H i is minimum for 11 cbpt=zi
(11.64)
to which Eqn 11.45 again corresponds. In Fig. 11.3.d Hi(c'1, given by Eqn 11.63, is also plotted, for k = l . Therefore we can immediately look for the minimum value of Hi: ckpt is determined from Eqn 11.64 and then any couple of values of b' and p' which satisfy Eqn 11.50 yield such a minimum. The choice of such values makes the selection of the bearing dimensions easier. For couples of variables that are different from b and p, for example b and h, the search for the minimum value of Hi can only be approximate: for a series of values of one of the two variables, the optimum values of the other variable are searched for, together with the corresponding minimum values of H i , of which the absolute minimum is evaluated (as a check, the trial can be repeated, starting with the other variable). The case of three variables, if they are b, p and h, can be reduced to that of two variables using Eqn 11.50. Efficiencv losses, Equations 11.55 and the first of Eqns 11.18 yield Tw(L')=rk(L')=const., while Eqn 11.56 and Eqns 11.19 give
rw=23
1
1-6' + 2 k2 1+b' '
r, = r;,
(11.65)
rw and r k are minimum for (11.66) In Fig. 11.6.a r i and r k are plotted for k=O, 1,2. It should be noted that for k = l , while the value of Hi for b'=O exceeds that for b'=bApt by only 15% (Fig. 11.3.b)) the value of for b'=O exceeds that for b'=bAPt by 74%. b&(k) is plotted in Fig. 11.6.b. Finally, Eqn 11.58 and the first of Eqns 11.20 yield
381
OPTlMlzATION
-b-
-a-
E
Fig. 1 1.6 a- Efficiency losses rp and r h versus mess width b’ for certain values of speed factor k; b- optimal values bLPl of recess width versus speed factor k.
In Fig. 11.7 r k is plotted as a function of h’ for k=O, 0.5, 1, 1.5, 2, 2.5,3. rk increases very slowly in the range of commonly used values of h’.
11.5.3.3Given load. Total dissipated power is 41
-
w2
Ht = 3 E h 3 L (B b) (B + b)2 +
1 u2h tB
b,
(11.68)
H t , expressed in dimensionless form, as a function of L’, is given by: (11.69)
H i is minimum for (11.70)
382
HYDROSTATIC LUBRICATION
0
15 2 h' Fig. 11.7 Efficiency loss r~versus film thickness h' for certain values of speed parameter k.
0.5
1
to which Eqn 11.45 again corresponds. HI is plotted for k = l in Fig. 11.4.a. Considering the dependence on b', Hi can be expressed as 4
1
II
TIT.
In Fig. 11.4.bHi is plotted for k = l . The condition for its minimum dH;/db'=O yields a fifth degree equation. In Fig. 11.4.b the calculated values of bbpt are also presented a s a function of k . As k increases, b' rapidly approaches unity and Eqn 11.71 approaches the form of Eqn 11.56. The value of bhpt can also be calculated with the semiempirical formulae 1 bbpt =j+ 0.33745k2 - 0.16818 k 3 + 0.024475k4
,
for 0 5 k c3 (11 72)
with a maximum error of 0.85% for k=0.4 in the first formula and one of 0.042%for k=3 in the second. Hi can be expressed as the following function of h' (11.73)
OPTlMlzATlON
383
H i is minimum for hbpt =
4
(11.74)
to which Eqn 11.60 again corresponds. Hi is plotted for K = l in Fig. 11.4.d. Finally, considering the dependence on p', the following expressions are obtained: (11.75)
H i is minimum for P&t
2 1 =az
(11.76)
to which Eqn 11.45 again corresponds. Hi is plotted for k = l in Fig. 11.4.d.
As regards the variation of H i as a function of B', given by Eqn 11.49, it can be immediately deduced from that of Hl(6'). If we put
g'=L'p'
(11.77)
Eqn 11.68 can be written as (11.78)
In Fig. 11.4.a H;(g'), given by Eqn 11.78, is also plotted, for k=1. Hi is minimum for (11.79) to which Eqn 11.45 again corresponds. Therefore the search for the minimum value of H i , as a function of L' and p', is immediate, as already specified in a previous case (Eqn 11.63). Putting 4(1-6') in place of (l-b')(l+b')zin the first of Eqns 11.71, the minimum occurs for 6' given by Eqn 11.57 with K given by the second of Eqns 11.71. The values of Hi, obtained from the first of Eqns 11.71, thus modified, differ from the exact values by less than 10%for k23. Therefore, putting q' = L' (1 - 6') p'
(11.80)
384
HYDROSTATIC LUBRICATION
in Eqn 11.68, modified and expressed in dimensionless form, (11.81)
is obtained. Hi is minimum for qApt =
1 1
(11.82)
to which Eqn 11.45 corresponds. So for k23 the approximate minimum of Hi of Eqn 11.68, considered as a function of L', b', and p' is given by Eqn 11.82; it then allows a wide choice for the three optimum values of L', b', and p'. The search for the minimum of Ht given by Eqn 11.68 as a function of h and L, or b, or p can be carried out in an approximate way, as already seen in the case of constant pressure. Efficiencv losses. Since W is now assigned, the variation of Ht is the same as r, and r, except for T'(h), for which dividing Eqn 11.73 by the first of Eqns 11.28 yields
r;,=
4 h'4 +
(11.83)
k2
Figure 11.8, which is very similar to Fig. 11.7, shows as a function of h', for k=O, 0.5,1, 1.5,2,2.5,3. ri increases very slowly in the range of commonly used values of h'.
0
0.5
1
1.5
L
h' Fig. 11.8 Efficiency loss rKversus film thickness h' for certain values of speed parameterk.
OPTlMlzATlON
11.6
385
OPTIMIZATION
The minima of total power H,, previously specified, have been determined considering Ht a s a function of only one of the variables L , b, h and p and having taken B as a reference quantity. The cases in which it was considered as a function of two or more variables was actually reduced to the case of only one variable. We shall now go on to the determination of the minima of Ht as really a function of two variables which is a rather frequent case in practice since o h n the values of two variables are assigned, and as a function of three variables which is a frequent case in practice since a t least the value of one variable is assigned, and finally as a function of all four variables. Optimization is also carried out in the presence of constraints of both the variables and of other quantities such as, for example, stiffness K.This is done for the three cases previously mentioned in section 11.2: given flow rate, given pressure and given load. Its application t o this last case, that is when the load is assigned, is particularly important since this is the condition most frequently encountered in design. An outline of optimization methods, that is for the determination of the unconstrained or constrained minima of a function, is presented for example in ref. 11.1. Here sufice i t to say that optimization has been carried out with the techniques of non-linear optimization of the "objective function" Ht, for which the Adaptative Random Search Method proved to be especially suitable. Constraints involved "penalty functions", for which the Schuldt's Functions proved to be particularly suitable.
11.6.1
Given flow rate
As shown in Fig. 11.2, Ht only has a minimum as a function of L', whereas with the other variables it merely increases or decreases.
11.6.1.1 Ht=f(L,b).Equation 11.41 can be expressed in dimensionless form as (11.84)
where k is given by Eqn 11.43. In Fig. 11.9 H;(L',b')is plotted in the range O G ' G and O
HYDROSTATIC LUBRlCATlON
588
H;
1.88
188.88
8.67
66.67
8.33
33.33
8.88 8.88
1.
588
8.7,
5 88
a
Fig. 11.9 Total power H,' versus pad length L' and versus recess width b; or viscosity p', or c'=(l-b')p', for certain values of speed factor k.
Figure 11.10 shows the results of the optimization of Hi,for OIk112 with the following constraints: lsL'12, 0.2
387 60
1Q
= const.
1
5c Him
!.4
6
?
5
Copt
4c bopt
4
Hb
f‘
Pbpt
p’,
30
C’
opt
3
20
a‘
2
H;k
K 10
0 0
1
3
I
2
4
8
6
10
1
k Fig. 11.10 Values of minimum total power Hi,,,, and corresponding optimal values of pad length or of viscosity pA or of c&=(I-b&)p+ versus speed factor k. Values of pumping power of friction power of recess pressure pr, of load capacity W’, of stiffness K‘ and of friction coefficientf’ are also represented (note that the values of W’, K’andf’ ). are valid only for the first case, that is for Hi,,,=f(L&f,b~pl
4,,,
LApf,and of recess width bi
fib, f,
where k is given by the second of Eqns 11.43,if it is inside the boundaries (Eqn 11.45 is then verified); otherwise it takes the boundary value nearest to it. The couple L&, 6APt allows the calculation of the other quantities. It should be noted that for these values the efficiency losses are also minimum.
11.6.1.2Ht=f(L,p).Equation 11.41 can be expressed as (11.86)
where k is again given by Eqn 11.43.On the analogy of Eqn 11.86with Eqn 11.84,in Fig. 11.9Him(L&,t,p&) is also presented, now in the range OSL’<5 and OSp’S1.For Hj,(LApt,p&,,pt)the considerations concerning H i ( L ‘,6‘) are still valid. I t must be borne in mind solely that p’ is the complement of 6‘ to 1.
388
HYDROSTATIC LUBRICATION
Similarly Fig. 11.10 may be used to obtain the results of the optimization of H;m(L&,,pt,~&,t) with the constraints lILY2,0.41~'10.8.Figure 11.10 also shows Hjk, Hi andp; ( which coincides with Hi), whereas W and K' are given by the second of Eqns 11.15 andf'by the following equation
It is obvious that p' has always taken the lower of the two boundary values, that is p&0.4. To obtain Him(L&,p&), a procedure similar to that for H;m(L;Zpt,b&,,pl) is followed.
11.6.1.3H,=frL,p,b). Equation 11.41 can be again expressed a8
Hi=--
Ht
POQ2
1-6' 1 - 3 p ' r + @p' (1- 6')L'
(11.87)
jp
where k is again given by Eqn 11.43. After having introduced Eqn 11.50, we can also express Eqn 11.87 as C'
Hi = 3 L'; + k2 c' L'
(11.88)
On the analogy of Eqn 11.88 with Eqn 11.86, H~m(L&,,pt,c~p,pt) is also presented in Fig. 11.9, now in the O G ' S 5 and O
as well as K' which coincides with W'. After Eqn 11.50 has been introduced, Eqn 11.89 becomes 3 W = 3 (1+ 6') c'
(11.90)
that for c'=constant increases linearly with 6'.
cAPt has always taken the lower of the two boundary values 0.4 and any couple of values of b' and p' satisfying (1-6')p'=0.4 is an optimum couple. To obtain Him(L&&,pt), for Him(L',6') is followed.
that is H~m(L~p,pt,6~p,p,,&,p,pt), a procedure similar to that
389
OPTIMIZATION
11.6.1.4 Ht=KL,h). Equation 11.41 can be expressed then as
HI=
Ht 1 L' 1B-b=3s+k2,=Hi+Hjk P Q 2 j p B h
,
with k=-
U B ho
Q
(11.91)
H;(L',h') is presented in Fig. 11.11 ,in the OIL'<5,O
Fig. 11.1 1 Total power H;versus pad length L' and film thickness h', for certain values of speed factor k.
In Fig. 11.12 the results of the optimization ofH&L',p')are also presented, with the constraints 15L.12 and 0.3
390
HYDROSTATIC LUBRICATION
250
2.5
200
0
Hkl
W' .5
150 HP
K'
p;
100
10 .
50
!.5
H;k
1
0 0
' 2
4
8
6
I0
12
k Fig. 11.12 Values of minimum total power Hi,,,, and corresponding optimal values of pad length L& and of film thickness h&. versus speed factor k. Values of pumping power H i , of friction power Hjk, of recess pressure p,!, of load capacity w',of stiffness 1y' and of friction coefficientf' are also represented.
with Hi,, W' and K given by the second and third of Eqns 11.13, respectively. The friction coefficient is given by
(11.92)
It should be noted that hAp,Pt has always taken the higher of the two boundary values: 0.6. Therefore, to obtain Hi,(L&,t,hAp,Pt), the maximum value of h' must be selected for h&, while LApt is determined from Eqn 11.44,that is
d51
LApt =--k h&
(11.93)
If this last value is outside the range of possible values for L', the nearest boundary value will be assumed to be LApt.
391 11.6.1.5 H,=f(L,b,p,h). Equation 11.41 can be expressed finally as (11.941
where k is again given by the second of Eqns 11.91. Figure 11.13.a shows the results of the optimization of Hi(L',b',p',h'), with the constraints lIL'i2
,
0.21b'I0.6
,
0.31h'l0.6
,
0.81p'11.6
(11.95)
Fig. 11.13 also shows H@, H i ,p; coinciding with HbJ and 3 , 1 - b'' W ' = ph '3 ,
K'T w
(11.96)
(11.97)
It should be noted that bbpt as well as hApt and PApt always taken boundary values: the higher for b' and h', the lower for p', that is bApt=0.6,hApt=0.6 and pApt=0.8. Therefore, to obtain H ; , ( L ~ p t , b ~ p t , p ~ p t , hthe ~ p maximum t), value of b' and h' must be selected for bbpt and hApt, while the minimum value of p' must be selected for PApt. LApt is determined from Eqn 11.93, if it is inside the boundaries; otherwise i t takes the boundary value nearest to it. With Eqn 11.50, Eqn 11.94 can also be expressed as C' L' c' H i = 3 -L , h,3 + k 2 F
(11.98)
In Fig. 11.13.a, therefore, Hi,,, also represents Hi,(LApt,hApt,c~pt),obtained with the constraints lSL'I2
,
0.32Ic'11.28 ,
0.3Ih'10.6
(11.99)
cApt has always taken the lower of the two boundary values, i.e. 0.32. Any couple of values of b' and p' satisfying ( 1 - b') p' = 0.32
( 11.100)
is a n optimum couple. To obtain Hi,(L~p,pt,h~pt,cApt) the procedure is the same as in the previous cases but now, since we always have cbpt=chin and (from Eqn 11.50) c ~ i n = ( l - b ~ , , , ) p h i n , we obtain bApt=b& and pLpt=phin.
392
HYDROSTATIC LUBRICATION
-a -
100
2.5
10
Copt
80
8
2 Gopt
H; m
w'
hbpt
60
1.5
Hb p; 40
1
H;k
cbpt
6 K'
Pbpt
4
0.5
20
2
f'
0
0
0
2.5
10
k
-b100
Copt
2
80
8
bopt
Him
W'
hbpt
1.5
60
H'P
6
K'
PLbpt
p; 40
1
4
C'
opt
H;k
20
0.5
2
f' 0
0
0
k Fig. 11.13 Values of minimum total power Hirn,and corresponding optimal values of Lhpl, of film thickness h C I , of recess width bhpr,and of viscosity & or of cipl=(l versus speed factor k. Values of pumping power Hi, of friction power &h,of recess pre load capacity W , of stiffness K and of friction coefficientf are also represented.
,,
I length pt)Phpr*
'e p,!, of
OPTlMlZ4TlON
393
In Fig. 11.13.b the results are presented with the same constraints for L' and p', but with 0.2 5 b' 1 0.9
,
0.3 5 h' I0.4
(11.101)
We must point out that in this latter case load capacity is the same as in the former case, but stiffness is 50%higher (this can be immediately deduced from Eqns 11.961, even with lower dissipated power and a much lower friction coefficient. The same figure also shows the results obtained with the constraints llL'12
0.08l~~51.28
0.35h'l0.4
and now chpt=0.08.
EXAMPLE 11.1 The pad of Fig. 11.1, with width B=0.1 m and length LQ-B, is directly supplied by a pump as in Fig. 1l.l.a. The pump is assumed to supply a constant flow Q=5.10-6 m31s, that is a little higher than hydrodynamic flow rates. We want to evaluate the pad load capacity W and stiffness K in the condition of minimum total dissipated power Ht, for speed U=0.3, 1,3 mls. H t , can be calculated following the first procedure described in section 11.6.1.5. I f the reference value of the film thickness is assumed to be ho=lO-4 m, factor k, given by Eqn 11.91, for the three values of speed, takes the values k=0.6, 2, 6. If, for example, Constraints 11.95 are adopted, the following results are obtained (also directly from Fig. 11.13.a): 1) k=0.6 Lhpt=2, bApt=0.6, h Apt,pt=0.6, IJ ApFO.8 H;,=2.61, Hi=p; =2.23, Hb=0.384 W'=3.56, K'=5.93, f'=o.18 from which, i n dimensional form, LoPt=0.2m, bopt=O.06m, hopt=0.6.10-4m and, taking as reference viscosity &=0.1 Nslm2, hpt=0.08Nslm2 and Ht,=6.52 Nmls, Hp=5.56 Nmls, H ~ 0 . 9 6Nmls, p,=11.1.105 Nlm2 W=1.78.104 N, K=8.89.1@ N l m , f=1.8.104. It should be noted that the value of pressure is not much higher than the values of hydrodynamic supply pressure and the very low value of the friction coefficient. 2) k=2 bOpt=0.06m, hOpt=0.6.1O4m, ~pt=0.08A?slmz Lopt=O.145 m, Ht,=15.4 NmIs, Hp=Hf =7.7 Nmls, p,.=15.4*1@ Nlm2 W=1.78-104N, K=8.89.1@ Nlm, fd.35.1Q4 It should be noted that assuming, for example, the recess depth to be h,=0.08 m, the power dissipated in it for friction would be Hfp=8.7.10-aN m l s (section 11.5.2.1), just
394
HYDROSTATIC LUBRICATION
a little higher than 1% of Hf. Assuming the lubricant density to be p=900 IQlrn3, the power dissipated to accelerate it in the film would be H1=5.63.10-4N m l s , less than 0.01 % of Hf. 3) k=6 Lopt=O.1 m, bOpt=0.06m, hOpt=O.6.104m, bpt=0.08Nslm2 Htm=59.1Nmls, Hp=ll.l Nmls, Hf=48Nmls, pr=22.2-105Nlm2 K=8.89.1@ N l m , f=9.104 W=1.78.104 N, We must point out that the value of load capacity, identical i n the three cases, is sufficiently high; that of stiffness, also identical i n the three cases, is high (comparable to that of the deformation of a roller slide-way substituting the pad). 4) I f constraints 11.95 are also considered, though in the form of Eqns 11.99, on the basis of the second method explained in section 11.6.1.5, or again from Fig. 11.13.a, we obtain, for example for the case in which k=2: L&=1.45* bhPt=O.32, h hpt=0.6 and for Hirn, Hb=p; and H b and the corresponding dimensional quantities, the same values obtained in such a case. According to Eqn 11.100 we can now put, for example, bhpt=0.8 and consequently pbpt=1.6,obtaining K=6.67, f'=0.16 W'=4, and W=2*104N, K=1@ N l m , f=1.6.10-4. Therefore, comparing this case to the second, for the same values of dissipated power and supply pressure, there is an increase in load capacity and stiffness of over 12% and a reduction in friction coefficient of over 16%. 5) Pad efficiency can be improved further by changing the constraints. So i f in Eqns 11.95 new upper boundaries are introduced for b' and he,for example: b'10.9 and h'10.4, that is, i f the constraints 11.101 are considered, according to the second above-mentioned method in section 11.6.1.5, or from Fig. 11.13.b, we obtain, for example, for the case in which k=2: L Apt= 1, bbpt=0.9, h &=0.4, pApt=O.8 from which Lopt=O.l m, bopt=0.09m, hOpt=0.4.1O" m, kpt=0.08 Nslm2 Htm=11.38 W, Hpz9.38 W, Hf=2 W, pr=14.1.105 Nlm2 W=1.78.104 N, K=1.34-1@Nlm, f=1.13.104. Therefore, comparing this case to the second, for the same load capacity, there is a 50% increase in stiffness, a 35% reduction i n dissipated power, a 9% reduction i n pressure, while the friction coefficient has become four times smaller. 6) It should be noted that without changing constraints the results can be modified by changing the reference values of the constrained quantities. So, i f the dimensional constraints allow it, we can put B=0.2 m. Therefore, for U=l m l s , for example, Eqn 11.91 gives k=4 and consequently:
OPTlMlzATlON
395
LoPt=0.2m, bOpt=O.12m, hOpt=0.6.104m, bpt=0.08Nslrnz Htm=32.4 NmIs, Hp=ll.l Nmls, Hf=21.3 NmIs, pr=22.2.105 N l m 2 W=7.12.1@ N, K-3.56.109Nl m, f=3.104. We must point out that, as compared to the second case, the dissipated power is almost doubled, pressure is increased by 50% but load capacity and stiffness are four times higher while the friction coefficient is decreased. IL on the contrary, because of considerable surface roughness and errors in parallelism, we want to increase film thickness, we can put, for example, ho=2.lO4 m, so that, again for U=l mls, Eqn 11.91 again gives k=4 and consequently: bopt=O.06m, hOpt=l.2.lO4m, kpt=0.08Nslmz Lopt=O.l m, Htm=4.06 Nmls, Hp=1.39Nmls, Hr2.67 Nmls, pr=2.78.105N l m 2 W=0.22-104N, K=0.185.108Nlm, f=12.10-4. It should be pointed out that, again as compared to the second case, total power and pressure are reduced to almost a fourth and to less than a fifth, respectively, while load capacity and stiffness are reduced to an eighth and to a little less than a fiftieth, respectively. The friction coefficient is greatly increased. Finally i f the reference viscosity is increased, for example, to the value po=0.2 Nsl m z , while we still have U=l mls, k is again that of the second case. Therefore all the dimensionless quantities remain the same, as well as Lop&bopt and hopb while popt and consequently Ht, Hp, Hf, pr, W and K are double. f is the same as in the second case.
11.6.2
Given pressure
As shown in Fig. 11.3, Ht has a minimum when it is considered to be a function of each variable except for L .
11.6.2.1 Ht=f(b,h) 11.6.2.1.1. Equation 11.54 can be expressed as (11.102)
In Fig. 11.14 Hi(b‘,h’)is plotted in the Ol and p>l. But investigation has also shown that (1,O)is a singular point:
396
HYDROSTATIC LUBRICATION
H;
I8 88
6.67
3.33
8.88
2.m
1
2.80
Fig. 11.14 Total power Hi versus recess width b', versus film thickness h' and versus viscosity p', or c'=( 1 -bl)p', for certain values of speed factor k.
therefoye i t would not be convenient, not even in theory, to choose HI corresponding to (1,O)or to points very close to it. On the other hand, in practice, for obvious constructive reasons, b'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
0
2
4
6
k
a
10
12
14
Fig. 11.15 Values of minimum total power H;,,,, and corresponding optimal values of film thickness h&,, and of recess width b+, or of viscosity &, !, or of c&,r=(l-b+r)p&,l versus speed factor k. Values of pumping power H and flow rate Q', of friction power Hfi of load capacity W', of stiffness K'and of friction coeffhentf' are also represented (note that the values of W', K' andf' are valid only for the first case, that is for H~,,,=f(h&,t,b~p,)).
(11.104)
which yield very high values because the value of bApt is high and the relevant values of hipt are small (for example, for k = l , h&0.1581 andK'=6.247), and
f"&=zk--
1 1-b' h'l+b'
which is still small; for example, for k = l , f'=O.l60.
398
HYDROSTATIC LUBRlCAT/ON
k
0
hbt
0
0.01 0.025 0.05 0.075 0.1 0.2 0.4 0.6 0.8 0.0158 0.0249 0.0353 0.0433 0.0501 0.0707 0.0999 0.1224 0.1415
Therefore it is to be noted that, for the values of bApt and hApt thus calculated, the condition of minimum total dissipated power is satisfied as well a s that of maximum load capacity with very high stiffness and a low friction coefficient. Therefore, if pressure in the recess is assigned, it is convenient to design the pad with the recess as wide as possible. Obviously b’=0.975 is a limit value. Practical values are lower: b’=0.95 to 0.75 and bbpt will take on such values. However the corresponding values of hApt and power increase while load capacity and stiffness decrease; the friction coefficient also increases. This is proved by the results reported in Fig. 11.16.a, obtained with the constraints OSb’S0.9 and O
ash’
( 11.105)
where a is not too small; this is because of surface roughness and errors in planarity and parallelism, etc., even if the present-day technologies make it possible to achieve increasingly smaller values of film thickness. Figure 11.16.b contains the results of the optimization of Hi(b’,h’),with the constraints OSb’10.9 and 0.9Sh’SZ (that is a=O.9). For k28.1 the diagrams coincide with those in Fig. 11.16.a. For 8.bk24.68 only bApt and consequently W’ still coincide while hbpt=0.9 and consequently H b and K become constant; Hik decreases more rapidly while Him decreases more slowly than in Fig. 11.16.a. For 4.68>k10.468, we still have hApt=0.9, but W‘ approaches 0.5 and K’ approaches 0.5. Moreover H@=Hb, that is we again find Eqn 11.45. For 0.468>k20, we still have hAPt=0.9and bbpt=O. On the basis of the above considerations, in the presence of constraint 11.105, b’ is again chosen as large as possible and is assumed to be bApt;then from Eqn 11.103 h‘ is determined; if h’ satisfies condition 11.105 i t is h&. If this does not occur, let-
399
OPTIMIZATlON
Fig. 1 1.16 Values of minimum total power Him, and corresponding optimal values of film thickversus speed factor ness h;,,,, and of recess width b; [, or of viscosity & [, or of c&,l=(l-bL k. Values of pumping power H F of friction power i j k , of flow rate of load capacity W', of stiffness K' and of friction coeff&entf' are also represented (note that the values of W', K' andf' are valid only for the first case, that is for H;m=f(h&t,b~pr)).
d,
T A B L E 11.2 Optimal values of film thickness for &&,=0.9 (a) and of recess width for h&,=0.9 (b), versus speed factor (see also Fig. 11.16, Eqn 11.103 and Eqns 11.106). 0 0.01. 0.025 0.05 0.075 0.1 0.2 0.4 0.6 0.8 a k 0 0.0316 0.0500 0.0707 0.0866 0.1000 0.1415 0.2001 0.2448 0.2830 h&,
b k bbpt
0.4 0
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.1125 0.2210 0.3272 0.4151 0.4814 0.5323 0.5750 0.6100 0.6387
ting hApt=a,H i and H@ are calculated and if the condition H @ > H i is satisfied, b& is still the one initially chosen. If this condition is not satisfied, keeping hAPt=a,bbt is determined from Eqn 11.45, that is
(11.106)
400
HYDROSTATIC LUBRICATION
-b-
-a-
14
I.4
12
1.2
1.4 14/
p, = const.
121
p
Him
bopt
10
1
Hb Q'
tiopt
1.8
8
Hi
PLpt
Gpt
6
1.6
K6
W' 4
1.4
f'
1 0
2
0
fl
2
1.2
0
1
6 k
k
Fig. 11.17 Values of minimum total power H b , and corresponding optimal values of film thickness hip,, and of recess width bGI, or of viscosity & or of c&,,=(l-b~ ,)&,,versus speed factor k. Values of pumping power Hp, of friction power of flow rate Q? of load capacity W', of stiffness K'(with constraint K'21 in Fig. 11.1 7.b) and of friction coefficientf' are also represented (note that the values of W , K'andf' are valid only for the first case, that is for H;,=f(hApl,bLp,)).
,,
T A B L E 11.3 Optimal values of film thickness versus speed factor, for b&,,=0.8 (see also Fig. 11.17 and Eqn 11.103). 0.2 0.4 k 0 0.01 0.025 0.05 0.075 0.1 It& 0 0.0447 0.0707 0.1000 0.1225 0.1414 0.2001 0.2828
11.6.2.1.3. The optimization procedure examined in section 11.6.2.1.1 yields high stiffness especially for high values of bhpt and low values of k. If a higher value of K is required, for example for functional reasons, a lower limit can be imposed on it.
OPTiMiZ4 TloN
40 1
For Eqn 11.104 where W' is already maximum, since bApt always takes the maximum value, it all comes down to imposing a n upper limit on h'. Indeed, letting K'2 y
W' Y
, we have h ' s -
(11.107)
Figure 11.17.b contains the results of the optimization of Hi obtained with the constraints OIb'10.8,O l . Employing Eqns 11.107, the last two conditions are reduced to O4.05:H i remains constant, Hjk approaches Hi increasing more rapidly, and hipt=0.9and K'=l. This is true for any other constraint on b' and K'. On the basis of the above considerations, in the presence of the further constraints 11.107 on stiffness, b' is still chosen as large a s possible and is assumed to be bhpt; then from Eqn 11.103 h' is determined which is hipt if i t satisfies conditions 11.107; otherwise, hipt=W'/y. 11.6.2.1.4. From the two previous sections, it follows that if, for practical reasons, film thickness satisfies condition 11.105 and, for functional reasons, stiffness must satisfy conditions 11.107, the search for the minimum total power is constrained by the following conditions:
Osb'
W' ash'sY
(11.108)
This search is carried out according to the methods described in the foregoing sections.
11.6-2.2Ht=f(uh) 11.6.2.2.1.Equation 11.54 can also be expressed as (11.109)
On the analogy of the first of Eqns 11.109 with the first of Eqns 11.102, Hi(p',h') is also presented in Fig. 11.14, in the Osp'S1 and Osh'12 range. As for Hi(p',h') the considerations regarding H'(b',h') are still valid. Similarly in Fig. 11.15 Hirn,phPt and hApt are also plotted, with the constraints 0.0251p'sl and 0 4 ' 1 2 . Figure 11.15 also shows H@, H i and Q' coinciding with H i . W' and K' are easily determined, since W'=1/2 and K=W'lh&,t, while f ' can be expressed in the form
HYDROSTATIC LUBRICATION
PApt has always taken the boundary value p’=0.025; hence, h’ is again given by Eqn 11.60. that is h&?t=
dmg
(11.110)
Then Him(p&,t,h&t) is also presented in Fig. 11.16.a, with the constraints O.l
The considerations and conclusions presented in section 11.6.2.1.2 are still valid; only bApt must be replaced by its complement to one p&, and Eqn 11.103, the second of Eqns 11.56 and Eqn 11.57 must be replaced by the corresponding Eqn 11.110, the second of Eqns 11.61 and Eqn 11.62. 11.6.2.2.3.H;,(pAPt,h&) is presented in Fig. 11.17.b, with the constraints 0.2Sp’11, (kh’12 and K‘21, where the latter two are reduced to O
The considerations and conclusions deduced in section 11.6.2.1.3 are still valid; only (l-b&) must again be replaced by p& 11.6.2.2.4. Finally, if the search for H~m(p&,trh&,,pt) is constrained by the conditions 11.108, the second of which comes from conditions 11.105 and 11.107, it is carried out according to the methods described in the two previous sections.
11.6.2.3Ht=f(b,p,h) 11.6.2.3.1. Equation 11.54 can also be expressed as
With Eqn 11.50 Eqn 11.111becomes
403
OPTIMlZ4TION
(11.112) On the analogy ofEqn 11.112 with the first of Eqns 11.109, H&‘,h’) is also presented in Fig. 11.14, in the O
, 0ch’<2
(11.113)
Hjk, Hi,Q’ coinciding with H i are also plotted. Fig. 11.15 also shows W’, K (given by Eqns 11.104)and f‘ which is f‘
- f - ho/B
’L X+ -- ‘ h ’ 1 b‘
C’
h’ (1 + 6’)
in the case of b’=0.975=bApt,that is phpt=l. It must be pointed out that chp,pt=0.025and any couple of values of b’ and p‘ satisfying (l-b’)p’=0.025is always a n optimum couple. Since chpt has always taken on a boundary value, hAPt is given by h hpt = 4z-
d k phPt (1 - bbpt)
(11.114)
H;,(cApt,hApt) is presented In Fig. 11.16.a, with the following constraints: O.l
(11.115)
W’, K‘ and f ‘ are also presented as well as the other quantities, in the case of b’=0.9=bhpt,that is phpt=l. Finally, Hi,(chpt,hhp,t) is presented in Fig. 11.17.a, with the constraints 0.21c‘
To find the minimum of H t , in the presence of condition 11.105, c‘ is chosen as small as possible. Letting c’=cbptrit is introduced in Eqn 11.114, thus determining
404
HYDROSTATIC LUBRICATION
h‘. If h’ satisfies condition 11.105 it is h&; otherwise, letting h&=a, H i and H)k are calculated. If H ) p H I ; , cApt is still equal to the value initially chosen; otherwise, still letting h&=a, c& is determined from Eqn 11.45, that is (11.116) Any couple of values of b‘ and p’ satisfying (l-b’)p’=c&,t and hAPt form the optimum combination yielding Hi,. 11.6.2.3.3. H&&t,h&,t) is presented in Fig. 11.17.b, with the constraints 0.21c‘
11.6.2.3.4. The search for Ht, constrained by conditions 11.108, the second o f which comes from conditions 11.105 and 11.107, is carried out according to the methods described in section 11.6.2.3.2 and section 11.6.2.3.3.
11.6.2.4 Ht=f(L,b,p,h). As seen in section 11.5.3.2, from Eqns 11.55 and Eqns 11.18 we can deduce that the efficiency losses are independent from L; nevertheless the values of L cannot be too large or too small, for many practical reasons. As for real pads, as mentioned in section 3.3.1, L is chosen so that B
EXAMPLE 11.2 The pad in Fig. 11.1, with maximum dimensions B=O.l m and L=0.15 m, is directly supplied by a pump, as in Fig. 1l.l.a. The supply and recess pressure is assumed to be p,=106 Nlm2, that is a little higher than in hydrodynamic lubrication. We want to evaluate load capacity W and stiffness K in the condition of minimum total dissipated power Ht, for speed U=0.3, 1,3 m l s . H t , can be calculated following the procedure described in section 11.6.2.3.1. Then i f ho=10-4m, and po=O.l Nslm2, the parameter k, given by the second of Eqns 11.111, for the three values of speed, takes on the values k=0.3, 1, 3. I f for example
OPTlMlZ4TlON
405
constraints 11.115 are adopted, the following results are obtained (also directly from Fig. 11.16.a): 1) k=0.3 and choosing bApt=0.9, pApt=l h bp,pt=O.173, c;pt=O.l Him=0.0693, Hi=Q'=O.O173, Hb30.052 W=0.95, K'=5.48, f '=O.182 from which bopt=0.09m, hOpt=0.173~1O4 m, k p t = O . l Nslm2 Hf=0.779Nmls, Q=0.260.106m31s Htm=1.04Nmls, Hp=0.26Nmls, f=1.82.10.4 W=1.43.1@N, K=2.47.108 Nlm, 2) k=I h0pt=0.316-10-4 m, k p t = O . 1 Nslm2 bopt=0.09m, Hf=4.74Nmls, Q=1.58.106m3ls Ht,=6,32 Nmls, Hp=1.58Nmls, W=1.43-1@N, K=1.35.108 N l m , f=3.33.104 3) k-3 bop,=0.09m, hOpt=0.548.1O4 m, kpt=O.lNslm2 Ht,=32.9 N m Is, Hp=8.22 Nmls, Hf324.6 Nm Is, Q=8.22.106m31s W=1.43.1@N, K=0.78-108Nlm, f=5.77.10-4. It must be pointed out that the value of load capacity, identical in the three cases, is acceptable as well as that of stiffness, even i f this decreases from the first case to the third one. It should also be noted that, even in the third case, flow rate, though higher than i n the second case and thirty times higher than in the first case, is still not much higher than in the hydrodynamic range. 4) Since, for k=0.3, hopt is too small, again letting ho=10-4 m and, for example, h'20.4, the method presented in section 11.6.2.3.2 can be followed. Furthermore, considering cipt=O.lr from Eqn 11.112 we obtain H~=O,O255<0,213=Hi.hbpt=0.4is then introduced into Eqn 11.116 obtaining chpt=0.308.Also letting bbpt=0.9, we obtain p'=3.08 and kpt=0.308Nslm2 bopt=0.09m, h0pt=0.4.10-4m, Htm=2.08Nmls, Hp=Hf=l.04Nmls, &=1.04.106 m3/s W=1.43.1@ N, K=1.07.108 Nlm, f=2.43.10-4. However it should be noted that the total power is almost doubled, stiffness is reduced to less than a half, while viscosity is too high. A s far as viscosity is concerned, we can make up for it by letting bApt=0.7;then p'=1.03 and we obtain bOpt=0.07m, hOpt=0.4.1O4m, kpt=0.103Nslm2 Htm=2.08Nmls, Hp=Hf=1.04Nmls, &=1.04.1@6m31s W=1.275.1@N, K=0.956.108 Nlm, f=2.72.10-4. As already stated in point 6 in example 11.1, the results can be modified without changing the constraints but by modifying the reference values of the constrained quantities.
406
HYDROSTATIC LUBRICATION
11.6.3
Given load
As also shown in Fig. 11.4, Ht has a minimum as a function of each variable. 11.6.3.1Ht=f(b,h) 11.6.3.1.1.Equation 11.68 can be expressed as
In Fig. 11.18 H;(b',h')is plotted in the OSb'S1 and OIh'12 range, for k=0.1, 1, 10. The variation of Hi is quite similar around the minimum to that in Fig. 11.14, so the same considerations can be made. That is also proved by the results of the optimization of H ; in the range O
Hi
Ht
1.00
100.60
0.67
66.67
8.33
33.33
8.88
2.w
0.88
w = COflSt. Fig. 11.18 Total power H;versus recess width b' and film thickness h', for certain values of speed factor k.
407
OPTlMlzATION
-a-
O
-
1.6
8
14
1
b-
v
2
6
Him 5
HP 4
H;k
3
K’ i
1
0 0
2
4
6
0
2’
6
k
k Fig. 11.19 Values of minimum total power Him, and corresponding optimal values of recess width bAp, and of film thickness h&, versus speed factor k. Values of pumping power H’, of friction power Hh, of flow rate Q’, of recess pressure p;, of stiffness K’and of friction coehicientf’ are also shown. T A B L E 11.4 Optimal values of film thickness for b&,,=0.9 (a) and recess width for h&=0.6 (b), versus speed factor (see also Fig. 11.19). k 0 0.01 0.025 0.05 0.075 0.1 0.2 0.3 0.4 0.5 a hbt 0 0.0308 0.0487 0.0689 0.0844 0.0975 0.1378 0.1694 0.1949 0.2180 b b& 0.3331 0.3333 0.3345 0.3386 0.3447 0.3523 0.4090 0.4844 0.5602 0.6247
408
HYDROSTATIC LUBRICATION
Hirn is plotted with the corresponding optimum values of bbpt and hhpt. Figure 11.19 also shows H@ and H i , as well asp;, K', Q' and f ' which are given by the following formulae: (11.118)
(11.119)
It should be noted that, for any value of I t , bhPt has always taken the boundary value b'=0.9; therefore hhpt is found to be the one which satisfies Eqn 11.60, that is ( 11.120)
From the above results and with reference to section 11.6.2.1, we deduce that the optimization with an assigned load must be carried out as follows: b' is chosen a s large as possible: it is bhpt; it is introduced into Eqn 11.120, thus determining h& The couple bbpt and hhpt makes it possible to evaluate the other quantities. It should be noted that since bLpt is as high as possible, consistently with any other design constraint, p ; is the lowest and that since hbpt is small, especially for small k , K is high. 11.6.3.1.2. If hhpt is too small, condition 11.105 can be introduced. Figure 11.19.b shows the results of the optimization of Hi, with the constraints OIb'I0.9, 0.65h'12. For k23.79 the diagrams are coincident with those in Fig. 11.19.a. For 3.79
In conclusion, as in the case of optimization with a n assigned load and satisfying condition 11.105, bhpt is chosen as large as possible, it is introduced into Eqn 11.120, giving h'. Then, if condition 11.105 is satisfied, such a value of h' is an optimum value with the chosen b'. As a proof, Eqn 11.60 must be satisfied. If condition
OPTlMlZ.4 TlON
409
11.105 is not satisfied, hbpt=aand if condition Hh>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
HYDROSTATIC LUBRICATION
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 O
It must be noted that for any value of k, bhpt and PApt have always taken on the boundary values 0.9 and 0.1, respectively. Therefore optimization is carried out as follows: b' is chosen as large as possible while p' is chosen as small as possible; they are bhpt and PApt, respectively. The optimal value of the film thickness is then given by the following equation:
41 1 15
-b-
45
-.a45
I
"(o.91
40.35
!0.4
10
4
35
3.5
- 0.35 I
106 66 2,Q14~<2,or0.l
t
Him 30
- 030
3
Hb
!5
1'
2:
0.25
Q' hT4
<'
p; 20
- 0.20
2
k 15
l.!
- 015
POP
% 10
:o.lo
1
c
0
k
4
5
0 :
- 0.05
3
0
-0 6
6
0 '
* k '
Fig. 11.20 Values of minimum total power Hirn, and corresponding optimal values of recess width b&, of film thickness h; and of viscosity or of gAPl=L;,+hp, versus speed factor k. Values of pumping power H;, offriction power ~ hof ,flow rate Q',of recess pressure p;, of stiffness K' and of friction coefficient7 are also shown (note that in the last case, i.e. H;,=f(h;,,,b~,,.gL,), diagrams of pi and Q'are valid only when LAPl=l).
,
&,.
41 2
HYDROSTATIC LUBRICATION
T A B L E 11.5 Optimal values of film thickness for b&,=0.9 (a) and of recess width for hAPl=0.4 (b), versus speed factor (see also Fig. 11.20). k 0 0.01 0.025 0.05 0.075 0.1 0.2 0.3 0.4 0.5 a hbt 0 0.0098 0.0154 0.0216 0.0267 0.0309 0.0434 0.0533 0.0619 0.0689 b b& 0.3331 0.3365 0.3593 0.4235 0.5125 0.5939 0.7751 0.8477 0.8848 0.9000
11.6.3.5.2. If hhpt is too small, condition 11.105 can be considered. Figure 11.20.b shows the results of the optimization of H i , with the constraints 056'50.9, 0.4Sh'S2 and 0.3Sp52. For k25.61, bApt takes on the upper boundary value 0.9 while PApt takes on the lower boundary value 0.3 and H@=3Hi;for 5.6bk23.24 hApt also takes on the lower boundary value 0.4; for 3.24>k>0.487 PApt increases to the upper boundary value 2 and H+=Hi. Therefore, optimization is carried out as follows: b' is chosen as large as possible while p' is chosen as small as possible; they are bApt and PApt, respectively. They are introduced into Eqn 11.131, obtaining h'. Then, if constraint 11.105 is satisfied, h'=hhptand, as a check, relation H@=3Hi must be satisfied; otherwise, h&=a and phpt is calculated from Eqn 11.45, that is (11.132) It must be checked that PApt is compatible with the range assigned; in particular, when i t is too high, the highest possible value is assumed to be PApt and bApt is evaluated from either Fig. 11.4.b or Eqns 11.72, in which k is given by the second of Eqns 11.71, that is
11.6.3.5.3.If for higher values of k , K is too low, we can introduce the further constraints 11.122 on stiffness and go on as in section 11.6.3.1.3, now choosing boundary values for both bApt and pApt, that are the uppermost and the lowest, respectively. 11.6.3.6Ht=flz,b,h).Equation 11.68 can also be expressed a s
OPTlMlZATlON
41 3
Eqn 11.134 is similar to Eqn 11.129, so its characteristics can be deduced from all the considerations regarding Eqn 11.129 to be found in section 11.6.3.5 and from corresponding Fig. 11.20.
11.6.3.7 H,=f(L,b,h,p) Finally Eqn 11.68 can be expressed as
(11.135) In this dimensionless form, Q' is given by the first of Eqns 11.130, K' by the second of Eqns 11.118 and ( 11.136) Substituting Eqn 11.77 in Eqn 11.135, it becomes (11.137) Eqn 11.137 is similar to Eqn 11.129, so its characteristics can be deduced from all the considerations regarding Eqn 11.129 to be found in section 11.6.3.5 and from corresponding Fig. 11.20. In particular, Fig. 11.20.a shows the results of the optimization of Hi, in the following ranges: Olb'10.9
,
Och'52
,
O.lSg'12
( 11.138)
while Fig. 11.20.b shows those in the following ranges: O
0.41h'52
,
0.31g'52
( 11.139)
Once the set of optimum values gApt,bLpt, hipt, is determined, any couple L ' , j i ' satisfying gApt=L'p' forms the required combination of four optimum values, together with bhpt and h&. Obviously, Eqn 11.77 increases the number of possible design choices. If the further constraints 11.122 are present, the reader is referred to section 11.6.3.1.3. EXAMPLE 11.3 The pad of Fig. 11.1, with size constraints on width B10.2 m and length L12.B, must be designed so as to carry fa load W=40000N, for speed U=O.S, 2.4,7.2 m Is. We want to evaluate pad stiffness i n the condition of minimum total dissipated power. Ht, can be calculated following the indications in section 11.6.3.7about Eqn 11.137,
41 4
HYDROSTATIC LUBRICATION
and then with the procedure described in section 11.6.3.5 (but replacing p' with g'). So, i f B=0.2 m, ho=2.10-4m and &=0.2 Nslm2, parameter k, given by the second of Eqns 11.135, takes on the values k=0.8, 2.4, 7.2. Then, with constraints 11.138, for example, the following results are obtained (also directly from Fig. 11.20.a): 1) k=0.8 bLPt=O.9, h bp;p,=0.0872, gAp;p,=O.l. So letting for example LLpt=l, p~pt=O.l, Him=O.0979, Hb=0.0245, Hjk=0.0734, Q'=0.0232, pi=1.05 K =11.5, f'=0.0918 from which, in dimensional form, LoPt=0.2m, bopt=0.18m, hOpt=O.174.104m, kpt=0.02Nslm2 Q=0.930.106 m 3 / s Htm=3.92 N m l s , Hp=0.979Nm Is, Hf=2.94 N m Is, pr=1.05.1@Nlmz, K=68.8.l@ N l m , f=0.918.104; 2) k=2.4 LoPt=0.2m, bopt=0.18m, h0pt=0.303~10-4 m, kpt=0.02 Ns I mz Htm=20.4Nmls, Hp=5.09Nmls, Hr15.3 Nmls, Q=4.83.106 m 3 / s pr=1.05.1@ Nlm2, K=39.7.1@ N J m , f=1.59.104; 3) k=7.2 bopt=0.18m, hOpt=0.523.lO-4 m, kpt=O.02 Nslm2 LoPt=0.2m, Htm=105.7Nmls, Hp=26.4Nmls, Hr79.3 Nmls, Q=25.1.106m31s pr=l.05.106 N l m2, K=22.9-108N l m, f=2.75.104; It must be pointed out that in this case and even more so in the previous ones, the value of stiffness is very high. 4) Since in the first case, that is for k=0.8, hoptis too small, the lower limit of h' can be increased. There would be no benefit in increasing the reference value ho because we would obtain the same results. Then, with constraints 11.139, for example, the following results are obtained (also directly from Fig. 11.20.b): k=0.8 b&O.9, h hp,pt=0.4, g bpt=1.22 so letting LApt=2 (so as not to have too high viscosity), pApt=0.608 H im=0.389, Hb=O. 194, Ha=O.l94, Q'=0.37, p;=0.527 K=2.5, f'=0.243 from which, in dimensional form, LoPt=0.4m, bopt=0.18m, hOpt=0.8.104m, kpt=0.122Nslm2 Htm=15.6N m l s , Hp=7.78Nmls, H~7.78 Nm/s, Q=14.8.106 m 3 / s pr=0.526.106 N l m2, K=15.1@ N l m, f=2.43.104. This value of h certainly makes the influence of roughness negligible, and reduces that of errors on parallelism, while stiffness is still quite high. It should also be noted that flow rate and supply pressure are not much higher than those
found i n hydrodynamic lubrication, while the friction coefficient is at least one hundred times lower.
11.7
REAL PADS
Rectangular pad
11.7.1
We refer to the pad with continuous film and characterized by the relation (B-b)/2=(L-l)/2=a, shown in Fig.6.25, whose outer corner radius is r,=O, as i t generally is in actual practice. For the sake of simplicity and considering that often the inner radius is also small, ri=O is assumed.
11.7.1.1 Given flow rate. In this case, referring to sections 5.2 and 5.3.5: ( 11.140) 3 1 2BL-(B+L)(B-b) W = 3 /I Q Q (B2- b2) ( L - B + 2 b)(B+ b) '
K = 3 hW
( 11.141)
If the pad is in motion, the fluid is subjected to inertia forces; nevertheless, since the variation of average pressure is negligible (Fig. 6.25), that is also true for W; a s regards flow rate, its decrease on one side is partially compensated by its increase on the other. The fluid is also subjected to recirculation in the recess but the phenomenon is negligible a t fairly low speed. Therefore, from Eqns 5.106 and 5.108 we obtain
F =p U
1
(L + b)(B- b)
(11.142)
Total power is
1 B-b H t = 3 P Q 2 P ~ - ~ + 2 b + C2'(L Lh U + b)(B- b)
( 11.143)
In dimensionless form, as a function of L' and b', Eqn 11.143 becomes
(11.144)
where k is given by Eqn 11.43. As a function of L', Hi is minimum for
Lbpt = 1 - 2 b'
+ d ( 1 -2 b')2 - 4 b'(1- b') - 1 + 3/12'
( 11.145)
41 6
HYDROSTATIC LUBRICATION
but obviously Lhpt21-b’must be satisfied. Lbpt,given by Eqn 11.145 as a hnction of b’, approaches zero as b’ approaches unity. As a function of h’, H i can also be expressed as in Eqn 11.47 (with obviously a different reference power) and it approaches zero a s h’ approaches infinity. k is given by the second of Eqns 11.47 multiplied by (L + b)(L - B + 2 b) LZ
(11.146)
As a function of p’,H i can also be expressed as in Eqn 11.48 (with obviously a different reference power) and i t approaches zero as p’ approaches zero. k is given by the second of Eqns 11.46 multiplied by factor 11.146. As a function of all four variables, HI can also be expressed as (11.147)
where k is still given by the second of Eqns 11.91. 9 Its optimization, that is the determination of the values of the independent variables that make it minimum, can be carried out as in section 11.6.1.5, taking into account the changes described in this paragraph a s regards the above-mentioned variables. Therefore, the maximum value of b’ and h’ must be chosen as bhpt and hhpt, while the minimum value of p’ must be chosen as p&. Lhpt is given by Eqn 11.145 (where b’=bhptand k is given by Eqn 11.43 with h=hhptho)if it is inside its boundaries, otherwise it takes on the boundary value nearest to it. Other useful information concerning the optimization of H , can be found in section 11.6.1.
As regards efficiency losses, as a function of L‘ and b‘, Eqns 11.2 yield ‘W”
L’-1+2b’ 2L’- (L’+ 1)(1+ 6‘)’
k2
(L + ‘b’)(L’-1 + 2 b’) 2L’- (L’+ 1)(1+ b‘)
’
rK=
rw
( 11.148)
As function of L’ they are minimum for 1 1 LhPt= 1.1 + 2-
which is an approximate relation valid only in the 0.51b’Sl and Osk
41 7
OPTIMIZATION
11.7.1.2 Giuen pressure. In this case:
Moreover, (11.149)
In dimensionless form, as a function of L’ and b’, Eqn 11.149 becomes (11.150)
where k is given by Eqn 11.56. HI decreases with L’, while generally it has a minimum when i t is considered to be a function of b’. As a function of h‘, HI can also be expressed as in Eqn 11.58 but k must be multiplied by (11.151) As a function of p’, H i can also be expressed as in Eqn 11.61, but k must be multiplied by factor 11.151. As a function of h’ and p’, Hi can finally be expressed as in the first of Eqns 11.109, where k is given by the second of Eqns 11.109 multiplied by factor 11.151.
Its optimization can then be carried out as in section 11.6.2.2.1. Therefore, p’ is chosen as high as possible: it is PApt; it is introduced into Eqn 11.110 (together with the value of k obtained from the second of Eqns 11.109 and multiplied by factor 11.151), thus determining hAPt.Useful informations can generally be found in section 11.6.2. As regards efficiency losses Tiy(L’) and r i ( L ’ ) , they decrease a s L ‘ increases, while T h ( b ’ )and Ti((b’) generally have a minimum. As for Tiy(h’) and r k ( h ‘ ) , what has been stated in section 11.5.3.2 is still valid. 11.7.1.3 Giuen load. In this case: Pr=
W
L(B + b) 2BL - (L + B)(B - b)
’
2h3 W
(L-B+2b)(B+b)
Q = 3 F m2BL - (L + B)(B - 6)
Moreover
In dimensionless form, as a function of L’ and b’, Eqn 11.152 becomes
41 8
HYDROSTATIC LUBRICATION
(11.153)
where k=(pUB3)/(h2W).HI(L') always has a minimum, but for much higher values than those given by Eqn 11.70: for instance, for b'=0.9, the optimal value of L' goes from 6.8 to 1.05 when k goes from 1 to 10. H;(b') always has a minimum a s well. Such a value can be approximately evaluated, for L ' = l and k>l, from Eqns 11.72 and increases with L'. As a function of h', H i can also be expressed as in the first of Eqns 11.73 but k now is (11.154)
As a function of p', HI can also be expressed as in the first of Eqns 11.75, but k must be multiplied by the expression between square brackets in Eqn 11.154. As a function of L', b', h ' and p', HI can be expressed as (11.155)
Po B4 where k is given by the second of Eqns 11.135. Its optimization, for a given value of L ' , can be carried out a s in section 11.6.3.5.1, taking into account the changes described in this section. Therefore, b' is chosen as large as possible while p' is chosen as small as possible; they are b&,t and phpt, respectively.The optimal value of film thickness follows from Eqn 11.60, that is (11.156) When the above value is too small we can go on as in section 11.6.3.5.2. Namely, we assume the minimum possible value to be hApt and obtain phPt from Eqn 11.45, that is (11.157)
If phPt proves to be too large it will be necessary to assume the highest possible value to be pApt and to reduce b'; bhpt may be approximately evaluated from Eqn 11.72, in which k is given by Eqn 11.133.
As regards efficiency losses, what has been stated in section 11.5.3.3 is still valid.
OPTlMZ4TlON
419
EXAMPLE 11.4 Let us consider a pad with width B=O.l m, which must carry a load of 45000 N at a speed of 0.2 mls. Recess pressure must be lower than 4-106N l m 2 and the friction coefficient must be lower than 10-4; the lubricant viscosity is 0.03 Nslm2. They are the same data as in example 5.12 but now ri=O is also assumed. Letting ho=0.75.10-4 m and p0=o.03 Nslm2, the speed parameter, given by Eqn 11.135, is k=0.0237. For this value, from Eqn 11.156, we would get a ualue of h which is too small. So let us impose a lower limit on h and consequently on h': for instance h'10.4. If we select L'=2 and b'=0.9, Eqn 11.157gives an excessively high value of optimal viscosity; then we may assume pApt=l and obtain an approximate value of bAptfrom Eqn 11.72 (after having calculated k by means of Eqn ll.133), that is bApt=0.36. Then we obtain, in dimensional form, bOpt=0.036m, hOpt=0.3lO4 m, L=0.2 m, p=0.03 Ns I m2 and the last three values are coincident with those in example 5.12. Moreover, p,=4.33.1@ Nlm2, &=3.48.106m3/s, Hp=15.1Nmls, Hf=O.6Nm/s, Htm=15.7Nmls, f=0.67.104. Recess pressure is a little higher than the maximum value given above, whereas Q and Htm are a little lower than the values obtained in example 12.5. I n order to reduce recess pressure one may increase the recess width or, more efficiently, the pad length. For instance, stating L'=2.5, the following values are obtained: LoPt=0.25m, popt=O.024Nslm2, which is a very common value and bOpt=0.037m, p,.=3.21,1@ Nlm2, &=3.44.106 m3/s, Hp=ll.l Nmls, HrO.7 Nmls, Htm=11.8Nmls, f=0.80.10-4. Now p r is lower than that in example 5.12.
11.7.2
Other types of pads
The methods presented above are a useful reference for the optimization of other types of pads, such as, in particular: cylindrical pads with rectangular recesses (Fig. 5.30) multipad bearings made of several cylindrical pads, separated by grooves (Fig. 7.32.a) multirecess bearings, without grooves (Fig. 8.l.a), in which more complex phenomena are involved. The optimization methods described above have shown that often bearings with wide recesses (high values of b') are more convenient. This confirms what is to be found in the literature regarding multipad and multirecess bearings. In ref. 11.2 an average value of about 0.78 is proposed while in ref. 8.17 the values suggested by other authors range from 0.45 to 0.85.
420
11.7.3
HYDROSTATIC LUBRlCATlON
Circular pad
The reader is referred to the bearing in Fig. 5.1.a. If the bearing rotates, the lubricant is subjected to inertia forces (section 5.3.61, with a pressure peak a t the inner edge of the recess (Fig. 5.8). This causes a loss of flow rate, slightly hindered by the inlet (section 5.3.4) and turbulence effects (section 5.3.51, but largely compensated by the increase of load capacity. That justifies the assumption of negligible inertia forces. The friction torque in the recess is also disregarded.
11.7.3.1 Given flow rate. Assuming a linear pressure drop in the film (r2-rl)/r2 in place of the logarithmic drop ln(r2/rl), the pumping power of the bearing can be written as (11.158)
which yields lower values with errors decreasing from 30% to 3% as r'=r1/r2 increases from 0.55 to 0.95. The friction power H f can approximately be written as Hf -/! - h Up2 (2E r2)(r2- rl) ,
with Up = 1.7 f2 r2
which yields values higher than the exact ones with an error of 30% for r'=0.55 but lower by 3% for r'=0.95. Thus total power is
which is similar to Eqn 11.41.
As regards its optimization as a function of one or more of the three variables rl, h and p , with r2 as the reference quantity, what has been stated about the Ht of the indefinite pad, as a function of 6, h and p , with B as the reference quantity, can be applied, approximately, replacing L with (rr12)r2 (given datum) and U with U p . In particular, the remarks made in section 11.6.1.5 are still valid, but now L' can no longer be considered to be an independent variable, since we still have L'=x12.
11.7.3.2Given pressure. In this case
OPTIMIZATlON
421
which now yields higher values than the exact ones, with the same error as in Eqn 11.158. Total power is then 5c -
H =-- h3 r2 + !, U 2('r2)(r2 - r l ) , t 3 p p F G h P 2
with Up = 1.7 R r z
(11.159)
which is similar to Eqn 11.54. The remarks regarding the previous case can be extended to this one.
11.7.3.3 Given load. In this case
with the same error as in Eqn 11.158. Moreover the minimum of H p now occurs for rhpt=113,which is much lower than the exact value 0.529. H f can be written as
H f = f Up2 ( 2 n r2)(rz- rl) ,
with Up = 0.85 Rrz
Total power then becomes (11.160) which is similar to Eqn 11.68. The remarks regarding the previous cases can be extended to this one, but now L must be replaced by 2xr2 (that is we have L ' = ~ Kand ) U by Up=0.85Rr2.Moreover, after the evaluation of bhpt for the equivalent infinite pad, the corresponding value of rApt can be calculated from the following equations: 159
rhpt = 100 + 59 k0.04 bhpt
rbpt = bhpt ,
,
for
r;
(11.161)
for k > 1
which, approximately, take into account the above-mentioned error corresponding to the minimum of H p .
EXAMPLE 11.5 Let us consider a circular pad with diameter D=2rz=0.1 m, which must carry a load W=50000N at an angular speed R=25n rad I s (same data as in example 5.2). Letting ho=O.75.10-4m, 1.(~=0.009 Ns Jmz,L=2mz and U=Up=0.85Qrz, the speed pa-
422
HYDROSTATIC LUBRICATlON
rameter, given by the second of Eqns 11.129, is k=0.0839. For it, from Fig. 11.20.a or from Table 11.5.a we would get a value of h which is too small. So, imposing a lower limit to h and consequently to h’ (for example h’20,4), from Fig. 11.20.b or by interpolation from Table 11.5.b we obtain bApt=0.540;then from the first of Eqns 11.161 rhpt=0.560, h hp,pt=0.4, C( hpt=2, from which, in dimensional form, rlopt=0.028 m; hoPt=0.3.104 m, which is much lower than the value in example 5.2; popt=O.018Nslm2, equal to the value in example 5.2. Then, utilizing of course exact equations, we find pr=10.8.106 Nlm2, Q=14.6.106m31s, Hp=157Nmls, Hr32.8 Nmls, Ht,=190 Nmls, f=2.1.10-4. It should be noted that, as r’ decreases to r’=0.529, Htm remains virtually constant. EXAMPLE 11.6 Let us again consider a circular pad with diameter D=2r2=0.1 m, but which must carry a load W=lOOOON at an angular speed l2=300nradls (the same data as in example 5.4). Letting ho=1.5.104 m, p,,=0,05 Nslm2, L=2m2 and U=Up=0.85f2r2,the speed parameter, given by the second of Eqns 11.129, is k=6.99. For it, with the method described in section 11.6.3.5.1 with the constraints ~(’20.3and b’10.9, we obtain rApt=bhpt=0.9, h hpt=0.4#7, &=0.3. Thus rlopt=0.045m; hopt=0.67.10-4m, hpt=O.015Nslm2; the two latter values are both lower than the values i n example 5.4. Moreover, pr=1.41.106 N l m 2 , the same value as in example 5.4, Q=141.10-6m31s, Hp=199N m l s , Hf=671 N m l s , Htm=870 N m l s and f=1.5.103. The power ratio is HfIHp=3.34, not much higher than the value 3 indicated in section 11.6.3.5.2. This proves that the optimization method described is sufficiently valid. Obviously, more approximated values could be obtained by introducing the inertia corrective factors, as i n example 5.4. In any case the values of powers and of the friction coefficient are about half those in that example.
11.7.4
Annular pad
The reader is referred to the bearing in Fig. 5.13. If the bearing rotates, the lubricant is subjected to inertia forces, with pressure variations also occurring inside the recess (Fig. 5.18).They have little effect on average pressure, on flow rate, especially for common values of the inner radius (that are never small) (Fig. 5-17],and on load capacity (section 5.4.5).Therefore the inertia forces can be assumed to be negligible. The friction torque in the recess can also be disregarded.
11.7.4.1Given flow rate. The relation (r2-rl)=(r4-r3)=a is assumed. The closer rl is to r4, the closer the former relation is to rl / ~ 2 = r 3 r4 / corresponding t o equal flow rates through the films. Linear pressure drops in the film (r2-r1)/r2and (r4-r3)/r4are also assumed in place of the logarithmic drops 1n(r2/rl)and ln(r4/r3).Finally 1t(r4+r2) is replaced, roughly, by ~ ( r 4 + r l )The . pumping power of the bearing can then be written as (11.162)
which yields lower values with errors decreasing from 16% to 3% a s r ’ = r l / r 4increases from 0.5 to 0.9. In Eqn 11.162 ~(r4+rl)and 2(r4-r3)correspond to L and (B-b) in Eqn 11.4. The friction power H f can also be written approximately a s
which yields values lower than the exact ones with an error decreasing from 8% to 0.2% as r’ increases from 0.5 to 0.9. As regards approximations, it should be noted that as rl goes to infinity the bearing turns into an infinite pad. Then total power is
which is quite similar to Eqn 11.41.
As regards its optimization as a function of one or more of the three variables r3 (or r2),h and p, with B=(r4-r1) as a reference quantity, what has been stated concerning the total power of the infinite pad as a function of b, h and p, with B as a reference quantity, can be applied here too, replacing L with x(r4+rl) (given datum) and U with U p .As far as r3, or r2 are concerned, they are correlated in a n elementary manner to 6’; we have for example r3 =
r,
+ rl + (r4- rl)b’ 2
11.7.4.2Given pressure. In this case
which now yields higher values than the exact ones, with the same error as in Eqn 11.162. Total power is then (11.163)
424
HYDROSTATIC LUBRICATION
Equation 11.163 is similar to Eqn 11.54 and the considerations regarding the previous case (i.e. section 11.7.4.1) can be extended to this one.
11.7.4.3 Given load. In this case 4 h3 W2 H =-P 3 p n (r4+ r l )(r4- r3) [(r4- r l ) + (r3 - r2)12
with the same error a s in Eqn 11.162. In i t n(r4+rl)/2,2(r4-r3),(r4-rl) and ( r 3 - r ~ ) correspond to L , (B-b),B and b in Eqn 11.25, respectively. The friction power can be written as
which yields values lower than the exact ones, with a n error decreasing from 11% to 3% as rl increases from 0.5 to 0.9, which is virtually equal to that of H p , especially for high values of rl. Total power then becomes
Equation 11.164 is similar to Eqn 11.68 and what has been stated in the previous cases can be extended to this one, but now L must be replaced by n(r4+r1)/2and U by Up=1.390(r4+r1)/2. EXAMPLE 11.7 Let us consider an annular pad with an outer radius r4=0.05 m and an inner radius rl=0.03 m (r'=0.6), which must carry a load W=20000N at an angular speed 0 = 4 n r a d / s (the same data as in example 5.6). Letting ho=0.75.10-4 m, h = 0 . 0 5 N s 1m2, L=n(r4+r312, B=r4-rland U=Up=1.390(r4+rJ12, the speed parameter, given by Eqn 11.129, is k=O.0156. For it, from Fig. 11.20.a or from Table 11.5.a, we would get a value of h which is too small. So imposing a lower limit to h and consequently to h', for example that in Fig. 11.20.b, from the same figure or from Table 11.5.b, we obtain bhpt=O.345,hApt=0.4 and pApt=2,from which r3,,=0.0435 m and rZoPt=O.0366m; hopt=0.3.lO-4 m, lower than the value in example 5.6; popt=O.l Nslm2, equal to the value in that example. Moreover, p,.=5.92.1O6 Nlm2, Q=10.2.10-6m3/s, Hp=60.3 N m l s , H ~ 3 . 0N m l s , Ht,=63.3 N m / s and f=3.0.10-4. Furthermore, letting p=900 Kglm3 and c=1900 JIKgOC, AT=3.6 "c.
OPTlMlZATlON
425
Flow rate, pumping and total power are much lower than those in example 5.6, but this is due to the lower value of h, obtainable with an accurate construction and assembling of the bearing. EXAMPLE 11.8 Let us consider an annular pad with an outer radius r4=0.03 m and an inner radius r1=0.02 m (r'=2/3), which must carry a load W=2000 N at an angular speed Q=GOOzradjs (the same data as in example 5.10). Letting ho=0.75.10-4 m and pt,=0.005Nsjm2, the speed parameter, given by Eqn 11,129 with the above-mentioned substitutions of L and U,is k=0.229 and for this, what has been stated in example 5.10 can be applied here too. Thus, we obtain bApt=0.8, hApt=0.4, pipt=2, from which r30pt=0.029m, then a'=O.l, r2,t=0.021 m; hOpt=0.3.10-4m, lower than the value in example 5.10; popt=O.O1 Nslm2, the same value as in example 5.10. Moreover, pr=1.41.106 Nlm2, &=100.106 mais, Hp=141 N m l s , Hf=255 N m l s , Ht,=397 N m l s and f=2.71-10-3. Furthermore, letting p=870 Q l m 3 and c=1930 J I K g V , AT=2.4 "c. Flow rate, pumping and total power are definitely lower than those in example 5.10. It should be borne in mind that the above results are based on an approximate equation (that is Eqn 11.164); indeed it could be proved that for a'=0.1 and hOpt=0.3.10-4m the optimal viscosity is rather lower, being hp,=0.0074 N s lm2; nevertheless, the total power which tallies with this new value of viscosity is only slightly lower (Ht,=380 Nmls).
11.8
COMPENSATED SUPPLY
Let us consider the infinitely long pad in Fig. l l . l . c , with width B , supplied by a pump through a compensating element (Fig. 1l.l.b) across which pressure drops from the supply value ps, kept constant by a relief valve, to that in the recess pr The performance of a portion of length L is studied.
11.8.1
Capillary tubes
If the compensating element is a capillary tube with diameter d and length L, equating expression 4.66 to expression 4.48 yields (11.165)
where
426
HYDROSTATIC LUERICATiON
R = -128 p-
n
r
1 d4
(4.66 rep.)
is the capillary tube hydraulic resistance and
is the hydraulic resistance of the pad. From Eqn 153 we obtain (6.11 rep.)
is the characteristic ratio of all compensated bearings. We also obtain (6.19 rep.)
11.8.2
Steady pad
11.8.2.1 Given pressure. Substitutingpp, forp, in Eqn 4.47 yields
With simple operations (see also section 6.31, we also obtain 3
K = 5 (1 - B) B 11
Q = --P 3P
Ps
L ( B + b) L
pSh3 ~
H = -1- 1
- 6
L
( 11.167)
(11.168)
(11.169)
It should be noted that Hp = Hpc+ Hpb
(11.170)
where Hpc=Q(ps-pr) is the power dissipated in the capillary tube, which can be written as (11.171)
and Hpb=&pris the power dissipated in the bearing, which can be written as
OPTlMlZ4 TlON
427
(11.172)
Comparison of the foregoing equations, for the typical value p=0.5 of the pressure ratio, with the analogous equations in section 11.5.1.2 (direct supply) immediately shows that, when supply pressure is the same, load capacity is halved, as well as recess pressure, flow rate and pumping power, and stiffness is reduced to one fourth. When recess pressure (that is load capacity) is the same, supply pressure and pumping power are double that found in the case of direct supply, whereas stiffness is halved. In our investigation into the performance of the pad as p vanes because of the variation in the dimensions of the capillary tube, diameter d a n d o r length 1, we assume that the dimensions of the pad are given, as well a s film thickness h and viscosity p; the above equations may hence be written in a dimensionless form as follows (11.173)
(11.174)
The ratio of Hic and Hib may also be of interest: (11.175)
which coincides with Eqn 6.19. The results are plotted in Fig. 11.21.a. From it, the characteristics of a bearing with assigned dimensions, film thickness (equal to the reference value) and supply pressure can be determined a s the capillary tube geometry varies. It must be pointed out that K is maximum for p=0.5 (as clearly follows from dKldb=O) and that for p=0.3 and 0.7 it still reaches 84% of its maximum value. For this reason i t is commonly recommended to adopt p=0.5 or values close to it. R , / R is also plotted in Fig. 11.21.a. From that curve, for given bearing dimensions and h, the dimensions of the capillary tube can be determined. Eqn 6.19 can also be written in the form (11.176)
Then, in the investigation into the performance of the pad a s p varies, now assuming the dimensions of the pad and capillary tube are constant (or, better, the ratio
428
HYDROSTATIC LUBRICATION
-a-
1.8-
-b-
1.6 -
H;
d>l+ 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
HYDROSTATIC LUBRICATION
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 O
(11.192)
Since the foregoing nondimensional equations are identical to those obtained in section 11.6.2.1.1, most of the remarks made there still retain their validity for the compensated supply system, too. 11.9.1.1.2, In practice the condition Osh' is replaced by condition 11.105, for the general reasons mentioned in section 11.6.2.1.2. The results of the optimization of Hi are presented in Fig, 11.16.b, with the constraints OIb'10.9 and 0.91h'<2: we refer the reader to section 11.6.2.1.2 for the relevant remarks. 11.9.1.1.3. Figure 11.17.b shows the influence of adding the further constraints 11.107, namely a minimum value of stiffness, on the optimal values of b' and h ' (see also sections 11.6.2.1.3 and 11.6.2.1.4). 11.9.1.2 Ht=f(uh) Equation 11.184 can also be expressed as
On the analogy of the first of Eqns 11.193 with the first of Eqns 11.190 and the first of Eqns 11.102, Fig. 11.14 also presents Hi(p',h'), in the O
PApt always takes the boundary value p'=0.025, whereas the optimal value of h' is given by Eqn 11.110. H ~ r n ( p ~ P , p t ,has h ~ Palso t ) been plotted in Fig. 11.16.a with the constraints O.lSp'11, k h ' a and in Fig. 11.17.a with the constraints O.21pu'11and 0 4 ' 1 2 . Finally, in Fig. 11.16.b the effect is shown of a constraint of type 11.105 (minimum film thickness), whereas Fig. 11.17.b refers to the case of a further constraint on minimum stiffness. 11.9.1.3 H&'b,b
h)
11.9.1.3.1.Equation 11.184 can also be expressed as
436
HYDROSTATIC LUBRICATION
Substituting Eqn 11.50 into Eqn 11.194 leads to an equation which is formally identical to Eqn 11.112; Hi(c’,h‘) is plotted in Fig. 11.14, in the Osc‘ll and OIh’52 range. H ~ , ( C ~ ~ ~ , ~is&presented J in Fig. 11.15, with the constraints 11.113. H i , Q’=Hi and H h are also plotted. There are also W and K‘ given by Eqns 11.191 and f ’ which is given by the following equation
f ‘ =1 fho_ = ,h, gl +rb: __ 4FB Since W’, K’ and f ’ depend explicitly on b‘, they have been plotted in Fig. 11.15, it having been assumed that b’=b&=0.975, that is p~pt=c&l-b~pt)=l(bear in mind that any couple of values of b’ and p’ satisfying (l-b’)p’=c& is an optimum couple). Since chpt has always taken on a boundary value, the optimal value of h‘ is given by Eqn 11.114.
H;m(c&&,p,pt) and the other related functions are also plotted in Fig. 11.16.a and 11.17.a, with a different choice of constraints (see also section 11.6.2.3.1). 11.9.1.3.2. The effect of introducing a constraint for minimum film thickness is again shown in Fig. 11.16.b. As pointed out in section 11.6.2.3.2, in order to find the minimum of H,, in the presence of condition 11.105, c’ is initially chosen as small as possible. Letting c’=c&, h‘ is calculated from Eqn 11.114; if h‘ satisfies condition 11.105 it is h&, otherwise, letting h&=a, H i and H@ are calculated. If H j k > H i , c&t is still equal to the value initially chosen; if not, still letting h&,,pt=a,c&,,t is calculated from Eqn 11.116. Any couple of values of 6’ and p’ satisfying (l-b’)p’=c&,t and h&,t form an optimum combination yielding Hirn. 11.9.1.3.3. The effect of introducing a constraint for minimum film thickness is again shown in Fig. 11.17.b. In order to find the minimum of H,, in the presence of conditions 11.107, c‘ is initially chosen as small as possible. Letting c’=c&, h’ is calculated from Eqn 11.114; if h’ satisfies conditions 11.107 it is h&,, otherwise h &=W/
EXAMPLE 11.9 The pad examined in example 11.2 (direct supply) is considered again. This time, however, it is supplied by means of a capillary tube, as shown in Fig. 1l.l.b;
the supply pressure, upstream of the capillary tube, is still the same ps=106 N l m 2 . The values of speed are also the same: U=0.3, I , 3 m ls. So, assuming hO=lO4 m, a pressure ratio @=0.5and h = O . l Nslm2, from the second of Eqns 11.194, k=0.424, 1.414, 4,24. If constraints O.le'51, OIh'12 (as in the case of direct supply) are adopted, the following results are obtained (also approximately from Fig. 11.16.a): 1) k=0.424 h bpt=0.206, c bpt=O.1 and choosing bbpt=0.9, pAp+ Him=O.l17, Hb=Q'=0.0291, H@=0.0874 and H@lHi=3 W=0.95, K'=4.61, f'=0.217. From which: bopt=0.09m, hopt=0.206.10-4m, kPt=O.1Nslmz, p,=0.5.106 Nlm2 Hf=0.655Nmls, Q=0.218.106 m3/s Htm=O.874 Nmls, Hp=0,218Nmls, W=7125N, K=0.519.109 N 1m, f=3.07.1@4 It should be noted that stiffness is nearly five times lower than the value relative to direct supply and load capacity is halved, wAereas power and flow rate are slightly smaller. 2) k=1.41 bopt=0.09m, hopt=0.376.10-4 rn, popt=O.1 Ns I m2, pr=0.5.106 N I m2 HtmS.32 N m Is, Hp=1.33N m Is, H f d . 9 9 Nmls, &=1.33.106 m3/s W=7125N, K=0.284.109 Nlm, f=5.60.10-4. 3) k d . 2 4 bopt=0.09 m, hopt=0.651.10-4m, kpt=0.l NsIm2, p,=0.5.106 Nlm2 Htm=27.6 Nmls, Hp=6.91Nmls, Hf=20.7 Nmls, &=6.91.1@6m 3 / s W=7125N, K=O.164.109 N 1m, f=9.70.10-4. For all the cases considered above the efficiency of direct supply as compared to capillary compensation is clearly greater. 4) Load capacity may be improved by selecting a higher value for the pressure ratio, such as p=O.7. For the lowest speed we now have k=0.359 and then hopt=0.189.10-4m, popt=O.l Nslm2, pr=0.7-106Nlm2 bopt=0.09rn, Hf=O.71Nmls, H@lHb=3 Htm=O.95 Nmls, Hp=0.24 Nmls, K=0.474.108 Nlm, f=2.38.104. Q=2.38.106 m3Is W=9975N, I n this way load capacity is 40% greater (but stiffness is slightly smaller); the efficiency loss rw is decreased from 123.106 mls to 95.10-6 m l s and friction coefficient is also lower. For the highest speed we have k=3.59 and then bopt=0.09m, hopt=0.599.10-4m, kpt=0.l Nslm2, p,=0.7.106 Nlm2 Htm=30.1 Nmls, Hp=7.5Nmls, Hf'22.5 Nmls, Q=7.52.106 mats W=9975N, K=O.150.109 N l m, f= 7.53-104.
438
11.9.2
HYDROSTATICLUBRICATlON
Given load
11.9.2.1 Ht=f(b,h) 11.9.2.1.1. Equation 11.187 can be also expressed as
(11.195)
The analogy with Eqn 11.117 is clear. In Fig. 11.18 Hi(b',h') is plotted in the Ogb'll and Osh'12 range, for k=0.1, 1, 10. What has been said in section 11.6.3.1.1, concerning Fig. 11.18, still holds good. Figure 11.19.a shows the results of the optimization of Hi, for OIkI6, with the constraints Osb's0.9 and Osh's2. Him is plotted with the corresponding optimum values of bApt and h& H@and H; are also plotted, as well as p; (given by the first of Eqns 11.1181, Q' (given by the first of Eqns 11.119)and (11.196)
(11.197)
It should be noted that, as in the case of direct supply, bApt has always taken the boundary value b'=0.9; therefore hAPt can be determined from Eqn 11.120. 11.9.2.1.2. In practice the condition Osh' is substituted with condition 11.105 for the reasons mentioned in section 11.9.1.1.1. The results of the optimization of Hi, with the constraints Od~'10.9,0.61h'<2, are presented in Fig. 11.19.b (see section 11.6.3.1.2 for the relevant remarks). For the optimization with assigned load and constraint 11.105, bAPt is initially chosen a s large as possible, and h ' is calculated by means of Eqn 11.120. Then, if constraint 11.105 is satisfied, such a value of h' is the optimum one; otherwise, a is chosen a s hhpt and bhPt is calculated from Eqns 11.72 (or Fig. 11.4.b) in which (11.198)
If, on the other hand, the value of b' thus calculated is greater than the upper limit selected for b', the latter has to be chosen as bhPt.
OPTlMlzATION
439
11.9.2.1.3. For high values of k the above procedure may lead t o a high film thickness and, hence, to poor stiffness. In the presence of constraints 11.122 on minimum stiffness the optimization procedure should be modified a s follows: the upper limit of b ' is firstly selected as bApt and h ' is calculated from Eqn 11.120. If constraints 11.122 are not satisfied, i t must be assumed that h&=l/y.
11.9.2.2 H,=f(Uh) Equation 11.187 can also be expressed as
(11.199) The nondimensional coefficients p i , p i and K can still be calculated by means of Eqns 11.196, whereas for Q' and f ' we have:
Optimization may be performed choosing the lowest available viscosity and calculating h& from Eqn 11.124. As usual, for small values of k, film thickness will be too low; in this case, in the presence of constraint 11.105, we shall assume h&=a and calculate p& from Eqn 11.125.
11.9.2.3 H,=f&,h) Equation 11.187 can also be expressed as
The considerations regarding Eqn 11.199 are still valid, if only p' is replaced by L'. 11.9.2.4 H,=f(L,p,h) Equation 11.187 can also be expressed as
(11.201) If we substitute Eqn 11.77 into Eqn 11.201, it becomes
(11.202)
440
HYDROSTATIC LUBRICATION
The remarks regarding Eqn 11.199 are still valid, if only p' is replaced byg'. 11.9.2.5H,=flb,h,p) Equation 11.187 can also be expressed as
(11.203) Since HI is identical to that given in Eqn 11.129, the remarks made in section 11.6.3.5 can be repeated word for word. In particular, in Fig. 11.20.a the results of the optimization of HI are presented, in the O
- f = k $ (1- 6')
f ' - 1 ho
Figure 11.20.b, on the other hand, shows the effect on optimal values of constraint 11.105 on minimum film thickness. Optimization may be carried out as follows: the largest and the smallest possible values are selected for bhpt and p&, respectively. The optimal value of film thickness is then given by Eqn 11.131.
If the film thickness is too small (that is, constraint 11.105 is not satisfied), we must assume h&a instead of the above value; if we still have H@>Hbno other change is needed. Otherwise, the optimal value of viscosity needs to be recalculated from Eqn 11.132. Again it must be checked that this last value is compatible with the constraints. Should it be too high, the maximum allowable value has to be selected for p& and b& has t o be evaluated from either Fig. 11.4.b or Eqns 11.72, in which (11.204)
On the other hand, when speed is high, the film thickness obtained from Eqn 11.131may be too high to ensure sufficient stiffness (constraints 11.122). In this case we have to select h&,,pt=llyandthe greatest and the lowest available values for b& and p&, respectively. 11.9.2.6 H,=f(L,b,h). Equation 11.187 can also be expressed as
441
OPTIMIZ4TION
(11.205) The analogy of the above equation with Eqn 11.203 is clear and the remarks made in section 11.9.2.5 can be repeated here, apart from simply substituting p' with L'.
11.9.2.7 H,=f(L,b,h,p). Finally Eqn 11.187 can be expressed a s
with k =
@PO U B
(11.206)
Furthermore, p; is given by the first of Eqns 11.136, K is given by the second of Eqns 11.196, Q' by the first of Eqns 11.130, whereas forpi and f' we have
Substituting Eqn 11.77 in Eqn 11.206, we again obtain Eqn 11.137, which, on the other hand, is similar to Eqn 11.129. All the remarks made in sections 11.6.3.5 and 11.6.3.7 may then be repeated.
EXAMPLE 11.10 The pad examined in example 11.3 is considered again. This time, however, it is supplied by means of a capillary tube, as shown in Fig. 11.1.6; it must carry the same load W=40000 N, for the same values of speed U=0.8, 2.4, 7.2 m l s . So, again assuming B=0.2 m, ho=2.10-4m, &=0.2 Nslm2 and P=0.5, from the second of Eqns 11.206, k=0.566, 1.70, 5.09. If constraints 11.139 are adopted, the following results are obtained : 1) k=0.566 bAp,pt=0.9, h Ap,pt=0.4, g Ap,pt=l. 72. Then, letting LApt=2,as in the 4th case (k=O.8) in example 11.3, ~Ap,t=O.859, Him=0.275, Hi=O.138, Hb=O. 138, Q'=0.261, pi=0.526, K=2,5, f'=0.243. From which, in dimensional form: Lapt=0.4m, bapt=0.18m, hapt=0.8.104m, hpt=0.172 Nslm2 Htm=22.0 N m J s , Hp=ll.ONmls, Hp11.0 Nmls, Q=10.4.10-6 m3/s
442
HYDROSTATE LUBRlCATloN
p,=1.05.106 Nlm2, p,.=0.526.106 Nlm2, K=7.50.1@ N l m , f=3.44.104. Comparing the above values with those obtained in the fourth case in example 11.3 (direct supply), it may be noted that stiffness is halved, supply pressure is doubled and power is 32% greater. 2) k=1.7 Letting now LApt=l and using constraints 11.138, as in the second case in example 11.3, bopt=0.18m, hOpt=0.254.1O4 m, kpt=0.02Nslm2 Lopt=0.2m, Hp18.1 Nmls, Q=2.87.1Q6 m31s Htm=24.2 N m Is, Hp=6.1N m Is, ps=2.10.106 N 1 m2, p,=1.05.106 N l m2, K=23.6-1@N l m, f= l.89.104. Compared to the second case in example 11.3, Htm is 20% greater and supply pressure is doubled, whereas K is 40% lower. Moreover these results require a film thickness which is notably smaller. If the same film thickness as i n example 10.3 (case 2) were used, stiffness would be further reduced. 3) k d . 0 9 Again letting LAPt=l,as in the third case in example 11.3, bopt=O.18 m, hopt=0.440-104m, kpt=0.02 Nslmz LOpt=0.2m, Hp=31.4 Nmts, Hp94.3 Nmls, Q=14.9.106 m3ts Htm=126Nmls, p,=1.05.106 Nlm2, K=13.6.108 Nlm, f=3.27.104. p,=2.10.106 Nlm2, 4) In all three cases stiffness is clearly lower than in the case of the pad which is directly supplied, in spite of the smaller values of optimal film thickness. It should be noted that the only way to improve stiffness (apart the obvious solutions of further reducing clearance or increasing load) is to reduce p. In reference to the first case, still letting L=0.4 m and h'20.4, the following results are obtained, now for p=0.3 (that is k =0.438):
h Ap,pt=0.4, g Apt'2. bApt=0.89, From which, in dimensional form: bopt=O.179 m, hopt=0.8-104m, kpt=0.2 Nslm2 Lopt=0.4m, Hp=15.0Nmls, Hp13.5 Nmls, Q=8.55.106 m31s Htm=28.5 Nmls, p,=l.76.106 Ntm2, p,=0.528.106 Nlm2, K=10.5.1@ N l m , f=4.2-104. It is clear than the 40% increase in stiffness has been paid for with a notable increase in power and supply pressure. Compared with the fourth case i n example 11.3, K is still much lower, even though Htm is now much higher. A further decrease i n pressure ratio would lead to a higher stiffness (although the stiffness of the pad that is directly supplied cannot be reached anyway), but with very high values of supply pressure and power consumption. For instance, for p=O.l we would obtain: Lopt=0.4 m, bopt=O.163 m, hopt=0.8.104 m, kpt=0.2Ns I m2 Htm=51.7 N m IS, Hp=28.3Nmts, Hr23.4 Nmls, Q=5.14-1O6 m31s ps=5.50-106N I m2, p,,=0.550.106 N l m2, K=13.5.1OB N I m, f= 7.3.104.
OPTlMlZ4TlON
443
Although stiffness is still lower than in the case of direct supply, supply pressure is now more than ten times larger and total power is more than three times larger.
11.10 11.10.1
OTHER TYPES OF COMPENSATING ELEMENTS
Orifices
If the compensating element is a sharp-edged orifice, equating expression 4.76 to expression 4.48 again leads to Eqn 11.165, where now the hydraulic resistance of the restrictor is
which depends on supply and recess pressures and where R is still the hydraulic resistance of the clearances of the pad. Solving for p,, we obtain
11.10.1.1 Given pressure. Proceeding as in section 11.8.2.1, for the capillary tubes, the expressions of the various quantities are obtained (see also section 6.5.2); in particular W, &, H p , H f andHt are still given by equations 11.166, 11.168, 11.169, 11.30 and 11.184, respectively, whereas stiffness is now:
The remarks made in section 11.9.1 regarding the capillary tubes can be repeated here. Indeed, the equations for all nondimensional parameters except stiffness remain exactly the same. As far as R is concerned, all that is needed is to substitute P(1-p)with 2P(l-p)/(2-p).
11.10.1.2 Given load. Proceeding as in section 11.8.2.2, the same equations can be obtained, except for stiffness which now is:
and remarks similar to those regarding the capillary tubes can again be made.
444
11.10.2
HYDROSTATlC LUBRlCATlON
Flow-control valves
If the compensating element is a flow-control valve, the flow-rate Q through the pad is constant. The hydraulic resistance of the restrictor must hence be
where R is the hydraulic resistance of the pad. Thus
11.10.2.1 Given pressure. Proceeding as in section 11.8.2.1 the expressions of the various quantities are obtained (see also section 6.5.3); in particular Eqn 11.166 and all the equations from 11.168 to 11.172 are still valid, as well as Eqn 11.184, whereas stiffness is now
The remarks made in section 11.9.1 regarding the capillary tubes can be repeated here. Indeed, the equations for all nondimensional parameters except stiffness remain exactly the same. As far as K is concerned, all that is needed is to substitute p(1-PI with p.
11.10.2.2 Given load. Proceeding as in section 11.8.2.2, we can obtain again the same equations, except for stiffness which is now given by Eqn 11.23: namely, it is identical to that obtained in the case of direct supply (naturally, within the operating range of the valve). As regards the operating range, the smaller P is, the wider the operating range is, as shown in Fig. 6.10.
11.11
REAL PADS
The formulae of the total power for the infinite pad in the case of compensated supply differ from those of the directly supplied pad for p,. replaced by @ps in the case of constant pressure, and for W replaced by in the case of constant load. As for the rest they are unchanged.
WIG
This also holds good for the other types of pad: rectangular, circular and annular, the formulae of which are the following: Eqns 11.149 and 11.152 for the rectangular pad,
OPTlMlZATlON
445
Eqns 11.159 and 11.160 for the circular pad, Eqns 11.163 and 11.164 for the annular pad. Consequently the optimization procedures in sections 11.9.1 and 11.9.2, which are good for the infinite pad, can be followed for the above-mentioned pads, bearing in mind the approximations introduced in sections 11.7.1, 11.7.3, 11.7.4. When the optimum values of L , b, h and p for the rectangular pad, rl, h and p for the circular pad, r3 (or rz),h and p for the annular pad, have been determined, the values of the other quantities are calculated, obviously, by means of the exact equations to be found in Chapters 5 and 6. The optimization procedure of the rectangular pad can be a useful reference for that of the cylindrical pad with a rectangular recess and of the multi-pad and multirecess bearings. Finally, what has been said a s regards the capillary-tube supply can be extended t o the supply by means of orifices, flow-control valves, etc.
EXAMPLE 11.11 Consider a rectangular pad, compensated by means of a capillary tube, with a width 3=0.3 m, which must carry a load W=60000 N at a speed U=0.05m l s . Film thickness is h20.4.104 m, stiffness is K>2.5.1@ 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<0.1 m, which must carry a load W=35000N (the highest load in example 6.4) at an angular speed Q=628 rad Is. Supply pressure must be p,=4.lO6 Nlm2; friction torque Mf must be smaller than 5 Nm. Putting r4=0.09 m, ho=0.3.104 m, h = O . O l Nslm2, the speed parameter, given by the second ofEqns 11.194 in which B=r4-rland U=R(r4+rl)12(see section 11.7.4) and 8=0.5,is k=6.91. Then from Fig. 11.16 C ~ ~ ~ (from = O . which, ~ putting bApt=0.9, pAPt=l) and, also from Eqn 11.114, h&=0.83. Therefore, r30pt=0.088m, a=0.002 m and rzopt=O.052m; hOpt=0.25.1O4 m, &,pt=O.Ol Ns I m2. The effective area (Eqn 5.66) is A,=0.0167 m2 and thus Eqn 6.11 shows that a slightly larger value of the pressure ratio, such as 8=0.524, is needed to obtain the required load capacity: however the optimal parameters calculated above are still very largely valid; thus for 8=0.524, the following results are obtained (from the relevant equations in chapters 5 and 6): p,=2.O9.1O6 Nlm2, W=35000 N, K=2.109 N l m , Q=120.106 m3/s, Hp=479Nmls, M ~ 2 . 6 4Nm, Hf=1661Nmls, Ht,=2140 N m l s . And i f p=920Kg/m and c=1850 JIIQC, AT=10.5%. Note that the power ratio isgreater than 3: this is due to the approximations introduced in section 11.7.4; indeed, it could be shown that the true optimal value for film thickness is hOpt=0.259.10-4m; however, this would lead to a minimum total power of Htm=2136 N m / s which is only slightly lower.
REFERENCES 11.1 Siddal J. N.; Optimal Engineering Design; M. Dekker, N.Y., 1982, 523 pp. 11.2 Michelini C., Ghigliazza R.; Optimum Geometrical Design of Multipad Externally Pressurized Journal Bearings; Meccanica, 3 (19681, 231-241.
Chapter
12
THERMAL FLOW
12.1
INTRODUCTION
In the previous chapters, all the heat produced in the lubricant of hydrostatic bearings by friction from viscous drag was assumed to remain in the lubricant itself (adiabatic flow). In this chapter the thermal flow in a hydrostatic system is studied, taking into account heat transfer by the various parts: bearing, supply pump, compensating element, and so on, showing how cooling can occur especially in the supply pipelines and in the reservoir, provided that they are suitably dimensioned.
12.2 12.2.1
TEMPERATURES IN THE BEARING Temperatures in the film
As already seen, with the assumption of adiabatic flow, the elementary relationship 5.7 holds good, but a further analysis involves the introduction of the energy equation 4.37 in the mathematical model, as happens in ref. 5.14 in the case of the circular pad, assuming known temperatures on the facing surfaces (ref. 5.15). In this chapter the thermal flow in another elementary pad is studied: the infinitely long hydrostatic pad in Fig. 11.1, of width B, recess width b and of which a finite part of length L is considered. The mathematical model is quite simple (ref. 12.1),if the variation of viscosity with temperature is disregarded, a s in this case. On the other hand, heat transfer between lubricant, pad and ambient is not disregarded.
448
HYDROSTATIC LUBRICATION
Assuming that heat in the lubricant is transferred only by conduction, the energy equation 4.39 becomes (12.1)
where Al is the thermal conductivity coefficient of the lubricant. If Eqns 4.12 and 4.13, suitably simplified, are introduced into Eqn 12.1, and the following boundary conditions are assumed: T = T , for y=O,
T = T 2 for y = h
we obtain
Then, if Tl=T2=Toand Td=T-T0,and taking into account that in the pad
with y'=y I h, in dimensionless form, Eqn 12.2 becomes
(12.3)
and k is the speed parameter. Figure 12.1 shows the diagrams of T i a n d of T i a = y ' ( l -3 ~ ' + 4 ~ ' 2 - 2 ~ ',3 ) Tja=y'(l -y')
- b-
-a-
0
.2
.4
T i , T+T&
.6
.8
0
.2 T i ThqT;d I
- c-
.4
.2
0
Ti
I
,Ti&
Fig. 12.1 Temperatures T'in the clearance of a pad bearing, for certain values of speed parameter k.
THERMAL FLOW
449
for k=1/3, 1, 3. As far as the intermediate case is concerned, it should be noted that, even if Hp=Hf for k = l , Ths is always smaller than Ti8 because of the different patterns of u and w ,the former being linear, the latter parabolic. This still holds good for k=1/3 and obviously for k=3, too. The diagrams in Fig. 12.1 partly correspond to those in Fig. 5.10, obtained by numerical analysis for the circular pad, without the simplifying assumptions of Eqn 4.39.
As may be seen in Fig. 5.10, which also takes into account the variation of viscosity with temperature, the average temperature of the lubricant increases (virtually linear) with the radius of the circular pad. Consequently the average temperature of the film may be assumed to increase with z while this cannot be established from Eqn 12.3. As a h r t h e r approximation the temperature at the film inlet, where w=O (see section 4.8 for what concerns the "inlet length), is assumed to be equal to TFs.
12.2.2
Temperatures at the film outlet
Naturally the temperature variation in the lubricant is related to the heat energy transferred &om i t to the pad and then to the ambient. The phenomenon can be described briefly as follows.
Ht is the total power dissipated because of friction from viscous drag in the film (it is assumed to be equal to zero in the recess), H , and H s are the heat energies transferred between the lubricant and the pad in the recess and in the film, respectively; T, and T, are the temperatures a t the recess inlet and at the film outlet, respectively (Fig. 12.2). Then (12.4) On the other hand, using Ti to denote the temperature at the film inlet and Tr and Fs t o indicate the average temperatures in the recess and in the film, respectively, and putting
- T,. + Ti T,.=- 2
1
Ts=-
Ti + T, 2
(12.5)
we have (12.6) where
450
HYDROSTATIC LUBRlCATlON
Ti3
h
S
. Te
h
----_j
B
~
~
.
Fig. 12.2 Temperatures in a pad bearing.
Rir and Ri, are the thermal resistances (in series) relevant to the lubricant, the pad, etc. in the recess and in the film.
In Eqns 12.6 coefficient 2 is due to heat transferred to the two pad elements (in parallel), assumed to be identical; indeed, the recess is also carved in the upper element (dashed line in Fig. 12.2);anyway, this is a safe condition. If we substitute Eqns. 12.5 in Eqns 12.6 and the latter in Eqns 12.4, this yields an expression of T , containing the unknown temperature Ti.Once Ti is determined, simply considering that it is the temperature a t the recess exit and, a t the same time, that a t the film inlet, and putting
we obtain
(12.8)
THERMAL FlO W
45 1
where AT is given by Eqn 5.7. Equations 12.5 might yield Te
T, = Te is reached, Eqn 12.8 becomes T, =
[(~+GR)R,+(~-G~)R,]T~+(~+G~)GR,R,AT (l+G~)(l+GRs)R,+(l-G~)(l-GRr)Rs
(12.9)
and T , is then the maximum temperature reachable by the lubricant: a t that temperature all the heat produced in i t is transferred. Equation 12.8 and especially Eqn 12.9 give useful informations on the thermal flow in the pad. Resistances Ri, and Ri, are, respectively,
(12.10)
R1, and R1, have been obtained from the formulae of forced convection with reference to the laminar flow in the pipelines (Appendix 21. In the expression of Rzs, if (B-b)/2
& 2 9
The above-mentioned coefficients can reasonably take on the following values
A1 = 0.15 J/ms°C (oil), ac = 3+150 J/m2s°C,
& = 45 J/ms°C (steel), aj = 4*7 J/m2soC
(12.11)
)
therefore, overall unitary conductance is a=7+157 J/m2soC. EXAMPLE 12.1 11 The pad shown i n Fig. 12.2 has the following dimensions: L=0.2 m, B=O.l m, b10.08 m, h,=0.0175 m, h,=0.0025 m. The recess is also carved i n the upper element.
452
HYDROSTATIC LUBRICATION
The bearing operating conditions are W=31180 N, U=8.66 m l s , and the lubricant properties are: p=0.025 Ns I m2,p 8 9 7 Kg I m3, c=1930 JIKg "y: (at 35 "c). According to what has been specified in section 11.6.3.1.1, we put bAPt equal to 0.8, thus bOpt=0.08m and putting the reference value ho=104 m, the second of Eqns 11.73yields k=0.144 which substituted in Eqn 11.74 yields hhpt=0.5,thus hOpt=0.5.10-4m. Moreover p,=1.73.106 Nlm2, Q=28.9.10-6 m3/s, Hp=50 NmIs, F=17.3 N, Hf=150 NmIs, Ht=200 N m / s , HflHp=3, K=1.87.109 N l m , f=5.5.10-4. Then, from Eqn 5.7, AT=4"C. These values are given in Table 12.1, together with those obtained in the same conditions but for other values of U.
-
'able 2.1 b' U
0.8
-
F N
Hf W
H, W
0.962
1.925
1.852
51.85
1.6 2.887 5
3.3 5.774 10
5.5
55.1
16.6
66.6 100
m/S
HP
W
-
1/27
1.038
0.617
1l9 113 1
1.112 1.334
1.069 1.852 3.208
4.002
5.5
10.01 28.01
9.623 16.67
24.91
1/27
0.958
0.296
2.669
26.69
0.514
32.03 48.04
1l9 113 1
1.026
8.006 24.02
1.23 1 1.847
0.889 1.541
72.06 216.2
96.08 24.02
3.694 9.235
2.669 4.623
648.5 24.96 --
672.5
3 9 27
50 150
15 25.98
30 51.96
450 1350
1400
0.924
0.890
1.6
1.601
2.887 5
2.773 4.804 8.32 14.41
25.98
.lo4
2.001
17.32
8.66 15
f
Oc
3 9 27
8.66
50
-
0.962
0.95
AT
Hf%
24.02
200 500
25.86 8.006 -
Figure 12.3 shows, as straight continuous lines, the values of T , obtained from Eqn 12.8, as a function of T , and for various values of a, for HflHp=1/3and AT=1.33C (Fig. 12.3.a), for H f / H p = land AT=2"y:(Fig. 12.3.b), and for HflHp=3and AT=4oC (Fig. 12.3.c). In all three cases T,.=35%', that is the temperature at the pad inlet is assumed to be constant. It should be noted that, for H f l H p = l for example, the corresponding adiabatic temperature rise AT=2oC is reduced, for ambient temperature Ta=300C and for global conductance a=5 J/mzs°C, to AT=1.6g0C, that is only 16% less, while for Ta=0T and -150 Jlm%"y:, it is reduced to AT=O.O46Y!, that is 98% less. Figure 12.4 shows the values of T, obtained from Eqn 12.9, that is where T,=T,, as a function of a and for various values of Ta, for H flHp=lI3 and AT=1.33Y!, for
453
THERMAL FL0W
-a39
-b-
-c -
39 39 I Hf/Hp = 1 , Tr =35 OC
38 Te
37
I
L
38
b'= 0.8 , AT =133OC-
38 6 = 0 . 8 , A T = 2OC __
Te
b'=0.95 , ~ l T = 1 , 8 5 ~ C_- _
OC 37
t
36
@
35
20
30
36
150
34 10
OC 37
35
// / 0
Te
34 0
10
20
30
0
10
20
Ta
Ta
Ta
OC
OC
OC
30
Fig. 12.3 Temperature at bearing exit T, versus ambient temperature T ,for certain values of global T,: temperature at recess entry; A% adiabatic temperaturerise; conductivity a and power ratio HflHp. b':recess width.
HfIHp=l and AT=2.001 "c, and for HflHp=3and AT=4.002"c. As may be seen in the diagrams, T , can reach excessive values so that some action must be taken on the system in order to reduce it, that is to reduce T,. The plotted values of Te are to be considered as examples, since they have been determined assuming p=constant. Such a relation is true only for small temperature increments, this being, on the other hand, one of the objects of the present investigation. Figure 12.5 shows the values of T, obtained from Eqn 12.8 as a function of Tw for a=50 Jlm2s "c, for various values of H f l H p , for controlled temperatures Tr=25OC (Fig. 12.5.a), Tr=35"C (Fig, 12.5.b), and Tr=45"c (Fig. 12.5.~).The straight line relevant to HfIHp=lI 9 has not been plotted to avoid overlapping. The values of the adiabatic rise in temperature AT, corresponding to the various values of HfIH,, are given in Table 12.1. It should be noted that, compared to the case of HflHp=land AT=2, in the case of HfIHp=l127 we have a AT which is almost a half, while in the case HflHp=27 we have a AT which is 14 times higher.
454
HYDROSTATIC LUBRICATION
-a-
-b
100
-c
-
100
100.
Te
Te
-
0.8 ,dT=1,33OC_ Te
0.95,dT=1.230C--
OC
80
OC 80
OC 80.
60
60
60
40
40
40.
20
20
0
40
80 J
120,
160
0
40
80
m-2s-bC-l
J
120
,160
0
40
m-2s-l~-l
80
120,
160
J m-2S-’oc-1
Fig. 12.4 Temperature at bearing exit Te versus global conductivity a,for certain values of ambient temperature To and power ratio Hf/Hp. T,.:temperature at recess entry; A T adiabatic temperature rise; b’xecess width.
- b-
-a-
6
7 65-
0
-1
Te
oc 55
60 -
.
b’= 0.8 b‘=095
_____
_ _ _ - - - - k; - 3 __---__---
50 -
-_40 -
5
n1
20
0
20
10
Ta OC
0
10
20 Ta OC
30
0
10
20
30
40
Ta OC
Fig. 12.5 Temperature at bearing exit T , versus ambient temperature Ta, for certain values of power ratio Hf/Hp T,.: temperature entering recess; a: global conductivity: b’:recess width.
455
THERMAL FLOW
Finally Fig. 12.6 shows the values of T,obtained from Eqn 12.9 as a function of T,, for a=100J/m%oC, for various values of HfIH,. It must be noted that in spite of the high value of a,for Hfl Hp-3, Te would reach excessive values so that the system must be acted upon in order to reduce it.
0
10
20
40
30
50
Ta
OC
Fig. 12.6 Temperature at bearing exit T, versus ambient temperature To, for certain values of power ratio Hf/Hp. T,: temperature at recess entry; CE global conductivity;&recess width.
2) ZL for the pad being considered, b10.095 m, put bhpt=0.95, thus b=0.095 m. Again assuming that ho=10-4m, with the same operations we get k=0.0677 and h &=0.26, thus hOpt=0.26.10-4m. Moreover p,=1.6.106 Nlm2, &=15.10-6m3/s, Hp=24 N m / s , F=8.32 N, H ~ 7 2 . 1N m l s , Ht=96.1 N m l s , HflHp=3,K=3.59.109 N l m and f=2.67.104. Compared to the previous case, pr is a little lower, K is about double and the other quantities are about half. Viscous friction should not be disregarded now. Finally, the adiabatic rise in temperature is AT=3.69"C, 8% lower than in the previous case. The above-mentioned values are given in Table 12.1, together with the others obtained for different values of H f l H p that is of U. Figures 12.3 to 12.6 show, as dashed lines, the values of Te relevant to the pad with b'=0.95, obtained from Eqns 12.8 and 12.9 (putting h,=hr in R2&,in the same conditions defined for the pad with b'=0.8. Figure 12.3, in particular, shows that the pad with b'=0.95 is convenient, especially for high values of a and for increasing H f IH, while it is not convenient for increasing T,. Even in the case of Fig. 12.4, the pad with b'=0.95 is still conveniently used, especially for low values of a now, and
456
HYDROSTATIC LUBRICATION
again for increasing HfIH,,. For example, for HflHp=l,e 3 0 Jlm%"C, Ta=20"C,in steady thermal conditions Te=61.60C which is still a n acceptable value, while for b'=0.8, Te=97.6"c which is unacceptable. Figure 12.5, and Fig. 12.6 even more, confirm the convenience of the pad with b'=0.95 for high values of Hf IH,. The fact that Ht is at a minimum does not necessarily mean that AT is at a minimum as well, i f Q is small. See, for example, the case where U=8.66 m l s and Ht is at a minimum for HflHp=3. Then it may be convenient to increase Q, not by increasing h which would mean a decrease of K but by increasing b. This is a further proof of the usefulness of pads with wide recesses. The results presented above can be extended approximately to real pads. For example, in the case of rectangular pads, in Eqn 5.7 H f increases because of friction in the frontal sills, but H p also increases because of lubricant losses from them. From the results presented above, it may be deduced that, in general, the cooling of the lubricant in any bearing is moderate, about 25%, except for very high values of a,that is of R3,. and R3,, often not achievable in common practice. Some further modest benefit could be achieved by reducing h, (for example to hr=50h), that is R1,,and h,, that is R p , and then h, and thus R3, are also reduced.
12.3
SUPPLY PIPELINE
As has been pointed out, it is convenient to control the thermal performance of the bearing by acting on temperature T,. at the recess inlet. This can be achieved by designing a suitable supply pipeline, i.e. by also using it as a heat dissipating element. In this case the mean temperature difference AT between the lubricant in the pipeline and the ambient is (ref. 12.2):
m=
AT
(12.12)
where AT is still given by Eqn 5.7 and Ta is the ambient temperature. On the other hand, the heat power transferred from the lubricant to the ambient is (12.13) where R is the total thermal resistance, that is (Appendix 2) (12.14)
457
THERMAL FLOW
where 1, D , d are the tube length and the outer and inner diameters. A1, can take on the values given in Eqns 12.11 while now cl, = 8+320 J/m2s°C.
&, and Glj
From Eqns 12.12,12.13 and 12.14, solving for 1 (12.15)
EXAMPLE 12.2 1) Consider the pad in Example 12.1 with b'=0.8. The temperature at the pad inlet is required to remain equal to Tr=350C in the operating conditions HflHp=1/3,1,3 with the corresponding adiabatic temperature increments AT=1.33, 2, 4. Figure 12.7.a shows the values of 1 as a function of T,, and for various values of AT and of a=cl,+ajfor D=O.Ol m, d=0.007 m, thus (D-d)l2=s=O.o015m. For example, for AT=2"c (HfIHp=l),Ta=280Canda=80 JlmzsoC, ll12.36 m, while for Ta=l4oCand e l 6 0 JlmzsoC, 1=3.57 m. For AT=4cY:(HflHp=3),Ta=14cY:and -160 JlrnzsoC, 1=6.84 m. For the given values, Rll=0.584 m s V l J; R21=0.00126 m s v l J; R31=3.18 to 0.0998 ms@lJ, for a=10 to 320 Jlm2sV. Therefore the second term of the third member of Eqn 12.14 is quite small compared to the first and also on an average as compared to -b-
-a15
15
1
I m
m
10
10
T, = 35OC 5 5
0
0 0
10
20
Ta
30
0
10
20
Ta OC
30
A Tf /=H4 O H p= C31- - -
Fig. 12.7 Supply line length I versus ambient temperature T,, for certain values of power ratio H#Hp. of adiabatic temperature rise AT for a fixed value of temperature at recess entry T, and for: a- certain values of global conductivity a and a fixed value of outer diameter D (and of inner diameter d) of the line; b- for certain values of D (and d) and a fixed value of a.
458
HYDROSTATIC LUBRICATlON
the third. Zt certainly becomes negligible for tubes with high thermal conductivity, for example, for copper tubes for which &=300 J l m s V . Therefore the influence of d is also often negligible and the results presented are also valid for different values of d, for lower values (that is for higher values of thickness s) and even more so for higher values (that is for lower values of s). Figure 12.7.6 shows the values of 1 as a function of T, and for various values of AT, D and d, for a=100 J I m 2 s T . The values of D and d are: D=0.002, 0.004, 0.008, 0.016 m; d=O.001, 0.0025, 0.0055, 0.011 m; s=0.0005,0.00075,0.00125,0.0025 m; the smaller values correspond to capillary tubes. For example, for AT=2 "c (HflH,,=l), T=14"c and D=0.016 m, still 1=3.57 m, while for D=0.032 m and d=0.028 m (not plotted) 1=3.11 m. As regards d and s the above considerations still hold good. 2) Pad with b'=0.95. Since AT and especially Q are lower than the values relevant to the pad with b'10.8, length 1 is also lower. For example, for AT=1.85"C (HfIHp=l), Tr=35"c, T,=14"c0, for a d 6 0 JJm2s°C, D=O.Ol m, d=0.007 m, the length is 1=1.72 m; for -100 Jlm2s"c, D=0.016m, d=0.011 m, still 1=1.72 m; for -100 Jlm2s"C, D=0.032 m, d=0.028 m, 1=1.50 m.
12.4
COMPENSATING ELEMENTS
The heating of the lubricant occurs in the bearing and in the regulation devices a s well, that is fixed ones such as capillaries and orifices and variable ones such as flow control valves, etc. The rise in temperature is still given by Eqn 5.7, putting H+Hp=O and replacing p s with the pressure drop Apc in the regulator, that is
APC ATc = -
(12.16)
P C
The tube length required to keep the inlet temperature Tc a t a given value, is then again given by Eqn 12.15, replacing AT with ATc and T , with Tc.
EXAMPLE 12.3 1) Pad with b'=0.8 in example 12.1. Zf it is compensated by a capillary with p=0.5, Ap,=1.73.106 Nlm2, thus ATc=l"c; in this case, for example for Tc=35"c, Ta=14"C, @lo0 J/m%"C, D=0.016 m, d=0.011 m, the length is 1=1.83 m. 2) Pad with b'=0.95. In this case for 8=0.5,Apc=1.6.106 Nlm2, thus ATC=0.924'C and in the same conditions as in the previous example, the length is 1=0.879 m.
12.5
PUMP
The heating of the lubri'cant also occurs in the pump. If q is its efficiency, the temperature rise of the lubricant may be put in the form
THERMAL FLOW
1-71PiT AT,= -71 P C
459
(12.17)
where pz=Apc+pr. The tube length required to keep temperature T , entering the pump at a given value, is then again given by Eqn 12.15, replacing AT with AT, and T, with TT
EXAMPLE 12.4 1) Pad with b'=0.8 in example 12.1. It is supplied by a pump with an efficiency (for example) q=0.8. If the pad is supplied directly, p,=p,=1.73.106 N l m z and ATZ=0.25"c;in this case, for T,=35"c, Ta=14"c,-100 Jlm%"c, D=0.016 m, d=O.011 m, the length is 1=0.464 m. If the pad is compensated, for p=0.5, p,=3.46.106 Nlm2, AT,=0.5 "c and 1=0.923 m. 2) Pad with b'=0.95. It is supplied by a pump with an efficiency q=0.8. If the pad is supplied directly, now p,=16,105 Nlm2 and AT,=0.231 "C;in this case, in the same conditions as in the previous example, the length is 1=0.223 m. I f the pad is compensated, for p=O.S, p,=3.2.106 Nlm2, ATz=0.462"cand 1=0.444 m.
12.6
COOLING PIPELINES
Cooling can occur in the supply tubes between the pump and the bearing, as already seen in section 12.3, as well as in the return tubes between the bearing and the pump. If the bearing is directly supplied, and cooling occurs in the supply tube, its length is still given by Eqn 12.15, where AT is substituted by the total temperature rise
ATT = AT,
+ AT
(12.18)
If cooling occurs in the return tube, 1 is still given by Eqn 12.15, where again AT is substituted by ATT in Eqn 12.18, and T, by (T,-AT,). Consequently I is a little larger than in the previous case because the mean temperature of the lubricant in the tube is a little lower. If the bearing is compensated and cooling occurs in the supply tube, 1 is still given by Eqn 12.15, where AT is substituted by the total temperature rise ATT = ATz -I- ATc -I- AT
(12.19)
and T, by (T,-AT,).
If cooling occurs in the return tube, in Eqn 12.15 AT is substituted by ATT in Eqn 12.19, and T, by (Tr-ATc-AT,).
460
HYDROSTATIC LUBRICATION
In common practice, temperature T , at the inlet of the pump is often taken as a reference value. In this case, with compensation and cooling in the supply tube
(12.20) where T,=T, referred to T, is T , = T , + AT. If, on the other hand, cooling occurs in the return tube
(12.21) where T , referred to T, is now
T, = T, - ATc - AT,
(12.22)
In the case of direct supply, the previous expression without ATc still holds good. If the return tube is used as the cooler, larger diameters and small thicknesses may be selected while on the contrary they must be avoided if cooling occurs in the supply tube because tube elasticity could have a negative influence on the dynamic stability of the bearing. If cooling occurs in the return tube, however, a recirculation pump is almost always necessary. Therefore there is a further temperature rise ATnp, still expressed by Eqn 12.17,but generally quite small since pressure head ATnp,even with filters, is small. Further temperature increments may occur in other elements of the circuit, in particular in the filters, for which ATp can be calculated with Eqn 12.16,substituting Apc with the pressure drop App in the filter, and also in the relief valve, for which AT,, is again given by Eqn 12.16,but only for the lubricant escaping from it. In the case of compensation and cooling in the supply tube, in Eqn 12.20,AT,, and ATq are added to ATT ; if cooling occurs in the return tube, in Eqn 12.21 ATnpis added to AT, as well as ATv and ATp while i t must be subtracted in Eqn 12.22. In the case of direct supply what has been said above still holds good but the expressions lack the term ATc.
EXAMPLE 12.5 Compensated supply with cooling in the supply tube. 1) Pad with b'=0.8; H f I H p = l , AT=2OC (see example 12.1). For 8=0.5, dpC=1.73.1O6 N l m 2 and ATc=l "c; for App=0.3.106Nlm2 (roughly), ATp=0.173"c;the temperature rise in the valve is assumed (very roughly) to be AT,,=0.05°C. p,=1.73.106 N l m z ,
THERMAL FLOW
461
therefore, for q=0.8, ATn=0.544"c. The total temperature rise is then d T ~ 3 . 7 7 " C . Putting Tr=35"c, Tn=370C.For Ta=14"c,e l 0 0 J/m2s0C,D=0.008 m, d=0.0055 m, Eqn 12.20 yields 1=8.49 m. 2) Pad with b'=0.95; H f I H p = l ,AT=1.85"C. Ap,=1.6,106 Nlm2 and ATC=O.924"C, AT,=0.173°C, ATv=0.050C;pr=16.106Nlm2 and ATn=0.505V; then A T ~ 3 . 5 " cand Tn=36.850C. For the former values of T,, a, D and d, L=4.11 m. As seen above, the length is almost unchanged even for lower values of d. EXAMPLE 12.6 Cooling in the return tube 1) Pad with b'=0.8. Compared to the previous case we still have AT=BOC, AT,=l "C, AT,=O.l 73"c, ATv=0.O5"C, but ATz=O.5OC and including the recirculation pump with q=O.8 and discharge pressure pnP=0.3.1O6N / m 2 , ATnp=0.0433"C. Then AT~3.77"cand Tp33.6V. For the former values of T,, 4 whereas D=0.032 m and d=0.028 m, Eqn 12.21 yields 1=6.06m. 2) Pad with b'=0.95. Now ATn=0.4620C, thus A T ~ 3 . 5 " cand Tn=33.50C. For the former values of the other quantities, 1=2.93 m. It must be noted that heat losses may occur in the bearing as well as in the other elements of the circuit: in the pumps (for which efficiency losses also depend on lubricant losses), in the compensator, especially in the capillary tubes, in the filters, especially at high pressures, in the valves, etc. The lubricant may be cooled further in the reservoir if this is reasonably sized (Appendix 2) and finned (ref. 12.2).Moreover it should also be noted that the set of supply tubes of a hydrostatic system can work as the tube bank of an air-oil heat exchanger. Assuming that, on average, 50% of the cooling rate pertains to all the other elements of the system, the length of the cooling tube would be reduced by more than 50%' approximately.
12.7
SELF-COOLING CAPILLARY TUBE
Consider a capillary tube through which a pressure drop Ap, occurs. Its length is easily determined from Eqn 4.66:
(12.23)
On the other hand, the pressure drop produces heat, therefore there is a rise in temperature AT, of the lubricant flowing through it, given by Eqn 12.16.Heat, however, can be dissipated by a tube of the following length: (12.24)
462
HYDROSTATIC LUBRlCAT/ON
where Tc is the temperature a t the inlet of the tube. The tube may even be the capillary itself. The diameters of the capillary must be related, in accordance with the typical equation of thick pressurized tubes:
(12.25)
where p is the maximum nominal pressure in the tube, o the material limiting stress and m the longitudinal deformation modulus. Actually, for construction requirements, Eqn 12.25 is already satisfied for small diameter tubes. Solving the system of three equations yields the dimensions of the self-cooling capillary. As may also be seen in Fig. 12.8, i t will be much longer than the length given by Eqn 12.23 for very small values of d. Anyway, d must be large enough to avoid clogging (d>0.5.10-3m).
EXAMPLE 12.7 The capillary is made of drawn steel with the following characteristics: 0=3.5.107 Nlm2, so that elastic deformations are negligible, m=10/3. 1) Pad with b'=0.8 in example 12.1; B=0.5, thus Apc=1.73.106 N l m 2 and ATc=l "C. The operating pressure is at a maximum at the inlet of the capillary tube and it is p ~ = 3 . 4 6 . 1 0 6N l m z ; for the maximum nominal pressure in Eqn 12.25 the value assumed is p=4pM=13.9'106N l m2, also considering possible dynamic overloading; Tc=35"c. Figure 12.8.a shows length 1, capillary tube inner diameter d and outer diameter D, as functions of ambient temperature T,, for a given value of overall conductance a=lOO Jlm2s"C. It can be seen, for example, that for T,=lO"C, 1=2.6 m, d=0.0026 m and D=0.0042 m; while for Ta=2O"C,1=4.1 m, d=0.0029 m and D=0.0047 m. In any case l l d is always much greater than 100, as usual for capillaries. In Fig. 12.8.b 1, d and D are plotted as functions of a for Ta=15V. It may be seen, for example, that for a=60 Jlm%"C, 1=4.2 m, d=0.0029 m and D=0.0047 m; while for a=120 J/mZs"C, 1=2.9 m, d=0.0027 m and D=0.0043 m. Capillaries may be wound in large pitch spirals. 2) Pad with b'=0.95; /3=0.5,thus Apc=1.6.106 N l m z and ATc=0.9240C;p=12.8.106 N l m 2 is assumed; TC=35"c.In Fig. 12.8 1, d, and D are plotted as functions of Ta (a) as functions of a (b). Length and diameters are smaller than those in the previous case. The results obtained apply even more to copper capillary tubes; however, it is advisable to increase the outer diameter by 25% with no detriment to cooling which
463
THERMAL FLOW
-a-
-b-
~~
350C , T,
T,
=
b‘
= 0.8
0.0075
= I 5 OC
D, d
___
m 0.005
---__ 3.0025
- -_ _
-- - - _
1
3
0
10
30
20
0
40
80
120
Ta OC
J
m-* s-’
a
160
oc-1
Fig. 12.8 Length I, outer diameter D and inner diameter d of a self-cooling capillary tube of a bearing with a recess width b‘,versus: a- ambient temperature T,, for a fixed value of temperature entering tube T,, and global conductivity a; b- a, for fixed values of T, and T,.
is still higher than in steel capillaries because of the much higher conductivity of
copper as compared to steel. In general, the design of a self-cooling capillary, without solving the system of three equations, may be carried out as follows: assign d (for instance, with the aid of Fig. 12.8); from Eqn 12.23 determine I and introduce it into Eqn 12.24 without the generally negligible term (1/2)(1/&)ln(D/d); thus determine D; 22D/d21.5 should be satisfied, instead of Eqn 12.25, otherwise: the procedure is repeated until the result is achieved.
12.8
VISCOSITY AND TEMPERATURE
Up to now viscosity has been assumed to be constant as temperature varies, especially in the pad film. This can be considered to be true if heat dissipation in the various elements of the hydrostatic system keeps the increment in the lubricant temperature low, especially if the supply and return tubes and the reservoir are designed as coolers. Otherwise the new value of viscosity in the film must be evaluated according to the mean value of temperature, and calculations must be repeated until the difference between two consecutive values of viscosity is sufficiently small. In the case of compensated supply the actual value of viscosity in the compensating
464
HYDROSTATIC LUBRICATION
elements must be taken into account. When lubricant temperature is under control, the hydrostatic system operates properly and thermal deformations of the bearings, detrimental to the smooth running of the machine and the accuracy of products in the case of a machine tool, are avoided.
REFERENCES
12.1 Bird R. B., Stewart W. E., Lightfoot E. N.; Transport Phenomena; Wiley and Sons,1960;780 pp. 12.2 Kreith F.; Principles of Heat Transfer; Intext Educational Publisher, N.Y., 1973;650 pp.
Chapter
13
EXPERIMENTAL TESTS
13.1
INTRODUCTION
An important aspect of the study of hydrostatic bearings is testing them, consisting in the measurement of a considerable number of input and output variables of the hydrostatic system. This, together with the large number of types of bearings, has led to the assembly of a considerable number of test rigs, only a few of which are equipped for the testing of more than one type of bearing. In this chapter, after some brief notes on the most important input and output variables and testing procedures, a number of test rigs are described, chosen among a host of equally good rigs, and details are given of the tests performed on a few particular types of bearing.
13.2
HYDROSTATIC SYSTEMS; INPUT AND OUTPUT VARIABLES
There are many variables affecting a tribological system and a hydrostatic system in particular, a s shown in Fig. 13.1. The behaviour of the mechanical system made up of a pad and a slide separated by lubricant and placed in the atmosphere, depends on numerous input variables and i t is also characterized by the output variables; the validity of the tests largely depends on the correct experimental measurement of such variables. The following input variables are examined: the type of motion, which is sliding in hydrostatic lubrication; speed, which can be linear o r angular and the control and measurement of which is carried out with electronic tachometers, which are especially necessary for very low or very high speeds;
466
HYDROSTATIC LUBRICATlON
load. The devices for measuring loads or torques can be quite different: leverisms, springs, hydraulic jacks and electromagnetic systems, the latter being especially suitable for dynamic loads. The measurement of the load is now generally obtained by means of strain gauge bridges; lubricant flow rate measured in volume by variable area and turbine flow-meters; supply pressure, measured by means of manometers and capacitive and piezoresistive transducers; lubricant supply temperature measured by means of common and infrared thermometers, o r thermocouples allowing continuous measurement; ambient temperature; the physical and chemical characteristics of the lubricant, such a s viscosity, density, specific heat, etc., generally measured with standard test devices and methods.
I
Input variables
1
output variables
T RIB 0-SYST EM
Typeof motion
m
4
I
LoadFN
f T i G m - l
I
r
Velocity v
I
Type of lubrication
1
I
recess and in the film
t
Supply pressure of lubricant
D
{Filmthickness)
i 1234-
/
1
First body Second body Third body: lubricant Atmosphere
Base lubricant
Characteristics of lubricant Additives Fig. 13.1 Hydrostatic system.
Friction
1
L Thermal increase of lubricant
I Modified charatcteristics of lubricant
EXPERIMENTAL TESTS
467
The following output variables are examined: type of lubrication. It is known that often, together with hydrostatic lubrication, hydrodynamic lubrication takes place and can become prevalent in radial bearings so that hybrid lubrication occurs; recess and film pressures. Pressure in the recess equals supply pressure in the case of direct supply while it is lower, sometimes much lower, in the case of compensated supply; film thickness, measured by contact micrometers and displacement transducers, the latter being eminently suitable for dynamic tests; viscous friction, measured by strain gauge load cells or torque meters; these also are of the inductive type; rise in temperature of the lubricant, due to viscous friction, which can cause changes in viscosity and in the other characteristics of the lubricant.
For the validity of the experimental results, it is necessary to check, before testing, that the macro and micro geometry of the experimental model complies with the design requirements (in tolerance), such as the following: dimensions of the bearing; dimensions and location of the recess; planarity and parallelism of the sliding surfaces; cylindricity of the rotating elements; parallelism of the axes of the rotating elements; orthogonality of axes, of planes and of axes and planes; dimensions and location of the compensating elements. All this is due to the fact that, for example, flow rate through a film varies with the third power of its thickness and that flow rate through a capillary tube varies with the fourth power of its diameter. If some of the requirements are not complied with, the experimental test can still be carried out but the actual values of the geometrical variables mentioned must be introduced in the calculations. Considering the complexity of the tribo-system presented in Fig. 13.1, it is advisable, in testing, to collect input and output data systematically. The reader is referred to ref. 13.1. In general, basic tests are carried out, that is on elementary systems (pin-disc, slide-way, etc.) and the results may be insufficient for a n adequate study of the phenomenon. So the following categories of tests may be carried out, possibly in varying combinations: 1-field tests, that is tests performed on the machine 2- stand tests on the machine 3- stand tests only on that part of the machine which the tribo-system is part of 4-stand tests on that part a t a reduced scale
468
HYDROSTATIC LUBRICATION
5-tests on the tribo-system taken off the part of machine containing it 6-basic tests, already mentioned.
13.3
EXPERIMENTAL RIGS
13.3.1
Electric analog field plotter
The apparatus shown in Fig. 13.2.a is not a n experimental bearing but a device for the determination of the characteristics of the bearing through the use of an electric analog field plotter which was being used some decades ago in the study of hydrostatic bearings of various shapes, giving satisfactory results. It is based on the fact that Reynolds equation, which allows us to calculate pressure distribution in the bearing clearance, is analogous to the electric field equation which makes it possible to calculate voltage distribution in a conducting sheet with the shape of the clearance. Figure 13.2.a contains an outline of the circuit of the electric analog field plotter (ref. 5.34). -b-
-a-
SILVER
E l.ECTRODES-
POTENTIOMETER
-
-A 0 4
6 V DC+
8
80
7
70
6
60
5
50 n
DETECTOR
I
' a
r.
4
40
3
30
2
20
I
1
10
10 1 0 30--100
,
0
c
11OV AC
]1(,6VAC
0
0
0 0.2 0.4 0.6 0.8 1.0 1 ' 1'2
Fig. 13.2 Electric analogy: a- Circuit of the electric field plotter; b- Performance factors of a circular thrust bearing
Figure 13.2.b gives the values of the pressure in the recess, of the flow rate and the pumping power a s functions 'of the radius, as obtained with the plotter for a circular bearing (ref. 13.2). As can be seen, there is a remarkable agreement between these and the theoretical results (solid lines).
EXPERIMENTAL TESTS
13.3.2
469
Axial bearings
i) Device simulating a hydrostatic p a d . Of the numerous types of experimental thrust bearings, a circular and particularly simple one (ref. 13.3), shown in Fig. 13.3, is examined. Pad 1, axially free to move in seat 2, is subjected t o a load produced by spring 3. The load can be varied by varying the spring tension by means of screw 4. The load is balanced by the pressure in the recess and in the film, the thickness of which is measured with micrometer 5 . The friction between pad 1 and seat 2 is lowered by the lubricant between them. Though lacking precision mainly because of friction, the device is quite useful for investigating circular pads with ratio r21r2 equal to the experimental one.
Fig. 13.3 Device simulating a hydrostatic pad.
With devices similar to that in Fig. 13.3, though less precise, differently shaped pads (rectangular, etc.) can be tested. If pads are supported radially by aerostatic bearings, there is almost no friction, thus the results are more precise. ii) Turn-table simulating a sliding table. Figure 13.4.a shows a (tilted) circular pad tested (ref. 13.4) with the apparatus outlined in Fig. 13.4.b. The test rig consists of a turn-table substituting a sliding table; this does not imply there are large errors because the turn-table is much larger (about 1.2 m) than the test pad (about 0.1 m) placed a t its periphery. The turn-table is supported and located by three pairs of opposed hydrostatic thrust pads equally spaced out around the structure. The test pad is attached to the base of a plunger which is located in a cylinder rigidly mounted on the base of the test rig. Free vertical movement of the plunger is assured by two rows of air bearings in the wall of the cylinder. Loads are applied to the test pad by adding weights to the plunger, and the tilt of the test pad with respect to the table is introduced by shims. In Fig. 13.4.c the variation of the dimensionless flow rate Q=Qp/@oh$is plotted as a function of the "dynamic term" S=6pRz(Ua+2V)l(pohg),where U and V are the sliding and the squeeze velocities, respectively, for certain values of the angle of tilt
470
HYDROSTATIC LUBRICATION
-a
-c
-
-
S positive
u_
-b-
SyEbol
a
+
u
0 I A A v v a2 04 a5 OB 065 07 a75 0.8
Fig. 13.4 Test rig with turn-table: a- tilted pad; b- apparatus; c- flow rate versus dynamic term. -
a=ar2/hoand for two different values of the ratio of radii iZ=rl/r2.The load being equal, flow increases linearly with the dynamic term.
iii) Transparent pad. In Fig. 13.5 a rectangular pad, studied in ref. 13.5 and tested in ref. 13.6, is presented. It is characterized by the relationship bIB=lIL and it has non-rounded corners, that is ri and r,, are equal to zero (Fig. 5.25). The pad is made of transparent material (Plexiglas) so that the fluid streamlines in the film can be visualized by introducing coloured liquid (ethylene glycol). Obviously, with Plexiglas, tests have been carried out with relatively low loads. In Fig. 13.6 lubricant velocities a t certain points [ of the diagonals are plotted. In Fig. 13.5.b the glycol fluid line is indicated with an arrow a t point in Fig. 13.6. The line, almost tangent to velocity Rz in the same diagram, is continuous, with no breaks; this has also occurred a t the other points of the diagonals.
c2
iv) Flexible pads. In ref. 2.15 an all-metallic flexible hydrostatic thrust bearing is investigated both theoretically and experimentally. Figure 13.7.a shows the pad being tested while Fig. 13.7.b contains an outline of the experimental apparatus. In this rig the flexible bearing being tested (a) is fitted into the adaptor (b) which is in
EXPERIMENTAL TESTS
471
Fig. 13.5 Visualization of streamlines: a- Plexiglas pad; b- ethylene glycol fluid lines.
its turn mounted on a heavy, rigid base plate (c). This base plate is attached to a free standing frame (d) and loads are applied to the upper member of the bearing (e), by the cross-head (f) through a ball (g). A lever system connected to the link. (h) enables loads to be applied by means of dead weights. To ensure that the lower surface of the block (e) remains parallel to the outside edge of the bearing, a system of four flexible restraints are fitted between it and the pillars (k). When properly adjusted these flexible elements make the block (e) move parallel to the bearing edge without offering any significant resistance to vertical movement. The lubricant is supplied to the bearing through the flexible pipe (1) and returns to the supply unit through the drain pipes (m).
c)
In Fig. 13.7.c dimensionless pressure in the bearing film F=p/pr, pr being the recess pressure, is plotted as a function of the ratio of radii r=r/r2; ratio r1/r2 is 0.25. The variation of is quite different from the logarithmic variation of the rigid pad. The diagram is obtained by assuming that film thickness a t the outer edge of the bearing is h&h&/(p,r$)=0.0045, with D=Et4/[12(1-v2)],where t is the plate thickness, E the elastic modulus of the material and v its Poisson ratio. In the same diagram the experimental results for t=2.08 mm, E=209000 N/mm2 and v=0.3 are
HYDROSTATIC LUBRICATION
472
Fig. 13.6 Velocity of fluid at certain points of a diagonal.
also given. It should be noted that the effectiveness of flexible bearings decreases as the recess increases.
13.3.3
Radial bearings
i) Test rig for static loads. In Fig. 13.8 a test rig for radial bearings subjected to static loads (ref. 10.9) is shown. A shaft driven by a variable-speed motor, is supported a t both ends by roller bearings and a block of the hydrostatic bearing floating on the middle of the shaft. In both sides of the floating block there are hydrostatic thrust bearings. A static load is applied by pulling up the floating block, using a ring. With such an apparatus it is possible to investigate the occurrence, in the presence of static loads, of forced vibrations caused by the fluctuations in recess pressure due to aeration of the working fluid, as well as self-excited vibrations of larger amplitudes, i.e. oil-whirl. The appearance of recess pressure P,. lower than ambient pressure is considered as the boundary of the allowable operating range. This boundary, for the upper pad of a four-pad capillary compensated bearing, is given in Fig. 13.9, as a function of the eccentricity ratio E=elhoand of the speed parameter or Sommerfeld hybrid number S = p o fP J c f D)2 (where c is the diametral clearance and D is the journal diameter), for certain values of load F. ii) Experimental apparatus for static and dynamic loads. Figure 13.10.a shows a test rig for journal bearings subjected to both static and dynamic loads (ref. 13.7). As
473
EXPERIMENTAL TESTS
-a-
-c -
I4
1
0.3 04
0.5 06 0.7 0.8 0.9 1.0
Dimensionless radius, T
Fig. 13.7 Flexible hydrostatic thrust bearing: a- pad; b- experimental apparatus; c- pressure versus radius.
in the previous case, a shaft (A) driven by a motor (B) through a n elastic joint (C) is supported by two roller bearings mounted in a trunnion assembly (D)and the block (El of the hydrostatic bearing is floating on the middle of the shaft. Eight symmetrically-placed flexible restraints (F) ensure that the bearing bIock moves perpendicularly to the shaft without imposing any significant restraint. Steady loads are applied to the bearing by the screwjack ( G ) through the calibrated spring (H). Oscillatory loads are applied to the bearing by the electromagnetic vibrator (I) which is mounted in a space provided in the base (L) mounted on four flexible supports (M). A strain gauge force transducer (N)measures the dynamic loads applied to the bearing. ' The dynamic tests have been carried out on a four-recess orifice-compensated bearing both with static loads and without them. Figure 13.10.b concerns the first type of tests performed with the bearing a t rest and shows the flexibility, or receptance, fu as a function of the forcing frequencyaf=Qf/SZ of a sinusoidal load (f2 being the undamped frequency of the bearing) for an equal amplitude of oscillation 6, around the centre, for two v a l ~ e sof the supply pressure and with the pressure ratio p=0.5. There is a good correspondence between the experimental and theoretical
474
HYDROSTATIC 1UBRlCATlON
t transducers
Jounal bearing block
'
T~~~~~~~~~~~~
Fig. 13.8 Experimental apparatus for journal bearings.
1.0
&
0
50
s
100
150
Fig. 13.9 Effeg of eccentricity ratio and speed parameter on the appearance of recess pressure (dimensionless P,=P,JP, where P, is supply pressure) lower than ambient pressure.
results up to +0.65, above which they differ because cavitation takes place in the recess. iii) Experimental apparatus for static and dynamic magnetic loads. The test rig (ref. 13.8) shown in Fig. 13.11.a consists of a symmetrical rotor with a flywheel (1) equipped with transformer steel sheets. Compressed air from a blower drives a light turbine wheel (2). Symmetrically, there is an equivalent disc (3) for triggering and sampling the signals measured. The rotor is supported by two identical hydrostatic bearings (4). The shaft is axially positioned by two air bearings (5). External vertical radial forces are applied by magnetic shoes (6). In order to induce basic harmonic sinusoidal forces alone, a static component must be superimposed on the top magnetic shoe. In this way the rotor is lifted up to the concentric position in the bearings. A bottom magnetic shoe is used to apply higher static load downwards.
475
EXPERIMENTAL TESTS
With this magnetic force system it is possible to vary the force frequency independently of shaft rotation frequency. An appropriate set of instruments, such as dynamometer (7) and eddy current probes (8),makes it possible to measure the various variables. -b-
-aC
supply pressure. lb/in2 50 0 66
B
o
n
N
. 0
0
'
04
'
08
Dimensionless forcing trequency,
12
-
a
Fig. 13.10 Experiments on journal bearings: a- Experimental apparatus; b- Bearing response curve.
-a-
b-
Fig. 13.1 1 Experiments on journal bearings: a- Experimental apparatus; b- orbit of the shaft centre.
476
HYDROSTATIC LUBRICATION
Because of the hydrodynamic component, which is almost always also present in hydrostatic bearings, a bearing subjected to both static and dynamic loads moves on an orbit (characteristic of hydrodynamic bearings) around the static equilibrium position. This is clearly shown by both the theoretical (solid line) and the experimental (dashed line) results contained in Fig. 13.11.b for a compensated four-recessbearing with 8=0.5, subjected to a static load and to a sinusoidal dynamic load giving rise to elliptical orbits. There is good agreement between the theoretical and the experimental results. In the diagram E is the static eccentricity, S=pn/p,(c / 0 2 ) the speed parameter, n the angular velocity and fF the dynamic load frequency.
13.3.4
Spherical bearings
Figure 13.12. shows an experimental apparatus (ref. 5.25) for spherical bearings (either the fitted or the clearance type). Pad (11, rigidly connected to the rotating shaft (2), is supported by the central recess bed (31, which is placed on the bed carrier (4). The bed carrier and the base (5) are adjusted by screws. The apparatus is equipped with a set of instruments for the measurement of some of the variables involved, in particular with mercury-in-glass manometers for the measurement of pressure in the recess and in the film.
477
EXPERIMENTAL TESTS
-b-
Fig. 13.13 Experiments on spherical bearings: a- outline of the bearing; b- Comparison between experimentaland theoretical pressure distribution.
Fig. 13.14 Test-rig for screws and nuts,
In Fig. 13.13.b the theoretical pressures in a directly supplied bearing are plotted, according to the isothermal assumption (solid line; the temperature of the fluid remains constant because of heat transfer to the surrounding environment) and according to the adiabatic assumption (dash-dot line; no heat transfer); PI is the
478
HYDROSTATIC LUBRICATION
supply pressure. The agreement between the theoretical and the experimental results is remarkable in the latter case. The results are ghen for a pad with a sphere radius R=59.31 mm with a vertical film thickness of 100 pm for 8=0,with 81=5", (&=15O,03=750and with an inertia parameter S=0.15pR2R2/Pl=2.
13.3.5
Screws and nuts
In Fig. 13.14 a test-rig for hydrostatic screws and nuts (ref. 13.9) is shown. The test rig allows the screw (S) axial movement, while the nut (N) is a t rest and is -a-
-c
;@
-
4 o po=lOxlO'~grn~'
po=20110'Kg m-'
3-
P' 2-
o
0.1
a2
03
04
05
06
&
-b 5'
Fig. 13.15 Test-rig for screws and nuts: a- self-regulated nut; b- dynamometer for loads and torques; c- load capacity P' versus eccentricity E, for two values of supply pressure po.
EXPERIMENTAL TESTS
479
engaged in the dynamometer (5) which hides it. The base (2) carries the support (41, then the dynamometer and then the nut; i t also carries the four capillary compensated bearings (3), which support the nut. A variable speed motor (6) moves the screw, which is loaded axially by the hydraulic jack (12). The various joints eliminate the effects of construction and assembly errors and those resulting from strains in the structure under loads. The screw and nut being tested are selfregulated. Figure 13.15.a shows the nut (2) with its two lateral seals (1) and (3) which cannot entirely prevent leakages. The nut is supplied through the dynamometer (Fig. 13.15.b) by means of two lateral pipes (5') of small diameter and thickness equipped with strain gauges. So these pipes are supply lines and also part of the dynamometer. In Fig. 13.15.c the screw and nut load capacity P' is plotted as a function of eccentricity E, with the screw at rest. For &=0.5 (and for higher values not shown in the diagram but very frequent in common practice) the experimental values are re-
Fig. 13.16 Test-rig for slide-way: (1) slide-way, (2) frame, ( 3 ) hydraulic jack.
480
HYDROSTATIC LUBRICAJlON
markably lower than the theoretical ones. This may be due to the leakage mentioned from the nut seals and to construction and coaxiality errors of the screw and nut.
13.3.6
Slide-way
Figure 13.16 shows a hydrostatic slide-way (1) (ref. 13.10) and its frame of H-beams (2) (ref. 13.11) loaded statically and dynamically by hydraulic jacks (3). It is a slide-way of a boring machine under which the upper pads are obtained (pads 1 in Fig. 13.18) by setting two ledgers, the front one of which can be seen partially in Fig. 13.17. The other two pads, the lower and lateral ones (pads 2 and 3 in Fig. 13.18) are located in the L-blocks (4) fixed underneath the ledgers. In the illustration the micrometric screws (5)for vertical and horizontal film calibration can be seen. In Fig. 13.18 the film thickness hl of the upper pads (1) is plotted as a function of the static load Fz.Its theoretical and experimental variation is almost linear.
Fig. 13.17 Test-rig for slide-way: (1) ledger, (2) L-block, (3) micrometric screws.
481
EXPERIMENTAL TESTS
Fig. 13.18 Film thickness h , versus normal load Fz.
kg
REFERENCES
13.1 Czichos H.; Tribology;Elsevier, 1978; 400 p. 13.2 Loeb A. M., Rippel H. C.; Determination of Optimum Proportion for Hydrostatic Bearings; ASLE Trans, 1 (1958),241-247. 13.3 Meo F.; La lubrificazione Zdrostatica Realizzata con Alimentazione Attraverso Resistenze Zdrauliche e le Sue Applicazioni a i Cuscinetti Piani (ZZZ parte); Lubrificazione Industriale e per Autoveicoli, 1968, N. 8; p. 19-26. 13.4 Howarth R. B., Newton M. J.; Investigation on the Effects of Tilt and Sltding on the Performance of Hydrostatic Thrust Bearings; Instn Mech Engrs, C20 (1971), 146-156. 13.6 Bassani R.; Calcolo Numeric0 delle Grandezze Caratteristiche dei Pattini Zdrostatici; Automazione ed Automatismi, Anno XIV (1970), N. 3; p. 20-30. 13.6 Bassani R.; Ricerca Sperimentale sui Pattini Zdrostatici; Automazione ed Automatismi, Anno XIV (1970), N. 4;p. 3-14. 13.7 Leonard R., Davies P. B.; An experimental Investigation of the Dynamic Behaviour of a Four Recess Hydrostatic Journal Bearing; Instn Mech Engrs, C29 (19711,245-261. 13.8 Vermeulen M.; Dynamic Behaviour of Hydrostatic Radial Bearings; Vibration and Wear Damage in High Speed Rotating Machinery; proc. NATO/Adv. Study Inst., Kluwer Acad. Publ., Dordrecht, 1989; 16 p.
482
HYDROSTATIC 1UBRlCATlON
13.9 Bassani R.;The Self-Regulated Hydrostatic Screw and Nut; Tribology International, 12 (19791, 185-190. 13.10 Bassani R., Culla C.; Progetto e Costruzione di una Slitta di Macchina Utensile, a Lubrificazione Zdrostatica; Atti 1st. Mecc. Appl. Costr. Macch., Univ. di Pisa, Anno Acc. 1973-74, N. 47; 69 pp. 13.11 Bassani R.,Culla C.; Progetto e Costruzione di una Attrezzatura per Prove di Carico su una Slitta Idrostatica di Macchina Utensile. Primi Risultati Sperimentali; Atti 1st. Meccanica, Univ. di Pisa, AIM 7612, 1976; 51 p.
Chapter
14
APPLICATIONS
14.1
INTRODUCTION
In the first chapter we have already mentioned that hydrostatic lubrication has been successfully applied in many branches of mechanical engineering, from large, slowly rotating machines to small and fast machines. In this chapter, a number of applications will be briefly described, beginning with the very important field of machine tools. Certain types of hydrostatic tilting pads used to build bearings for large machinery, such as telescopes, air preheaters, ore mills, debarking drums, and so on will then be examined. Lastly, after having mentioned a few applications of a different kind, a number of supply systems will be described, with particular reference to constant-flow systems making use of flow dividers or multiple pumps.
14.2 14.2.1
MACHINE TOOLS Spindles
Machine tool spindles form one of the most common fields of application of externally pressurized lubrication, since a high degree of stiffness and damping (i.e. precision characteristics) is required. Hydrostatic spindles may be supported by separate journal and thrust bearings, a s well as by a couple of opposed conical bearings; in certain cases other configurations may prove to be suitable: for instance, conical bearings may be substituted by spherical bearings, o r an opposed-pad bearing and a journal bearing may be com-
484
HYDROSTATC LUBRlCATlON
bined in a Yates configuration. Lubricant may be supplied directly, by means of multiple pumps, or a t a constant pressure (restrictor-compensated). The latter method is generally preferred because it is simpler. As a matter of fact, the compensating restrictors may be easily incorporated in the spindle housing; i t is therefore possible to build compact standardized units with only one inlet and one outlet port for lubricant: the supply system has merely to deliver lubricant at a given constant pressure and a t a temperature varying in a reasonably narrow range. Examples of spindles equipped with separate radial and axial bearings are to be found in Fig. 14.1 and Fig. 14.2. Figure 14.3.a shows how a combined journal and thrust bearing (see also section 8.7) may be used in a spindle, instead of conventional rolling bearings, Fig. 14.3.b. In this connection i t must be remembered that attempts have been made to produce ranges of hydrostatic bearings with outside and inside diameters following
V
6 Fig. 14.1 Hydrostatic spindle with journal and thrust bearings (compensating restrictors are not shown). (Reference 14.1).
U Fig. 14.2 Hydrostatic spindle with journal bearing and combined journal and opposed-pad thrust bearing. (Reference 14.1).
485
APPL /CATIONS
-a-
-b-
Fig 14.3 Hydrostatic spindle with a combined journal and thrust bearing (ref. 14.1).
the IS0 series for rolling bearings. In particular, the bearings depicted in Fig. 14.4 (ref. 14.21,mainly intended for use in machine tool spindles, follow the IS0 "0" series (their main dimensions are given in Table 14.1).After this first experimental range, another range was produced with similar dimensions and performance, but without the built-in seals, as shown in Fig. 14.5. -a-
-b-
Fig. 14.4 Standardized hydrostatic bearings: a- journal bearing; b- combined journal and thrust bearing (ref. 14.2).
In all the above units the journal bearings, of the multirecess type, with four recesses, are characterized by narrow lands: this has been done in order to obtain the greatest load capacity, while reducing the friction area and rise in temperature in the lubricant. The thickness of the film can be chosen in a small range of values, dependending on the stiffness and speed required, while the viscosity of the lubricant should be chosen, as usual, bearing minimum power consumption in mind. Journal bearings may also work without the inner ring.
HYDROSTATIC LUBRICATlON
T A B L E 14.1 Standardized hydrostatic bearing un d
D
(mm)
(mm)
(mm)
(mm)
50 60 70
80 95 110 125 140 150 170 180
60 70 80 90 105 115 130 140
75 90 100 110 130 140 160 170
80 90 100 110 120
-a-
dl
(see Fig. 14.4). B (mm)
Dl
Journal b. 68 80 88 98 110 118 126 140
-b-
Combined b. 75 85 95 106 115 125 136 152
-C -
Fig. 14.5 Standardized hydrostatic bearings: a- journal bearing; b- thrust bearing; c- combined journal and thrust bearing (ref. 2.2). Bearing units are usually fed at constant pressure and for this reason can be provided with laminar-flow restrictors, made up of a stack of special discs fitted in proper holes in the outer ring, very near the recesses of the bearing (Fig. 14.6.a). These discs are of two types: one is plain with a hole in its centre, whereas the other has a rectangular groove on both sides. Restriction is obtained i n the grooves, since they are shallow (however, not less than 80 pm). The total hydraulic resistance of the restrictor may be changed by varying the number of stacked discs. Another type of variable restrictor is shown in Fig. 14.6.b: in this case, a setscrew is used to adjust the hydraulic resistance. Bearings can also be feed a t a constant flow rate: this may be convenient especially for thrust bearings, in order to increase stiffness, at the cost of a slightly more complicated supply system.
487
APPLlCATlONS
t
-b-
-a-
t
1
2 3 2
3 4
c
.c
t
Fig. 14.6 Variable restrictors. a- Laminar-flow disc restrictor: 1-locking ring, 2-spacer disc, 3-restrictor disc, 4-bottom disc (ref. 14.2). b- Laminar-flow screw restrictor.
A typical application of the aforegoing standardized units is shown in Fig. 14.7. Another example is shown in Fig. 14.8: a spindle for a vertical grinding machine supported by two journal bearings and an opposed-pad thrust bearing (ref. 14.3). In the latter example a high degree of axial stiffness was required: for this reason it was decided to feed the thrust pads a t a constant flow rate, by means of a flow divider; the radial stiffness of the spindle, measured a t the nose, was found to be 180 N/pm under a 300 N load (the spindle diameter was 80 mm, the supply pressure 5 MPa), whereas axial stiffness was 500 N/pm under a 800 N load; maximum axial load was 14 KN, since the maximum supply pressure of the thrust bearings was limited to 8 MPa. Figure 14.9 shows a different type of spindle, used in a plane grinding machine, borne by a journal and an opposed-pad thrust bearing. Note that this type of combined bearing may be made to support large tilting moments using multirecess thrust pads (section 8.4) instead of the simpler annular-recess pads.
Fig. 14.7 Hydrostatic spindle with a journal bearing and a journal and thrust bearing (ref. 2.2).
488
HYDROSTATIC LUBRICATlON
E Fig. 14.8 Hydrostatic spindle for a grinding machine; the thrust bearing is fed at a constant flow rate (ref. 14.3).
Fig. 14.9 Hydrostatic spindle with a journal and a double-effect thrust bearing. (Reference 14.1).
For tapered-bearing spindles the most common configuration seems to be that to be found in Fig. 14.10, although different types of spindles have been built, for instance with cones arranged as in Fig. 8.19.a. Standardized spindle units are cur-
A PPLICA TIQNS
489
Fig. 14.10 Hydrostatic spindle with conical bearings. Ring R is used to adjust film thickness. (Reference 14.1).
rently produced, which are interchangeable with rolling-bearing o r hydrodynamic units produced by the same firm; of course, the main spindle dimensions comform with international standards for machine tools (ref 14.4). An example of a standard spindle unit is shown in Fig. 14.11and the main relevant data are to be found in Table 14.2(ref. 14.4).
L
a
-
Fig. 14.11 Standard spindle unit with cylindrical housing for boring, turning or milling (ref. 14.4).
Selection of the main hydrostatic parameters (number of recesses and their dimensions, film thickness, lubricant viscosity and so on) is generally made caseby-case by the manufacturer, on the basis of the operating conditions for which the spindle is designed (mainly load and velocity) and also on the basis of particular requirements, concerning stiffness and damping. Comparing the data in Table 14.2 with data concerning the equivalent spindle units equipped with ball bearings (ref. 14.41,it should be noted that the hydrostatic units show greater radial stiffness (although units equipped with special roller bearings are much stiffer). It should be borne in mind, however, that the stiffness of
490
HYDROSTAX LUBRlCATION
Size
D
a
d
(mm)
(mm)
(mm)
120
350
40
130
450
50
200
550
70
330
650
90
550
8
150 180 230 300
850
110
11
380
1050
150
3 4 5 6
CR
tl
t2
nmax
(W
@m)
(rpm)
0.5
0.5
8500
750
0.5 0.5 0.6 0.8
0.5 0.5 0.6 0.8
1000
1
1
7000 5500 4000 3000 2000
the hydrostatic units is proportional to supply pressure, and may be considerably affected by large axial loads (see section 8.5.2). A distinguishing feature of hydrostatic spindles is their very good running accuracy: values of tl and t 2 are always smaller than 1 pm, whereas the values of similar ball-bearing spindles range from 2 to 4 pm for tl and from 1.5 to 2 pm for t 2 (these values may even double int he case of roller-bearing spindles). Figure 14.12 shows an opposed-cone multirecess bearing that may be used to build hydrostatic spindles (ref. 2.11, as in Fig. 14.13. Note that, in this case, only the right-hand bearing sustains axial loads, whereas the other is used as a pure radial bearing.
Fig. 14.12 Hydrostatic opposed-cone bearing.
APPLICATIONS
491
Fig. 14.13 Hydrostatic spindle with a pair of opposed-conebearings. 14.2.2
Steady rests
Mounting of long and heavy rotors (e.g. turbine rotors, steel mill rolls, calenders, etc.) on lathes or other machine tools often requires the use of steady rests in order t o relieve the headstock and tailstock spindles from excessive loads and to reduce the bending of the axis of the workpiece. On the other hand, conventional steadies are characterized by high friction, with the relevant wearing and heating of the rubbing surfaces: these problems can be completely eliminated by means of hydrostatic lubrication.
A steady for a heavy machine tool may easily be built with a couple of self-aligning shoes (ref. 14.5)of the type shown in section 14.3.Each shoe must be mounted on a radially adjustable support to allow exact positioning of the workpiece. In the application described in ref. 14.6,steadies for sustaining rubber-coated cylinders (up to 600 KN in weight) on a grinding machine have been built. The hydrostatic shoe is provided not only with a spherical seat allowing tilt in all directions, but also with a screw and nut assembly for easily adjusting the radial position of the shoe (Fig. 14.14).I t should be noted that the intermediate piece of the bearing is fitted in a hydraulic cylinder which is widened in the base piece; pressure in the cylinder is the same as in the recess: in this way the fillets of the screw and nut are loaded with only a fraction of the force acting on the bearing. In this case the cylinder to be machined does not lean directly on the shoe bearings since intermediate rings are fitted on the necks of the cylinder: the same steadies can hence be used with different workpieces without needing to change the shoes, but using different rings, all of which have the same external diameter.
492
HYDROSTATIC 1UBRlCATION
Fig. 14.14 Adjustable hydrostatic shoe bearing (ref. 14.6).
14.2.3
Feed drives
Modern high precision machine tools require feed drives with high feeding accuracy, freedom from backlash and low friction. For these reasons recirculatingball lead screws and nuts are widely used. Hydrostatic lead-screw nuts meet the same requirements and also have other advantages as compared to recirculatingball nuts. In particular, they are inherently free from backlash (without the need for mechanical preload) and from wear (which ensures continuity of performance) and have better damping properties. This last feature has a certain importance in machines with roller-bearing or hydrostatic guideways, since the intrinsic lack of damping in the feed direction of frictionless guides can lead to poor stability against chatter in the same direction (ref. 14.7). Moreover, construction of the lead screw should be simpler in the case of hydrostatic nuts, since a very high degree of surface hardness is not required. Nevertheless, hydrostatic nuts are much less used than recirculating ball nuts (at least in small and medium-size machines). The main reasons are, probably, the following: recirculating-ball nuts are well proven and perform satisfactorily; hydrostatic lubrication requires a high-pressure lubricant source; construction of hydrostatic nuts is much more difficult and critical than other types of hydrostatic bear-
493
APPLlCATlONS
ings. This last is also obviously true for recirculating-ball units, but does not constitute a drawback in this case, since they are easily available in the stock of specialized manufacturers. Since some firms have recently begun to produce a wide range of standardized screw and nut assemblies, this type of feed system is expected to spread in the future. Data concerning a range of hydrostatic screws are to be found in Table 14.3 (ref. 14.8). The nut constitutes a compact unit, with built-in restrictors and seals, a n inlet port and an outlet port, requiring only a n adequate but fairly simple supply system. Feeding accuracy depends mainly on the pitch error of the male screw, but owing to the levelling effect of hydrostatic lubrication the manufacturer claims that actual feeding inaccuracy is less than one third of the pitch fluctuations of the male screw. Lo
T A B L E 14.3 Hydrostatic screw and nuts (ref. 14.8).
63
36.5
108
173 205 167 207
134 166 124 164
D
20
2.79 3.72 2.26 3.39
181 241 147 220
1.733 2.300 0.755 1.133
494
HYDROSTATIC L UBRICATlON
As already noted, the manufacture of hydrostatic nuts is somewhat difficult, either because of the relatively inaccessible position of the recesses, or because a small pitch difference in relation to the male screw can lead to a considerable loss of loading capacity (see section 7.3). Both problems can be easily overcome by means of a clever technique consisting in coating the inner surface of the nut with a thick layer of plastic, which is cast while the lead screw is held in position; recesses are obtained by means of patterns temporarily fixed to the flanks of the screw with an adhesive (note that hydrostatic nuts are in general of the multirecess type rather than of the continuous recesses type). The gap is obtained because of the shrinkage of cast plastic (ref. 14.9).
In large machine tools it may be preferable to substitute the screw-nut feed drive with rack and worm systems, which permit runs of practically any length, with a high degree of stiffness; furthermore, stiffness proves t o be independent from run length and the position of the slide. These systems can also obviously be assisted with hydrostatic lubrication. An example is that of the so-called hydrostatic "Johnson drive" (ref. 14.9)shown in Fig. 14.15.In this case, a short worm drives a long rack firmly fixed to the slide. The
Fig. 14.15 Hydrostatic Johnson drive (Ingersoll). 1-Slide, 2-rack, 3-pump pressure, 4-capillaries, 5-cells, 6-worm, 7-external gear teeth, 8-oil supply for forward flanks, 9-bed.
A PPLICA TIONS
495
worm is supported by means of hydrostatic thrust bearings; its circumference is toothed and is in mesh with a pinion driven by the feed gear. Recesses are hollowed in the flanks of the rack. A simple distributing device is needed to deliver lubricant only to the recesses covered by the worm, hence avoiding a considerable waste of power. In other applications (see, for instance, ref. 7.1 o r ref. 14.11) the rack is fixed to the bed, whereas the worm is supported by the slide, together with the relevant feed gear, which drives it by means of a toothed gear, fitted near the worm on the same shaft. Lubricant is supplied through ducts drilled in the worm; recesses may be hollowed in the flanks of the rack (as in Fig. 14.16)as well a s in the flanks of the male screw.
Fig. 14.16 Hydrostatic rack and worm;diarnete-270 mm,pitch=60 mm (INNSE).
In this case, too, a distributor is needed in order to cut off the high-pressure supply of lubricant to the ducts not ending on the flanks of the rack. When speed is high (speeds up to 750 rpm can be used) the centrifugal force may empty inactive ducts and that may cause aeration of the lubricant: hence the supply distributor should incorporate a pre-filling device whose task is to pump lubricant at low pres-
496
HYDROSTATIC LUBRICA TlON
sure into the inactive ducts, just before they become active again (a similar device is described in ref. 14.12). Hydrostatic worms are generally built with a pitch of between 36 and 60 mm and a n outside diameter of between 150 and 300 mm; load capacity may vary between 50 and 180 KN. The rack may be of virtually any length since it is built in sections (for instance, 1000 mm in length) that are bonded and firmly bolted to the slide bed after having been adjusted in relation to one another and measured to verify the pitch error (ref. 7.1). Accuracy may be about 70i-80 pm on a length of 25 m. Owing to this accuracy and to the very high degree of stiffness this feed system can also be used for monitoring the position of the slide during normal operation (by means of electronic compensation the relevant error can be further reduced to a very small value).
14.2.4
Guideways and rotating tables
Hydrostatic lubrication proves to be particularly suitable for guideways of modern high precision machine tools (especially those equipped with numerical control), because of their intrinsic characteristics: very low friction (and proportional to speed); freedom from stick-slip; freedom from wear (which means constancy of performance for an indefinite time); thickness of the oil film independent of the sliding speed (whereas for lubricated plain bearings it increases with speed); high damping capacity fin directions perpendicular to guide); levelling ability: the fairly high film thickness (commonly a few hundredths of a millimeter) allows the hydrostatic lubrication to compensate, a t least partially, for small geometric inaccuracies and deformations of the guides; possibility of building guides of virtually any length (which is difficult with roller guides). On the other hand, it should be noted that the virtual elimination of friction can enhance the effects of the flexibility of other parts of the machine and in particular of the feed drive. For instance, consider the experimental diagrams in Fig. 14.17 (see ref. 14.13 for further details): they refer to a milling machine and show that the displacement due to loading in the direction of the guides (mainly due to the flexibility of the ball screw and nut and of the relevant thrust bearing) is greatly reduced by the friction of the sliding ways. Diagrams in Fig. 11.18 (ref. 14.71, obtained with a similar experimental rig, show the reduction in damping connected with the use of frictionless guides (either
497
APPLlCATlONS
0
10000
20000
kN
load P
Fig. 14.17 Influence of hydrostatic ways on static stiffness, compared with sliding ways. a- Hydrostatic system in action; b- without the hydrostatic system.
plain or ball screws were used as feed drives, without leading to any notably different behaviour). Problems of this kind are easily eliminated by means of simple clamping devices when feed rate is null, whereas in other cases they may be solved by stiffening the feed drive (for instance, in heavy machines, by selecting a worm and rack feed drive instead of screw and nut), by eliminating any backlash and increasing damp-
I
0
I
200
I
I
400
I
V
*
mmlmin
Fig. 14.18 Influence of feed rate V on maximum vibration amplitude A, (at resonance frequency) along feed direction for: a- sliding guideways; b- hydrostatic guideways; c- roller guideways.
498
HYDROSTATIC LUBRICATION
ing (for instance, introducing hydrostatic lubrication in the feed drive) or by means of external dampers (ref. 14.13). A number of different examples of layout for slideway guides are presented in Fig. 14.19; type ‘c’ and ‘d‘use an opposed-pad design: this is necessary when great stiffness and damping are required for a large range of loading conditions. The lower pads are in this case much smaller than the upper ones, in order to compensate for the weight of the slide.
-a-
-b-
-c-
-d-
Fig. 14.19 Sample layouts of hydrostatic guideways.
A compromise, often used in rotary tables, may consist i n substituting the preloading effect of the hydrostatic recesses on the underside of the guide with a spring force applied by means of rolling bearings (in practice, this is a trick for increasing the weight of the slide without increasing its mass). At least two recesses must be used on each guide to absorb torque, but a larger number of smaller recesses (each fed independently) provide greater compensating ability for the geometric inaccuracies of guideways; moreover, since the load is more evenly distributed on the guides, better results should also be obtained from the point of view of elastic distortion. Recesses may be either of the conventional fully-hollowed type, or be reduced to narrow grooves, as in the guides i n Fig. 14.20. From the point of view of static load capacity both designs perform i n the same way, but the narrow-groove recesses have greater damping ability and a larger bearing area in the absence of lubrication (hence, they are less prone to damage in the event of failure of the supply system). On the other hand, friction is also much higher and this type of recess proves to be adequate only for low-sliding velocities. The ability of hydrostatic lubrication to even out inaccuracies due to manufacturing errors or deformations caused by external forces is limited by the thickness of the lubricant film. Especially in the case of very large and heavily loaded slides
APPLlCA TlONS
499
Fig. 14.20 Rototraversing table equipped with hydrostatic lubrication of the guides (INNSE). In a the thrust bearing of the rotary table is shown; the pinions of the feed drive are also visible, as well as four clamps that may be used to fix the angular position of the table and the laminar-flow restrictors. In b the same table is shown from another angle: the linear guideways are visible, as well as two clamps.
500
HYDROSTATIC LUBRICATION
and rotary tables (such as the rotary table of a large vertical lathe), elastic deformation might even force the designer to select an excessively thick film to avoid metalto-metal contact. A solution may be to build the guideway with self-aligning tilting pads, as will be shown in section 14.3. The geometric inaccuracies of the slideways (e.g. waviness) might be completely compensated by controlling recess pressure: the principle is outlined i n Fig. 14.21 (see also ref. 14.10). Pressure in each recess is controlled by a valve, piloted by a regulator which compares a reference signal with the signal produced by a transducer. This last is, for instance, a pneumatic sensor which monitors the position of the slide in relation to a reference straight edge, or a photoelectric sensor, which uses a laser beam a s a reference "guide".
1
2
Fig. 14.21 Scheme of compensating bearing control. 1-Guide, 2-reference guide, 3-distance transducer, 4-regulator, 5-set value, 6-controlled valve, 7-supply pressure.
An example of hydrostatic lubrication applied to guideways is presented in Fig. 14.20,in which details are shown of a hydrostatic rototraversing table: one of a wide range of such equipment, suitable for indexing and contour milling (ref. 14.14)with a load capacity varying from 400 to 5000 KN. A similar range of rotary and rototraversing tables is also suitable for turning operations, with a turning speed of up to 2565 rpm, depending on the diameter of the table (2.5+10m).
The rotary table in Fig. 14.20 has a circular thrust bearing (with a mean diameter of 1400 mm) made up of 12 pads, all fed independently through a set of laminarflow restrictors. These are made by cutting small-diameter (111.5mm) pipes to the appropriate length and are also visible in the photographs. With a supply pressure of 6 MPa, the table can bear loads of up to 600 KN. The radial forces are sustained by a tapered roller bearing, which also exerts a preloading force (150KN) on the hydrostatic thrust bearing, in order to increase its stiffness. The photographs also show
APPL ICATlONS
501
clamps that are able to hold the table firmly in any position, without affecting the film thickness of the hydrostatic bearings. Rotary motion is obtained by means of two controlled-preload pinions meshing with a helical crown gear, whereas a ball screw is used for linear axis transmission (the largest members of the same family of tables use hydrostatic worms and racks for axial feed drive). Hydrostatic lubrication is often also applied to the guides of ram-type milling arms (Fig. 14.22). The design of the guides is of course different from that of the guides of horizontal tables: in this case the ram is supported by two rows of eight recesses (two for each side). The recesses in the lower end of the guide are generally larger since they must support higher loads (in other applications there are three rows of recesses, two of which are set at the lower end of the guide). The supply system is made up of a set of multiple pumps (each pump directly feeds one recess),
Fig. 14.22.a- Hydrostatically lubricated milling arm (Pensotti).
502
HYDROSTATC LUBRlCAT/ON
Fig. 14.22.b- Hydrostatically lubricated milling arm:detail showing hydrostatic pads.
which are fed at constant pressure (-2.5MPa) by a larger pump. In this case, too, the recesses of the pads (which are made of bronze) are reduced to narrow grooves.
It is interesting that hydrostatic lubrication has also been used to compensate for the deflection of the ram due to the cutting force. The geometric adaptive control system described in ref. 14.15 measures the displacement of the milling head by means of a laser gun fixed to the milling arm, which emits a laser beam parallel to the undeformed axis of the ram, and a photoelectric scanner attached to the milling head. The signal produced by the measuring equipment is taken a s its input by a control unit which varies accordingly the speed of a servo-motor driving a further set of pumps. The flow produced by these compensating pumps is directed towards the appropriate recesses and added to the normal flow in order to produce a displacement of the milling head, realigning it with the laser beam.
A P P l ICA TlONS
503
A particular application of externally pressurized lubrication to the ram guide of a gear-shaping machine is described in ref. 14.10. A cross section of the guide is shown in Fig. 14.23: the ram is shaped like a spur gear with every third tooth removed. The accuracy of the internal bore of the sleeve is obtained by casting with a plastic material (this technique is briefly described in section 14.2.3).
Fig. 14.23 Hydrostatic ram guide of a gear-shaping machine (Liebherr). 14.3
LARGE TILTING PADS
Hydrodynamic bearings for very large rotating machine-members have been equipped for many years now with high-pressure hydrostatic pockets, used as jacking devices at starting (hydrostatic lifts). More recently, i t has been found to be expedient to retain the hydrostatic effect in normal running and then to substitute the hydrodynamic bearings completely with hydrostatic (or hybrid) bearings, in the case of slowly rotating machines in particular, or when irregularities in load or speed are expected. One problem connected with this type of bearing in certain machines (such as ore mills) is that the elastic deformation of the runner, due to the pressure of the lubricant, may greatly reduce the effectiveness of hydrostatic lubrication (Fig. 11.24.a). This problem may be overcome by foregoing the "optimum" design, obtained by assuming rigid surfaces and uniform film thickness, and displacing the recesses from the centre of the bearing (Fig. 11.24.b); separate pads may even be used instead of a multirecess bearing (ref. 14.16).
A further improvement in design, able to eliminate most of the problems connected with elastic deformation, machining tolerance, thermal expansion and so
504
HYDROSTATIC LUBRICA TlON
Fig. 14.24 Trunnion deformation due to bearing pressure: a- bearing as designed; b- improved concept; e- most effective concept (ref. 14.16).
on, consists in supporting the large journal by means of a set of self-aligning hydrostatic shoes, as shown in Fig. 14.25 (ref. 14.17). Each shoe is split up into two parts: the upper part rests on a spherical seat and hence can tilt in all directions. The underside of the upper part is shaped like a piston which fits into a cylinder in the base: since the piston area, on which the recess pressure acts, is slightly smaller than the effective area of the pad, the load on the spherical seat is quite low during normal operation.
Fig. 14.25 Arrangement of tilting-pad hydrostatic bearings (ref. 14.17).
505
APPLICATIONS
The spherical rest of each inner shoe (slave shoe) is pushed against the runner by a further piston on which, thanks to a hydraulic connection, the recess pressure of the relevant outer shoe (master shoe) acts. Clearly, if the sum of the two piston areas equals the effective pad area, the slave shoe must necessarily have the same film thickness, and thus the same recess pressure, a s the relevant master shoe (each pad is fed by the same flow rate). Thus when the load direction is vertical all four shoes have the same film thickness and recess pressure, regardless of the deviation of the runner from the ideal circular shape. When the load deviates from the vertical direction the two shoes on each side have an equal part of the load component falling along the line between the two shoes (ref. 14.17). The shape of the recess is also of particular interest. It is known that when a cylindrical pad with a simple recess (as in Fig. 5.30) is tilted from the concentric configuration the pressure field on the land surface is altered and produces a moment that tends to realign the pad; however, this self-aligning capacity is too small to ensure the stability of the shoe in all conditions and i t is hence necessary to use multirecess pads. In Fig. 14.26 the main recess is surrounded by four auxiliary recesses, situated in the corners of the pad, which are fed with the lubricant which passes from the central recess over the bearing lands and through small drilled ducts (this is a compromise aimed a t avoiding dependence upon the direction of rotation: for the greatest stability the auxiliary recesses on the trailing side should only be supplied over the lands). The hydrostatic system described in ref. 14.17 supported a large tube mill for crushing ores: each bearing runner had a diameter of 2700 mm, the maximum
-.
I
I
L Fig. 14.26 Improved recess pattern (ref. 14.17).
W
I
-
506
HYDROSTAT C LUBRlCAT/ON
load was 3500 KN and the velocity was 0.24 reds. Each shoe was 640 mm long and 500 mm wide and was fed at 25 Ym with a lubricant whose viscosity was 0.1 Ns/m2 at 50°C. Film thickness in normal operation was 0.1410.15 111111.
A range of hydrostatic shoes based on the foregoing working principles is currently produced by the same firm (ref. 14.5): a sketch of them is to be found in Fig. 14.27 and their main dimensions are given in table 14.4. The recess pattern is similar t o that shown in Fig. 14.26, but the main recess is now annular in shape, in order t o increase the bearing area a t rest (in the absence of hydrostatic lubrication) virtually without affecting bearing performance during normal operation. Hydrostatic shoes may be used to support horizontal as well as vertical rotating equipment. In the first case the rotating drum may lean on the shoes by means of trunnions (Fig. 14.28.a) or by means of girth rings (Fig. 14.28.b). The latter arrangement, which is often inapplicable with rolling bearings due to their size limits, permits large feed openings and a simplified (and less expensive) design. Each -b-
-a-
Fig. 14.27 Bearing shoes: a- master shoe; b- slave shoe. T A B L E 14.4 H
410 SO0 600
530 640 756
(mm)
Master
Slave
300
180+190 2601270 2951305 320+330 4201430
425
507
APPLlCA TlONS
ring (or trunnion) is supported by two master shoes, to each of which one or two slave shoes may be added to boost the load-carrying capacity (Fig. 14.29). The suggested ring diameter D varies between 500 and 5400 mm, with a load capacity F ranging from 480 to 12000 KN, depending on pad size and the total number of pads. -a-
-b-
Fig. 14.28 Horizontal rotating arrangements: a- trunnion arrangement; b- girth ring arrangement.
D
IMaster
shoe: O S l a v e shoe
Fig. 14.29 Shoe arrangements for horizontal rotating cylinders.
In the case of vertical equipment three master shoes are obviously required in order t o obtain a statically determined load distribution; to each master shoe a slave unit can be added, thereby doubling the load capacity. A typical arrangement is shown in Fig. 14.30, in which two alternatives are also proposed for the radial guidance of the runner: a rolling bearing mounted on the shaft, or a set of hydrostatic guiding pads (see below). For six pad arrangements the load capacity ranges from 1900 t o 14000 KN (depending on the pad size) and correspondingly the minimum pitch diameter D varies from 800 to 2400 mm. Besides the hydrostatic shoes described above, the same firm produces a range of smaller tilting pads of simplified design (see Fig. 14.31 and table 14.5). These still retain a self-aligning capacity, since they have a spherical seat and multiple recesses, but are not equipped with hydraulic cylinders. They are mainly proposed (ref. 14.5) as guiding pads for the axial location of a girth ring (Fig. 14.32) or for the
508
HYDROSTATIC LUBRICATION
D
r=
ALT I
I
ALT II
Fig. 14.30 Shoe bearing arrangement for vertical rotating equipment. radial guiding of platforms (Fig. 14.30). Compact assemblies are also available consisting in a master shoe bearing with two guiding pads (in an opposed-pad configuration) mounted on the fixed part of the shoe (Fig. 14.32.b). Another type of tilting pad (ref. 14.18) can be used to build spherical thrust bearings with a very large diameter. In practical terms, it consists of a circular recess pad laid on a spherical rest whose position can be adjusted by means of a wedge. In Fig. 14.33 a set of twenty pads is used to build a large bearing (with a mean diameter of 5000 mm) for a large parabolic antenna. The dimensions of the bearing and angle a depend on the value of the axial and radial components of the load: bearings with an external diameter of up to 8000 mm can be built.
APPLlCATlONS
509
Fig. 14.31 Guiding pad.
TABLE 14.5 Dimensions of guiding pads (ref. 14.5).
Fig. 14.32 Axial guiding pads: a- separate axial guidance; b- axial guidance integral with a master shoe.
A further type of tilting shoe is shown in Fig. 14.34(ref. 14.19):it can tilt around the cylindrical rib on the underside and align itself thanks to the multiple recesses (two or four) which are fed independently through capillary restrictors. These pads can also be used to sustain radial loads as well as the axial thrust of a large rotating platform. In the latter case, bearings with diameters exceeding 5000 mm may be built, which sustain thrusts greater than 5000 KN and rotating a t more than 20 rpm. These pads prove to be particularly suitable for building rotary tables for large machine tools (e.g. for vertical lathes): an example is given in Fig. 14.35.A different application is described in ref. 14.20,concerning the supporting ring of a 3.5 m telescope.
510
HYDROSTATIC LUBRICATION
P I
Fig. 14.33 Spherical pad arrangement,
Fig. 14.34 "Hydro-tilt"shoe bearing (ref. 14.19).
APPLlCA TlONS
51 1
Fig. 14.35 "Hydro-tilt" shoe arrangement.
Lastly, Fig. 14.36 shows a large-size spherical bearing (ref. 2.2); it has three recesses fed a t a constant flow rate. Bearings like this can sustain heavy loads (up to 10,000 KN) and in general their rotating speed is low. For instance, the bearing depicted in Fig. 14.36 was made to support the rotor of an air preheater weighing 800 KN and rotating at 2 rpm.
14.4
OTHER APPLICATIONS
Apart from those quoted in the foregoing sections, hydrostatic lubrication has a number of different applications. For instance, let us consider the pump in Fig. 14.37 (ref. 14.21): the pistons (1) lean on the tilted plate (2) by means of the spherical pads (3) which are hydrostatically borne by the same circulating fluid. Another special application is quoted in ref. 14.22, that is the lower journal bearing of the main pump of the Super-PhBnix nuclear power plant. This bearing has a
51 2
HYDROSTATIC LUBRICATlON
L.L.1
r't'i
Fig. 14.36 Large spherical bearing.
Fig. 14.37 Piston pump.
diameter of 0.85 m and a width of 0.3 m; it has twelwe recesses carved in the shaft. In this case, too, the lubricant is the fluid circulating in the plant, i.e. liquid sodium. Hydrostatic lubrication has been successfully used in a number of testing rigs. An example is shown in Fig. 14.38:an experimental rig for testing rolling bearings (ref. 14.23).The bearing being tested (1) is made to rotate by a motor (2)by means of a
APPLICATIONS
513
Fig. 14.38 Experimental rig for rolling bearings.
belt drive (3) and is loaded by a jack (4) through the hydrostatic bearing (5);this last leans on a cell (61, which measures load Fa, by means of a spherical seat. Friction moment M R is measured by means of dynamometer (7)and the angular speed n by means of thacheometer (8).
14.5 14.5.1
HYDRAULIC CIRCUITS
Simple layout
A typical supply system for hydrostatic spindles, such as the one in Fig. 2.24, is shown in Fig. 14.39 (ref. 2.2). The bearings are fed at a constant pressure, which is usually in a range between 3 and 7 MPa. A gear pump supplies lubricant at a rate which is 30% greater than the calculated value: the surplus flows back to the reservoir through the pressure regulating valve. Lubricant is pushed through two filters, the first of which is coarser (15 pm), while the other is narrower (5-10pm). A pressure switch prevents the spindle from running until the pressure reaches the established value and stops it when pressure falls: in the last case, an oil accumulator
51 4
HYDROSTATIC LUBRICATION
Fig. 14.39 Supply system for a hydrostatic spindle: 1-oil tank; 2-pump; 3-motor; 4-pressure regulating valve; 5-pressure filter; 6-pressure switch; 7-check valve; 8-piston accumulator; 9-pressure gauge; 10-cooler; 11-thermostaticsystem; 12-heater.
1
2
Fig. 14.40 Supply system for the hydrostatic bearing of an air preheater: 1-pump; 2-motor; 3-pressure filter; 4-pressure switch; 5-check valve; 6-flow divider; 7-piston accumulator; 8-pressure-limiting valve; 9-cooler.
515
APPLICA TlONS
keeps on feeding the bearings for the time needed for the spindle to come to a complete stop. A thermostatic system keeps the temperature of the lubricant close to the design value. Sometimes a further pump may also be needed (generally inserted upstream from the cooler) to bring the lubricant back from the spindle to the reservoir.
Flow dividers
14.5.2
Figure 14.40 shows the supply circuit for the three-recess preheater bearing in Fig. 14.36.The flow rate produced by the main pump is divided into three equal streams by means of a flow divider made up of three equal gear pumps mounted on
8
1
1
13
9
\ 5
6 4
3
2
17
15
16
14
1
Fig. 14.41 Supply system for the hydrostatic bearing of an ore mill: 1-oil tank; 2-pump; 3-pressure switch; 4-pressure filter; 5-check valve; 6-pressure limiting valve; 7-pressure gauge; 8-piston accumulator; 9-nitrogen gas bottle; 10-flow divider; 11-shoe bearing; 12-circulation pump for cooling circuit 13-throttle valve; 14-oil cooler; 15-water flow control valve; 16-temperature-sensing device.
51 6
HYDROSTATIC LUBRICATlON
a common shaft. To ensure continuous operation a second pump is ready to be started up automatically when supply pressure drops below a safe value. A further spare pump is available for replacement, to permit maintenance operations. In the case of a n electric mains failure a diesel generator can provide power for the motors of the pumps. The last emergency device is a set of oil accumulators which can supply lubricant to the bearings for a short time. The hydraulic circuit for the bearing arrangement in Fig. 14.25 is shown in Fig. 14.41. The flow rate produced by the main pump is divided into four equal streams by means of a flow divider. To ensure continuous operation a second pump is ready to be started up automatically in case of failure of the other one and a set of piston accumulators (driven by pressurized nitrogen bottles) can feed oil to the bearings for a certain time in case of power failure, allowing the runner to stop without damaging the bearings.
14.5.3
Multiple pumps
The constant-flow supply circuit of the guideway presented in Fig. 14.22.b is shown in Fig. 14.42. The pre-feeding pump (1)delivers lubricant at a pressure of 25 bar to two multiple pumps (2). Each pump can feed ten recesses independently, each with a 0.33 m3/s flow rate, a t a pressure of 40 bar.
Fig. 14.42 Supply system, with multiple pumps, of a hydrostatic slide.
A PPLlCATlONS
517
Figure 14.43 shows the supply circuit of the pad arrangement in Fig. 14.33. Each pad is directly fed a t a constant flow rate; that is, a set of five multiple pumps is used and each pump delivers four equal streams which are supplied to four pads situated a t 90 degrees from each other. Thanks to the layout mentioned, emergency operation of bearing system is possible even if a pump fails.
Ic
Fig. 14.43 Supply system for the bearing system of a large-beam antenna: M-motor; P-multiple pump; 1-20 pads.
REFERENCES
- Herzstuck Leistungsfahiger Werkzeugmaschinen; FAG publ. 02-113DA (1985);68 pp. Hallstedt G.;Standardized Hydrostatic Bearing Units; Instn. Mech. Engrs., C48 (1971),420-430. Lewinschal L.; Contributo dei Cuscinetti Zdrostatici allXumento d i Produttivitb delle Rettificatrici; La Rivista dei Cuscinetti/SKF, 196 (19781,24-27. FAG Spindeleinheiten fur das Bohren-Drehen-Frasen; FAG publ. 02-1OW3 DA (1985);12 pp. Hydrostatic Shoe Bearing Arrangements; SKF Publication 3873 E (19881,28 PP. Bi l dt sh C., Htillnor G.; Problema Risolto con 1'Adozione di Pattini Idrostatici;La Rivista dei CuscinettYSKF, 181 (1974),18-20. Polseck M.,Vavra Z.; The influence of different types of guideways on the static and dynamic behaviour of feed drives; Proc. 8th Int. MTDR Conf. (19671, pt. 2, p. 1127-1138.
14.1 Die Arbeitsspindel und Hire Lagerung 14.2 14.3 14.4 14.6 14.6 14.7
518
HYDRCSTATIC LUBRICATlON
14.8 Catalog B1025E; Nachi Corp., Japan, 1984; 4 p. 14.9 Weck M.; Handbook of Machine Tools, Volume 2 (Construction and Mathe-
matical Analysis); J. Wiley & Sons, 1980; 296 pp. 14.10 Rohs H. G.; Die Hydostatische Bewegungspaarung i m Werkzeugmaschinenbau; Konstrudion, 22 (1970); 321-329. 14.11 Andreolli C.; Eliminazione dell'Attrito e dei Giochi nelle Macchine Utensili;
Controlli NumericiIMacchine a CN/Robot Industriali, anno XI1 (19791, n. 7, p. 32-45. 14.12 Appoggetti P.; Perfezionamento negli Accoppiamenti Vite-Cremagliera a Sostentamento Zdrostatico; Patent IT 51829 N69; Bollettino Tecnico RTM n. 9, 1969; p. 47-51. 14.13 Umbach R., Haferkorn W.; Some Examples and Problems in Zmplementation of Mwlern Design Features on Large Size Machine Tools; 10th Int. MTDR cod., Manchester, 1969; paper 34; 30 pp. 14.14 Rototraversing Tables for Indexing, Milling and Turning; INNSE Publication DMU/27 (1985),4pp. 14.15 Weck M.; Handbook of Machine Tools, Volume 3 (Automation and Controls); J . Wiley & Sons, 1980; 451 pp. 14.16 Rippel T., Hunt J. B.; Design and Operational Experience of 102-Znch Diameter Hydrostatic Journal Bearings for Large Size Tumbling Mills; Instn. Mech. Engrs., C16 (1971), 76-100. 14.17 Arsenius H. C., Goran R.; The Design and Operational Experience of a SelfAdjusting Hydrostatic Shoe Bearing for Large Size Runners; Instn. Mech. EWS., C303 (19731,361-367. 14.18 Supporti idrostatici FAG; FAG Publication 44109 IB (19711, 8 pp. 14.19 Andreolli C.; Guida Circolare Idrostatica Assiale per Tavola Portapezzo Rotante; Patent IT 2353CA, 1975; 15 pp. 14.20 Andreolli C.; Sopporto Zdrostatico per 1'Asse Azimutale del 3.5 m New Technology Telescope (NTT) dell %SO; Convegno AIM-AMME (Tribologia-Attrito, Usura e Lubrificazione), Sorrento, 1987; p. 421-430. 14.21 Giordano M., Boudet M.; Thermohydrodynamic Flow of a Piezoviscous Fluid Between Two Parallel Discs; J. Mech. Eng., 1980. 14.22 F r h e J., Nicolas D., Deguerce B., Berthe D., Godet M.; Lubrification Hydrodynamique; Edition Eyrolles, Paris, 1990; 488 pp. 14.23 Martin F. J.; Prove Funzionali e di Qualificazione nello Sviluppo dei Cuscinetti Volventi; La Rivista dei CuscinettiBKF, 224 (1986),28-36.
APPENDICES
A.l
SELF-REGULATED PAIRS AND SYSTEMS
The principle of self-regulating flow, applied to circular bearings and screws and nuts, can also be applied to pads of infinite length. See the pair of pads shown in Fig. Al.l.a, clearly similar, from the functional point of view, to the bearing in Fig. 7.25. The formulae and the corresponding diagrams are also quite similar (see ref. 7.5). Naturally the pad of finite length needs lateral seals. Consider, for example, those shown in Fig. Al.l.b, made of two shaped plates 3, in the peripheral grooves of which internal (static) 3.1 and external (dynamic) 3.2 seals are housed. The former are elastomeric seals while the latter are made of a material with a very low friction coefficient (PTFE), which can be lowered even further by allowing small side leakage.
A further development of the principle of self-regulation is its application to mutually orthogonal pairs of opposed pads of infinite length (ref. 2.23). Figure A1.2 schematically shows that application. A purely vertical load has almost no effect on the gaps of the horizontal pairs while it makes the vertical ones bearing it work as self-regulating. Similarly a purely horizontal load makes the horizontal pairs work a s self-regulating bearings. The simultaneous presence of a vertical load and a horizontal one, each supported by the corresponding self-regulating pair, involves the self-regulation of total flow Q, which is subdivided into two equal partial flows Q12 which are again subdivided into two equal partial flows Ql4. For the formulae and the relevant diagrams the reader is referred to ref. 2.23. The hydrostatic system in Fig. A1.2 which has the same advantages as self-regulated pads, in particular very high stiffness, is directly applied in hydrostatic slideways. Obviously, lateral seals of the type shown in Fig. Al.1.b will be necessary.
520
HYDROSTATIC L UBRICAT I m
I?
-a-
- b-
Fig. A l . l Self-regulating opposed-pad hydrostatic bearing: a- theoretical bearing; b- actual bearing with side seals.
A further application of the principle of self-regulation is that of a system made up of a self-regulating screw and nut in series with the above-mentioned slideway. Self-regulating circular bearings, screws and nuts, pads and mutually orthogonal pairs, can be supplied at constant pressure as well as with constant flow rate. As for circular bearings, the matter has been discussed in section 7.4.2 and in ref. Al.1, where it is pointed out that the efficiency of the self-regulating bearing is comparable to that of a conventional one fed through two flow-control valves. For equal flow rate and pumping power in particular, the load capacity of the self-regulating bearing is generally higher than that of the conventional bearing, especially if the latter is fed through capillaries or orifices. What has been said above also holds good for self-regulating screws and nuts, compared in ref. A1.2 with the conventional ones fed through two flow-control
APPENDICES
521
Fig. A l . 2 System of self-regulating opposed-pad hydrostatic bearings.
valves. What has been said above also holds good for self-regulating pads, compared in ref. A1.3 with conventional pads fed through capillaries or orifices. Finally for systems made up of mutually orthogonal pairs of opposed pads, it is pointed out, ref. A1.4, that the self-regulating system bears higher loads than the conventional system with fixed compensators and the phenomenon is more marked as the load increases. Again, a further application of self-regulation consists in a screw and nut assembly in series with a slideway.
A.2
DYNAMICS
In Chapter 10 the dynamic behaviour of hydrostatic bearings has been studied, using linear mathematical models and Laplace transform. For a bearing with a circular recess (Fig. 5.11, directly supplied by a pump with constant flow Q,carrying a static load W and subjected to an instantaneous overload LW,the non-linear mathematical model yields the following equation (ref. A2.1)
522
HYDROSTATICLUBRICATION
(A2.1) where h'=h/h,, h, being the film thickness under the final static load may be due to gravity),
W+AF (which
which can be considered a damping constant, C s = 2 m , where M is the bearing mass and K,=9pr~(1-r'2)Qlh4,its stiffness; C, can then be considered to be the critical damping;
is another damping constant;
is the fluid stiffness, K, the apparent bulk modulus c the fluid and \ the volume of the fluid in the supply tube and in the recess. The initial conditions in Eqn A2.1, a t the time t'=O, are the following
where ho is the film thickness under load W and t'=t/ts,with t s = 2 n m s (period). Film and recess pressures are
(A2.2)
respectively, where p ' = p / p s ,p;=p,lp, and p s is the recess pressure under load W+AF. In Fig. A2.1, A2.2 and A2.3 the variations of film thickness h' and of recess and film pressures p i and p ' , versus time are plotted. The former are determined by solving Eqn A2.1 numerically, the latter by introducing the values of h', h' and obtained from Eqn A2.1 in Eqns A2.2. The results concern the cases defined in Table A2.1.
x'
523
APPENDiCES
'2
case
m
1 2
0.05 0.05
3
0.025
r' 0.9 0.9 0.875
p.102
W
AF
Ns/m2
N 1000
430
5.4 5.4 5.4
500 1000
N 1500
430
h,.104 K , . I o - ~ K ~ . I o - ~K ~ I K ,
m 0.9 0.95 0.9
N/m
N/m
48 63 48
77 77 7.7
1.6 1.2 0.16
In the first case the bearing is stable (Fig. A2.l.a). After a few oscillations it stops in the equilibrium position h'=l. Figure A2.1.b shows an over-pressure in the inner part of the bearing clearance followed, however, by a small depression in the outer part with possible cavitation and development of air bubbles. -a-
- b-
Fig. A2.1 Stable pad: a- oscillation of film thickness h' and recess pressure $; b- recess pressure
p i and film pressure p'. Start of cavitation in the film.
In the second case the bearing is the same but the initial load is lower while the instantaneous over-load is higher. Anyhow the bearing is still stable (Fig. A2.2.a) but the initial oscillations are larger and the bearing comes to a stop after a greater number of oscillations (interrupted in the diagram). Figure A2.2.b shows a considerable over-pressure followed by a remarkable dangerous depression. Of course, in such a case, cavitation must be considered in the numerical solution otherwise it yields meaningless results. In the third case the bearing is smaller but with the same loads as in the first case. It is unstable (Fig. A2.3): film oscillations ("relaxation oscillations", characteristic of a self-exciting system (ref. A2.2) with positive damping) settle a t very high
524
HYDROSTATIC LUBRICATION
n ~
P\,
-1.6
5
I
,
I
(
I
'
I I
: 1.4 i1 12
1
I I
- -- -- .
1.8
:,
I
2.0
I
- - --
I
~
1.0
I
! I 0.8 ! \ I 06 P' T-
i1 \
IL 0.4
I I
;0.2
8
I OA
0.9
I I
0 ,I
1, I
1- -02
i
I
!
;
,'
-a4
! -46 \, ; ! ; ..-0.8
,'
"
"
i'
-1.0
amplitude values, as well as recess pressure and in the recess initial signs of depression can be seen (ref. A2.3).
.-
Putting h'=l+&,with E <el,Eqn A2.1 becomes the following linear
equation
with the initial conditions (at t'=O) E = Q
,
E=O
,
..
4
E =p2-
AF
Adopting Routh's elementary criterium for stability, mentioned and applied in section 10.5.2, the following relationship must be satisfied
525
APPENDICES 4
3
h’ 2
pd 1
0 0
1
2
3
4
5
6
7
a
9
t‘
Fig. A2.3 Unstable pad: oscillation of film thickness h‘ and recess pressure pi. Start of cavitation even in the recess.
(A2.3) where
It must be noted that, but for high values of r’, often required in order to have minimum total dissipated power H , (Chapter 111, pslpf approaches unity while (p,-pf)pf approaches zero (e.g. for r’=0.6,ps /pf=1.09and (ps-pf)pf=0.115).If also (ps-pf)pfc2/(KdM<<1, Eqn A2.3 is reduced to Kd/Ks>l. Referring to the cases shown in Fig. A2.1, A2.2, A2.3, Kf lKs=l. 6, 1.2, 0.16, respectively, consistently with the diagrams. The viscosity of the fluid has always been assumed to be constant. Actually, the viscous squeeze of the film causes a n increase in fluid temperature, thus a reduction of viscosity and of the effectiveness of the squeeze itself. When the rise in temperature becomes quite high, for high values of film thickness h, a decrease in load capacity might even occur as h decreases (ref. A2.4). The results given refer to direct supply. In the case of supply with restrictors, there are differences, some of which are considerable. In particular: the volume of fluid influenced by compressibility is reduced to the volume down-
526
HYDROSTATIC LUBRICATION
.
stream from the restrictor; if the restrictor is rigid, i t works as a damper. This does not mean that compensated supply is better than direct supply from this point of view, because, on the contrary, the latter shows higher damping, as can be seen in Chapter 10; if the restrictor is variable, it may produce a negative effect because the number of degrees of freedom increases.
As regards degrees of freedom, a bearing, and a hydrostatic system in general, is part of a bigger system: a machine made up of various elements (each with its own stiffness and internal damping), therefore with various degrees of freedom. It should also be noted that hydrostatic systems, because of the fact that their films work as vibration attenuators (ref. A2.5), may be preferred to other low friction systems in those machines in which forced vibrations are expected, especially if resonance is possible, and with several degrees of freedom. In this connection, it must be pointed out that very stiff films are not always convenient because they would behave in practice as stiff elements with no damping properties.
A.3
THERMAL EXCHANGE
A.3.1
Resistances
In the case of laminar flow with forced convection in a tube of diameter d and length 1, that is for Red=Vd/ v<2000, the Nusselt number is
where Pr=pccll is the Prandtl number and I fluid thermal conductivity. Its thermal unit conductance is
(A3.1) For fluids with high Pr, such as lubricating oils, in long tubes, with good approximation Nud=3.66, thus 'yc=3.66(Ild). The lubricant thermal resistance in the tube is
where A is the wet surface. Considering the recess as a rectangular tube, its equivalent diameter is
APPENDICES
527
for L>>h, (in the examples given L=40h,.).So R1, becomes
Its value may be increased without causing any problem (almost doubled). R1, can be determined in the same way. For turbulent flow, that is for Re&6000,
and the values of NUd can be much higher than those relevant to the laminar flow. In transient conditions, that is for 2000dZedC6000, the evaluation of NUd is very difficult.
A.3.2
Coefficients
A.3.2.1 coefficient ac i) Forced convection on an infinite plate of width B with fluid lapping a t one face, a t speed V , far from the face.
NUH= 0.664Re~l"PrIf3 and since Pr=0.72 for air
NUH= 0.595ReBlI2
(A3.2)
that is for Rt?B=V,B/ ~ 4 5 . 1 0 for higher This formula is true for laminar values, not easily achievable, flow becomes turbulent, and
NUH= 0.036 (ReBo.8- 23,200)Pru3 and for P-0.72
NUH = 0 . 0 3 2 R e ~- ~ 748 .~ Forced convection on an infinite plate of width B , for Re~C5.105,with flow perpendicular to one face. The relationship
NuBa = o . 1 5 R f ? ~ ~ ~ ~ can be used with a good approximation. Trail of a n infinite plate of width B, for R e ~ C 5 . 1 0 ~ .
(A3.3)
528
HYDROSTATIC LUBRICATION
(A3.4)
With equations A3.4, A3.2 and A3.3, the conductance a, of a prismatic structure (slideway) of infinite length can be roughly evaluated. For example, for B = l m and thickness H=0.2 m, and for V,=50 d s , with v=1.6.10-5 m2ls for air, it is Re~,=1768, Nu~,=427.5,so %~,=7.16 Reg=7N.6, Regp21370. Therefore NU~,=265.2,Nu~=470.4, J/ms2s°C, acB=63.5 J/ms2s0C, acB,=115.4 J/ms%"C and the average value is a,=(acga+2ac~+ 0&!,)/4=62.5J/ms2s°C.
ii) Natural convection around a square plate of side B and thickness H much smaller than B , with flow perpendicular to one face:
NUB= 0.45 (Grg pr)'I4
where Grg is the Grashof number which in natural convection replaces Re,
where deq=2[BH/(B+H)], j? is the coefficient of volume expansion, T , surface temperature and T, the air temperature far from the surface. For Pr=0.72, NUB= 0.414 (Gr#4
This formula is true for laminar flow, that is for G r ~ < S . l ofor ~ ; higher values of Grg, that is for turbulent flow, NUB= 0.083 (Grg Pt')y3
The corresponding values of spy, obtained from Eqn A3.1, substituting d with deq, are small anyhow. For example, for B=0.15 m, H=0.015 m, T,=5OoC, Tm=2OoC,since (gp)/v2=108l/m3"C for air, G r ~ = 6 . 1 . 1 0thus ~ , NUB=6.5, and since k 0 . 0 2 7 J1ms"C for air, from Eqn A3.1 in which d is replaced by deq,we have a,=6.4 J/ms2s°C. In the case of a rectangular plate of width B and length L, B may be substituted by (L+B)/2.In the case of a disk of diameter D,B is substituted by 0.9D.
iii) Horizontal rotating disk of diameter D,with flow perpendicular to a face.
wD2 v2 NuD = 0.18 ( 7)
APPENDICES
529
The boundary layer is laminar if Re~<5.106;for higher values
where D , is smaller than D and decreases with w, and flow is laminar from 0 to DJ2 and turbulent from DJ2 to 012. For example, for w=500 s-1, DpO.127 m. For D=0.2 m flow is laminar from 0 to 0.0635 m and then it becomes turbulent. Therefore N u ~ = 3 . 5 . 1 and 0 ~ q = 4 7 Jlms2sOC. In ref. 5.15 the cooling of the lubricant is due to the high speed (-628 rads) and the large diameter of the rotating disk (D=0.3 m) and mostly to the “ducted fan” effect.
iv) Horizontal rotating cylinder of diameter D,with flow perpendicular to its axis. For rotational speed w lower than the critical value, that is for R e ~ < 2 . 5 . 1 0see ~ , the following case of the motionless cylinder; for higher values, the flow becomes turbulent and
This formula can be used for journal bearings. Natural convection around a horizontal cylinder of diameter D , with flow perpendicular to its axis.
and for P r 4 . 7 2
This formula is true for laminar flow, that is for G r ~ < 5 . 1 @for ; higher values, that is for turbulent flow,
and for Pr=0.72
For example, for D=O.OOl m, T,=50°C, T,=20°C, we have G r p 3 . 1 0 3 , thus N u ~ = 3 . 6and q=9.62 J/m2soC.With the simplified formula
530
HYDROSTATIC LUBRlCATfON
the value q = 8 . 5 J/mspsOCwould be obtained. Forced convection around a horizontal cylinder of diameter D, with flow perpendicular to its axis.
where the first term between square brackets concerns the laminar boundary layer on the front part of the cylinder, the second concerns the partly turbulent trail. The term (pU,/p,)O.25takes into account the influence of temperature variations on the physical properties of the fluid. This relationship is true for l I R e ~ 1 l OAs ~ .for air, in a hydrostatic system, (p,/&0.25=0.98, and putting also Pr=0.72
Table A3.1 contains the values of ReD, NuD and ac.They have been determined for D=O.Ol m and for increasing values of V,, putting ~ = 1 . 6 . 1 0m2/s - ~ for air. It should be noted that for V,=O.l m the value of q is almost equal to that obtained in the previous case of natural convection for AT=30°C. ‘ABLE A 3 1
v, m 0.1 0.5 1 5 10 20 50 100
Re,
a,
6.25.1 0 3.125.102 6.25.102 3.12510’ 6.25.103 1.25.104 3.125.104 6.25.1 04
J/m*s°C 9.23 22.1 32.3 79.7 117 182 295 442
3.42 8.19 12 29.6 43.3 67.4 109 164
Forced convection around banks of tubes with perpendicular flow. For more than 10 rows of in-line or staggered tubes,
Nub = 0.33Reb0.6Prv3
This formula is true for turbulent flow, that is for Re>6000, and
where Vm,, is the velocity reached by air in the minimum available cross-section.
APPENDICES
531
For laminar flow (Reb<200) and also for transient flow (200dEeb<6000) the relationship
NUb = J Reb Pru3 is true. This formula is complicated because j , which is Colburn's dimensionless factor, is a function of R e , of the number of rows of the tubes and of their arrangement.
A grid upstream from the bank of tubes makes turbulence possible even at low Reynolds numbers. Barnes in the bank make the air move in a winding way, thus increasing the actual surface of heat transfer.
A.3.2.2 Coefficient aj. Coefficient aj for air is:
to 0.9 is the geometric form factor, 0=5.7 J/m2s°K is the Stefanwhere !F11.2=o.7 Boltzman constant, TIand T2 are body and ambient temperatures, respectively. For temperatures included in the hydrostatic range, aj=4.5 to 7.5 J/m2s°K, approximately. Such values must be added to ac in order to obtain the global unit conductance a. What has been described above can be useful for the design of a n air-oil heat exchanger with the tube bank made of the hydrostatic system supply pipelines, bent more times to a U shape. The approximate value of the coefficient of global heat transfer for air-oil exchangers is a=30 to 180 J/m2s°C.
For high lubricant temperature rise it may be advisable to use water-oil heat exchangers for which a=120 to 350 J/m2s°C approximately (ref. A3.1). Water heat exchangers, as well as air exchangers, can be placed in the return line; however, the absence of water or air leakage in the lubricant must be carefully checked. Finally, when heat exchangers are placed in the supply line, a more effective control of the temperature of the bearings may be obtained.
REFERENCES
A l . l Bassani R.; The Self-Regulated Hydrostatic Opposed-Pad Bearing in a Constant Pressure System; ASLE Trans., 25, (1982),95-100. A1.2 Bassani R., Piccigallo B.; Vite-Madreuite Zdrostatica Autoregolata Alimentata a Pressione Costante; Tribologia e Lubrificazione, AMO XIV (1979),98-109.
532
HYDROSTATlC LUBRlCATfON
A1.3 Bassani R.;Pattini Contrapposti Zdrostatici Autoregolati, Alimentati a Pressione Costante; Oleodinamica-Pneumatca, 24 (19831,18-25. A1.4 Bassani R.; Sistema di Pattini Zdrostatici Autoregolati, Alimentati a Presswne Costante; Scritti per L. Lazzarino; Pacini Editore, Pisa, 1986;p. 235-250. A2.1 Bassani R.;Cuscinetti Zdrostatici di Spinta Sottoposti a Variazioni Istantanee del Carico; 2nd AIMETA Congr., Napoli, 1974;vol. 3,p. 225-236. A2.2 Nayfeh A. H.,Mook D. T.; Non linear oscillation; J.Wiley & Sons Inc., 1979; 704 pp. A2.3 Hell H., Savci M.; Bynamische Eigenshaften Hydrostatischer Axiallager bei Kleistmilglichem Gesamtleistungsaufwad; Konstruction, 27 (19751,137-144. A2.4 Pinkus O.,Wilcock D. J.; Thermal Effect in Fluid Film Bearing; Mech. Engineering Publ. Ltd, 1980;p. 3-23. A2.5 Wilcock D. F., Bevier W. E.; Externally Pressurized Bearings. - Vibration Attenuators; ASME Trans., J. of Lubrication Technology, 90 (19681,614-617. A3.1 Wilcock D. F.,Booser E. R.; Lubrication Technique for Journal Bearings; Machine Design, June 25,1987;p 84-89.
Author index
For each author this index shows the relevant reference numbers, as well as the pages on which each reference is cited. For example Fuller D. D. 1.5- 4,141,143
The reference is the fifth item in the reference list of chapter 1and is cited on pages 4,141and 143.
Andreolli C. 14.11- 495 14.19- 5@?,510 14.20- 509 Anwar I. 8.2- 237 Aoyama T. 8.29- 261,263 Appoggetti P. 14.12- 496 Arsenius H.C. 14.17- 504,505 Arsenius T. 2.2- 17,486,487,512,613 Artiles A. 8.15- 245 Barrett L. C. 10.7- 322,339 Bassani R. 2.10- 23 2.19- 28 2.23- 30,519 4.11- 81,101 5.1- 94 5.2- %, 98 5.4- 98, loo 5.5- 98,100 5.9- 1 M 5.10- 105,107 5.14- 107,111,447 5.15- 107,laS,447,529 5.16- 111 5.18- l22 5.31- 135
Bassani R. (continued) 5.32- 135 5.42- 145 5.43- 146,227 7.5- 221,223,619 7.6- 223 7.7- 29,339 7.8- a.2 13.5- 470 13.6- 470 13.9- 478 13.10- 480 13.11- 480 Al.l- 520 A1.2- 520 A1.3- 521 A1.4- 521 A2.1- 521 Berthe D. 14.22- 511 Bettini B. 8.8- 239 Bevier W. E. A2.5- 526 Bildtsh C. 14.6- 491,492 Bird R. B. 12.1- 447 Booser E. R. 3.4- 40,42 A3.1- 531 Bottcher R. 10.1- 301,304
Boudet M. 14.21- 511 Boyd J. 3.2- 37 8.5- 238 Brzeski L. 2.17- 27 Bucciarelli A. 3.9- 47 Cameron A. 4.4- 69,81 Casely A. L. 2.8- 22,169 Castelli V. 5.28- 135 Chang T. S. 5.37- 142 Chen B. 8.25- 2-54 Chen C. R. 5.36- 142 Chen K. N. 10.4- 302,320,358 Chen N. N. S. 8.13- 245 8.14- 245 Chen Y.S. 10.14- 356 Chong F. S. 9.2- 292,233,296 10.15- 356 Colsher R. 8.2- 237
534 Cowley A. 8.6- 238 Culla C. 5.32- 135 13.10- 480 13.11- 480 Czichos H. 13.1- 467 Davies P. B. 2.15- 26,470 8.7- 239 8.11- 240,242,244,354,355 13.7- 472 De Gast J. G. C. 2.11- 23 De Shepper M. 10.13- 356 Decker 0. 7.10- 233 7.9- 233 Deguerce B. 14.22- 511 Dorinson A. 3.1- 36,37 Dowson D. 2.14- 26 5.6- lm, 104,110 5.21- 128,132 Dumbrawa M. A. 8.17- W , 250,254,419 8.18- 254,288 Effenberger W. 10.1- 301,304 El Hefnawy N. 8.20- 250 El Kayar A. 9.3- 292 El Sayed H. R. 5.41- 144 El Sherbiny M. 8.20- 250 El-Efnawy N. 4.8- 74 El-Sherbiny M. 4.8- 74 Ernst P. 7.1- 218,495,496 Ettles C. H. M. 4.6- 74 Fowle T.I. 3.7- 44 Fr6ne J. 14.22- 511 Fuller D. D. 1.5- 4,141,143 Ganesan N. 5.17- l20,125 5.20- 128 8.30- 262,268 8.32- 262 10.5- 307
HYDROSTATIC LUBRICATION
Geary P. J. 2.20- 28 Ghai R. C. 8.3- 238 8.9- 240 Ghigliazza R. 11.2- 419 Ghosh M. K. 10.10- 354 10.11- 354 10.17- 358,360 Giordano M. 14.21- 511 Girard L. D. 1.1- 4,35 Godet M. 14.22- 511 Goldstein S. D. 4.1- 54 Goran R. 14.17- 504,505 Haferkorn W. 14.13- 496,498 Hallnor G. 14.6- 491,492 Hallstedt G. 14.2- 485,487 Hegazy A. A. 9.3- 292 Hell H. A2.3- 524 Heller S. 5.35- 142 Hessey M. F. 8.33- 264 Iiirai A. 5.24- 132 Hirs G. G. 2.13- 25,172 Ho Y. S. 8.13- 245 Hooke C.J . 2.12- 25 8.1- 237 8.26- 255 Hornung V. G. 4.5- 74,247 Howarth R. B. 2.8- 22,169 13.4- 469 Hunt J. €3. 14.16- 503,504 Ichikawa A. 9.1- 290,292 Ikeuchi K. 2.24- 23 Inasaki I. 8.29- 261,263 10.9- 348,472 Ives D. 9.5- 298
Ives D. (continued) 9.6- 298 Jain S. C. 8.10- 240 Kapur V. K. 5.12- 107,110 Karelitz M. B. 1.4- 4 Katsumata S. 8.2- 237 Kazimierski 2. 2.17- 27 Kennedy J. S. 5.13- lW,125 Khalil F. 5.19- 125 5.25- 132,476 Khalil M. F. 9.3- 292 Khataan H. A. 5.41- 144 Kher A. K. 8.6- 238 Kong Y.C. 8.14- 245 Koshal D. 8.19- 247,293 9.4- 293,294,296,377 Kreith F. 12.2- 451,456,461 Kubo M. 2.18- 27 7.4- 220 Kundel K. 2.2- 17,486,487,511,513 Langhaar H. L. 4.9- 79 Lansdown A. R. 3.11- 52 Leonard R. 10.12- 356 13.7- 472 Lewinschal L. 14.3- 487,488 Lewis G. K. 5.23- 131 Lightfoot E. N. 12.1- 447 Lingsrd S. 8.14- 245 Loeb A. M. 5.34- 139,468 13.2- 468 Lombard J. 5.40- 144 Lord Rayleigh 1.3- 4 Ludema K. C. 3.1- 36,37 Lund J. W. 8.37- 282
535
AUTHOR INDEX
Lund J. W. (continued) 10.16- 356 Majumdar B. C. 10.11- 354 Makimoto Y. 2.18- 27 7.2- 218 Manea G. 8.16- 246 Martin F. J. 14.23- 512 Martin H. R. 3.5- 44,47,48 Massa E. 5.27- 135 Masuko M. 10.2- 301 Matsubara T. 2.18- 27 7.2- 218 7.3- 220 7.4- 220 Mayer J. E. 2.9- 23 5.11- 107,110 McCloy D. 3.5- 44,47,48 Meo F. 13.3- 469 Merritt H. E. 2.7- 22,78,so,82,85 Michelini C. 11.2- 419 Mizumoto H. 2.18- 27 7.2- 218 7.3- 220 7.4- 220 Mohsin M. E. 2.6- 21,29,167 Moisan A. 5.40- 144 Mook D. T. A2.2- 523 Mori H. 2.24- 23 5.24- 132 Morsi S. A. 2.5- W,165 10.6- 320 Morton P. G. 9.6- 298 Moshin M. E. 8.28- w6 10.6- 320 Mote C. D. 5.3- 98 Mueller-Gerbes H. 7.1- 218,495.4s N a k a h a r a T. 10.2- 301
Nayfeh A. H. A2.2- 523 Neale J. M. 3.10- 50 Newton M. J. 13.4- 469 Nicolas D. 14.22- 511 O'Connor J. 3.2- 37 ODonoghue J. P. 2.12- 25 4.6- 74 5.23- 131 5.26- I 3 3 8.1- 2-37 8.26- 255 8.31- 2 M 8.33- 264 8.35- 279 8.36- 282 Ogate K 10.8- 324 ogiso
s.
8.27- 255 Ohsumi T. 2.24- 23 Okamura S. 2.18- 27 Okasaki S. 7.3- 220 Opitz H. 1.7- 4 2.3- 19,173,376 10.1- 301,304 Piccigallo B. 5.42- 145 5.43- 146,227 7.7- 229,339 A1.2- 5 W Pinkus 0. 4.2- 54,68,81 A2.4- 525 PolsEeck M. 14.7- 492,496 Prabhu T.J. 5.17- 120,125 5.20- 128 8.30- 262,268 8.32- 262 10.5- 307 Radkiewicz Cz. M. 5.13- 107,125 Ragab H. 5.22- l28,131,132 Raimondi A. A. 8.5- 2-38 Raznyevich K 3.8- 46,50 Recchia L. 7.8- 232
Reddi M. M. 5.29- 135 Rippel H. C. 1.6- 4 5.33- 139 7.11- 233 13.2- 468 Rippel T. 14.16- 503,504 Rohs H. G. 14.10- 500,503 Rowe W. B. 1.9- l0,89 8.1- 237 8.12- 241,244,353,355 8.19- 247,293 8.24- 253 8.26- 255 8.34- 276,277,278 8.35- 279 8.36- !B2 9.2- 292,293,296 9.4- 293,294,296,377 9.5- 298 9.6- 298 10.12- 356 10.15- 356 Royle J. K 2.8- 22,169 Rumbarger J. H. 5.39- 144 Safar Z. S. 5.3- % Salem E. 5.19- 125 5.25- 132,476 Salem E. A. 9.3- 292 Salem F. 4.8- 74 8.20- 250 Sasaki T. 5.24- 132 Sato Y. 8.27- 2-55 Savci M. A2.3- 524 Shapiro W. 5.28- 135 5.35- 142 7.9- 233 7.10- 233 8.4- 238 8.15- 245 Sharma S. C. 8.10- 240 Shaw H. C. 2.9- 23 Shen F. A 5.8- 106
536 Shinkle J. N. 4.5- 74.247 Siddal J. N. 11.1- 385 Siebers G. 2.1- 17,490 Singh D. V. 8.3- 238 8.9- 240 Sinha P. 5.13- 107, I25 Sinhasan R. 8.3- 238 8.9- 240 8.10- 240 So H.
5.36- 142 5.37- 142 so0 L. s. 5.7- 105 Speich H. 3.9- 47 Stansfield F. M. 1.8- 6 , B , 115,133 Sternlicht B. 4.2- 54,68,81 Stewart W. E. 12.1- 447 Stout K. J. 8.19- 247,293 8.24- 253 8.34- 276,277,278 Straccia P. F. 5.38- 142,239,261,264 Streeter V. 4.10- 81 Szeri A. 2. 5.30- 135
HYDROSTATIC LUBRICATION
Taylor C. M. 2.14- 26 5.21- rzS,l32 Ting L. L. 6.11- 107,110 Tipei N. 4.3- 54 4.7- 74 Tully N. 2.16- 27 Umbach R. 14.13- 4%, 498 Usuki M. 7.3- 220 Vavra 2. 14.7- 492,4% Verma K 5.12- 107,110 Vermeulen M. 8.21- W0,Wl 10.13- 356 13.8- 474 Viswanath N. S. 10.17- 358,360 Vogelpohl G. 1.2- 4 Walowit J. 8.15- 245 Wang X. 10.4- 302,320,358 Wearing R. S. 8.35- 279 8.36- 282 Weck M. 14.9- 4M 14.15- 502 Wertwijn G. 5.39- 144
Weston W. 9.2- 292,293,296 9.6- 2M Wiener H. 2.4- 20 Wilcock D. F. 10.3- 302 A2.5- 526 A3.1- 631 Wilcock D. J. A2.4- 625 Wills J. G. 3.3- 39 Wong G.S. K. 2.22- 30 Wu H. Y. 10.14- 356 Wylie C. R. 10.7- 322,339 Xie P. L. 10.14- 356 x u s. 8.25- 254
xu s. x.
9.2- 292,293,296 Yang G. P. 10.4- 302,320,358 Yang H. H. 10.4- 302,320,358 Yates S. 2.21- 28,279 Yonetsu S. 8.29- 261,263 Yoshimochi S. 2.18- 27
Subject index
Bold page numbers indicate that the item is the main topic of a section. Symbol u->” means “see”.
additives 50,51,52 adiabatic flow 89,107,110,132,151,225,248,295 air entrainment 44,245 air preheaters 511 annular clearance 75-’33 annular-recess pads 7,112123,133,907,422-425 antennas 5,508 apparent bulk modulus -> equivalent b. m. attitude angle 292 Bernoulli equation 83 boring machines 6 boundary layer 79 bulk modulus 42,45,46,47 bulk modulus (equivalent) -> equivalent b. m. capillaries -> restrictors (laminar-flow r.) cavitation 48,104,108,133,245,262,288 characteristic equation 322,323 circular-recess pads 7,76-77,91111,123,128, 305-307,4210 -422,469,521 clearance -> film thickness clearance (radial) -> radial clearance compensated supply 16,17-30,31,88,91,153 172, 173,180-186,192-213,220-221,312320, 334-339,415-433 compensating devices -> restrictors compressibility 42,47,326 conical bearings -> tapered bearings conical pads -> tapered pads constant-flow supply -> direct supply constant- pressure supply -> compensated supply constitutive equations 55 continuity equation 54- 58,59,61,64,77,81,107, 111 contraction coefficient 83 cooler 460 correction factors 96,98,99,100, 101,105,111,118, 119,122,126,127 Couette flow 69,246,288 critical speed 73,74,356
cryogenic fluids 245 cylindrical pads 9, 136-141,151,233 damping coefficients (journal bearings) 354, 356,360 damping factor 321,342,356 density 42-49,310 design hybrid bearings 296-297 multirecess journal bearings 251-!HI multirecess thrust bearings 263 opposed-pad bearings 213-218 single-pad bearings 172-1% spherical bearings 278 tapered bearings 269 Yates bearings 283-285 diaphragm bearings 27 direct supply 16,17,SO,32,88,91,148153,173, 177-180,188192,219,ZS- 227,230,311312,
333-334,988-381 discharge coefficient 84,85 displacement (nondimens.) --> eccentricity dynamic viscosity --> viscosity 6% dynamics 301-361,472,474,521eccentricity hybrid bearings 292 multipad journal bearings 233 multirecess journal bearings 240,242 opposed- pad bearings 187 direct supply 192 flow dividers 205 laminar-flow restrictors 195 screw-nut assemblies direct supply 219 laminar-flow restrictors 220 single-pad bearings 89,302 Yates bearings 279 effective area 87 annular-recess pad 113,118,120 circular-recess pad 91,96,101,105,111 cylindrical pad 139
538
HYDROSTATIC LUBRICATION
effective area (continued) flow rate (continued) sing1e - pad bearings (continued) infinite-length pad 369 multirecess thrust bearings 261 infinite-stiffness valves 169,171 rectangular pad 133 laminar-flow restrictors 156,157 screw-nut assembly 145 spool valves 165,167 self- regulating bearings 223 slideways 231 spherical pad 129,131,133 spherical bearings 276 tapered pad 124,125,127 spherical pad 129,131,132 Yates bearings 280 tapered bearings 266,271 efficiency losses 365,377,380,384,416,417,433 tapered clearance 124,127 electric analog field plotter 468 tapered pad 125 electronic compensators 23 Yates bearings 283 electronic control 23,502 foam 44,48,51,52 energy equation CW-69,107,448 frequency response 304,328-321,357 equivalent bulk modulus 46,47,310 friction area experimental tests 485-482 cylindrical pad 141 feed drives 492-433 multirecess journal bearings 247 film thickness rectangular pad 137 multirecess thrust bearings 261 Yates bearings 283 opposed-pad bearings friction coefficient direct supply 191 annular-recess pad 115 orifices 201 circular-recess pad 94 single-pad bearings 89 infinite-length pad375,431 compensated supply 154 friction force 68 constant-flow valves 161,162 infinite- length pad 72,373,431 diaphragm- controlled restrictors 169 infinite-length strip 71 direct supply 150,152 recess 74 laminar-flow restrictors 156,157 rectangular pad 136,415 orifices 158,159 friction moment 68 spool valves 163,167 annular clearance 76 finite-differencemethod 96,107,128,135,236 annular-recess pad 115 finite-elementmethod 135,139,142,236,239,261, circular-recess pad 77,92 264 cylindrical pad 141 flash-point 50,52 hybrid bearings 294 flexible-plate bearings 26,470 multirecess journal bearings 247 flow dividers -> restrictors multirecess thrust bearings 262 flow rate 66 - 67,87.309 spherical pad 131,132 annular clearance 75 tapered pad 124,127 circular-recess pad 76,91,98 Yates bearings 283 hybrid bearings 289,294 friction power 87,88,90 infinite-length pad 71,369,371,426 annular-recess pad 115,423,424 infinite-length strip 70 circular-recesspad 94,111,420,421 inherently compensated bearings 172 cylindrical pad 141 multipad journal bearings 233 hybrid bearings 291,294,295 multirecess journal bearings 245 infinite-length pad373,431 multirecess thrust bearings 261 multipad journal bearings 233 opposed-pad bearings 186 multirecess journal bearings 246,247 flow dividers 205,207,210 multirecess thrust bearings 262 laminar-flow restrictors 193,195 opposed- pad bearings orifices 198 direct supply 189,191 orifices 84 flow dividers 205 pipes 77,78 laminar-flow restrictors 195 rectangular pad 133,417 orifices 198 screw- nut assemblies 220 rectangular pad 137 self- regulating bearings screw - nut assembly 146 compensated supply 229 self-regulating bearings 224 direct supply (constant pressure) 227 direct supply (constant flow) 225 single- pad bearings direct supply (constant pressure) 227 compensated supply 154 single- pad bearings constant-flow valve8 161 compensated supply 154 diaphragm- controlled restrictors 169 constant-flow valves 161
SUSJECT lNDEX
539
journal bearings 9- 10,13,31,89,472-458 friction power (continued) multipad 9,233- 234,348- 349 single- pad bearings (continued) multirecess 10,11,236,239-& ! O, 349-360,485 direct supply 150,153 laminar-flow restrictors 157 kinematic viscosity 38,40 spherical bearings 277 Laplace equation SS-aS,133,135 spherical pad 131,133 lathes 6,491 tapered bearings 266,267 load capacity 66,87,281 tapered pad 124,125,127 annular-recess pad 113 Yates bearings 283 circular-recess pad 76,91,98 gas solubility 43,44 hybrid bearings 292,293 Grashof number 528 hydrostatic lift 142 grinding machines 6,491 infinite-length pad 71,366,369,426 hybrid bearings 7,8,10,14,89,105,142,250,288inherently compensated bearings 172 288 multipad journal bearings 233 hydraulic circuit 31-32 multirecess journal bearings 240,242,244,249 hydraulic circuits 613-617 multirecess thrust bearings 261,262 hydraulic diameter 82 opposed- pad bearings 186 hydraulic resistance 87,88,90,151 constant-flow valves 201 annular clearance 76 direct supply 188,191,192 annular-recess pad 113,118,119,120 flow dividers 205,207,210 laminar-flow restrietors 193,195 circular-recesspad91,98,101,105,111 cylindrical pad 139 orifices 198 diaphragm- controlled restrictors 167,168 rectangular pad 415,417 infinite-length pad 71,426 screw - nut assemblies infinite-length strip 70 direct supply 219 laminar-flow restrictors 155,290,426 l a m i n a r - flow restrictors 220 multirecess journal bearings 238 self- regulating bearings 223 multirecess thrust bearings 261 compensated supply 229 orifices 157,443 direct supply (constantflow) 225 pipes 77 direct supply (constant pressure) 227 rectangular pad 133 single - pad bearings screw-nut assembly 145 compensated supply 153 self- regulating bearings 223 constant-flow valves 161 spherical pad 131,133 diaphragm- controlled restrictors 169 spool valves 162,165,167 direct s u p ~ l y150,152 tapered pad 124,125,127,128 infinite- stiffness valves 171 Yates bearings 279 laminar-flow restrietors 156 hydrodynamic load capacity 89,244,254 orifices 158,159 hydrostatic lifts 141-143,298 spool valves 163,167 inertia effects slideways annular-recess pad 120-122 direct supply 231 circular-recess pad 109-108 1ami n ar-flow restrictors 231 multirecess journal bearings 245 spherical bearings 276,277,278 multirecess thrust bearings 262 spherical pad 129,131 spherical pad 132 tapered bearings 265,268,269,271,273,275 tapered pad 125-128 tapered pad 125 inertia parameter 103,108,133 Yates bearings 279,281 infinite-length pad 71-72,362,366,425,447 lubricants 35 - I infinite-length strip 69 mineral oils 35,36-51 inherently compensated bearings 16,26-28,31, synthetic lubricants 35,38,52 172 lumped resistances -> thin-land method inlet length 79-80,100 machine tools 6,483-Soc inlet losses 80 measuring instruments 6 annular-recess pad 118 mechanical models 301,320,322 circular-recess pad W-100 milling arm 501,502 hybrid bearings 290 milling machines 6 instability 47 mills 5,505 interface restrictor bearings 30 misalignment annular-recess pad 117-118 IS0 classification of lubricants -> viscosity system for industrial lubricants circular-recess pad 95-98 Johnson drive 494 multirecess journal bearings 245, 255
540
HYDROSTATIC LUBRICATION
misalignment (continued) power ratio (continued) screw-nut assembly 145 infinite-length pad 376 tapered pad 128-128 multipad journal bearings 234 mixing length 81,106 multirecess journal bearings 248,254 momentum equations 55 opposed- pad bearings momentum torque 247,295 constant-flow valves 203 multipad journal bearings --> journal bearings direct supply 189,191 multiple pumps 501,616-617 flow dividers 205 multirecess bearings 7,8,236-285 laminar-flow restrictors 195, 196 multirecess journal bearings --> journal orifices 198,201 bearings self- regulating bearings multirecess thrust bearings --> thrust bearings compensated supply 229 50 naphthenic oils 36,39,41,42,49, direct supply (constant flow) 225 natural frequency 321,342,356 direct supply (constant pressure) 227 Navier- Stokes equations 54-58,61,64,77,81,107, single-pad bearings 174 111 compensated supply 154 Newtonian fluids 37 direct supply 150 Nusselt number 526,527,528,529,530 l a mi n a r - flow restrictors 157 Nyquist method 323 spherical bearings 278 oiliness 41,50,52 tapered bearings 268 oils --> lubricants Yates bearings 284 opposed- pad bearings 7,8,9,14,31,186218,331- Prandtl number 526 339 prediction- correction method 98 optimization 362- 446 preheaters 5 annular-recess pad 113,445 pressure 40,42,44,55 given flow rate 423 pressure ratio 87,90,154,193,196,198,203,205, given load 424 211,231,240,242,254,271,272,279,293,426, given pressure 423 427,434,443 circular-recess pad 92,445 pumping power 87,88 given flow rate 420 annular-recess pad 113,423,424 given load 421 circular-recess pad 91,420,421 given pressure 420 cylindrical pad 140 cylindrical pad 140 hybrid bearings 291,294 hybrid bearings 296 infinite- length pad 366,369,371,426,429 infinite- length pad 585-416,434443 mukipad journal bearings 233 given flow rate 3 8 5 - S multirecess journal bearings 245 given load 406-415,438,443 multirecess thrust bearings 261 given pressure 395-406,434-437 opposed- pad bearings multipad journal bearings 234 direct supply 189,191 multirecess journal bearings 251,252,253,254 flow dividers 205 opposed- pad bearings 213,214 laminar-flow restrictors 195 rectangular pad 135,445 orifices 198 given flow rate 416 rectangular pad 135 given load 418 self-regulating bearings 224 given pressure 417 compensated supply 229 self- regulating bearings 224 direct supply (constant flow) 225 single-padbearings 173,174,175 direct supply (constant pressure) 227 tapered bearings 270 single- pad bearings Yates bearings 284 compensated supply 154 orifices --> restrictors constant-flow valves 161 ovality 255 direct supply 150,153 oxidation 50,51,52 laminar-flow restrictors 156,157 paraffinic oils 36,39,41,42,49,50 spherical bearings 278 parallelism error --> misalignment tapered bearings 266 pitch error 146,218,219,220 Yates bearings 283 plastic throttle --> restrictors (elastic pumps 511 capillaries) r a c k - worm assemblies 494 Poiseuille flow 69,246,288 radial clearance 253,272,284 pour-point 50,51,52 recess flow recirculation 73-74,137,245,246,288 power ratio 87,89,90 recess pressure hybrid bearings 291,292,296 circular-recess pad 76,91
SUBJECT INDEX
recess pressure (continued) hydrostatic lift 143 infinite- length pad 71,366,371,429 opposed- pad bearings direct supply 188,191,192 laminar-flow restrictors 192 rectangular pad 415,417 self-regulating bearings Compensated supply 228 single- pad bearings compensated supply 153 constant-flow valves 161 direct supply 149,150 infinite- stiffness valves 169 slideways direct supply 230 laminar-flow restrictors 231 Yates bearings 280 rectangular pads 7,133-137,416419,470 reference bearings 29-30 restrictors 16,18,19,30,468 constant-flow valves 21,30,32,160162,201-
541
Sommerfeld hybrid number --> velocity parameter specific heat 49 speed enhancement factor 244 speed parameter --> velocity parameter spherical bearings 8,11,275-279,4%-478 spherical pads 8,128133,151 spindles 6,13,483-490,513 squeeze coefficient 302,304,306-3EB annular-recess pad 307 circular-recess pad 305 opposed- pad bearings 333 rectangular pad 309 screw-nut assembly 308 self-regulating bearings 339 tapered pad 307 squeeze parameter 305,321 stability 301,304,322- 326,342,363,356 steady rests 6,491 stick-slip 4,6 stiffness 87,89,90 infinite-length pad 366,369,371,426,429,443, 444 204,444 diaphragm- controlled restrictors 20,23,29, lubricant 310,351 31,167-169,209,316318,337 multirecess journal bearings 242,255 elastic capillaries 19 multirecess thrust bearings 261 elastic orifices 20 opposed-pad bearings 187 flow dividers 17,23,24,31,204-213,336-339, constant-flow valves 201 487,515-516 direct supply 189,191,192 infinite- stiffness valves 22,24,169172,318. flow dividers 206,209,211 320 laminar-flow restrictors 195 laminar-flow restrictors 18,30,32,78,156orifices 198,201 157,192197,240,261,266,276,290,313, rectangular pad 415 425-426,461-463,486 screw-nut assemblies orifices 18,30,8385,167-lS0,lm-201,313,443 direct supply 219 spool valves 20,23,162-167,204,207,315 318 laminar-flow restrictors 221 Reynolds equation 58-65,70,75,95,98,E!O,W, self- regulating bearings 126,128,133,138,144,236,238,261,264,276, compensated supply 229 289,305,308 direct supply (constant flow) 226 Reynolds number 80 direct supply (constant pressure) 227 annular clearance 76 single-pad bearings annular-recess pad 118 compensated supply 155 circular-recess pad 99,101,105 constant-flow valves 161,162 infinite-length strip 71 diaphragm- controlled restrictors 169,318 multirecess journal bearings 245,247 direct supply 150,153,312 pipes 78,290 laminar-flow restrictors 156,157,314 recess flow 74,247,267 orifices 158,314 rotary tables 496-603 spool valves 165,167,318 roughness 4,255 tapered bearings 266,269,271,273 Routh criterion 322,323 supply pressure 88 SAE classification of lubricants 38 self-regulating bearings screw- nut assemblies 6,8,30,31,144146,218 direct supply (constant flow) 225 222,308,418 - 4so,492,493 single- pad bearings self-aligning pads 491,503-511 compensated supply 153 self-regulating bearings 7,9,28,30,222-229, direct supply 149 519-621 tapered bearings 8,11,13,263275,488 shear stress 67,131 tapered pads 8,123-lzS,907 shoe bearings (self- aligning) --> self- aligning telescopes 4,s pads temperature 39,42,44,151153,157,159,165,192, single-pad bearings 89,149-186,320-322 197,198,204,447 - 466 slideways6, 14,31,229-233,314-348,480,496-603
542
HYDROSTATIC LUBRICATION
temperature rise 88 total power (continued) hybrid bearings 295 infinite-length pad375,378,381,432,433 multipad journal bearings 234 multirecess journal bearings 251, 252 multirecess journal bearings 248 opposed- pad bearings 214 opposed- pad bearings rectangular pad 415,417 direct supply 189 single- pad bearings 173,174 flow dividers 205 transfer function 304,326 laminar-flow restrictors 195 multirecess journal bearings 357 orifices 198 opposed- pad bearings 333 self- regulating bearings direct supply 334 compensated supply 229 flow dividers 338 direct supply (constant flow) 225 laminar-flow restrictors 334 direct supply (constant pressure) 227 self- regulating bearings single- pad bearings constant flow 341 compensated supply 154 constant pressure 342 constant-flow valves 161 single- pad bearings 320-322 direct supply 151 controlled restrictors 318 tapered bearings 268 direct supply 312 testing rigs 512 laminar-flow restrictors 314 thermal conductivity 49,448 orifices 314 thermal decomposition 50 tribological system 465,467 thermal effects turbulence 80-82 annular-recess pad 123 a n n u l a r - recess pad 118 120 circular-recess pad 107-111 circular-recess pad 101-103 thermal flow 447-434 multirecess journal bearings 245,246 thin -land method 238,255 recess flow 74 thrust bearings 7-9,488-472 valves --> restrictors multirecess 260-a83,487 velocity parameter 243,244,289,292,376 opposed- pad -> opposed- pad bearings vibration attenuators 6 single-pad --z single-pad bearings viscosity 36-41,44,45,55,15% 153,157,159, 192, tilting error --> misalignment 198,212,215,262,297,483-484 tilting pads -> self-aligning pads viscosity index 39,50,52 tilting stiffness (multirecess thrust b.) 262 viscosity parameter 110 tolerances 191,196,198,203,211,215,227,253,297viscosity system for industrial lubricants 38 total power 87,88 whirl instability 349,353,356,357 annular-recess pad 423,424 worm-rack assemblies 501 circuIar-recesspad420,421 Yaks bearings 12,279-285
