FLUID FILM LUBRICATION OSBORNE REYNOLDS C ENT ENARY
TRIBOLOGY SERIES 11
FLUID FILM LUBRICATION OSBORNE REYNOLDS CENTE NARY edited by
D. DOWSON, C.M. TAYLOR, M. GODET AND D. BERTHE
Proceedingsof the 13th Leeds-Lyon Symposium on Tribology, held in Bodington Hall, The University of Leeds, England 8-1 2 September 1 9 8 6
ELSEVIER Amsterdam - Oxford - New York - Tokyo 1987 For the Institute of Tribology, Leeds University and The lnstitut National des Sciences Appliquees de Lyon
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Libran of Congress Catalogingin-Publication Data
Leeds-Lyon Symposium on Tribology (13th : 1986 : University of Leeds). Fluid film lubrication--Osborne Reynolds centenary. (Tribology series ; 11) Bibliography: p. 1. Fluid-film bearings--Congresses. 2. Reynolds, Osborne, 1842-1912--Congresses. I. Dowson, D. 11. University of Leeds. Institute of Tribology. 111. Institut national des sciences a p p l i q d e s de Lyon. IV. Title. V. Series. TJ1073.5.L44 1986 621.8'22 87-19944 ISBX 0-444-42856-9
ISBN 0-444-42856-9 (Vol. 1 1 ) ISBN 0-444-4 1677-3 (Series)
0Elsevier Science Publishers B.V., 1987 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./ Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the USA - 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 USA. All other copyright questions, including photocopying outside of the USA, should be referred t o the copyright owner, Elsevier Science Publishers B.V., unless otherwise specified. For pages 3-13,27-47,
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CONTENTS Introduction Session I Session I1
Session I11
Session IV
Session V
Session VI
Session VII
.................................................................................................................................................................................
ix Keynote address Osborne Reynolds A. CAMERON ........................................................................................................................................ History The contribution of the Department of Scientific and Industrial Research to the study of hydrodynamic lubrication -The work of N.P.L. and N.E.L. F.T. BARWELL .......................................................................................................................................... 17 Historical aspects and present development on thermal effects in hydrodynamic bearings M. FILLON, J. FRENE and R. BONCOMPAIN .................................................................................... 27 Michell and the development of tilting pad bearings 49 J.E.L. SIMMONS and S.D. ADVANI ...................................................................................................... Journal bearings An approximate global thermal analysis of journal bearings D.F. WILCOCK ........................................................................................................................................... 59 Negative pressures in statically and dynamically loaded journal bearings S. NATSUMEDA and T. SOMEYA .......................................................................................................... 65 Mixing inlet temperatures in starved journal bearings H. HESHMAT and P. GORSKI ................................................................................................................ 73 Starvation effects in two high speed bearing types D.T. GETHIN and J.O. MEDWELL ............................................................................................ Thrust bearings (1 ) Three dimensional computation of thrust bearings 95 C.M.McC. ETTLES .................................................................................................................................... Parametric study and optimization of starved thrust bearings H. HESHMAT, A. ARTILES and 0. PINKUS ....................................................................................... 105 Tilting pad thrust bearings tests - Influence of three design variables W.W. GARDNER ....................................................................................................................................... 113 An experimental study of sector-pad thrust bearings and evaluation of their thermal characteristics T.G. RAJASWAMY, T. MURALIDHARA RAO and B.S. PRABHU .................................................. 121 Hard-on-hard water lubricated bearings for marine applications P.J. LIDGITT, D.W.F. GOSLIN, C. RODWELL and G.S. RITCHIE ................................................. 129 Thrust bearings ( 2 ) Inlet boundary condition for submerged multi-pad bearings relative to fluid inertia forces A. MORI and H. MORI .............................................................................................................................. 141 Pressure boundary conditions a t inlet edge of turbulent thrust bearings 149 H. HASHIMOTO and S . WADA ............................................................................................................... Dynamic analysis of tilting pad thrust bearings A. BENALI, A. BONIFACIE and D. NICOLAS ..................................................................................... 157 Hydrodynamically lubricated plane slider bearings using elastic surfaces C. GIANNIKOS and R.H. BUCKHOLZ .................................................................................................. 165 Elasto-hydrodynamic lubrication (1) Solving Reynolds’ equation for E.H.L. line contacts by application of a multigrid method A.A. LUBRECHT, G.A.C. BREUKINK, H. MOES, W.E. ten NAPEL and R. BOSMA ...................175 The use of multi-level adaptive techniques for E.H.L. line contact analysis 183 R.J. CHITTENDEN, D. DOWSON, N.P. SHELDRAKE and C.M. TAYLOR .................................. Solutions for isoviscous line contacts using a closed form elasticity solution 191 R. HALL and M.D. SAVAGE .................................................................................................................... Elastohydrodynamic lubrication ( 2 ) Parametric study of performance in elastohydrodynamic lubricated line contacts B.J. HAMROCK, R.T. LEE and L.G. HOUPERT ................................................................................. 199 Elastohydrodynamic lubrication of point contacts for various lubricants G. DALMAZ and J.P. CHAOMLEFFEL ................................................................................................. 207 A numerical solution of the elastohydrodynamic lubrication of elliptical contacts with thermal effects A.G. BLAHEY and G.E. SCHNEIDER ................................................................................................... 219 A full E.H.L. solution for line contacts under sliding-rolling condition with a non-Newtonian rheological model S.H. WANG, T.Y. HUA and H.H. ZHANG ............................................................................................. 231
VI
Session VIII
Session IX
Session X
Session XI
Session XI1
Session XI11
Elastohydrodynamic lubrication (3) Elastohydrodynamic lubrication of grooved rollers G. KARAMI, H.P. EVANS and R.W. SNIDLE ...................................................................................... The lubrication of elliptical conjunctions in the isoviscous-elastic regime with entrainment directed along either principal axis R.J. CHITTENDEN, D. DOWSON and C.M. TAYLOR ....................................................................... Effect of surface roughness and its orientation on E.H.L. D.Y. HUA, S.H. WANG and H.H. ZHANG ............................................................................................. The elastohydrodynamic behaviour of simple liquids a t low temperatures C. WATERHOUSE, G.J. JOHNSTON, P.D. EWING and H.A. SPIKES ........................................... Elastohydrodynamic lubrication (4) A method for estimating the effect of normal approach on film thickness in elastohydrodynamic line contacts N. MOTOSH and W.Y. SAMAN .............................................................................................................. Transient oil film thickness in gear contacts under dynamic loads A.K. TIEU and J. WORDEN ..................................................................................................................... A full numerical solution for the non-steady state elastohydrodynamic problem in nominal line contacts Y.W. WU and S.M. YAN ............................................................................................................................ The lubrication of soft contacts C.J. HOOKE ................................................................................................................................................ Lubricant rheology Pressure viscosity and compressibility of different mineral oils P. VERGNE and D. BERTHE .................................................................................................................. Measurement of viscoelastic parameters in lubricants and calculation of traction curves P. BEZOT and C. HESSE-BEZOT ........................................................................................................... High-shear viscosity studies of polymer-containing lubricants J.L. DUDA, E.E. KLAUS, S.C. LIN and F.L. LEE ................................................................................. Properties of polymeric liquid lubricant films adsorbed on patterned gold and silicon surfaees under high vacuum M.R. PHILPOTT, I. HUSSLA and J.W. COBURN ............................................................................... B e a r i n g dynamics (1) Identification of fluid-film bearing dynamics: time domain or frequency domain? J.E. MOTTERSHEAD, R. FIROOZIAN and R. STANWAY ................................................................ The influence of grooves in bearings on the stability and response of rotating systems P.G. MORTON, J.H. JOHNSON and M.H. WALTON ......................................................................... Theoretical and experimental orbits of a dynamically loaded hydrodynamic journal bearing R.W. JAKEMAN and D.W. PARKINS .................................................................................................... The effect of dynamic deformation on dynamic properties and stability of cylindrical journal bearings Z. ZHANG, Q. MA0 and H. XU ................................................................................................................ Bio-tribology Development of transient elastohydrodynamic models for synovial joint lubrication T.J. SMITH and J.B. MEDLEY ............................................................................................................... An analysis of micro-elasto-hydrodynamic lubrication in synovial joints considering cyclic loading and entraining velocities D. DOWSON and Z.M. J I N ....................................................................................................................... Lubricating film formation in knee prostheses under walking conditions T. MURAKAMI and N. OHTSUKI ......................................................................................................... S u p e r l a m i n a r flow i n b e a r i n g s A review of superlaminar flow in journal bearings F.R. MOBBS ................................................................................................................................................ Frictional losses in turbulent flow between rotating concentric cylinders C.G. FLOYD ................................................................................................................................................ Turbulence and inertia effects in finite width stepped thrust bearings A.K. TIEU .................................................................................................................................................... A theory of non-Newtonian turbulent fluid films and its application to bearings J.F. PIERRE and R. BOUDET .................................................................................................................
239
247 261 267
279 285
291 299
309 317 325
333
339
347 355
363
369
375 387
395 403 41 1
417
VII
Session XIV
Bearing analysis
Session XV
A new numerical technique for the analysis of lubricating films. Part I: Incompressible, isoviscous lubricant C.H.T. PAN, A. PERLMAN and W. LI .................................................. The boundary element method in lubrication analysis D.B. INGHAM, J.A. RITCHIE and C.M. TAYLOR .................................................. Thermohydrodynamic analysis for laminar lubricating films H.G. ELROD and D.E. BREWE ............................................................................................................... The lubrication of elliptical contacts with spin D. DOWSON, C.M. TAYLOR and H. XU ................................................................................................ Bearing dynamics (2) Oil film rupture under dynamic load? Reynolds’ statement and modern experience ................................................................. O.R. LANG .............................................. efficients in a hydrodynamic The influence of cavitation on the non-li journal bearing ............................................................................ R.W. JAKEMAN .................................................. Effects of cavity fluctuation on dynamic coeffici .............................................. K. IKEUCHI and H. MORI ................................ Investigation of static and dynamic characterist T. HUANG, Y. WANG and S. WEN .........................................................................................................
Session XVI
451
467
473 481 487
Oil film instability Instability of oil film in high-speed non-contact seal M. TANAKA and Y. HORI ............................................................................................ Instability of the oil-air boundary in radial-grooved bearings A. LEEUWESTEIN .......................................................................................................... An experimental study of oil-air interface instability in a grooved rectangular pad thrust bearing D.J. HARGREAVES and C.M. TAYLOR ................................................................................................
Session XVII
443
505
Gas bearings The performance of an out-of-balance rotor supported in self acting gas bearings .................................................................................................................... H. MARSH ................ Comparison of theoretical characteristics of two types of externally pressurized, gas lubricated, compliant surface thrust bearings ........................................................................................ 525 K. HAYASHI and K. HIRASATA .... ate performance of a compliant surface aerostatic An experimental investigation of the s thrust bearing ..................................................................... 533 D.A. BOFFEY, G.M. ALDER ............. iment The effect of finite width in foil bearin ..........539 J.G. FIJ NVANDRAAT .....................................................................................
Session XVIII
Session XIX
Session XX
Seals The influence of back-up rings on the hydrodynamic behaviour of hydraulic cylinder seals ......................................................... H.L. JOHANNESSON and E. KASSFELDT ....................... Study on fundamental characteristics of rotating lip-type oil M. OGATA, T. FUJI1 and Y. SHIMOTSUMA ............................................................................... Influence of pressure difference and axial velocity on a spiral-groove bearing for a moving piston F. BREMER, E.A. MUIJDERMAN and P.L. HOLSTER ..................................................................... Elastohydrodynamic lubrication of an oil pumping ring seal G.J.J. van HEIJNINGEN and C.G.M. KASSELS .................................................................................. Machine elements (1) - Ring oiled bearings Thermal network analysis of a ring-oiled bearing and comparison with experimental results D. DOWSON, A.O. MIAN and C.M. TAYLOR .......................................... ................................. Performance characteristics of the oil ring lubrica K.R. BROCKWELL and K. KLEINBUB ............ ................. ............................................... Machine elements ( 2 ) - Cams and tappets Mixed lubrication of a cam and flat faced follower D. DOWSON, C.M. TAYLOR and G. ZHU .............................................................................................. Elastohydrodynamic film thickness and temperature measurements in dynamically loaded concentrated contacts: eccentric cam-flat follower .............................. H. van LEEUWEN, J . MEIJER and M. SCHOUTEN .........................
545
561 569
579 587
599
611
VIII
Machine elements ( 3 ) - Rolling bearings Study of the lubricant film in rolling bearings; effects of roughness P. LEENDERS and L.G. HOUPERT ...................................................................................................... The prediction of operating temperatures in high speed angular contact bearings R. NICHOLSON ......................................................................................................................................... Study on lubrication in a ball bearing T. FUJII, M. OGATA and Y. SHIMOTS Special lecture Continuity and dry friction: An Osborne Reynolds approach M. GODET and Y. BERTHIER .............................................................................................. Written discussions and contributions ............................ List of authors ........................................................................ List of delegates .. Session XXI
629
639
IX
INTRODUCTION The thirteenth Leeds-Lyon Symposium on Tribology was held from 8th to 12th September, 1986 at Bodington Hall, The Univerisyt of Leeds. It was particularly pleasing to welcome a strong contingent from our co-sponsors a t the Institut National des Sciences Appliquees de Lyon led by Professors Maurice Godet and Daniel Berthe, and in total some 180 delegates from over twenty countries participated in the Symposium. On this rather special occasion the meeting was devoted to the topic of “Fluid Film Lubrication” in celebration of the centenary of the publication of the classical paper of Professor Osborne Reynolds in which he identified the mechanism of hydrodynamic lubrication and which was entitled,
“On the theory of lubrication and its application to M r Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil ”. The range of subjects addressed by the series of Leeds-Lyon Symposia since their inception in 1974 is now considerable, as the list of titles below testifies: 1. 1974 ( Leeds) -Cavitation and Related Phenomena in Lubrication 2. 1975 ( Lyon - Super Laminar Flow in Bearings 3. 1976 (Leeds) - The Wear of NonMetallic Materials 4. 1977 (Lyon) - Surface Roughness Effects 5 . 1978 ( Leeds) - Elastohydrodynamics and Related Topics 6. 1979 (Lyon) - Thermal Effects in Tribology 7. 1980 (Leeds) - Friction and Traction 8. 1981 (Lyon) - The Running-In Process in Tribology 9. 1982 ( Leeds) - Tribology of Reciprocating Engines 10. 1983 (Lyon) - Numerical and Experimental Methods in Tribology
11. 1984 (Leeds) -Mixed Lubrication and Lubricated Wear 12. 1985 (Lyon) - Mechanisms and Surface Distress 13. 1986 (Leeds) - Fluid Film Lubrication - Osborne Reynolds Centenary The 1986 Symposium was given a lively send off by Professor Alastair Cameron, who presented the Keynote Address, “Osborne Reynolds”. His provocative paper, delivered in characteristic style, gave much food for thought for the delegates as they travelled immediately afterwards to the Symposium Dinner at The Old Swan Hotel, Harrogate. The Guest of Honour was Sir Derman Guy Christopherson, formerly a Professor of Mechanical Engineering a t the University of Leeds, who gave a stimulating after-dinner speech. Other distinguished guests were Dr. R. Delbourgo, Scientific Counsellor at the French Embassy in London, and Mr. A. Pattison, Director of the Regional Office of the British Council based in Leeds. There were twenty-one sessions at the Symposium, with some seventy papers presented. Sessions were devoted to History, Journal Bearings, Thrust Bearings, Elastohydrodynamic Lubrication, Lubricant Rheology, Bearing Dynamics, Biotribology, Superlaminar Flow in Bearings, Bearing Analysis, Oil Film Instability, Gas Bearings, Seals and Machine Elements. Because of the large number of papers it proved necessary to hold parallel sessions on the morning of Thursday, 1l t h September. The standard of presentations was once again high and the discussion lengthy and lively. Indeed, such was the enthusiasm that the delegates themselves organised an impromptu evening discussion to throw light on that enigmatic subject, the pressure spike in elastohydrodynamic lubrication. For this Symposium, written discussions and contributions were invited, and we are pleased to include these, with authors’ responses where appropriate, in the volume of proceedings. Professor Godet presented a special lecture on the Tuesday evening entitled
X
“Continuity and Dry Friction: An Osborne Reynolds Approach”, and this paper, coauthored by Dr. Berthier, is also contained in the volume. We are indebted to the distinguished visitors who chaired the session presentations and ensuing discussion. Our warm thanks go to Professor J.F. Booker, Professor Y. Hori, Professor C.H.T. Pan, Professor B.J. Hamrock, Professor F.T. Barwell, Professor M. Godet, Professor R. Bosma, Professor H. Christensen, Professor A. Cameron, Dr.-Ing. O.R. Lang, Professor K. Aho, Professor H. Block, Professor H. Marsh, Mr P.G. Morton, Professor B.O.Jacobson, Professor H.G. Elrod, Professor D. Berthe and Professor W.O. Winer. Delegates to the symposium were honoured by a reception given by the City of Leeds a t the Civic Hall on the evening of Wednesday, 10th September. The Lord Mayor, Councillor Rose Lund, welcomed delegates individually and presided a t a dinner held in the magnificent surroundings of the City Banqueting Hall. On the afternoon of Thursday, 11th September, delegates travelled to the City of York. A comprehensive series of visits had been arranged, and careful planning by Mr Brian Jobbins, described with military precision a t a well-received talk the previous evening, ensured that no delegates were mislaid. Four tours were organised to The National Railway Museum, The Palace of the Archbishop of York, York Minister and the Jorvik Viking Centre. The visits were most successful, and we would like to thank those who so kindly assisted, including Sister Catherine of St. William’s College, Miss D. Wood of the Archbishop’s Palace and Mr. F.J. Bellwood, Chief Mechanical Engineer at the National Railway Museum. Delegates were
able to enjoy an evening river cruise after the visits, many studying the benefical influence of lubrication against the chill air, and a pleasant occasion was suitably terminated by dinner at the Viking Hotel. We are very pleased to acknowledge the financial support for the Symposium generously provided by British Petroleum International Ltd., Sunbury-on-Thames;The Glacier Metal Co. Ltd., Alperton; Nippon Steel Corporation, Hokkaido, Japan; Shell Research Ltd., Thornton; SKF Engineering and Research Centre BV, The Netherlands; and the United States Army, European Research Office, London. The organisation and smooth running of such a large Symposium could not proceed without the contribution of many. We would like to express our sincere thanks to our colleagues who so generously gave of their time and enthusiasm - Mrs. Sheila Moore, Mrs. Cath Goulborn, Mr. Stephen Burridge, Mr. Ron Harding, Mr. Brian Jobbins, Mr. David Jones and Dr. Nigel Wallbridge. This is the first occasion on which the volume of proceedings has been published by Elsevier Science Publishers BV. We are grateful to the staff of Elsevier for their efforts, and we look forward to a long and rewarding association. The 14th Leeds-Lyon Symposium on Tribology will be held in Lyon, France from 8th-11th September, 1987, with the title “Interface Dynamics”. We already look forward to joining our French colleagues, renewing acquaintames with old hands at the Symposia and welcoming new delegates. Duncan Dowson Chris Taylor
SESSION I INTRODUCTORY SESSION Keynote address: Osborne Reynolds Paper I(;)
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3
Osborne Reynolds A. Cameron
SUMMARY
This paper aims at giving some idea of Reynolds 78 page long 1886 paper, the basis of the theory of hydrodynamic lubrication. It also tries to suggest solutions for some of the puzzles in it.
First the background is described, then its contents are sketched bringing out the innovations, i.e. Reynolds capilliary viscometer and the viscosity law, the idea of a clearance bearing, boundary conditions as well as the achievement of Reynolds equation, together with its integration for a tilted pad and for Tower's bearing. The puzzles inherent in the paper are listed. Why it is so full of misprints; why the viscosity of Tower's Heavy Machine Oil was not measured; why such a clumsy method of integration was used; why interest in hydrodynamic theory stopped so abruptly after this paper? Some light on these questions is shed by looking at Reynolds' life, using letters found in the collection of Stokes's correspondence in the Cambridge University Library, as well as other little known sources. A paper by Clerk Maxwell seems of importance. Finally, the reason for Reynolds holding a slightly curved bowl in his hands in the well known portrait by Collier, is mentioned. INTRODUCTION We are here to celebrate the centenary of Reynolds' 1886 paper (1) with which the whole theory of hydrodynamic lubrication began. The size of this conference and the number of contributions tells its own story. In this talk I will deal only briefly with his life and work since this was covered comprehensively at the Osborne Reynolds centenary symposium. This was held to mark his election to the Chair of Engineering at Owens College, Manchester. The 263 pages of proceedings ( 2 ) start with an 82 page article by Jack Allen of Aberdeen on his life and work. Then acknowledged experts write on various aspects of his achievements. M.J. Lighthill deals with Turbulence, J.E. Ffowcs Williams on Turbulence generated noise. There are three contributions on Heat Transfer, three on Hydraulics and finally one on Tribology by our own F.T. Barwell. Surprisingly this book is very little known. The copy in the Cambridge University Library was book-shop new and I was its first borrower. A more accessible though brief biography and survey of his lubrication work is in Duncan Dowson's The background to History of Tribology (3). journal bearing development is described in Chapter 11 of Principles of Lubrication ( 4 ) .
I will try and shed some light on several puzzles in the 1886 paper, and also refer to material in Cambridge which (as far as my knowledge goes) has not been published before. I am indeed grateful f o r the opportunity to talk about Reynolds. The first lubrication paper I published, with Mrs. Winifred (now Lady) Wood (5), on the solution of the finite journal
bearing, was a continuation of the 1886 paper. As I have now retired it is very nice for me to give this valedictory paper, as my interests have moved away from hydrodynamics. BACKGROUND Reynolds' paper gives the analysis of the discovery of Beauchamp Tower ( 6 ) that a partial journal bearing, when running correctly, was supported on an oil film. He measured the friction in a report published in 1883, and then in 1885 gave the oil pressures at 9 points around the bearing. This work had been commissioned by the Institution of Mechanical Engineers in 1878 and stirred up considerable interest in Universities. The background to Beauchamp Tower's work is given in a paper marking the centenary of the Mechanicals' decision to investigate the problem ( 7 ) . Reynolds' paper was received at the Royal Society on December 29th, 1885. Their records show that it was reviewed by Lord Rayleigh and by Stokes. Rayleigh's hand written report was received on January 8th, 1886. He wrote that he was "of opinion that the paper is worthy of being printed in the Phil. Trans., both on account of the importance of the subject and of the ability with which it is treated". He remarks that "exception may be taken to isolated passages. I do not see in what sense there can be said to be 'two viscosities"'. He also thought that "there is a tendency to underrate what had been done in this subject before the author took it in hand and his attention might well be called to certain passages in the writings of Professor Stokes (e.g. vol. 1. Reprint of papers p. 99 and footnote)". It does seem that Rayleigh, who
4
took only a week to review the paper, did not appreciate quite how important this contribution was. Stokes reported February 11, the day the paper was read, but no record exists of his comments. It was voted on March 4 and sent on March 22 to the Cambridge University Press who have no records of its progress prior to publication. Reynolds' papers are long; this one is 78 pages. All his work is very hard to read. Horace Lamb (8) of hydrodynamics fame, and a former student of Reynolds sums up his papers accurately, "of their originality and value there can be no question, but it cannot be said that they are always easy to follow. The leading idea is in most cases simple. But the involved style of exposition which he adopted had a tendency to perplex all but determined students". J.J. Thompson the physicist, also a former pupil, commented that his papers made "very difficult reading".
I will try and give some idea of the contents of the 1886 paper. CONTENTS
1.
surface roughness. He gives a f gure for the roughness of each surface as 10 inches, so the minimum separation is double that, i.e. 5 microns. This section has a most up to date ring about it. He then brings in a major feature of his work; the importance of frictional heattng.
-t
4. The Equations of Hydrodynamics as Applied to Lubrication This section is really the kernel of the whole paper and is only 4 pages long. He starts with the (Navier) Stokes equations and making what are now the standard simplifications, produces Reynolds equation in two dimensions with the squeeze film term. 5.
He integrates the equations for plane tilted surfaces quite simply. The tilt was always considered fixed. The concept of a pivotted pad, free to alter its inclination was left for Michell ( 9 ) to invent 20 year-slater though Hall and deGuerin (10) doubted whether Michell pads, as originally designed, ever did tilt; their action being ascribed to pad distortion. The paper then goes on to treat cylindrical surfaces.
viscosity 6.
The first ? pages of Reynolds' paper, called "Introductory", are really a summary. Then come 8 pages defining "the two viscosities". Here he draws on his earlier work on Turbulence to show that the flow in bearings is laminar and friction varied directly with velocity and not with velocity squared. Turbulence produces his "second" viscosity. He then devised a simple syphon capilliary viscometer and with it measured the viscosity of olive oil between 60' and 120°F.
It is unfortunate that the description of his viscometer, and the emphasis on laminar flow appeared too late to influence Redwood, Eagler and Saybolt who designed orifice viscometers in the mid 1880's. (Ref. 4 Chapter 2). He correlated the variation of viscosity with temperature by an exponential relation, now known as Reynolds viscosity law. The first puzzle is, why did he measure only olive oil rather than the Heavy Machine Oil which Beauchamp Tower used for his pressure measurements. 2.
Clearance Bearing
Next follow some 13 pages on a "general view of the action of lubrication" which consists of diagrams of the velocity profiles of fluid flow in various configurations of plates and cylinders under both full and starved conditions. Talking about Tower's bearing he states t at there is a difference in radii of 0.77 x 10- inches between the bearing and 4" diameter shaft, even though Tower says they were "beautifully fitted". I think this is the first reference to a "clearance" bearing, though 0.4 parts per mil is small, even by today's standards.
9
3.
Integration of the Equations
Roughness and Temperature Rise
Of considerable interest is his realisation that the minimum film thickness is determined by
Cylinders, Integrals and Boundary Conditions
In order to apply his equations to a partial journal beari g he needed the int grals of These he de/(l+a cose) and dO/(l+a cos 8) obtains by expansion leading to equations with 9 coeficients of horrifying complexity, and takes some 15 pages to do it. Why Reynolds did not integrate them directly is another puzzle discussed below.
s
5.
He describes the correct boundary conditions, but in such a complex way that they lay hidden until Mrs. Wood h I ( 3 ) disinterred them in 1947 and christened them "Reynolds Conditions". In the same paper the other two conditions, Full and Half Sommerfeld were named. 7.
Elasticity, Heat, Comparison with Tower
He considers the effect of heat and elasticity, and shows that the peak carrying power of the bearing is when the eccentricity ratio is between 0.5 and 0.6. In the final 22 pages he draws all this together and in Table V, page 230, compares his theoretical values with Tower's experiments. The agreement is unbelievably good, as Figure 1 shows. This is redrawn from Figure 3 of his Plate 8. As a piece of historical detective work we used (11) inverse hydrodynamic methods to find out from Tower's measured pressures what was the actual film shape of Tower's bearing. As may be expected it was nothing like the one Reynolds postulated. It would be most valuable to rework Reynolds' analysis of Tower's data to put clearly the scale of Reynolds' achievement. His presentation is so complex that I doubt if anyone has followed all his steps. It would seem to be an ideal final year project €or a student with detective tendencies.
5
PUZZLES
A close reading of the paper produces several puzzles. 1. Why does the whole paper appear so very hurried and full of misprints? Kingsbury (12) notes ten mistakes in two pages. For instance, on page 191 the load on a tilted pad W is defined (equation 35) as "load per unit of breadth" and on the next page, referring to the same equation, W is now "load in lbs per square inch". 2. Tower measured journal friction in the 1883 report using 5 oils; olive, lard, sperm, rape and mineral. The pressures, with which Reynolds compares his theory, were taken, in 1885, only with Heavy Mineral Oil. Why did Reynolds measure the viscosity of olive oil and not get a sample of Tower's mineral oil? 3. The standard puzzle is why did he use this clumsy method of integration? 4. Why did Reynolds publish when Stokes, in the 1884 British Association Meeting at Montreal, announced that he had solved the problem? 5. Why did all academic interest in hydrodynamic lubrication cease after this paper?
A possible, and very tentative solution to some of these puzzles may be found in a study of the relations he had with Stokes and Lord Rayleigh. A brief biography will perhaps throw some light on this. Some small details, not widely known, will be added in here. BIOGRAPHY
A full account of his life, especially his time in Manchester, can be found in the centenary volume (2); here I give only brief details. He was born on August 23rd, 1842, in Belfast but went to Dedham Grammar School in Essex where his father was headmaster. Before going to Queens' College, Cambridge, in 1863 to read Mathematics (like his father before him), he spent a year in an engineering workshop. A few side lights on his character come from an obituary in the Queens' magazine. "The Dial", written by a Queens' Fellow, A. Wright, who had come to Queens' the same term as Reynolds, and wrote of their time together 50 years earlier (13). Wright says that Reynolds "was required to pass the Previous Examination which necessitated him studying the greek author, Aeschines, which he would never require again". On passing it at the earliest possible date Reynolds invited Wright "to his rooms to witness a holocaust of his classical books. In vain I pleaded that some of them would be useful to me. He had registered a vow and insisted on keeping it". Also Wright says, "At the end of his first year he was upset in some canoe races and missed much of the Mays". The Mays are the summer rowing events at Cambridge. The upset must have been quite considerable if it involved missing the May races. Unfortunately, the records of rowing at Queens' start only in 1870, so I have no knowledge of his rowing career. He sat the Mathematical Tripos in 1867 passing 7th out of a field of 119. Such a large
number is surprising considering that there were then only 1632 undergraduates in Cambridge, of which about 30% did not bother to take a degree. For comparison in 1985 some 180 people read mathematics out of 84 thousand undergraduates. In those days if one did not care to read Classics or Theology, mathematics was the only reasonable alternative.
I looked up the paper which Reynolds took on "Wednesday 16th Jan. 1867, 1* to 4", and question 4. vi i asked for the integral of dx/(a+b cos x) The puzzle is therefore why he did not integrate this directly when he met it later in the hydrodynamic theory of lubrication.
3.
Wright then says he was elected to a non resident fellowship in June, 1867, only 6 months after graduating. This involved only 3 visits a year to the College to keep,Audit and attend general meetings. Such a fellowship lasted 13 years. He went to Owens College (later Manchester University) as the first ever full time professor of engineering in Britain, on 26th March, 1868. Full details of his outstanding career there are in the centenary volume (2). One booklet of his, which seems to have been prompted by a domestic problem, and which ran to two editions, was published in 1872 and is entitled, "Sewer gas and how to keep it out of houses. A handbook on house drainage". It describes how to fit simple earthenware water traps. In it he writes, "It would be a calamity if the widespread alarm caused by the recent illness of the Prince of Wales were allowed to subside without producing a beneficial effect". The effect of this booklet on health legislation is beyond the scope of this paper. His retirement from Owens was forced by ill health in 1905. He went to St. Decuman's near Wantage, Somerset, and died on 21st February 1912. The West Somerset Free Press records tht he had been living in the Vicarage there and that Horace Lamb was at his funeral. ROYAL SOCIETY PAPERS Reynolds had much to do with Stokes, Secretary of the Royal Society and Lucasian Professor of Mathematics. An important person at the time was Lord Rayleigh, a member of the Mechanicals committee which sponsored Tower.
I think some light can be thrown on several of the puzzles mentioned earlier by looking at Reynolds relations with Stokes, who was in close contact with Lord Rayleigh. Beauchamp Tower, the discoverer of hydrodynamic lubrication, was acquainted with Rayleigh as Tower's cousin lived at Weald Hall, which adjoined Rayleigh's estate. The Cambridge University Library has a large collection of Stokes's correspondence, given to them by Sir Joseph Larmor, who collected many of them in a memoir ( 1 4 ) . An index to all this was compiled in 1976 (15), so I have been able to trace much of Reynolds' correspondence with Stokes.
6
Lamb ( 8 ) says that Stokes was ''a pioneer in Cambridge in the use of the typewriter"; hence many copies of his letters are typed. In the University Library collection is a copy of a letter he wrote to Tower on April 21st, 1884. This is given in full by Larmor (14) but as it concerns us so closely I thought it worth reproducing in full (Figure 2). It shows clearly that Stokes had found the solution of hydrodynamic lubrication of a journal bearing. He finishes his letter by saying, "I cannot at present work the thing out on account of my lectures". Tower's reply, which has not previously been published, I reproduce (Fig.3). In passing, I would like to suggest that the Institution should press for a plaque to be placed on the wall of 19 Great George Street to commemorate Tower's residence there. The discovery of hydrodynamic lubrication is no mean event
.
Larmor says that Stokes was very thorough in the preparation of his lectures, even though he had given the course in previous years. I thought it interesting to see what his lecture load was by looking through the University Reporter. I n the April 15, 1884, issue it announced "The Lucasian Professor will deliver a course of lectures on physical optics in the Easter term. They will begin on Friday April 25th at 1 p.m. and will be continued on every weekday except Thursday. The lecture fee will be one guinea. It is requested that the fees be paid to Messrs. Deighton Bell & Co.". Deighton Bell was (and still is) a Cambridge bookshop. This arrangement seems to be an early example of privatisation. Lord Rayleigh also lectured on Optics in other years. Thus Stokes's lectures started in the same week as he wrote to Tower. Looking through the Reporter it is easy to get distracted. For instance on March 24th, 1874, there i s the notice that the "Cavendish Laboratory is open daily 10-6 under Clerk Maxwell, Professor of Experimental Physics, for the use of any member of the University who may desire to acquire a knowledge of Experimental physics or to take part in physical researches". Things have certainly changed in the last 100 years. Unfortunately, Stokes never did complete the calculations. In the British Association meeting that same September at Montreal Lord Rayleigh, in his presidential address, said, "We may, I believe, expect from Professor Stokes a further elucidation of the processes involved". What has tantalised all of us who are interested in the history of this subject, is a report in the Montreal Daily Witness of 3rd September, 1884, which said, "Professor Osborne Reynolds gave a paper on 'The Friction of Journals' which was entirely theoretical and only of interest to the initiated". Despite repeated attempts no one has been able to find anything further. Were there "0 other reports of the papers of what must have been a very prestigious meeting? It is clear then that Stokes, very much an establishment figure, had announced he was working on the analysis of Tower's discovery. In
those days one did not invade another academic's field of work, especially a very senior one, without permission. A further side light on all this appears from other letters in the Library's collection. MAXWELL'S PAPER I n Phil. Trans. of 1879 Clerk Maxwell wrote a paper on stresses in rarified gases and added an appendix in May 1879 (16). Reynolds had discovered a phenomenon in gases which he called Thermal Transpiration. Maxwell then extended his theory and wrote (p. 249), "I think that this method is in some respects better than that adopted by Professor Reynolds, while I admit that his method is sufficient to establish the existence of the phenomena, though not to afford an estimate of their amount". Reynolds was somewhat put out by this and wrote a rather sharp letter to Stokes from Owens College (although normally he wrote from his home), of which I reproduce the first page (Fig. 4). All subsequent pages are lost, but the first page is enough t o shown his displeasure. It is indeed surprising that a letter to the Secretary of the Royal Society should be sent in such a ragged form. He writes, "Professor Maxwell proceeds to criticize my work and compare it to its disadvantage". Stokes's reply of 5th November was rather startling. It begins, "A little after 12 o'clock today Professor Maxwell passed quietly away without apparent pain". He goes on to discuss a proposal from Reynolds, but without the subsequent pages of the Reynolds letter it is not possible to know what Reynolds proposed, but it may have been a joint statement. Stokes goes on to say, "I have the right to demand that my name should be removed from the title". Reynolds' paper on this Thermal Transpiration later appeared in 1879,
(17). After this exchange the correspondence stops. There is a rather formal letter from Reynolds on 14th November 1888 which acknowledges the news that he had been awarded the Royal Medal. Nothing further exists until he invited, on 29th May, 1897, Stokes and his wife to stay at his home during their visit to Manchester on July 2nd, 1897. This invitation seems to have been declined. SOLUTION TO PUZZLES 2
Reynolds needed th integrals dg/(l+a cos 0 ) and dO/(l+a cos ')0 for the solution of pressure in Tower's bearing. The first one he had seen in the maths paper he took in 1863. Why did he not integrate them directly as there were several text books containing them? One is tempted to speculate that he could not do the integral in his final maths paper and so thought expansion was the only way. This puzzle will have to remain unsolved. A possible solution to the other puzzles might follow from the evidence of the letters in the Library collection. This is that there appears to have been a coolness between Reynolds on the one hand and the establishment figures of Stokes and Lord Rayleigh on the other. Remember also that Tower was a family
7
friend of Rayleigh as well as working on a project of which Rayleigh was a member of the supervising committee. Perhaps because of the somewhat acrimonious correspondence with Stokes over Maxwell's paper Reynolds did not mind further upsetting academic protocol by infringing territory marked out, as it were by Stokes. He wanted to make sure his paper appeared first. He did not know how far Stokes had progressed, hence he rushed its preparation and proof reading. Reynolds also would be chary of asking Tower for a sample of the Heavy Machine oil used in the bearing tests for viscosity measurements, firstly because Tower was a friend of Rayleigh; and secondly such a request would reveal Reynolds' progress on the problem. It may also explain why no academic studied Tower's next two reports, numbers 3 and 4. These gave clear evidence of an oil film between nominally parallel surfaces, so flatly contradicting Reynolds' theory. It would seem that there had been enough to worry over the reception of the 1886 paper without embarking on anything further. The enormous physical insight which Reynolds had could well have resulted in his discovering the mechanism of "parallel surface" thrust bearings. As it was, this had to wait for the bearing to be re-discovered by Fogg (18) and then analysed by Mrs. Wood and myself (19). The correct solution to it was suggested by Swift in the discussion to Fogg's paper but printed out of place. Its mechanism was finally proved optically by Robinson and me (20).
Reynolds, Stokes and Beauchamp Tower. It has produced a possible explanation of why the 1886 paper looks so rushed, why Reynolds only measured olive oil, and why no one spent any time considering Beauchamp Tower's 3rd and 4th reports after their publication. No amount of historical research can reduce, or for that matter enhance, the importance of the 1886 paper which founded the theory of hydrodynamic lubrication, which we are all here to celebrate. ACKNOWLEDGEMENTS Without the help of the members of the Cambridge University Library this paper could never have been written. In particular Dr. Elizabeth Leedham-Green of the archives section answered cheerfully and almost instantly any question put to her. I have also to thank Dr. Ian Wright of Queens' College, who looks after their archives, for finding the relevant copy of the Dial for me. The Editor of the West Somerset Free Press told me of the report of Reynolds' funeral. The archivist at the Royal Society gave me the details of the reviews. All these people I would like to thank for their willing help. REFERENCES 1.
There is an interesting modern parallel. After the Russian researcher, Ertel, having produced what is known as the "Grubin" EHL solution (211, defected to the West, no one in Russia felt it sensible to work in such a sensitive area. Russian work on EHL, from being many years ahead of the West, then dropped behind.
Reynolds, Osborne. On the Theory of Lubrication and its Application to Mr. BEAUCHAMF' TOWER'S Experiments, Including an Experimental determination of the Viscosity of Olive Oil. Phil. Trans. Roy. SOC. 1886, 177, 157-234 with one plate.
2.
Osborne Reynolds and Engineering Science Today. McDowell, D.M. and Jackson, J.D. Editors. Manchester University Press. 1970.
GRANULARITY OF NATURE
3.
Dowson, D. 1979, History of Tribology, Longmans.
4.
Cameron, A. 1960, Principles of Lubrication, Longmans.
5.
Cameron, A. and Wood, Mrs. W.L. The Full Journal Bearing. Proc. Inst. Mech. Eng. 1949, 161, 59-69.
6.
Tower, Beauchamp. 1st Report on Friction Experiments. Proc. Inst. Mech. Eng. 1949, 161, 59-69. 2nd Report. ibid. 1885, 36, 58-70. 3rd Report. ibid. 1888, 39, 173-205. 4th Report. ibid. 1891, 42, 111-140.
7.
Cameron, A. Beauchamp Tower Centenary Lecture. Proc. Inst. Mech. Eng. 1979. 193, No. 25.
8.
Lamb, Horace. Obituary of Osborne Reynolds. 1912-1921 Dictionary of National Biography. O.U.P. Also Proc. ROY. SOC. 1913, A.88.XV-XXI.
9.
Michell, A.G.M. British Patent 1905, No. 875. Also The Lubrication of Plane Surfaces. Zeit Math und Physik. 1905, 52, 123-1 37.
Reynolds' final work, which he says had taken him 20 years to complete, concerned the "granularity" of nature. He gave the Rede Lecture in 1902 (22). This lecture is the oldest endowed lecture in Cambridge University, dating from 1524, and is chosen annually at the discretion of the Vice Chancellor. In it Reynolds laid out his views that an array of grains are the cause of the physical prope ties of matter. These grains have a diameter lo-' times the classical size of the electron. This lecture was then expanded into a very long and exceedingly complex 254 page paper to the Royal Society entitled "the sub-mechanics of the Universe'' (23). Reynolds obviously thought this last work of his to be of considerable importance. A photograph, Figure 3 of the Rede lecture, depicts a curved bowl slightly tilted which was a model of two layers of grains in conformity one with another. He is shown holding the bowl, copied exactly from this photograph, in the well known portrait by Collier now in Manchester University. CONCLUSION Whether all this study is of value is a question I do not ask. I have enjoyed tracking down letters, which are not widely known, from
8
10.
Hall, L.F. and de Guereln, D. Some Characteristics of Conventional Tilting Pad Bearings. Inst. Mech. Eng. Lub. Conf. 1957, Paper 82, 142-146.
11.
Ettles, C.M.Mc.C., Akkok, M. and Cameron, A. Inverse Hydrodynamic Methods Applied to Mr. Beauchamp Tower's Experiments of 1885. Trans. ASME (J.O.L.T.) 1980, 102, 172-181.
12.
Kingsbury, A. Optimum Conditions in Journal Bearings. Trans. ASME, 1932, 54, RP 54.7 123-148.
13.
Wright, A. Old Queens' Men. Obituary. Dial. 1912, Vol. 3, No. 13, 41-43.
14.
Larmor, Joseph. Sir George Gabriel Stokes. Memoir and Scientific Correspondence. C.U.P. 1907. 246-248.
15.
16.
The
Williams, David. Catalogue of the Manuscript Collections of Sir George Gabriel Stokes and Lord Kelvin. Cambridge University Library. 1970. All the letters mentioned are catalogued here. Maxwell, J. Clerk. On Stresses in Rarified Gases Arising from Inequalities of Temperature. Phil. Trans. Roy. SOC. 1878, 170, 231-252.
17.
Reynolds, Osborne. On Certain Dimensional Properties of Matter in the Gaseous State. Phil. Trans. Roy. SOC. 1879, 170, 727-845.
18.
Fogg, A. Fluid Film Lubrication of Parallel Surface Thrust Surfaces. Proc. Inst. Mech. Eng. 1946, 155, 49-67.
19.
Wood, Mrs. W.L. and Cameron, A. Parallel Surface Thrust Bearing. ASLE Trans. 1958, 1, 254-258. Also Cameron, A. New Theory for Parallel Surface Thrust Bearings. Engineering. 1960, 190, 904.
20.
Robinson, C.L. and Cameron, A. Studies in Hydrodynamic Thrust Bearings. Phil. Trans. Roy. SOC. 1975, 278, 351-395.
21.
Cameron, A. Righting a 40 year old Wrong. A.M. Ertel - the true author of "Grubin's ehl" solution. Tribology. 1985, 18(2), 92.
22.
Reynolds, Osborne. On an Inversion of Ideas as to the Structure of the Universe. Rede Lecture, June 10, 1902, C.U.P. 1902, pp 1-44.
23.
Reynolds, Osborne. The Sub-Mechanics of the Universe. C.U.P. 1903, pp. 1-254.
1.0
0
0 LOAD LINE
50 degrees around bearing
- REYNOLD s' THEORY x
TOWER'S EXPERIMENTAL RESULTS
FIGURE
1
9
FIGURE 2 . a
10
FIGURE
2.b
11
FIGURE
2.c
12
FIGURE
3
13
This Page Intentionally Left Blank
SESSION II HISTO RY Chairman: Professor D. Dowson
PAPER Il(i)
The contribution of the Department of Scientific and Industrial Research to the study of hydrodynamic lubrication - The work of N.P.L. and N.E.L.
PAPER Il(ii) Historical aspects and present development on thermal effects in hydrodynamic bearings PAPER Il(iii) Michell and the development of tilting pad bearings
This Page Intentionally Left Blank
17
Paper Il(i)
The contribution of the Department of Scientific and Industrial Research to the Study of hydrodynamic lubrication - The work of N.P.L. and N.E. L. F.T. Barwell
Bearing research at the National Physical Laboratory was initiated by the Lubricants and Lubrication Committee of the Department of Scientific and Industrial Research and was characterised by the work of Sir Thomas Stanton and his colleagues on pressure-distribution, temperature-rise and eccentricity of journal bearings. Aircraft turbines introduced high-speed bearings and much theoretical work was stimulated by the so-called 'thermal wedge' effect. With the formation of the Mechanical Engineering Research Organisation (now N.E.L.) the N.P.L. lubrication group was transfered to temporary Laboratories at Thorntonhall in Scotland and reinforced by additional staff to constitute a multi-disciplinary team devoted to research on Lubrication, Wear and the Mechanical Engineering Aspects of Corrosion This pauer summarises that portion of the work of the two laboratories which was concerned with hydrodyn%c and hydrostatic bearings. The Department ceased to exist in 1964. 1
THE D.S.I.R. COMMITTEE
Following the outbreak of war in 1914 it became evident that much of British industry was insufficiently based upon science. Accordingly the Department of Scientific and Industrial Research was set up to remedy this state of affairs. The work of the Department was monitored by an advisory council. In the year 1917 an application for a 'grant-in-aid' was received from the Bradford Association for Engineering Research to support a study of the relationship between the viscosity of lubricant and the load on a bearing together with the action of lubricants at high temperatures as applied to commercial methods of oil testing. The Advisory Council appointed a special committee to report on this application and after fully considering it they regretted that they could not recommenend support because they considered its scope to be too limited. They were however impressed by the need for thorough investigation of the problems of lubrication and suggested that they might be empowered to investigate the matter further. They pointed out that before the war the annual expen& iture on lubricants in Britain was f 6 million and that an annual saving of one o r two millions could be achieved if a systematic investigation of the subject were undertaken. The Advisory Council accepted this recommendation and set up a Lubricants and Lubrication Enquiry Committee which was charged with the duty of preparing a memorandum on the field of research and containing an analysis of the problems involved together with a suggested scheme of research which would be likely to lead to valuable results. The committee was composed of distinguished engineers and other scientists and had as consultative members The Right Honorable Lord Rayleigh and the Honorable Sir Charles Parsons. Dr. C. H. Lander (who became Head of Mechanical Engineering at the City and Guilds and eventually the first Dean of the Military College of Science) was Technical Officer. The Committee reported in 1920 to the effect that no research on lubrication of really fundamental importance except that by Michell had been undertaken during the past thirty years. They
recognised three stages of lubrication as follows JJnlubricated surfaces (dry friction = 0.1 to 0 . 4 ) 2. Partially lubricated surfaces (greasy friction = 0.01 to 0.1) and 3. Completely lubricated surfaces ( v i s c o u s friction 0.001 to 0.01 ) They made numerous recommendations relating to stages 1 and 2 but as far as our present subject i s concerned , they proposed that journal bearings should be used to study stage 3. They recognised that the hydrodynamic approach to lubrication problems was first applied by Reynolds to Beauchamp'Tower's results in a paper which they stated had been read before tl?e British Association at their 1884 meeting in Montreal. They noted that Rayleigh and Stokes appeared to have arrived simultaneously atasimilar result and that Reynold's maLhematica1 treatment had been improved and extended by Rayleigh, Martin and Sommerfeld. These treatments were confined to surfaces of infinite width and the Comittee considered it necessary, particularly with narrow bearings, to take into account the ratio of length to width and recommended that completely lubricated surfaces should be studied using a loaded journal. This was done at the National Physical Laboratory where Stanton and Hyde had already made many tests using the Lanchester worm gear.
p
1.
v=
2
NOTATION
b C
e h
P r
")
breadth clearance (radial) eccentricity film thickness pressure radius velocity in x, y and z directions
W
X
Y z
h'
E
position measured in direction of motion position, distance from reference surface position measured at right angles to motion heat capacity energy
18 NOTATION (continued)
&
eccentricity ratio
7
viscosity,
8
e/c
=
coefficient of friction
p
8'
angular position, density angle of maximum pressure
W
angular velocity
method of calculating the eccentricity ratio. An arbitrary zero was chosen as the point of nearest approach at 6' 15' and the osition of maximum pressure was estimated at 3g 30' s o that Q t 0 = 177' 15'. The pressure slopes at 100' and 170 30' were -52,700 and 13,280 psi per radian respectively so that by substitution in equation (1) & = 0.9946 a n d 7 = 0 . 3 2 (in c.g.s. units)
dilation dp/d8
0
function defined b y equation 3 . 3 NATIONAL PHYSICAL LABORATORY
3 . 1 Pressure distribution
As pointed out by Dowson ( l ) , Stanton was aware of the essential features of elasto-hydrodynamic lubrication which may have influenced his decision to use high values of clearance and thus of eccentricity ratio in his tests. In his 1922 report to the Committee he described aseries of tests on a steel journal one inch in diameter and three inches long which ran in a phosphor-bronze bush. Pressure was measured through a single hole in the bush which was rotated between successive readings. 1.f
1.5 1.4 1.3
1.2 1.1
6"
1.o
=
6 7 U r& (COB 8 - Cos 8' ) c ( 1 + & cos Q) 3
*
.(I)
The viscosity of the lubricant at operating temperature and atmospheric pressure was 0.23 s o that the calculated value of 0.32 represented an increase due to pressure of some 28 %. Thus evidence of the existance of elasto-hydrodynamic action was provided by this comparatively simple experiment. The separation of the surfaces at the point of nearest approach was calculated to be 0.000054 inch or 1.37 urn. Stanton's breadth of outlook is demonstrated by'the fact that in 1923 he published a book entitled 'Friction' in which he included the fundamentals of aerodynamic friction as well as fluid lubrication. In 1927 he reported to Section G of the British Association the results of a series of experiments in which the relative displacement of a bearing and journal were measured by electrical inductance. Up to an eccentricity ratio of about 0.5 the agreement between observation and theory was reasonably good but a further increase produced a deviation of the point of nearest approach towards the crown of the bearing instead of in the lateral direction as predicted by Sommerfeld's theory. This led to considerable controversy until it was accepted that the negative pressures postulated by this theory could n o t be withstood by lubricant. 3.2 Heating of journal bearings
0.9 0.8 0.7 0.6
*=
+13200
0.5
Failure of journal bearings in practice usually occurred by seizure which was preceded by overheating and the -use was often confused between breakdown of hydrodynamic effects and failure of boundary lubrication. Stanton ( 4 ) reported the results ofaseries of tests by Jakeman in which the temperature of a bearing under test was raised by means of a gas jet situated within a hollow journal. He reproduced curves which were similar to Stribeck curves excepting that the abscissa was temperature rather than speed. Fig. 2. shows a comparason between castor-oil and two
0.4
Load =SO0 Lb per Sq I n D m o f Journal = 2 0 I n s Length of = 225
0004
8,
0.3 E
of Load
0
; - 0003
\
0.2
L
LL c
0 c
0,l ANGLE DEGREES
0
: -
0002
c
P,
10"
5"
0"
o Observed pressures
-5" -10" -15" -20" x Pressure c a l d from e q c
Fig 1 Oil pressure distribution- after Stanton ( 2 ) Fig. 1. has been selected from several curves because it was used by Stanton to illustrate his
v 0 001
M
im
150 Temperature Deg Cent
zw
250
Fig. 2 Friction-temperature curves-after Stanton
19 mineral oils which demonstrates the important potential for hydrodynamic action even at higher temperatures. He summarised this conclusion as follows;"The importance of boundary lubrication to the engineer is therefore undoubted but it seems worth pointing out that to the engineer boundary lubrication is in all cases a pis aller, both on account of the relative loss of power involved when it exists, and the liability of the surface to seize if the heat cannot be conducted away rapidly enough".
3.3 Experimental exploration of the journal centre locus
lamp 5 was reflected onto the cross-wige by a glass plate 6 which was inclined at 45 to the axis. The parallel beam fromthe lens was divided by prism 7 secured to plate 8 which could be rotated in its own plane. Each half of the beam was reflected by mirrors 9 on to one mirror R of the bearing apparatus. Light frrm each mirror returned along its original path to form an image in the same plane as thecross-wires; careful adjustment of the inclination of the mirrors was necessary to ensure that the image was sufficiently close to the cross-wires to be within the field of the microscope. The tilt of the mirror R was measured by moving the micrometer 10 in the eyepiece of the microscope. ___
There was still a certain reservation regarding the shape of the journal centre locus until Clayton and Jakeman (5) carried out a masterly series of measurements which will be described in some detail because they indicate what could be done with mechanical and optical methods before electronic sensing and amplification became avaiL able. The machine used for the tests consisted of a main steel shaft 4 inches in diameter which was mounted in two ball-bearings and driven by belt. It was reduced to2inches diameter for a some 34 inches at one end t o form the test journal. A bronze bush 2% tnches long was supported by the shaft and loaded through-knife edges situated equidistant from the bearing centre. These were recognised as being a primary source of friction and frictional error. Bearing friction was therefore measured by connecting the knife edges to a beam which was loaded on the centre line of the bear,ing assembly but which was provided with an extension to which weights were added to counterbalance the bearing torque. The system was initially placed out-of-balance so that it was always necessary to add loading to obtain equilibrium. Each experimental point was determined as the mean value of torque measured from two directions of rotation. It was recognised that the coefficient of friction on the iournal differed from that determined on the bush-by the product of the applied load and the horizontal displacement of the bearing.
I
-__
I!
I 5
g-
o
I Clrornrira
(a) 0 0015, (I)) 0 003.5, ( c ) 0 IKJiBotd
(d) 0.0161 in
Figs4 Examples Of path followed by bearing centres- after Clayton and Jakeman (5).
n
1 n
CONDENSER
w
Fig. 3 Arrangement of optical measuring apparatus - after Clayton and Jakeman (5) Measurement of journal displacement was based on the inclination of spring hinges which were supported at one end onto the periphery of the shaft through bronze spheres. This inclination was reproduced by-mirrors,changes in inclination of which were measured by an optical system shown in Fig. 3. This consisted of a microscope 2 which was used to observe cross-wires which were at the Light from a 'pointolite' focus of a lens 4 .
The small values of negative clearance which were recorded were the subject of much concern to the investigators who found that the readings on the side mirrors agreed to within 0.00005 inch but that there was a discrepancy between the top and bottom mirrors which increased with load. This error was attributed to relative elongation of the bush as the machine changed from the running to the stop condition.
20
had not been recognised prior to Fogg's experiments.
THE THERMAL WEDGE
4
C/D
=
c/r
= 0.0038.
During a series of tests intended to evaluate the performance of taper-land bearings at high speed, Fogg introduced a set of plane parallel bearings as amlimiting case (10) within a set of examples 'of varying inclination. He discovered that these were effective in spite of the absense of any obvious wedge action. A selection from his results is given in Table 1.
Fig. 5. Measurements of bearing friction-after Clayton and Jakeman (5). Fig 5 shews the friction results. These are the measurements made on the bush corrected by using the horizontal component of the measured eccentricity to derive a calculated journal torque. is derived from bush measurements.
Table 1. Comparison of performance of fixed and tilted plane bearings - after Fogg (10) Speed (r.p.m)
Type of Bearing
FB
1 1
Tilting pad
I I
9.000 18,000
I
I I
I I
10.2 9.2
I
9,000 18,000
I
114 140
I
I
I
I Fixed pad
Bearing temperature ( OC.)
Load at failur (MN m-2)
+ .
8.1 7.5
I
100 115
I
*
The average of the inner and outer radii of the pads was 0.015 m giving mean sliding speed2 of 35 and 70 m/s respectively. Fig. 6 Variation of eccentricity with Sommerfeld variable- after Clayton and Jakeman ( 5 ) Some examples of measurements of eccentricityratio plotted against Sommerfeld number are shewn in Fig. 6 These were in qualitative agreement but there were quantitative differences for the various values of clearance.
.
3.4
Hydrodynamic bearing applications
Contributions to the 1937 General Discussion on Lubrication held at the Institution of Mechanical Engineers revealed continued progress in the experimental study of hydrodynamic bearings at the N. P. L. Thus Fogg and Hunwicks ( 6 ) demonstrated that considerable loads could be carried with very low friction using water in bearings made of rubber. As regards oil-lubricated journal bearings, a wide range of.confirmatory tests of the relationship between coefficient of friction and duty parameter were reported (7). The situation occurring immediately after the breakdown of a lubricant film was investigated by Clayton (8). 3.5
High-speed bearings
Work at the N.P.L. was very much influenced by war-time problems and the advent of jet propulsion of aircraft threw into relief problems relating to the operation of bearings at high speeds. Initialy work was divided between rolling contact and hydrodynamic bearings (9). Regarding the latter, tests on a two inch diameter journal bearing at 2,000 rpm emphasised the advantage of 'short' bearings (b/d ratio of 5 )in contrast to the longer bearings which were traditional and which featured in the tests of Stanton and his coworkers. Shorter bearings had become common in internal combustion engines from geometrical necessity but the hydrodynamicandthermal. ahantages
At the time of publication the subject was still classified and the actual speeds were not disclosed until later. These results have given rise to much theoretical and some experimental investigation. Fogg postulated the 'thermal Wedge' hypothesis to exThis induced Cope (11) to plain his results. undertake a mathematical investigation in which he simultaneously solved the continuity equation, the Navier-Stokes equation and an energy equation in the form shown in equation 2.
w h e r e n represents dilation. The first group of terms representstherate of change of internnal energy in any element of the fluid, the second the work done in compressing the fluid, the third governs the rate at which heat is conducted away whilst the fourth term denotes the rate at which work was done against viscosity. This took the form 2 Q = ll [2"&]
[g] 2
+
+
[$)
[ g + z] + [g+ g ]2
+
2
+
- 7
[ % + 212
"'I
.
(3)
A further equation was required to take into account variations in the physical properties of the lubricant, notably the change in viscosity Cope referred with temperature and pressure. to this as the'equation of state'and proposed applying the perfect gas law to air-bearings.
21
He recognised the complexity of mineral oils and in fact restricted his analysis to the variation of viscosity with temperature. The complexity of the array of equations was such as to render any rigorous solution impracticable and it became necessary to introduce a number of simplifying assumptions of which the most important was that because the interacting surfaces were close together and nearly parallel, flow was laminar. From this it could be assumed that pressure, density, temperature and viscosity were functions of x and y only. Thus (4) (5)
Fluid velocity normal to the surfaces as well as the body forces were neglected. Using these assumptions the equations became simplified as follows:Continuity
Moment um (7)
= 12rl
uz
;
[I1
4 +
{
7 11 12n u
which were solved by a process of successive approximation using the Hollerith machines in the Mathematics Department of the N.P.L. The results indicated that, except for extremely thin films, the variation of viscosity with temperature virtually negated the thermal wedge action.
5 TRANSFER TO SCOTLAND As a result of the formation of the Mechanical Engineering Research Orgar.isation, now N . E . L . , the N.P.L. lubrication team was transfered to Scotland and reiriforced by men and women qualified in physics, chemistry, metallurgy and electronics so as to constitute a multi-disciplinary team which was devoted entirely to the study of "Lubrication, Wear and the Mechanical Engineering Aspects of Corrosion". Although a large site had been reserved at East Kilbride for the construction of what is now the National Engineering Laboratory, the Lubrication Division was accomodated in a pilot laboratory which was located in the village of Thorntonhall, some 3; miles from East Kilbride.
Further simplification was achieved by an order-of-magnitude analysis wherein representative quantitiesasdisplayed in Table 2 demonstrated that the dissipation terms were several orders of magnitude greater than the dilation terms. The inertia terms and those relating to conduction in the x direction were also negligible. Table 2. Order of magnitude of derivitives. Fig. 7 Bishops House, Thorntonhall
I
1
Applyingtheboundary conditions y=O,u=Uandw=O
y = h, u = w = 0 the equations become
The laboratory was situated in the grounds of a country house, Fig. 7 ., in which a number of semi-permanent buildings had been erected by the London, Midland and Scottish Railway as an.energency wartime evacuation headquarters. Fig. 8 . They had never been used for this purpose and were readily adaptable to form laboratories, Fig 9 and workshops, Fig. 10. Although designated as 'temporary' these laboratories rendered good service for the remThe Thorntonhall laboraainder of the decade. tories were necessarily self-contained from the point of view of workshop facilities and other services and although much material was transfered from N.P.L., some critical components and important records were lost when the last of the
22
removal vehicles developed a tyre fault and was burnt-out at Penrith on the way to Scotland.
F i g . 8.
Temporary laboratory,exterior
6
DESIGN CONSIDERATIONS
T h e programme of work covered a wide field and included a section concerned with hydrodynamics. T h e Mechanical Engineering Research Board of the D.S.I.B. which had been set up to advise the department of Mechanical Engineering, was insistent that existing knowledge should be expounded for the benefit of designers irrespective of the recognised limitations of current theory. T h i s resulte d , among other things, in the preparation of a text-book by the present writer (12). Although the state of bearing theory was reviewed, a rigorous optimisation procedure was not regarded a s being feasible a t that time. However there appeared to be good practical reas o n s for operating a bearing at an eccentricity ratio of about 0.7 and calculation methods were proposed based on the recently enunciated narrow-bearing theory of Dubois and Okvirk (13). Subsequent developments have justified this decision (14) (15).
7.
HIGH SPEED BEARING TESTS
T h e advent of industrial and marine gas turbines provided a stimulus for the investigation of hydrodynamic journal and thrust bearings r'iitable for operation at high speeds. A serids of bearing test machines of realistic speed and load capacity were designed and built and measurements of the film thickness, temperature and friction were recorded with particular reference to the provisA certain amount of this ion of design data. work was published in the proceedings of the Conference on Lubrication and Wear held at the Institution of Mechanical Engineers in October 1957 (16-23). 7.1 Thrust bearings
Fig. 9
Fig. 1 0
Temporary laboratory, interior
Workshop a t Thorntonhall
Tests on a large plane bearing were carried out in a test rig which had been transfered from the N. P. L. and described by Fogs and Iiebber (24) It was driven by a 75 h.p. motor with a step-up gear box giving a speed range of from 5,000 to 15,000 r.p.m. The problem of withstanding the reaction to the thrust load appied to the test bearing was resolved by using two identical test bearings in a mirror image. The mean surface speed ranged ranged from 50 to 100 metres per second. Commercially designed tilting-pad bearings embodying six white-metal and bronze pads gave considerable power loss, pad temperature rising Removal of three of these pads resto 116' C. ulted in improved performance although local temperatures were still considered excessive. Removal of restrictor rings which were originally fitted to aid lubrication led to a marked reduction in friction and satisfactorily cool operation was finally secured by feeding oil into grooves in the pad surfaces. A new design of plane parallel pad bearing in which great emphasis had been placed on surface rigidity and dissipation of heat gave relatively poor performance and confirmed that the thermal wedge principal was unsuitable for appllcations where substantial speeds and forces were involved.
23 7 . 2 Journal bearings
In order to establish the validity of hydrodynamic theoretical projections at realistic speeds as well as to bring into account the effect of practical design features such as oil inlet arrangements, a set of machines was projected and initially the journal bearing test machine illustrated in Fig. 11 was constructed.
a relationship with theoretical analysis, After initial experience with air-gauging, attitude and eccentricity were measured by four inductance transducers spaced at 90' intervals. Friction torque on the bush was measured by unbonded strain-gauge. Thermo-couples were inserted axially in the test-bush at 0.015 inches below the bearing surface. The machine was driven by a direct current motor having thyratron and ignitron control of the field and armature voltages respectively. A tachometer on the motor shaft supplied a reference voltage for servo speed control. Loading was by dead weights which were applied or removed by a pneumatic cylinder. Analysis of the results threw into relief a number of feedback loops governing the behaviour of these bearings. The viscosity-temperature relationship was found to be the most powerful transfer function. The value of friction was found to be in accord with narrow-bearing theory and good correlations were found between friction and effective viscosity based on either the mean outlet temperature o r the load-line bush temperature. The most influential factor in the control of temperature of high speed bearings was shown t o bethe oil inlet arrangements and clearance ratio. Agreement with published theory such as Cameron and Wood (30) and DuBois and Ocvirk (13) was within experimental error. 7.3
Fig. 11 High-speed journal bearing test machine The test shaft of this machine was driven by a 75 h.p. motor by means of a 40 to 1 step-up gear The test box at speeds up to 60, 000 r.p.m. bearing arrangement as shewn in Fig 12 was chosen to simulate the degree and nature of restraint experienced by the shaft of a high-speed turbine. AIR
Oil film extent in journal bearings
Film extent and the disposition of fluid in the cavitated region of journal bearings were matters of conjecture until Cole and Hughes (31) devised a satisfactory method of photographing them. Glass bearings were used and the oil film made to fluoresce by ultra-violet irradiation. Typical inlet and outlet patterns are shown in Figs 13 and 14.
B E A R I N G HOUSING
OIL
kS.22 z.7Bssm
A1 FI
BASE-PLATE
BASE -PLATE
Fig. 12
HYDROSTATIC AIR BEARING PROVIDES FRICTIONLESS SUPPORT
Arrangement of journal bearing test rig
The test bearing was contained in a cartridge. which itself formed the inner portion of an air bearing which allowed the applied torque to be transmitted directly to measuring apparatus. The shaft was symmetrically supported in a slave bearing which was dimensionally similar to the test bearing but lacked the air bearing feature. Load was applied centrally to a slave bearing through a self aligning cross-strip system. Great importance was attached to the policy that all the important variables- friction, bearing centre locus, temperature and oil flow- should be measured simultaneously so as to provide as close as possible
Fig. 13 Visual study of :reformation boundary after Cole and Hughes (31). This work was extended to dynamically loaded bearings in which instantaneous pictures were obtained by flash illumination. 7 . 4 Turbulence
The increase in the size of electrical turbo alternators and the projected use of low viscosity fluids such aswater and liquid metals in atomic power stations led to the possibility that turbulent flow might occur within bearings. Taylor (35) investigated vortex conditions which
24
Fig. 14 Visual study of cavitation boundary after Cole and Hughes (31) occurred in an annular space between a rotating and a stationary cylinder. This geometry is analogous to that of a journal bearing except for the effect of eccentricity. This effect was explored by Cole (16) using model experiments from which he deduced that vortices were formed when the speed exceeded W n ,n
where
w=
15.28
(T//p) d-2
(r/C)
3/2 ( 1
+ 0.89f"
.
. (13) . , This expression was put forward as a warning to designers so that they could avoid turbulence; the inference being that vortex formation indicated the lower speed limit for turbulence. It is now realised that the onset of small-scale turbulence is determined mainly by the local Reynolas Number in the film. 7.5 Dynamic load
In order to ensure that conditions.af1oading and movement were realistjc, actual engines ranging from a Petter AV 1 to a Doxford marine diesel engine were instrumented to enable the actual shaft centre locus to be displayed on a c.r.0. It was shown that shaft and crankcase deflections played an important part in the distribution of load on the bearings. For more basic studies a machine was constructed in which a 64 mm diameter test shaft was rigidly supported and loaded by hydraulic rams which were programmed from a peg matrix board to reproduce actual engine conditions (37). 8
BASIC HYDRODYNAMIC THEORY
Recognising that a knowledge of the effect of variation in magnitude and direction of load was of fundamental importance in the development of bearings for high-speed turbines and internal combustion engines, Milne reviewed the experimental work then in progress at the N.E.L. and then developed a theoretical treatment which embodied a series of DEUCE computer programmes (32). He adopted the narrow bearing theory as his starting point and extended it to allow for the radial and tangential movement of the shaft. Cases of both complete and cavitated films were treated and produced different loci. For the complete film the locus usually of the 'half-speed' effect combined with a loop which closed within two revol-
utions of the shaft. With cavitation the loop did not always close but the system was very stable so that with a steady load equilibium was reached within two revelutions As the work developed it became possible to identify areas warranting extended mathematical treatment and a definite stage in the development of the subject was reached with the publication of a paper at the University of Houston symposium held in 1963 (33). This paper occupied more than one hundred pages and was sub-divided as follows? Section 1 concluded that inertia effects were negligible in laminar conditions but played a crucial role in determing the onset of Taylor vortices or turbulence. In Section 2 the various boundary conditions were critically examined by the folloing methods:Averaged inertia, Momentum integral, Iteration, Series Expansion and the step-by-step method. All these methods produced the same general form of expression for pressure gradient but the Momentum Integral method was less restricted. Section 3 presented an alternative method based on stream-functions. These yielded little advantage for thin films but were very useful for study of sharply converging films at the bearing entry. Section 4presented tabular results of integrating pressure and drag and confirmed that turbulence intervenes before inertia effects become significant in laminar hydrodynamic bearings. In contrast, Section 5 demonstrated that in the case of hydrostatic bearings, lubricant inertia could be important even when flowwaslaminar. 'Shock-wave' flow limitation could provide inher.ent compensation of gas bearings. Section 6 introduced transient effects and emphasised the importdnce of the precise mechanism governing the flow of lubricant into a bearing. Section 7 recognised that oil flow was a majordetermin nt of bearing performance and provided a basis for calculating the position and shape of the upstream boundary of a hydrodynamic pressure film. This approach was continued inaseries of internal reports one of which,issued in 1973 (38), explored the conditions governingthe formation of the two types of boundary separating the full film from the cavitated regions within a hydrodynamic or a hydrostatic bearing. These were known as the 'reformation' and'cavitation'boundaries respectively. The former has both form and position and depends on the transportation of fluid in bulk Cavitation boundaries on the other hand are essentially spacial, are dependent on pressure and have no intrinsic velocity. The functions of the two types of boundary may be interchanged during the loading cycle of a bearing. The analysis was extended to cover a wide range of bearing geometries where rupture and reformation boundaries co-exist or interchange (34). The finite-element method was shewn to be simple and convenient in operation (39) and was applied to the main and big-end bearings of 1 . in-line' and 'V' engines. Bearing loads were calculated from the cylinder pressures and the journal locus was determined from the bearing Loads. Thus the outcome of theoretical work was directly applied to the solution of an important but otherwise intractible practical problem.
.
25
References
9 AIR LUBRICATED BEARINGS Aerodynamic bearings were studied up to a speed of 100,000 rpm using 'Veridia' glass bearings one inch bore with clearance ratios from 0.007 to 0.0019 running on a steel shaft which was used as the rotor of a high-frequency induction motor. The experimental arrangement is illustrated in Fig. 15
DOWS0N.D. 'History of tribology', 1979, Longmans, London, pp 340-2. STANTON, T.E. 'On the characteristics of cylindrical journal lubrication at high values of eccentricity',Proc. Roy. Soc.A., vol cii. pp241-55, 1923 STANTON, T.E. 'Friction', Longmans,London, 1923. STANTON, T.E. 'The lubrication of surfaces under high loads and temperature', Engineering, vol. pp312-3.1927. CLAYTON, D. and JAKEMAN,C. 'The measurement of attitude and eccentricity in complete clearance bearings', Proc. 1.Mech.E. 134, 437-506.1936. FOGG, A. and HUNWICKS, S.A. 'Some experiments with water lubricated rubber bearings', Proc. General Discussion on Lubrication and Lubricants,Institution of Mechanical Engineers, October, 1937.101-6. JAKEMAN, C and FOGG,A. 'The performance of clearance bearings asaffected by load, speed, clearance and lubricant: ibid 138-
.,
Fig. 15 Air bearing test apparatus A , three phase stator; B, pedestal, C, case-hardened mild-steel shaft; D9glass test bush; E, 0 rings; F, spherical bush; G , clamp screw; H, bush support pedestal; 11capacity type displacement transducer; J, photoelectric cell, K, capacity plate for measurement of bush vibrations; L, vent to annular space between 0 - rings; M, base plate secured to spring-mounted concrete block.
The initial results were summarised by the statement that"1oads of about one pound per square inch per 1,000 rev/min could be carried" (21). An unexpected observation was that water-vapour condensed within the bearing clearance but this did not appear to affect performance. A revised expression as a basis for initial design (42) is (14.) This data relates to bearings having a b/d ratio of unity operating between the following limits of compressibility number. pa.is ambient pressure and compressibility number is defined as and lies between 6 and 12.
10 CONCLUSION Although the main emphasis in the foregoing contribution has been focussed on the two laboratories within the Department of Scientific and Industrial Research's own establishments which made prominent contributions to tribology, the department had throughout its existence supported much relevant research through the agency of numerous research associations as well as academic research, particularly through the medium of grants to post-graduate students. A particularly useful headquarters function was the provision of important translations Lor exampletthat of Grubin and Vinogradova (42) *
11 ACKNOWLEDGEMENTS Thanks are due to the Directors of the National Physical Laboratory and the National Engineering Laboratory for.permission to reproduce illustrative matter which is Crown Copyright Reserved. * The Department of Scientific and Industrial Research ceased to exist in 1964 its functions being taken over by the Ministry of Technology (now part of the Department of Trade and Industry) and the Department of Education and Science.
44. CLAYT0N.n 'The effect of seizure on the shape of the crown of a bush and the influence on siwhequent rubbing', ibid.59-65. FOGG, A. ' Proc 7th Internationa Congress of Applied Mechanics, 4. 181- .1948. FOGG, A. 'Fluid film lubrication of parallel thrust surfaces', Proc. I. Mech. E.,155
49-53, 1946. COPE, W.F. 'The hydrodynamic theory of film lubrication', Proc. Roy. SOC. A 197, 201-17, 1949. BARWELL, F.T., 'Lubrication of bearings: Butterworth, London, 1956. -~ DUBOIS, G.B. and OCKVIRK,F.W. 'Analytical derivation and experimental evaluation of short-bearing approximentation for full journal bearings', N.A.C.A. Report no. 1157, 1953. BARWELL, F.T. 'Bearing systems- Principles and practice', 1979, Oxford University Press. WOOLACOTT, R.G. and MACRAE, D. 'A design method for hydrodynamic journal bearings, Ministry of Technology, National Engineering Laboratory report No. 315, 1967. COLE, J.A.,'Experiments on the flow in rotating annular clearances', Proc. conf.on Lubrication and Wear, Inst. Mech. E.,Lono n , 1957.,16-19.
COLE, J.A. 'An experimental investigation of the temperature effects in journal bearings', ibid,lll-7. COLE, J.A. 'Experimental investigation of power l o s s in hinh-speed plane thrust bearings', i b i d , 158-63. COLE, J.A. 'Film extent and whirl in complete journal bearings, ibid.186-90. COLE, J.A. and HUGHES, C.J. 'Visual study of film extent in dynamically loaded complete iniirnal bearings', ibid, 147-50. COLE, J.A. and KERR, J. 'Observations on the performance of air-lubricated bearings: ibid 164-70. MILNE, A.A. 'A theory of rheodynamic lubrication for a Maxwell liquid,' ibid. 66-71 MILNE, A.A. 'On,grease lubrication of a slider bearing',ibid 171-5. FOGG, A and WEBBER, J.S. 'The influence of
26
References continued some design factors on the characteristics of ball bearings and roller bearings at high speeds', Proc.I.Mech.E., 169,716-31, 1953.
WOOLACOTT, R.G. 'Hydrodynamic journal bearing performance at high speed under steady load', Proc.1. Mech.E., 180. Pt3K.76-89, 1966.
WOOLACOTT, R.G. 'The performance at high speeds of complete plain journals with a single hole inlet', NEL Report No326 1965 WOOLACOTT, R.G. and MACRAE, D. 'The performance at high speeds of complete plain journal bearings with a chambered oil inlet: NEL Report No. 324, 1967. WOOLACOTT, R.G. and MACRAE,D. 'The performance at high-speeds of complete plain journal bearings with double axial groove inlet',NEL Report No. 326, 1967 WOOLACOTT, R.G. and MACRAEP. 'The performance at high speeds of complete plain journal bearings with a circumferential groove oil inlet', NEL Report No 338, WOOLACOTT, R.G. and MACRAE, D. 'The performance at high speed of complete plain journal bearings with a single hole inlet, Part 11', NEL Report No 359, 1968. CAMERON,A, and WOOD, W.L. 'The full journal bearing', Proc. I. Mech.E., 161,59-72, 1949.
COLE, J.A. and HUGHES,C.J.'Oil flow and film extent in complete journal bearings', Proc. I. Mech.E., 170.499-510,1956. MILNE, A.A. 'On the effect of lubricant inertia in the theory of hydrodynamic lubrication,' Trans. A.S.M.E., J.Basic Eng, 81,239-44.
MILNE, A.A. 'Inertia effects in self-acting bearing lubrication theory', Proc. conf. Lubrication and Wear, Houston, Texas 429-527,
MILNE, A.A. 'Variations in film extent in dynamic loaded bearings', 1st. Leeds-Lyon Symposium on Tribology, Cavitation and reated Phenomena in Lubrication, I. Mech. E. 79-90, 1975.
TAYLOR,SIR GEOFFREY, 'Stability of a viscous liquid contained between two rotating cylinders', Trans, Roy. S0c.A. 102, 541-2, 1923.
MILNE, A.A.'A contribution to the theory of hydrodynamic lubrication, A solution in terms of the stream function for a wedge shaped oil film: WEAR 1 , 32-39,1957. COOKE, W.L. 'Simulation of dynamically loaded bearings at NEL', Tribology,l, 1023 , 1968.
MILNE, A.A. 'Oil flow and film extent, transient variations in a hydrostatic system',NEL Properties of Fluids Internal Report No76, March 1973. MILNE, A.A. 'A finite element method of calculating oil flow in bearings', NEL Report No 569, 1974. MILNE, A.A. 'Diesel PTLP/FLOW',NEL Properties of Fluids Internal Report No 131 MILNE, A.A. 'Diesel PTLPLT/FLOW, A specific example on Report No 131. KERR, J. 'Air lubricated bearings for high compressibility numbers', Proc 1st Annual Meeting of Lubrication and Wear Group, Institution of Mechanical Engineers. 69-76, 1962.
(43)
GRUBIN, A.N. and VINOGRADOVA, I.E. 'Investigation of contact phenomena (friction, contact, stresses, etc)', Gozud Nauk Tekh. Izdat Mashin Lit Book No. 30 Moscow (in Russian) see D.S.I.R. Translation No. 337.
27
Paper Il(ii)
Historical aspects and present development on thermal effects in hydrodynamic bearings M. Fillon, J. Frene and R. Boncompain
Thermal effects in hydrodynamic lubrication w e r e recognized as very i m p o r t a n t s i n c e t h e f i r s t works in this field. This paper p r e s e n t s t h e evolution of
r e s e a r c h concerning t h e r m a l effects in hydrodynamic
bearings from t h e f i r s t s t u d y published by Hirn in 1854. A s u m m a r y of g e n e r a l thermoelastohydrodynamic theory for t h e journal bearing case and i t s comparison with e x p e r i m e n t a l r e s u l t s is t h e n presented.
I
INTRODUCTION
Thermal
effects
are
to
known
play
a
very
p = pressure Pa
The gth Leeds-Lyon Symposium on Tribology which
p. = inlet pressure Pa 2 = PC / p i w R 2 nondimensional pressure
was held
Q.
important
role
in lubrication, f r i c t i o n and wear.
in t h e "Institut
AppliquCes d e Lyon" entirely
devoted
symposium hydrodynamic [2]
many
[I]
to
National d e s S c i e n c e s in September 1979 w a s
this
papers
subject. were
During
concerned
the with
lubrication and Pinkus and Wilcock
presented
an
extensive
bibliography
on
d
3 i n l e t flow in t h e bearing groove m /s 3 Qr = r e c i r c u l a t i n g flow m /s 1
0
= Q/L C R
w
nondimensional flow
R = journal bearing radius m
-
R 2 = R 2 / R nondimensional outside bush radius
r=
r/R nondimensional radius
T = t e m p e r a t u r e "C
thermal effects in fluid film bearings.
T
1.1 Notation
a m b i e n t t e m p e r a t u r e "C
a
Tb = bush s u r f a c e t e m p e r a t u r e "C BibBis = Biot numbers f o r t h e bush, f o r t h e s h a f t C
radial c l e a r a n c e m
Ti = inlet fluid t e m p e r a t u r e in bearing grooves "C
C0 = specific h e a t of lubricant J/kg. "C DI = film z o n e w h e r e U D2
-
film z o n e w h e r e
u
> <
T r = t e m p e r a t u r e of t h e recirculating fluid "C
0
TS = t e m p e r a t u r e of t h e s h a f t
0
T
h = nondimensional film thickness
hb,s = convection h e a t t r a n s f e r c o e f f i c i e n t s W/m2
To = inlet film t e m p e r a t u r e "C
oc
= T I T O nondimensional t e m p e r a t u r e
W = load c a r r y i n g c a p a c i t y N - - -
u, v, w = nondimensional components of t h e fluid velocity in t h e x, y and z direction
k0,1,2 = viscosity c o e f f i c i e n t s
respectively, U = u/U,V = vR/CU,
K = t h e r m a l conductivity W/m "C KO
thermal
conductivity
of
the
lubricant
W/m " C Ka = t h e r m a l conductivity of t h e a i r W/m "C
r
G = w/u
e2=
nondimensional c o o r d i n a t e s
xyz = coordinates
- -
8 , y, z = nondimensional c o o r d i n a t e s
K b = t h e r m a l conductivity of t h e bush W/m "C K
S
= t h e r m a l conductivity of t h e s h a f t W/m "C
L = bearing l e n g t h m L' = a p p a r e n t l e n g t h m N = rotational s p e e d of t h e s h a f t r p m u i w 2 R 2/KOTi dissipation number 2 Pe = pCo W C /KO Peclet number
Nd
O C
y
r
e
= x/R,
= y/h, Z = Z/L
= boundary b e t w e e n D I and D2 = eccentricity ratio
6
+, = a t t i t u d e a n g l e
,, -
,
= inlet fluid viscosity in t h e bearing groove
Pas
=
1
nondimensional viscosity
28
lubricant density kg/m es(z)
3
50
the
abscissa of t h e a c t i v e zone end degrees
and
the
half
bearing,
to
the
was
lever.
measured
The
by
adding
friction balance is
described as a n extremely delicate and a c c u r a t e
BIBLIOGRAPHY
brake. The cast-iron drum was water cooled to
Earlier studies on thermal effects in
2.1
weights of
Friction
MI.
weights 2
dead
which includes t h e torque a r m and added masses
M
angular velocity of t h e s h a f t rad/s
kg
control t e m p e r a t u r e and t h e t e m p e r a t u r e rise of
lubrication
t h e cooling water was recorded. It
commonly
is
accepted
that
the
preliminary
effects in lubrication were undertaken in t h e forties. In f a c t thermal e f f e c t s
studies
on
thermal
had been analysed much earlier by G. Hirn, 0. Reynolds, N. Petrov and A. Kingsbury. Soci6td
Industrielle
de
Mulhouse"
a
study
entitled "Etudes sur les principaux phdnomgnes q u e prdsentent diverses
les
manigres
mdcanique des
frottements
des
de
m6diats
et
ddterminer
i.e.
"Studies
on
sur
la
matigres employ6es au
machines".
the
les
valeur graissage principal
phenomena presented by mediate friction and on the
various
means
efficiency of machines".
to determine t h e mechanical
t h e materials used to lubricate t h e
This work was submitted first to T h e
"Acaddmie des Sciences in Paris" in 1949 [4] and then to t h e Royal Society in London but neither body f e l t moved to publish t h e paper. a half
bearing
made out of
vegetal oil like
water
and
air.
running-in
He
upon
discovered
bearing
the
friction
effect
and
of
further
continuously
a
for
certain
time
before
an
equilibrium friction torque, lower than t h e initial one
is
reached.
regimes
exist,
He
found
the
that
direct
Itfrottement imm6diat"
in
two
different
contact
which
called
friction follows
Coulomb's law, and t h e lubricated c o n t a c t called "frottement m6diat" known today as hydrodynamic lubrication,
in which for a constant temperature,
t h e friction torque is directly proportional to t h e rotational
speed.
H e also noted
t h a t when t h e
speed is t o o low or when t h e load is too heavy the
is
friction
speed
to a
proportional
certain
to
power
the
lower
rotational 1. H e
than
wrote t h a t :
bronze loaded
against a polished cylindrical cast iron drum. T h e drawing of
animal and
sperm, olive and rape oils but also mineral oil,
Hirn presented experimental results obtained on
tested
pointed o u t t h a t lubricated bearings must be run
On t h e 28th june 1854 G. Hirn [3] presented at "La
Hirn
t h e apparatus as i t appeared in t h e
original paper is shown in figure 1. T h e bearing
"Pour
que I'eau
et
comme
lubrifiants,
il
tourndt
assez
pour
vite
Pair
fallait les
pussent
que
y agir
le
tambour
entrainer
sous l e
coussinet. Dds q u e l a vitesse diminuait jusqu'h un c e r t a i n degrd, les deux fluides, !Inon visqueuxtt, d t a i e n t expulsds par l a pression, les deux surfaces arrivaient e n c o n t a c t immddiat, et l e f r o t t e m e n t devenait t o u t d' un coup 6norme".
i.e.
[5] :
"For water and a i r to act as lubricants it
is necessary for t h e drum
to turn sufficiently
rapidly to drag t h e m into t h e bearing. When t h e speed
reduces
to
a
certain
value
the
two
ttnon-viscoustt fluids a r e expelled by t h e pressure and t h e surfaces c o m e into direct contact, and t h e friction at o n c e Hirn founders Fig. I : Drawing of t h e apparatus design by
mm,
to b e one of t h e applied thermodynamic science
and heat. H e showed t h a t friction produces h e a t and
were as follow
the
becomes enormous".
is known
was interested in t h e relationship between work
Hirn [3] characteristics
of
who
: Diameter
230
Lenght 220 mm Rotational speed beetween
45 rpm to 100 rpm. The bearing was loaded by
that
equilibrium
temperatures
depend
on
friction. Using t h e cooling system of t h e drum he maintained t h e bearing t e m p e r a t u r e within plus o r
29
minus 0.1"C out
of
and
the
he
measured
bearing
by
the
t h e heat carried coling
water.
He
evaluated also t h e h e a t c a r r i e d o u t of t h e bearing by
convection.
thus
measured
equals to 370 kg.m
kilocalorie 3.63
He
at
joules),
the
(1842)
time
but
independently Joule and Mayer found respectively that
1 kilocalorie equals 417 and 365 kg.m.
error
done
by
Hirn
could
be
The
to a n o v e r
due
evaluation of t h e h e a t c a r r i e d o u t of t h e bearing
"I
in his paper "On t h e t h e o r y of lubrication and i t s
to Mr
have
problem
Bauchamp Tower experiments"
machine
studied never in
noticed
that
and
those
when
on
railway
all
sufficient
cases
where
approaching
experimenters
who
this have
up to t h e p r e s e n t h a v e
m e d i a t e friction paid
a t t e n t i o n to t h e fact t h a t , the
friction
is e f f e c t i v e l y
m e d i a t e , t h e liquid film completely s e p a r a t e s t h e t w o solid surfaces". He
Thirty years l a t e r 0. Reynolds in 1886 [ 6 ]
bearing
journal bearing. In 1883 in his paper
originally w r i t t e n in Russian, h e w r i t e s :
by conduction a n d convection.
application
journal
wagon-axle
I
I calorie =
(i.e.
same
that
a
on
or
used
t h e words of
"frottement
qualify the
mediat"
hydrodynamic
hydrodynamic
" m e d i a t e friction"
proposed
by
lubrication.
pressure
is
Hirn
Assuming constant
in
to that the
was also concerned by t h e r m a l effects. H e states
film, h e showed t h a t t h e friction could b e given
that :
by :
"If
the
resistance
would
viscosity
was
constant,
the
uv Q
F =
i n c r e a s e d i r e c t l y as t h e speed.
XI
As this was not in a c c o r d a n c e with Mr T o w e r ' s experiments,
in which t h e r e s i s t a n c e increased at
a much slower r a t e , i t a p p e a r e d t h a t e i t h e r t h e boundary
actions became
sensible
or
that there
must have been a rise in t h e t e m p e r a t u r e of t h e oil which had e s c a p e d t h e t h e r m o m e t e r s used to That
there
would
be
some
of
excess
temperature in t h e oil film on which all t h e work
t h e d y n a m i c viscosity, v is t h e velocity of
t h e journal, Q is t h e a r e a of t h e friction s u r f a c e E
after
carefully
considering
the
would be a d i f f e r e n c e of s e v e r a l d e g r e e s b e t w e e n This
increase
of
take
into
account
would
be
an
e v e n t u a l slip between
= o
(3)
x2 Using
that
formula,
Petrov
showed
that
f o r a given oil and f o r a c o n s t a n t load friction coefficient
temperature
are
is e x a c t f o r a n unloaded bearing with :
escape of t h i s h e a t , i t s e e m s probable t h a t t h e r e the oil b a t h a n d t h e film of oil.
x2
and
t h e lubricant and t h e solid surfaces. T h a t relation
of
means
xI
is t h e m e a n film thickness and
t h e e x t e r n a l friction o n t h e s u r f a c e s introduced
of overcoming t h e friction is s p e n t is c e r t a i n ; and
x2
is t h e c o e f f i c i e n t of internal friction
where i.e.
to
measure t h e t e m p e r a t u r e of t h e journal.
(2)
J-+l
E +
f
divided
by
the
of
product
the
viscosity and t h e velocity remains c o n s t a n t :
attended by a diminution of viscosity, so t h a t as
(4)
the resistance and t e m p e r a t u r e increased with t h e velocity t h e viscosity would diminish a n d c a u s e a In order to e s t i m a t e t h e s e t h e r m a l e f f e c t s different viscosity
measured
t h e viscosity of
temperatures
and
value
of
t h i s c o n s t a n t depends on t h e oil.
T h e e x p e r i m e n t a l d a t a obtained by P e t r o v showed
departure f r o m t h e simple ratio". Reynolds
the
olive oil at
showed
that
t h a t C was n o t c o n s t a n t but t h a t i t s variations w e r e very small. Petrov
the
was
very
much
concerned
with
t h e r m a l e f f e c t s in bearings. H e m a d e e x p e r i m e n t s
of olive oil could b e given by :
on a wagon-axle bearing at a m b i a n t t e m p e r a t u r e between
-10,6"C
experiments where T is t h e t e m p e r a t u r e in d e g r e e s c e n t i g r a d e . He
also
explained
that
the
thermal
differential dilatation b e t w e e n t h e journal and t h e bearing
modif ied
the
radial
clearance
which
increases when t h e t e m p e r a t u r e increases. A t t h e s a m e t i m e N.
Petrov
numerous experiments, actually 627 of t h e m
[5],
and
26°C.
conducted
Another
set
of
on bearing machine
to m e a s u r e m e a n oil film t e m p e r a t u r e f o r linear s p e e d s b e t w e e n 0.5 m / s to 1.075 m/s. H e obtained oil
temperature
between
20.4"C
and
a m b i m t t e m p e r a t u r e b e t w e e n 13.4"C He
[7, 81 ran
were
proposed
a
graphical
method
f r i c t i o n and oil t e m p e r a t u r e of
61°C
for
and 17.8"C.
to
calculate
a given bearing
running at d i f f e r e n t speeds and d i f f e r e n t a m b i e n t
30
0
1.2
.f!
.4
160.
1.6
120'
1bO
100.
UTEMPERATURE RELATIVE RATE OF SHEAR Fig. 2 : S h e a r r a t e , velocity and temperature a c r o s s t h e film [ 9 ]
(OF)
t e m p e r a t u r e s and lubricated with d i f f e r e n t oils. T o
H t h e h e a t flow per unit a r e a and unit t i m e in
this
t h e y direction a c r o s s t h e film.
purpose
variation
he
used
of
curves
the the
viscosity-temperature oils
and
his
friction
to c a l c u l a t e f r i c t i o n a l work. H e assumed
formula
These conditions.
t u r n is conducted through t h e solid body to t h e
The
surrounding
air.
The
prediction
agreement
and
the
between
experimental
a
published
paper
lubricating films"
1933 A.
in
entitled
[9]
effects
in
which w a s both t h e o r e t i c a l and
slightly was
to
a given
for
experimental
viscosity-temperature
law.
The
r e s u l t s a r e p r e s e n t e d on f i g u r e 2 which shows t h e relative
shear
rate
variation,
the
velocity
variation a n d t h e t e m p e r a t u r e variation a c r o s s t h e
In t h e e x p e r i m e n t a l p a r t Kingsbury used a
shape
obtained
film.
rotational viscometer in which t h e f i t t e d s u r f a c e s approximately
was
s u r f a c e s , by a graphical integration method using
experimental.
were
solution
r o t a t i o n a l speed and given t e m p e r a t u r e s on both
Kingsbury
"Heat
the
the
an
recently
describe
data
w a s very good. More
correctly
t h e r m a l problem in t h e fluid under a x i s y m m e t r i c
t h a t a l l friction w a s c o n v e r t e d into h e a t which in
graphical
equations
conical,
a
with
I%
of
the
radius.
This
used
to
give
any
desired surface
t h e o n e calculated
He
wrote
"The
showed
load
f o r isothermal conditions. that
can
be
borne
by
a
s t r e s s t h a t c a n b e maintained, i t is obvious t h a t
itself
inner
of
Kingsbury
bearing being roughly proportional to t h e shearing
test oil
The
small
2000 rpm,
s t r e s s at t h e wall is only 39 %
was
the
film.
conical
r o t a t i o n a l s p e e d of t h a t t h e shear
25.4 p m and a
coaxial to t h e o u t e r s u r f a c e d u e to t h e a c t i o n of
clearance
the
of
taper
For a radial c l e a r a n c e of
in t h e c l e a r a n c e space. T h e
t o r q u e was measured on t h e o u t e r s u r f a c e which
t h e i n t e r n a l h e a t i n g of t h e film is a n i m p o r t a n t f a c t o r in limiting t h e possible load"
was maintained fixed by t h e t o r q u e m e t e r . A t low
Ten
to
analytically
rotational
speed
a
thermometer
was
used
m e a s u r e t h e oil film t e m p e r a t u r e . In t h e t h e o r e t i c a l p a r t
years the
later
problem
[lo]
Hagg of
Kingsbury
solved using
a
p a r t i c u l a r viscosity t e m p e r a t u r e relation.
Kingsbury proposed
2.2 T h e t h e r m a l wedge c o n c e p t
t h e t h r e e following equations : The P =
f(e)
load
carrying
of
capacity
parallel
surface
t h r u s t bearings has s t i m u l a t e d a l o t of research e f f e c t s . In 1919 Harrison
on thermohydrodynamic
[Ill
suggested
could
be
that
explained
this by
load
carrying
considering
the
effect role
of
c h a m f e r at t h e groove edges. where the
p
fluid
is t h e viscosity, velocity,
8 t h e temperature,
uo t h e fluid velocity
u
at t h e
point w h e r e t h e t e m p e r a t u r e is t h e highest in t h e film,
s
the
shear
stress
in
the
film,
k
the
coefficient of t h e r m a l conductivity of t h e oil and
In
1946
e x i s t e n c e of
Fogg
[I21
also
observed
the
load c a r r y i n g c a p a c i t y in a parallel
film t h r u s t bearing. H e a t t r i b u t e d this a c t i o n to the
thermal
expansion
of
the
fluid.
This
phenomenon w a s n a m e d t h e r m a l wedge by analogy
31
with t h e inclined
pad.
In
1947 Shaw
[I31
took
into account t h e density variation of t h e oil in a
bearings is d u e to t h e t h e r m a l dilatation of fixed This s t u d y p u t s a n e n d to a half century
pads.
new form of Reynolds equation. This new t h e o r y
of
did
knowledge of t h e r m a l e f f e c t s .
not
explain
the
surfaces bearing constant
load very
at
fluid
capacity high
characteristics
of
parallel
speeds.
Assuming
across
the
film
thickness and using a simplified Reynolds equation, Cope [I41 in 1949 showed t h a t t h e load carrying
of parallel s u r f a c e s could b e i m p o r t a n t
capacity
when t h e viscosity variation with t e m p e r a t u r e is low,
the
fluid expansion c o e f f i c i e n t is high and
the
film
is
thin.
Boussugues and C a s s a c c i
[I51
load
thickness
of
carrying c a p a c i t y
The
the
1956
density
in
Zienkiewicz
the
these
thrust
bearings
both t h e viscosity
direction of
[I61
year
proposed t h a t t h e
was due to t h e variations of and
same
showed
motion. In
that
the
fluid
density variation w a s n o t sufficient to explain t h e load c a p a c i t y of viscosity
parallel is
effect
s u r f a c e s and t h a t t h e
more
important
the load. L a t e r o n C a m e r o n
and
reduced
[17, 181 showed t h a t
thermal deformations of materials, and particulary those of b a b i t t whose c o e f f i c i e n t of expansion is large. could explain t h e load c a p a c i t y of t h i s t y p e
of
bearing.
He
also
noted
that
the
density
variation of t h e fluid is n e g l i g i b l e in comparison with
viscosity
solved
both
variation.
Reynolds
equation
for
the
bearing.
He
showed
experimentally
In
both
high
Young
and
parallel
flat
that
1962
equation
[19]
the
energy
surface
thrust
theoretically
load-carrying
and
capacity
required high t e m p e r a t u r e g r a d i e n t s and very thin oil
films
which
engineering that
cannot
practice.
In
hydrodynamic
obtained
by
the
b e achieved 1963 Neal
operating
thermal
in n o r m a l
[20]
showed
conditions
dilatation
of
were
material
when they c r e a t e a convergent-divergent film. H e concluded wedge"
"thermal
effects
of
operation remark
that
were
this
not
kind
of
was also f o r m u l a t e d
Dowson and Hudson of
wedge"
lubricant
bearing
and
"viscous
essential bearing.
to
the
This
last
t h e s a m e year
by
[21] who studied t h e effect
variation
performances.
on In
the
parallel
agreement
with
thrust Swift,
Cameron and Neal t h e y showed t h a t t h e t h e r m a l dilatation
of
the
fixed
phenomenon.
Their
recently
Taniguchi
by
pads
could
conclusions and
explain
were
Ettles
showed theoretically and experimentally
this
confirmed [22]
who
that the
load carrying c a p a c i t y of parallel s u r f a c e t h r u s t
r e s e a r c h which has considerably increased our
2.3 Theory The
importance
to
given
effects
thermal
in
hydrodynamic lubrication l e d to t h e development
of
elaborate
theories.
In
1937
Swift
[23]
proposed to u s e a n e f f e c t i v e t e m p e r a t u r e and t h e corresponding
effective
of
evaluation
viscosity
the
through
dissipated
power
an using
i s o t h e r m a l theory. This e l e m e n t a r y approach which is v e r y commonly used today, gives quick but not v e r y p r e c i s e results. In 1952 C h a r n e s et a1 [24]
of
fluid
proposed
the
same
Cope. They also studied
[25]
t h e r m a l e f f e c t s on
discussed
different
equations
and
forms
film
energy
equation
as
viscosity and i t s influence o n t h e p e r f o r m a n c e s of a pad
bearing
Assuming
with
a n exponential
adiabatic
conditions
temperature
across t h e
approximate
solution
viscosity
variation
film
and
constant
t h e y presented
which with
film shape.
takes
both
into
an
account
temperature
and
pressure. In 1957 Pinkus and S t e r n l i c h t [26] g a v e the
temperature
journal
bearing
distribution for
different
in
a
mid-plane
bearing g e o m e t r i e s
a n d o p e r a t i n g conditions. T h e s a m e y e a r Purvis et solved t h e s h o r t a n d wide bearing cases
a1 [27] using
the
equations
coupling proposed
of
by
c o m p a r e d reasonably
energy
Cope.
and
Reynolds
T h e o r e t i c a l results
well with e x p e r i m e n t a l d a t a
obtained by C l a y t o n and Wilkie [28]. I t m u s t b e noted t h a t Zienkiewicz [ 161 Cuillinger and Saibel [29]
and Hunter
and Zienkiewicz
[30] a r e t h e
f i r s t a u t h o r s to p r e s e n t simultaneous solutions f o r Reynolds
and
energy
temperature
variation
same
Tipei
time,
equations
across
and
Nica
the [31]
including
film.
At
suggested
the
a
s p e c i f i c t e m p e r a t u r e variation law a c r o s s t h e film thickness in o r d e r to o b t a i n a n a l y t i c a l solution to t h e t h e r m a l problem. In
1962
Dowson
[32]
modified
the
classical Reynolds e q u a t i o n in o r d e r to t a k e i n t o a c c o u n t lubricant viscosity variations and density variations obtained
both
along a n d
a new
equation
across t h e film.
He
called t h e generalized
Reynolds equation which when associated with t h e energy
equation
describe
properly
the
phenomena in hydrodynamic lubrication.
thermal
32 Thus for
the
i n the
early
sixties
thermohydrodynamic
basic equations
(THD)
lubrication
were known. However thermal boundary conditions solution s t i l l had to be defined.
and methods of
The equations form a system of non linear partial differential equations,
the non-linearity being due
to viscosity-temperature
variations.
Two kinds of
approaches were
proposed : the
first
analytical
semi-analytical
methods
and
simplif ied hypotheses to obtain fast second
one
employed
methods to
take
into
one
with
results.
sophisticated account
used The
numerical
precisely thermal
phenomena. et
Bupara [35]
a1
341
[33,
and
Pinkus
and
have proposed an energy equation i n
fluid
velocities
Reynolds equation
are
calculated
assuming a
constant
from
viscosity
across the f i l m ; the method i s justified because of i t s simplicity. Tipei and Degueurce studied THD problems for film
[36]
have
exponential lubricating
thicknesses. With this method, Reynolds and
energy
equations
Motosh
[37]
can
be
treated
solved
the
independently.
problem
from
a
variational point of view by considering that a t a given
bearing
which
load,
corresponds
the
shaft
takes
a
position
to
the
minimum
power
dissipated. More recently Suganani and Szeri [38] proposed to elliptical appear
solve
equation.
at
the
energy
Thus
equation
reversed
like
flows
an
which
high eccentricities can be taken into
account.
It
already
proposed
must
be noted this
that
Huebner
method
in
[39]
1974.
This
approach has been criticised as i n lubrication the energy equation i s not elliptical. Smith and Tichy 1401
gave
bearing
an
analytical
thermal
solution
characteristics.
to
calculate
They
used
the
energy equation formulated by Mc Callion et a1 [41]
and
constant.
postulated
that
Hansen and Lund
the
[42,
viscosity
431
is
proposed a
simplified analysis i n order t o reduce calculations. I n this work the energy equation i s written with velocities
deduced
from
isothermal
theory
i s neglected. Recently
and
et
independently and
Mitsui
Fr@ne [46,
thermohydrodynamic bearing
case
generalized
of
solution
including
a1
[44,
451
471
presented
for
the
recirculating
Thermo.elasto.hydrodynamic
was also presented by Bou-said [48].
and a
journal flow.
A
solution
the
conditions
hypotheses
are
equation.
concerning
associated
with
boundary
the
energy
The earliest hypothesis was limited to
considered
fixed
between the
temperature
film
at
the
interface
and the solids
[16,
301 and
34,
491. As
later
to
adiabatic
conditions
early
as
1958 Guillinger
[26,
and Saibel
[29]
took
into account the fact that heat generated i n the f i l m i s carried out of the bearing by the oil and conducted through the metal of 1963 Dowson and Hudson flux
interface
continuity
[50]
by
between the
the bearing. I n proposed to use
conduction
lubricant
and
at
the
the solids.
This condition imposes the solution of
the heat
equation
i n the solids and the definition of new
thermal
boundary
between
the
conditions
bearing
Dowson and March preceding
and
[51]
theoretical
bearing case.
at
the
the
interfaces
surrounding
air.
tried to generalise the
results
to
the
cylindrical
The heat flux continuity condition
at the interface between the f i l m and the bush is
not
imposed.
temperature
They
distribution
obtained with
a
linear
discontinuities
in
slope at the maximum and minimum values. This simplified
analysis
approximation
to
gives the
a
reasonnable
surface
temperature
Using Dowson' s hypothesis, Ezzat and
distribution. Rohde
[52]
solved
the
THD
problem
for
the
finite slider bearing case. The boundary conditions are specified by the continuity both
the
fluid-solid
of heat flux at
boundaries
and
the
solid
ambient boundaries. They also proposed a change i n variable i n order t o transform the f i l m shape in
a
simple
rectangle
boundary conditions.
which
simplified
writing
More recently Huebner [53]
gave the f u l l THD solution for the sector-shaped thrust
bearing.
He
showed
that
the
adiabatic
solution which takes temperature gradient across the
film
into
account
can
often
be
used to
accurately predict bearing performance.
and
axial temperature gradient Boncompain
Most
heat
Tipei which
2.4 Boundary conditions
Another difficult film fluid.
to
inlet This
boundary condition which i s more
define which
concerns the temperature at depends
problem
was
on
the
partially
recirculated taken
into
account by Mc Callion et a1 [41, 541 and more completely
by
Mitsui
et
a1
[44,
451.
These
authors introduced a mixing coefficient which i s the ratio of the recirculated flow to the flow of
33
Recently
flow. Very recently Jeng et a1 [70] presented a
et a1 [55. 561 studied a finite journal bearing assuming t h a t t h e h e a t transfer is
thermohydrodynamic solution of pivoted thrust pad bearing in laminar and turbulent flow. They used
three dimensional in t h e fluid, t h r e e dimensional
Boussinesq' s eddy viscosity model and Reichard' s
in t h e bearing, and two dimensional in t h e shaft.
wall formula for t h e turbulent shear stress. The
the
lubricant
at
applied
the
inlet.
Boncompain
Cavitation
and
lubricant recirculation were
taken into account. Pinkus
[57]
the groove in on
recently
Heshmat and
studied t h e mechanism of mixing in
temperatures. based
Very
also
t e r m s of They
extensive
determination
of
t h e relevant flows and
proposed
empirical equations
series
test
of
for
the
t h e inlet t e m p e r a t u r e for both
thrust bearings and journal bearings.
mentioned
earlier
wedge
differential dilatation could
largely
the
modify,
t h e load
This problem
the
carrying capacity
of
bearings.
which had already been mentioned
by Reynolds was studied by different authors [58, 591.
The
elastic
and
the
homogeneity
of
t h e turbulent
flow s t r u c t u r e which is possibly a too restrictive hypothesis. The solution is obtained by numerical techniques
using
equation and
finite
differences for pressure
Galerkin' s
method for
t h e energy
equation. This kind of analysis is undoubtedly t h e to
develop
in
future
thermohydrodynamic
turbulent flow studies.
thermal
concept
for
assuming
one
2.5 Elastic and thermal deformations As
final form of t h e momentum equation is obtained by
thermal
deformations
2.7 Experimental studies In
parallel
with
theoretical
studies
a
few
experimental studies were performed to determine heat
effects on
bearing
performance
in journal
and in thrust bearings. 2.7.1 Journal bearings
generally decrease t h e load carrying capacity of tilting pad thrust bearings as shown by Rohde and
In
Oh [60]. For this reason Huffenus and Khaletzky
t e m p e r a t u r e distribution in t h e bush of a journal
[61]
proposed to include cooling system in t h e
pads
to
reduce
thermal
deformations.
Results
obtained on a large turbine thrust bearing
were
very satisfactory and bring about a reduction in the size of
t h e pads. The s a m e idea was also
proposed
Kuhn
by
[62]
who obtained t h e s a m e
t h e journal bearing case t h e thermal
deformations
of
the
bearing
are
not
uniform.
Nevertheless Boncompain et a1 [56] showed t h a t the
bearing
circle.
could
be
represented
a
by
single
this case t h e radial clearance is t h e
In
only modified parameter carrying capacity
which changes t h e load
.
gave
for
studies
laminar
authors
[63
were
flow,
but
671
tried
to
mainly
since to
1973
include
different
temperature
a1
et
[71]
flow
in
the
conditions
bush
of
temperature They
presented
of
gave
the
and
the
an
showed
experimental
journal
steadily
equilibrium
loaded journal
isotherms c h a r t in
bearing.
the
bearing
shell and showed t h a t t h e temperature is nearly constant
on
the
direction
but
shaft
that
in
the
circumferential
i t varies slightly along t h e
axial direction. They also determined t h e quantity h e a t carried o u t of t h e bearing by t h e bush, shaft
and
t h e oil and they concluded t h a t
adiabatic
conditions
are
recently
Tonnesen et
a1
not
verified.
More
[72, 731 Ferron
[74]
and Mitsui [45, 751 added experimental d a t a for different kinds of journal bearings and different conditions.
studies
theoretical
results
They
also
compared
and
took
by
using
the
obtained
Pan
[68]
and
Very recently Gethin and Medwell [76] presented
Constantinescu et a1 [69] for isothermal turbulent
experimental d a t a for a journal bearing fed by
lubrication.
In these approaches i t is implicitely
two
axial grooves and
assumed t h a t t h e thermal transport phenomena a r e
and
non
basically t h e s a m e in laminar and in turbulent
circumferential
account
the
analysis
performed
turbulence by
effects
Ng
and
into
studied
t h e minimum film thickness zone. In 1966 Dowson
operating
adiabatic conditions
[28]
inlet
mapping
thermal effects in turbulent analysis. Most of t h e used
Wilkie
t h a t t h e maximum bush t e m p e r a t u r e is located in
the
Thermohydrodynamic various
for
and
(pressure, t e m p e r a t u r e and type of grooves). They
of
2.6 Turbulence
performed
bearing
investigation
type of results.
For
1948 Clayton
good
with
agreement
laminar
experimental between
data
these results.
operating under
conditions.
temperature
and
They variation
laminar
gave
the
on
the
34
bearing
center
line
for
different
loads
and
high
speed
thrust
large
amount
of
power is dissipated by oil churning between t h e
clearance ratio. T h e tilting pad journal bearing was studied
pad.
He
proposed
by D e Choudhury et a1 [77, 781 ; Booser et a1
with
a
drained
[79] and Fillon et a1 [ 8 0 ] . These authors showed
temperatures
that
developed
the
a
bearing
maximum
temperature
of
the
pad
is
to use a directed lubrication casing
and
by
power
New
losses.
bearing idea
compared
flooded,
t h e end of t h e pad at about 80 % of its length.
lubrication
They
fluid
author showed t h a t directed lubrication decreases
t e m p e r a t u r e is not representative of t h e maximum
t h e maximum pad t e m p e r a t u r e between 10 to I5 %
pad temperature.
when compared to t h e fully flooded bearing case.
noted
The
transition
laminar
flow
studied
by
showed
in
the
tilting-pad
the
outcoming
between
Gardner
that
slightly
that
laminar
non
journal bearings was
and
Ulschmid
maximum
decreases just
to
[81]
pad
who
temperature
a f t e r t h e transition. This
for
tilting
inlet pad
orifice
was fully
situated on t h e loaded pad in a zone located near also
and
This
who
[88]
directed
reduce
to
thrust
type
bearing.
of The
of non laminar flow on t h e t h r u s t bearing, was studied by
The effect temperature
in
C a p i t a 0 et gl These
[89, 901 and Gregory
authors
between
showed
laminar
the
transition
laminar
temperature
flow,
maximum
Masters [ 7 8 ] .
speed increases. This d e c r e a s e also observed with journal bearing
Thrust bearings
[81]
decreases
when
the
fact was recently confirmed by D e Choudhury and
2.7.2
pad
at
that non
to
191, 921.
the
C and was
c a n reach 20"
due to b e t t e r h e a t transfer convection conditions Thrust bearings which a r e usually larger than t h e journal
bearings
mounted
on
the
same
machine
have correspondingly higher s u r f a c e velocities and a r e more prone t o t h e r m a l problems. dissipated
that
the
power.
pad
The
authors also showed
temperature
decreases
when
the
number of pads decreases. This fact was recently confirmed by Neal
C o l e [ 8 2 ] showed t h a t
[74].
non
laminar
flow.
The
effect
of
lubricant
supplied methods on t h e maximum t e m p e r a t u r e of non
laminar
tilting
pad
thrust
bearing
was
recently studied by Mikula and Gregory [93, 941.
Early studies [82, 831 w e r e concerned with the
in
These
authors
showed
that,
in
conventional
a
flooded thrust bearing a large amount of power is
dissipated
leading
between
edge
the
pads
distribution
and
groove
that
bearing
the gave
lower maximum pad temperatures.
t h e inlet flow conditions have a n important effect on temperature. He used a n oil supply groove at
2.8 Conclusion
t h e inlet of
Since Hirn, 132 years ago, t h e thermal e f f e c t s in
power [85]
and
t h e pad to d e c r e a s e t h e dissipated the
pad
temperature.
measured
the
film
Baudry
thickness,
the
et
a1
babbit
fluid
lubrication
were
be
t h e pad thickness on a large-waterwheel-generator
performance.
From
constitutive
equations
pivoted-pad
thrust bearing. They showed t h a t pad could
temperature performed
be
due
to
gradients. the
the
Elwell
s a m e t y p e of
load et
and
a1
experiments
destroyer
USS Barry.
They
showed
that,
to describe t h e r m a l
for
a
and
work
industrial
bearings
t e m p e r a t u r e is noted.
t h r u s t bearings.
of
method
outlet
Additional
t e m p e r a t u r e but poor correlation with t h e babbit supply
the
the
plays
an
important role on t h e power dissipated in t h r u s t bearing. Bielec and Leopard [87] showed t h a t for
elaborate.
phenomena
in a c t u a l
is
needed
both
in
t h e o r e t i c a l and experimental sides : and
lubricant
that
function
more
conditions were written in
bearings.
-
largely
thermal
Then numerical methods and computing
solutions
is
bearing
1965
programs w e r e elaborated for different kinds of
oil
The
conditions,
bearings.
inlet
temperature
became
order
a
The
predict
to
1940
[86] on
to
account
1955 boundary
t h e bearing grooves may vary widely depending on flow
into
Since
constant inlet oil t e m p e r a t u r e , t h e t e m p e r a t u r e in the
taken
the
centrally pivoted s e c t o r t h r u s t bearing pad of t h e
extensively.
f i r s t works showed t h a t t h e r m a l e f f e c t s have to
t e m p e r a t u r e , and t h e t h e r m a l gradient throughout
distorsion
studied
-
Analysis
should
provide
performance
Thermal
should b e analysed
as
data
tilting
effects
in
pad
more for
actual complex
journal
superlaminar
and
flow
35
Experimental map
and
between
data
mixing the
including
temperature
pads
for
actual
in
temperature
nondimensional pressure, R is t h e radius and L is
grooves
t h e l e n g t h of t h e bearing.
bearing
or
must
In
be
the
classical
obtained.
active
form
of
zone
the
of
non
the
film,
dimensional
the
energy
equation used in lubrication is : 3
THERMOELASTOHYDRODYNAMIC ANALYSIS OF JOURNAL BEARINGS
t h e paper a s u m m a r y of
In t h e second p a r t of the
methods
of
solution f o r t h e journal bearing
(9)
[(-=) a; 2
= N
case presented in [ 5 5 , 561 is given.
h2
3.1
Basic equations
The
problem
is
formulated
with
the
following
-
the
film
-1 a% -2 -2 aY
h
T
fluid,
velocity 2/26
*
the
-
u,
v,
nondimensional
w
c o m p o n e n t s of
are
curvature
of
the
film
is
the
t h e fluid
introduced to c h a n g e
t h e s h a p e of film i n t o a rectangular field
no slip a t t h e i n t e r f a c e general
and
the
and t h e o p e r a t o r
negligible e x t e r n a l and i n e r t i a f o r c e s the
in
nondimensional
- s t e a d y and l a m i n a r operation
-
number
temperature
newtonian and incompressible fluid
aY
in t h i s equation Pe is t h e Peclet number, Nd t h e dissipation
assumptions
+ (=) ;a 2 +
ay
[52],
is given by :
neglected -
thickness
is
(10)
very s m a l l when
compared to t h e o t h e r dimensions. generalized
T h e i n a c t i v e z o n e of t h e film c a n b e represented
Reynolds equation in non dimensional f o r m c a n b e
Under
these
conditions,
the
as shown in f i g u r e 3 ; t w o d i f f e r e n t s zones have
written [32]
to b e considered :
symctncc! a m of mr b a r i n g
I, 1
where
t h e nondimensional functions F
F2 a r e defined by
0'
F I , and
2
: 3 I
5
(7) '
p
'0
F2 =
L'.
i14 (7 O l J
8,
the
nondimensional
F) F1
L' . L / 2
o
dy
0
film
thickness,
and
p
Fig. 3 : Developed half bearing : c a v i t a t i o n a r e a
the
viscosity which is assumed to b e
defined by :
pi
L
_ * 1
L12
Z a r e t h e nondimensional coordinates, F i is
nondimensional
6
In t h e fluid z o n e as t h e pressure gradient i s nil, t h e e n e r g y equation is w r i t t e n :
are, respectively, t h e nondimensional t e m p e r a t u r e
and t h e viscosities of
t h e lubricant
-
in t h e film
and at t h e inlet of bearing groove, p is t h e
In t h e gas zone, t h e power dissipation is negligible a n d
o n e c a n a s s u m e t h a t t h e energy
36
equation reduces to :
where Bb i
hbR2/Kb i s the Biot number for the
5a
the nondimensional ambient temperature,
bush,
and R 2 the
a2T
-=
(12)
-2 aY
-
where
Ts
outside bush radius.
On the surface between the bush and the fluid, the
This yields :
nondimensional
temperature
is
given
by
the
heat
flux
continuity condition which gives :
-
and
Tb
are,
respectively,
the
nondimensional shaft and bush temperatures. The heat transfer equation
where
Kb
the
is
thermal
conductivity
of
the
bush, and K(e) i s the thermal conductivity of the
i n the bush is
defined as follows :
and equal t o KO i n the
fluid which i s constant active zone and which
i s variable i n the inactive
zone of the f i l m and given on Fig. 4 , where
-a2T + - - +-1 - aT -2 ar
The transfer
r
same in
a2T 2 I,(+
i -2 r
aF
equation
the
R 2
ae
is
shaft,
a2T =
-2 az
used
assuming
temperature i s independent of
(14) -,
for
heat
that
the
4
___
i
E V E N I G A S ZONEI
i
ODD
(OIL
LONE)
e.
Temperature and pressure fields change the initial geometry of both shaft and bush. I n order t o determine thermoelastic
surface displacements,
the thermoelasticity laws are used :
z
Ka
a
w
(15)
X
cc
0 In
this
relation,
E
ij
and
'11 are respectively small strain and stress tensors, E and
y are
the
respectively Poisson's
the modulus of
ratio,
X
is
the
of
expansion and A T (MI i s the temperature variation at point M, considered with regard to a reference temperature. 3.2 The
K a i s the thermal conductivity
located on
Reynolds
the gas, and
L ' ( e ) i s defined as the apparent length of the fluid
zone
i n the
inactive arc by the relation
1
boundary
conditions
the
load line
for
a f u l l journal
multilobe bearing. The boundary conditions on the temperature On the outer
part of
the bush,
the free convection and radiation hypothesis gives :
L(Os5?)dy]dz
~~'2(~(Os~
are
bearing and a t the beginnning of each pad for a
are as follows.
of
(Fig. 3)
associated with Reynolds equation. An inlet groove is
inactive zone of the film
Boundary conditions classical
1 V L
Fig. 4 : Equivalent thermal conductivity i n the
elasticity and coefficient
1/2
0
L'(e)=L
1/2 h(0) 0
(18)
1
(j
u(8yg)dy]dz
0
On the surface between the fluid and the shaft, the temperature i s given by the heat flux continuity
conditions
assuming
temperatures are independent of 0
that
. This
the
shaft
gives :
37
Ks
where
the
is
thermal
conductivity
of
the
5= 2
1/2,
differential equation :
shaft. A t the ends of the shaft i.e., for a
free
convection
hypothesis
assumed which
is
gives : This
equation
is
then
integrated
using
finite
difference with the following boundary conditions :
where 8. = hS L/Ks i s the Biot number for the IS
shaft. In inlet
the
film
inlet
zone,
temperature
parabolically
across
film,
the
is assumed t o vary
%y)
-
Ts
between
the
-
and
Tb
which
are
respectively the shaft and the bush temperatures at the inlet. The third value needed t o determine the
temperature
profile
is
calculated
from
conservation equations by the following relation :
3.3.3
Energy equation
The energy equation (9) looks l i k e a differential
e plays the role of the
parabolic' equation where
time i n the non-stationary problems. An implicit finite
method i s used to
difference
equation employed positive
and
the
when i.e.,
solve
Richtmyer technique
the when
coefficient the
this
[95]
is
?T/a y2 i s velocity i is
of
fluid
positive. When
a
reversed
eccentricities for
The
example,
-
occurs,
at
large
u i s negative i n the
inlet zone of the f i l m and classical solutions do
3.3 Methods of Solution and Procedure 3.3.1
flow
not
converge.
In
this
case,
the
fluid
film
is
Pressure Distribution i n the F i l m and Heat
divided into two zones : these zones D1 and D2
Transfer i n the Solids
are shown on Fig. 5. I n the first one which
pressure
temperature
distribution field
both
in
the
i n the
film
film
and
the
and i n the
solids are obtained by iterative techniques. Finite differences
and
Gauss-Seidel methods
with
over
relaxation are employed to solve the generalized Reynolds equation. set
equal
to
zero
Negative when
pressure they
terms
appear
in
are the
iterations needed t o satisfy the Reynolds boundary conditions. methods
Finite with
differences
over-relaxation
and are
Gauss-Seidel also
used to
solve heat equations both i n the shaft and i n the bush.
3.3.2
Velocity Fluid i n the y Direction Across
9
the F i l m
M O V I N G SURFACE
Due to numerical uncertainties, direct computation of
the velocities in the f i l m yields a velocity
in the
7
direction
diffferent
from zero on the
bush. This result is physically impossible and could induce some difficulties i n the converging process. Thus the continuity equation i s differentiated with respect to
7
which gives a second order
Fig. 5 : Streamlines i n thrust bearing for hl/h2
= 4
38
corresponds to the zone where i s positive, the energy equation is solved using the iterative Richtmyer
technique.
values
the
are
The
initial
temperature
boundary conditions at the inlet
imposed
the
On
continuity surface.
moving
conditions
The
surface
are
and heat
used
on
the
flux fixed
f u l l lines are the isothermal lines
lines.
and the dash lines are the heat
boundary and a given value at points situated i n
D2
near
zone, the
the
the
boundary
same iterative
computing
boundary
(fig.
process
6a).
I n the second
technique started
is
i s used but from
the
using the initial temperature at points
situated i n Dl
near the boundary (fig. 6.b).
The
BOUNDARYlrl
Fig. 7 : Isotherms and heat flux lines i n the f i l m for hl/h2
3.3.4 Fig. 6a : Resolution i n zone D
1
= 4
Shaft and bush bearing deformations
Experimental
results
displacements
of
have
both the
shown shaft
should be taken into account. BOUNDARY
I rl
film
thickness
and
the
that
thermal
and the
bush
I n this case, the
depends on both the eccentricity
thermal
displacements.
But
the
axial
temperature variations i n both the shaft and the is
bush
small
displacement
and
effect
its
can be neglected.
on the
thermal
I n this case the
expression of f i l m thickness can be written :
h
where
=
1
+
( 8)
E
is
cos 8 + 6 ( 0 )
the
difference
between
the
non-dimensional displacements of the bush and the shaft. Fig. 6b : RCsolution i n zone D iterative
process
is
In
2
performed
relations alternatively
in
both zones and converges very quickly. Figure 7 shows,
as
an
example,
results
obtained
for
plane thrust bearing. A constant temperature i s
a
this are
case,
classical
used analytically
thermoelasticity for
the shaft i n
which the temperature
i s axisymmetrical due to
the
In
rotational
speed.
the
bush,
the
two-
dimensional thermal displacements are determined using finite element method. This computation i s made only i n the mid-plane of the bearing.
39 Table I : Operating conditions
3.3 Procedure The computational procedure is described by t h e flow chart of fig. 8. The global i t e r a t i v e scheme is as follows : a n initial value for the temperature field is given to calculate t h e fluid viscosity at each point along and across t h e film. Reynolds equation is solved and t h e fluid velocity vector is calculated at all points of t h e fluid. Energy equation and h e a t transfer equations a r e then solved simultaneously in t h e fluid and in t h e solids thus producing a new t e m p e r a t u r e field. The elastic displacements a r e determined in both t h e shaft and t h e bush which give new film thickness values. stopped when at
The each
iterative point on
procedure is t h e boundary
between t h e film and t h e bush, t h e relative difference between t w o successive s t e p s is less than 0.1 percent. The convergence of this process is fairly rapid.
-
Initialisation Temperature
-
Viscosity
*,
Journal radius R = 50 mm External bearing radius R2 = 100 mm L = 80 mm Bearing length Radial c l e a r a n c e at 200~ C = 145 Um speed range 1000
'
,.
Energy equation
Film thickness
1-1
p + L l
Therrnoelastic displacements
3.4 Results and discussion In
to
order
compare
experiments, t h e
the
theory
and
results were obtained
the
for t h e
operating conditions used by Ferron [74] in his experimental study. These conditions a r e given in Table 1. As t h e experimental bearing is housed in a protective box, t h e ambient t e m p e r a t u r e is above t h e inlet lubricant t e m p e r a t u r e (40°C). Depending on t h e operating conditions, t h e
5-
Bearing characteristics
ambient
temperature
varies between
47°C.
In
conditions
these
43°C
the
and
ambient
t e m p e r a t u r e is assumed to be constant and equal
to 4 5 ° C as shown in Table 1. Figures 9 and 10 produce t h e isothermal chart
Fig. 3 : Flow chart of t h e computational pr oceedu re
in
the
eccentricity
of
rpm
for
and
journal 0.8,
a
bearing
at
a
relative
a rotational speed of 2000 convection
heat 2
coefficient of t h e bush of 80 W/m "C.
transfer
40
D i f f e r e n t s c a l e s a r e used f o r t h e s h a f t , t h e bush
and
for
t h e film which a p p e a r s to b e of
first one, which corresponds to t h e coldest p a r t of t h e film t h e s h a f t and t h e bush give h e a t to
c o n s t a n t thickness b e c a u s e of t h e variables used.
t h e fluid.
Figure 9 which gives t h e isothermal c h a r t in t h e
to t h e h o t t e s t p a r t of t h e film t h e fluid gives
mid-plane shows t h a t t h e maximum t e m p e r a t u r e is
h e a t to t h e bush a n d to t h e s h a f t . T h e s h a f t and
situated
t h e bush c a r r y o v e r t h e h e a t f r o m t h e h o t t e s t
in
the
oil film
near
t h e bearing e d g e
downstream t h e minimum film thickness zone. T h e h e a t flux lines which a r e perpendicular
In t h e second one, which corresponds
z o n e of t h e film to t h e c o l d e s t one. This fact is clearly
shown
to t h e isothermal lines are deduced f r o m fig. 9
temperature
and a r e given in fig.
different
10. This f i g u r e shows t h a t
t w o d i f f e r e n t zones e x i s t in t h e bearing. In t h e
on
Fig.
I1
abscissae.
which
shows
the
the
film
for
across
variation For
smaller t h a n
190 d e g
t h e s h a f t is h o t t e r t h a n t h e bush ; b u t f o r
e
higher t h a n t h a t value, t h e s h a f t is colder t h a n the
bush.
These
conditions
can
results only
show
be
that
used
adiabatic
as
a
first
approximation e v e n though t h e g r e a t e r p e r c e n t a g e of t h e h e a t (about 90 p e r c e n t ) is c a r r i e d o u t of t h e bearing by t h e fluid.
2 5
w
a D
F
Fig. 9 : Isotherms in t h e mid-plane bearing f o r
4
a
a n e c c e n t r i c i t y r a t i o of 0.8 a n d rotational speed of 2000 rpm
oil supply groove
40 1
0 NONDIYENSIONAL
THICKNESS
Fig. I 1 : T e m p e r a t u r e variations in t h e film in mid-plane f o r d i f f e r e n t angular positions
N = 2000 rpm, Figure
12
gives
E
the
= 0.74
axial
temperature
variation on t h e s h a f t s u r f a c e a n d o n t h e bush internal surface for
e
= 220 deg. T h e waves noted
in t h e s h a f t t e m p e r a t u r e a r e due to t h e method of Fig. 10 : H e a t flux lines in t h e mid-plane f o r a n e c c e n t r i c i t y r a t i o of 0.8 a n d a rotational speed of 2000 rpm
solution applied in t h e c a v i t a t i o n zone. This
f i g u r e shows t h a t t h e axial t e m p e r a t u r e variation is very s m a l l a n d could b e o f t e n neglected.
41
ratio a r e t h e only modified parameters due to thermal deformations. Variation of film thickness at 45 degrees on both sides of load direction versus eccentricity
t
is
presented
in
Fig.
14.
Experimental
and
theoritical results a r e in good agreement : t h e observed discrepancies a r e less than 3
m. This
shows t h e very good quality of measurements on t h e one hand and validates calculation of bearing thermal deformations. From a geometrical point of view, good agreement between theoretical and experimental
results
is
obtained
if
thermal
deformations a r e taken into account.
Fig. 12 : Axial t e m p e r a t u r e variations on t h e
220t
shaft surface and t h e bush internal surface ( The displacements
shaft
E
= 0.8 and N = 2000 rpm)
and
calculated
the in
the
bush
thermal
mid-plane
are
presented in Fig. 13. Different scales a r e used on
look
Figure 14 : Film thickness variations at 2 45 degrees of t h e load line versus t h e eccentricity r a t i o Figures pressure
15 and
variations
in
16 give t e m p e r a t u r e and
the
mid-plane
of
the
internal bush s u r f a c e for a n eccentricity ratio of 0.8
and a rotational speed of 2000 rpm. Large
discrepancies Fig. 13 : Radial thermal deformations on both t h e shaft and t h e bush surface (
E =
0.8
before deformation, N = 2000 rpm) this figure and thermal displacements a r e reported from t h e undistorted s h a f t and bush circle. O n e can see t h a t unless t h e t e m p e r a t u r e around t h e bush is not constant, t h e internal distorted bush could be approximated by a single circle. In this case t h e radial clearance and t h e eccentricity
results
are
obtained
deformations
of
noted
between
with
and
the
shaft
theoretical
without and
thermal
the
bush.
Theoretical results including thermal deformations a r e in good a g r e e m e n t with experimental d a t a of Ferron [74]. of
Figure 17 shows t h e maximal temperature t h e bush internal s u r f a c e versus eccentricity
ratio. Good a g r e e m e n t is observed particularly for eccentricities above 0.5 : t h e discrepancies which d o not exceed 3"C, c o m e first, from modelling and from t h e difficulties in t h e definition
of
42
-0--
," 5 51 ---
E X P E R I M E N T A L RESULTS
0
THEORICAL R E S U L T S W i T n o u T
T HD T E HD
THERMAL DEFORMATION
THEORICAL RESULTS W I T H T H E R M A L DEFORMATIONS
A
z
0
I
w w
I
/
55 / #-'
4 0
I
--*
0
270
180
90
360
50
DEGREES
Fig. I5 : Temperature variations in mid-plane at 2000 rpm and for
= 0.8
E
1
,
,
'
,
,
,
,;
; ;;:,f ___--
,
45
0
0.2
0.4 0.6 0.8 1 E C C E N T R I C I T Y RATIO
E X P E R I M E N T A L RESULTS
T HD T EHD
THEORICAL RESULTS W l T n O U T THERMAL DEFORMATION THEORICAL RESULTS W I T H T H E R M A L DEFORMATIONS
Figure 17 : Maximum bush t e m p e r a t u r e versus
A
eccentricity ratio
L
DEGREES
Fig. 16 : Pressure variations in mid-plane at 2000 rpm and for
E
= 0.8
.
c o r r e c t thermal boundary conditions for t h e solids. The minimal temperature eccentricities near 0.4.
is
obtained
capacities and t h e nondimensional axial flow r a t e of t h e bearing versus t h e eccentricity r a t i o for speeds.
Good
agreement
is
thermal effect appears clearly on fig. gives
the
axial
isothermal solution.
flow
rate
19 which
obtained
2 0 0 0 R PM
01
0
1
a
0.2
-
'
0.4
'
'
0.6
'
'
0.8
ECCENTRICITY
'
*.C
1 RATIO
found
between experimental d a t a and theoretical results which include thermoelastic displacements. T h e also
4@00 R P M
Y'.,
for
Figures 18 and 19 show t h e load carrying
different
0
5000 R P M
from
Fig- 18 : Load carrying capacity versus eccentric i t y ratio for different speeds
43 Kef e r e n c e s
"Thermal E f f e c t s in Tribology" Proceedings of t h e 6 t h Leeds-Lyon Symposium on Tribology, 1980, (Mech. Eng. Publ. Ltd).
I
W k
2
0.8
PINKUS, 0. and WILCOCK, D.J. "Thermal e f f e c t s in fluid film bearings" Proc. of t h e 6 t h Leeds-Lyon Symposium, 1980, 3-23.
3
0 d k 0.6.
HIRN, G.A. "Sur l e s principaux p h k n o m b e s q u e prdsentent l e s f r o t t e m e n t s mddiats, et sur l e s diverses mani&es employkes au graissage d e s machines", Bull. SOC. Ind. d e Mulhouse, 1854, XXVI, 188-277.
4
< a
0
0.4 -
a
w
z
I
z
Compte-rendu d e 1' Acaddmie d e s Sciences, 1849, 28, 290.
2 000 R P M
0.24000 RPM
0
" 01
a
0.2
0
,
I
I
I
DOWSON, D. "History of Tribology", (Lonaman Grouo. London).
REYNOLDS, 0. "On the theory of lubrication and i t s application t o M. Beauchamps Towezr' s experiments, Phil. Trans. Roy. SOC. 1886, A 177, 157-234.
I *
1
0.4
0.6 0.8 1 E C C E N T R I C I T Y RATIO
Fig. 19 : Nondimensional flow r a t e versus eccentricity r a t i o for different speeds 3.5 Conclusion The results show t h a t : -
the
temperature
gradient
across
and
along t h e fluid film is important ;
of
h e a t recirculates from t h e h o t t e s t point
the
groove
area
in
the
fluid
due
to
convection, in t h e bush due t o conduction and in the shaft due t o s h a f t rotation ; - most of t h e h e a t g e n e r a t e d is evacuated
by fluid flow. Good
agreement
experimental
results
is
deformations
together
between obtained with
theoretical
and
when
thermal
differential
thermal
dilatation between t h e bearing and t h e s h a f t a r e considered.
4 This
ACK PI OWLEDCEMENT
work
Dklkgation et
was
supported
by
the
Gdndrale a l a R e c h e r c h e Scientifique
Technique
80.7.0658
partially Contracts
No
1979,
80.7.0657
and
and by Electricitd d e F r a n c e Direction
des Etudes et Recherches C o n t r a c t s no 2D 3085
P 33 D 15 and 1089-1191 and by CIT ALCATEL Division Graffenstaden C o n t r a c t N o 650 309.
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MITSUI, J. "A study of t h e lubricant film c h a r a c t e r i s t i c s of journal bearings (Part 3 : e f f e c t s of t h e film viscosity variation on iournal t h e d y n a m i c c h a r a c t e r i s t i c s of bearing;)", Bull. JSME, 2,no 210,' 1982, 2018-2029.
(1031 HUEBNER, K.H. "Solution f o r t h e pressure and temperature in thrust bearings operating in the thermohydrodynamic turbulent regime", ASME, Journ. Lub. Tech., 96, 1974, 58-68. [I041 GREGORY, R.S. " P e r f o r m a n c e of bearings at high o p e r a t i n g speeds", Journ. Lub. Tech., 96, 1974, 7-14.
KNIGHT, J.D. and BARRET, L.E; "An approximate solution technique for multilobe journal bearings including t h e r m a l effects, with comparison to experiment", ASLE Trans., 26, no 4, 1983, 501-508.
thrust ASME,
KIM, K.W., TANAKA, M. and HORI, Y. "A three-dimensional analysis of thermohydroof sector-shaped, dynamic performance tilting-pad t h r u s t bearings", ASME, Journ. Lub. Tech., 105, 1983, 406-413.
[ 1051 NEW,
N.H. "Experimental comparison of flooded d i r e c t e d , and i n l e t o r i f i c e t y p e of lubrication f o r a tilting pad t h r u s t bearing", ASME, Journ. Lub. Tech., 96, 1974, 22-27.
C. The development of a [I061 ETTLES, generalized c o m p u t e r analysis f o r s e c t o r shaped tilting pad t h r u s t bearings", ASLE Trans., 19, no 2, 1976, 153-163.
CHIEN, H.L. and CHENG, H.S. "T her mohydr odynam i c per for m a n c e of s m o o t h plane sliders", Proc. of t h e JSLE International Tribology C o n f e r e n c e , Tokyo, 1985, 559-564.
and MEDWELL, 3.0. "A 071 BOWEN, E.R. thermohydrodynamic analysis of journal bearings operating under turbulent 1978, 345-353. conditions", Wear, 1, 081 SAFAR, Z.S. "Thermohydrodynamic analysis f o r laminar flow journal bearing", ASME, Journ. Lub. Tech., 3,1978, 510-512.
[ 109
[I10
KAWAIKE, K., OKANO, K. and FURUKAWA, Y. " P e r f o r m a n c e of a l a r g e t h r u s t bearing with minimized thermal distorsion", ASLE Trans., 22, no 2, 1979, 125- 134. SUGANAMI, T. and SZERI, A.Z. "A of journal bearing parametric study p e r f o r m a n c e : t h e 80 deg. p a r t i a l a r c 101, bearing", ASME, Journ. Lub. Tech., 1979, 486-491.
J.A. and SMITH, R.N. "An [I111 TICHY, analytical solution f o r t h e r m a l behavior of t h e s t e p t h r u s t bearing", ASME, Journ. Lub. Tech., 102, 1980, 34-40.
TANAKA M., HORI, Y. and EBINUMA, R. "Measurement of t h e film thickness and t e m p e r a t u r e profiles in a tilting pad t h r u s t bearing", Proc. of t h e JSLE International Tribology C o n f e r e n c e , Tokyo, 1985, 553-558.
[ 122
GARDNER, W.W. " P e r f o r m a n c e c h a r a c t e r i s t i c s of t w o t i l t i n g pad t h r u s t bearing designs", Proc. of t h e JSLE International Tribology C o n f e r e n c e , Tokyo, 1985, 61-66.
[I23
MIKULA, A.M. "Evaluating tilting pad thrust bearing operating temperature", ASLE Trans., 2, no 2, 1986, 173-178. KHONSARI, M.M. and BEARMAN, J.J. "Ther mohydr odynam i c analysis of laminar incompressible journal bearings", ASLE Trans., 29, no 2, 1986, 141-150.
This Page Intentionally Left Blank
49
Paper Il(iii)
Michell and the development of tilting pad bearings J.E.L. Simmons and S.D. Advani
The major contribution of A.G.M. Michell to the science of lubrication through his three dimensional solution to Reynolds' equation is well-known. Much less known is the story of the industrial introduction of tilting pad bearings to Europe and the eventual establishment of a specialist bearing company carrying the inventor's name. The purpose of this paper is to relate something of this history and then to highlight some of the subsequent technological changes encountered by the company, Michell Bearings, in the 65 years of its existence.
1.
INTRODUCTION
The personal biography of A.G.M. Michell, figure 1, has been documented several times (1, 2, 3) and most recently and On the comprehensively by Dowson (4, 5). occasion of the Osborne Reynolds centenary it is perhaps appropriate to recall that Michell (6) in 1905, nineteen years after Reynolds had considered the lubrication of plane surfaces, obtained a three dimensional solution to Reynolds' equation which took account of side leakage. Nearly forty years later, on the occasion of the James Watt International Medal presentation in 1943, Michell's achievement even then was said to be the only really important extension of Reynolds' theory yet effected. 2.
FIRST USES OF MICHELL BEARINGS
Michell took out patents covering his tilting pad invention in Australia and Britain in 1905, the same year that his solution to Reynolds equation was published. At the time Michell was running his own small consultancy business specialising in hydraulic machinery. The first tilting pad bearing in service was probably that built under his guidance by George Weymouth (Pty) Ltd., for a centrifugal pump at Cohuna on the Murray River, Victoria The bearing, shown in figure 2, in 1907, (5). was described by G.B. Woodruff (7) in a lerture given to the Institute of Marine Engineers in October 1908. The same bearing was also described later by H.T. Newbigin in "The Michell Bearing Book", a private publication dated April 1916. According to Newbigin the total load was 13.3 kN (3000 lbf), equivalent to a specific bearing load of 1.5 MPa (220 psi) and the running speed was 200 revolutions per minute. When the bearing was stripped down after five months of continuous running hand scraper marks were still evident on the tilting pad surfaces such had been the absence of wear. Evidently the success of the installation was rewarded for Newbigin notes that 'a large
number of the bearings have since been fitted to pumps of this kind'. The early industrial development of Michell bearings took place through the granting of manufacturing licences to interested parties. In England Michell formed a partnership with H.T. Newbigin, who was a fellow civil engineer based in Newcastle-uponTyne, to promote his invention and administer the licensing arrangements. The actual date of the formation of the Michell Newbigin partnership is uncertain. However in the discussion following a paper presented by J.H. Gibson to the Institute of Naval Architects in 1919 (8). Newbigin claimed to have introduced the Michell bearing to Britain in 1905 and it is possible that he and Michell already had some working arrangement as early as this. Certainly Newbigin had co-operated with Woodruff in the preparation for the latter's 1908 lecture which has the distinction of being the first published account in Britain of Michell's invention other than in the 1905 patent application. In a paper delivered to the Institution of Civil Engineers in February 1914, Newbigin (9) describes a Michell bearing designed by him for Belliss and Morcom in 1910. The bearing which is shown in figure 3 carried a load of 6.7 kN (1500 lbf), corresponding to a pressure of 3.5 MPa (500 psi), and operated at 1750 revolutions per minute. Judging from the discussion, the paper must have raised considerable interest and particular attention was given to the question of the optimum position for the pivot upon which the pads tilt. In the discussion of another paper by Gibson ( l o ) , Newbigin gives details of tests carried out on the Belliss and Morcom bearing in June 1910 to determine the effect on pad temperature of different pivot positions while retaining constant operating conditions. Temperature was measured by a thermometer placed in a hole in the pad, close to the working face. At the duty conditions given above the following results were obtained: with the pad pivoted 3 mm ( 1 / 8 inch) behind centre, which was close to the optimum point according to the Michell theory, a temperature
-
of 51.1"C (124'F); was recorded; pivoted 1.5 mm (1/ 16 inch) behind centre the temperature was 52.2"C (126'F); pivoted at the centre, 55.6"C (132'F); pivoted 1.5 mm (1/16 inch) in front of centre, equivalent to an offset pivot operating in reverse, the This early temperature was 5 7 2 ° C (135'F). evidence of the considerable reverse running capabilities of offset pads is of contemporary interest for as a recent paper (11) makes clear this is a feature of tilting pad bearing operation which even now is not well documented. It seems that the adoption of tilting pad bearings for land based machinery proceeded in the next few years steadily if not spectacularly. In his 1914 paper, Newbigin ( 9 ) was able to state that considerable numbers of Michell thrust bearings had been made in Australia and England, and by 1916, in Britain alone, upwards of 800 Michell thrust bearings were in service ( 1 2 ) . They were in use for a range of applications from high speed steam turbines to low speed, high load grinding machines. In this latter case quoted by Newbigin the lubricant was thick grease and 1% ounces were required each day. It is interesting to note even at this early stage the principal bearing designers, Newbigin in England and Michell in Australia, were turning their attention to the problem of maintaining effective lubrication independently of external oil supply. Figure 4 , taken from Newbigin's "The Michell Bearing Book" of 1916, shows a vertical self-contained journal bearing which was fitted to two vertical circulating pumps at Newport Power Station in Melbourne, Australia. Circulation of the lubricant is achieved by a fixed curved nozzle projecting into a reservoir rotating with the shaft at 490 rpm. The bearing is formed by the four spherically seated, tilting ''bearing splints" shown separately in figure 4. In later years the design of self-contained systems became a major speciality of Michell Bearings which extends to the present day. 3.
MARINE THRUST BEARINGS
Up to 1914, the use of tilting pad bearings f o r the thrust blocks of ships had barely commenced although it was this application which was eventually going to lead to the real boom in interest in Mr. Michell's bearings. The published discussion which followed Woodruff's 1908 lecture gives some idea of the enormous scepticism Newbigin must have faced among marine engineers in the early stages. He persevered trying to interest shipbuilders in the Michell principle but was completely unable to get anyone to abandon the traditional multicollar thrust block with flat horseshoe surfaces. In truth though, Newbigin was not only competing against conservatism in shipbuilding and designers but also contending with a barrier caused by another technological change. The direct drive steam turbine developed for marine use was being increasingly In installed in vessels of the early 1900:s. these turbines practically all the thrust was balanced by steam pressure on a dummy piston
( 1 3 ) . A small thrust was indeed necessary on the end of the turbine spindle but its purpose was merely to register and maintain the fore and aft position of the rotor in the casing; its loading was negligible compared with the total thrust of the ship. H.M.S. Repulse, for example, launched on 8 January 1916 and in her day the most powerful warship in the world was fitted with the largest direct drive turbines ever built up to that time. In consequence Repulse, a four screw ship, had no requirement for main thrust blocks but f o r "only a comparatively small block on each line of shafting", ( 1 4 ) . Nevertheless each block had on 16 a thrust surface of 2.82m' (4,370 in') collars. Since the load was almost nothing it is not surprising that these bearings evidently performed perfectly well. With the arrival of the direct drive turbine, thrust bearing problems of wear and overheating which had tended to dog earlier ships fitted with reciprocating engines, had apparently gone away. Existing bearing technology was good enough and who wanted to risk a large and costly ship by replacing a large and satisfying multicollar thrust block with what must have seemed a ridiculously small single collar device? The situation might have remained like this for some considerable time were it not for another technological change which re-opened the way for the tilting pad bearing just as effectively as the direct drive turbine had blocked its progress. In the years from 1912 onwards the geared turbine was developed for marine use and in time superseded completely the direct drive turbine. Now with the turbine driving through a gearbox there was no longer steam available to balance the main thrust and once again shipbuilders were required to fit propulsion shaft thrust bearings capable of supporting major loads. At first standard multicollar blocks were installed with all their attendant difficulties only worse than before. It was suggested that the varying torque from reciprocating engines of the earlier generation had led to fluctuating thrust loads and this had allowed oil to be drawn in between the flat and parallel bearing surfaces. Now however the relentless drive of the turbine driven ships gave no such opportunity and marine engineers began to search urgently for a better means of absorbing thrust. Fortunately for all concerned, the solution in the form of the tilting pad thrust bearing had been in waiting for several years and it was not long before the first installations were made. Some experimental work was put in hand by C.A. Parsons and Co., in the early part of 1912 with a view to testing the suitability of single collar tilting pad thrusts for marine use. Prior to this Parsons had tried and tested a small block having a bearing surface per pad up to a pressure of 1290 mm' (2 in') The full scale of 24.1 MPa (3,500 psi). experiments commenced at Wallsend in August 1912 with loads of up to 368 kN (83,000 lbf) being applied to a test bearing at rubbing speeds which ranged between 0.3 to 15 ms-l The maximum load, 368 kN, ( 1 to 50 fts-'). actually applied was equivalent to a surface pressure of 20.6 MPa (3,000 psi) ( 8 ) .
51
Following the success of Parsons' Wallsend test work the first pivoted pad main thrust blocks were commissioned. These were for the first all-geared destroyers, H.M.S. Leonidas and Lucifer, built by Messrs. Palmer of Jarrow and engined by the Parsons Marine Co., and the thrust blocks for the channel steamer Paris, built by Messrs. Denny of Dumbarton. Of these vessels the Paris was first on trial in June and July 1913 (8) with twin thrust blocks taking 239 kN (54,000 lbf) each at about 300 revolutions per minute, ( 9 ) . At the time that the Paris was on trial Cammell Laird & Co. Ltd., of Birkenhead were having a great deal of trouble with the standard multi-collar thrust block in a geared turbine cross channel ferry built by themselves. J. Hamilton Gibson, then Chief Mechanical Engineer at Cammell Laird heard of the Paris installation and contacted Newbigin for his assistance. Very quickly a tilting pad design was produced and Gibson persuaded h i s company to allow him to install Michell bearings in two geared turbine ships then under construction for the Argentine Navigation Company to operate as passenger ferries between Buenos Aires and Montevideo (13). The 107 m (350 ft) long, 5635 shaft horse power vessels were launched on 12 May 1914 and 25 July 1914, and named Ciucad di Buenos Aires and Ciucad di Montevideo respectively. * Apparently Gibson encountered substantial opposition from within his company and in his own words; "When the first boat went on trials there were not wanting Jeremiahs who foretold dire disaster But for once, the prophets were mistaken, the end justified the means, and these blocks ran from the very first without the slightest trouble. Only the other day 1. saw the report from the engineer of one of the boats on her long voyage to the River Plate. He said 'Thrust Bearings ran quite cool with the chill not off!'". ** A typical very early Michell marine thrust bearing similar to those installed in the South American ferries is shown by figure 5. In this case the flat backed pads are supported by individually adjustable pivots. It was soon realised that this provision for axial adjustment of the pads was not only unnecessary but actually a source of potential calamity if errors were made in setting up. Accordingly the idea was soon dropped only to be resurrected and re-used in recent years in certain special applications such as very large vertical thrust bearings. The first ship of the Royal Navy fitted with tilting pad main thrust bearings ran her trials in August 1914. The trials, like those of the Paris and the two Camell Laird vessels
.......
*
We are indebted to Mr. A.C. Hackman of Cammell Laird and Co. Ltd. for this information.
**
From a paper on 'Geared Marine Steam Turbines' read by J. Hamilton Gibson before the Foreman's Mutual Benefit Society, Birkenhead, on 30 March, 1915, and quoted by H.T. Newbigin, The Michell Bearing Book, April 1916.
must have been completely successful for in the discussion of a paper by Ward (13), Gibson commented that the Admiralty had thereafter adopted the single collar thrust whole heartedly and universally. Thus during the First World War Michell Thrust Blocks were fitted in British Naval Ships to propulsion machinery totalling ten million shaft horse power, ( 8 ) . A figure which reflects the huge shipbuilding effort made at the time; for example no less than 280 destroyers were launched from British Yards between 1914 and 1918 (15). as well as many other vessels, smaller and larger. The largest thrust blocks made during this period were for the battlecruiser H.M.S. Hood Hood, launched on 22 August 1918 (16). was a four screw ship with 36,000 horse power transmitted per shaft and a maximum load of 1,000 kN (224,000 lbf) applied to each thrust bearing. Figure 6 shows models of the Michell single collar thrust block as installed and the multicollar version originally proposed. A photograph of one of the actual bearings with its casing top removed is shown by figure 7. One of the two similar but much smaller thrust blocks fitted to the destroyer H.M.S. Mackay, on 2 1 December 1918, is shown by figure 8. The casing top has been raised and two pads positioned to show respectfully the white metal face and the offset spherical pivot. Gibson ( 1 7 ) states that the spherical pivot arrangement did not provide entirely satisfactory results and was soon abandoned in favour of line pivots of the type still in principal use today. The Mackay was the first ship to incorporate Michell tilting pad journal bearings in her port-side turbines and around the periphery of the collar to support the weight of the shaft as figure 8 indicates. This expensive arrangement understandably did not find wide favour and the usual means of supporting the shaft in a marine thrust bearing remained journal bearings fitted on each side of the collar as shown for example by figure 5. The influence of the Michell thrusts to the Navy of the first war was immense. Specific loadings rose by a factor of ten and typical coefficients of friction were reduced by a factor of twenty from 0.03 to 0.0015. When Michell applied for an extension to his patent in 1919 it was estimated that the Royal Navy saved coal to a value of €500,000 in 1918 alone as a result of fitting tilting pad bearings. Albert Kingsbury himself recorded in his autobiographical notes (18) that it was the use of Michell bearings by the British which influenced the United States Navy to adopt his similar bearing beginning in 1917. By the time the war was over the dominance of tilting pad bearings was complete. At the Summer meeting of the Institute of Naval Architects in 1919, in discussion of his own paper Gibson (8), reported that mercantile shipowners were 'clamouring' for Michell thrusts, not only in geared turbine vessels, but for reciprocating engines, and even in renewals of old vessels.
52 4.
THE FORMATION OF A BEARING COMPANY
The partnership between Michell and Newbigin was a tremendous success with many major companies becoming licencees and huge numbers of bearings being built. In "The Michell Bearing Book" of 1916, Newbigin lists 3 1 British licencees. Later the number of licencee companies grew to at least 66 in different countries of the world including 39 in Britain and 11 in France. However if the original mathematical inspiration and invention had been Michell's, any commercial success was wholly the work of Newbigin. From 1903 until 1925 Michell was chiefly occupied with engineering engagements associated with his consultancy business, (2) and with the Crankless Engine Company formed in August 1920." In 1919 Newbigin and Michell applied for an extension to the 1905 patent citing the great advantage to the Navy that the tilting pad bearings had been in the war. The application was successful and a seven year extension was granted. The report of the 1919 patent case in chancery reveals that the scale of royalties at that time was € 5 0 plus € 1 per However, even inch diameter of shaft, (19). as the patent extension was granted it must have become increasingly apparent to Newbigin that he and Michell could only count on a relatively few more years of royalty income and some alternative, more permanent business arrangement was necessary. Considerable assistance was provided at this stage by J.H. Gibson who had been largely instrumental in introducing tilting pad bearings for marine use and who had since lectured widely on the subject. Gibson now helped to convene a group of leading shipbuilders with whom negotiations were commenced with a view to establishing a limited company which would take over the existing patents from the Newbigin Michell partnership and have a manufacturing as well as licensing role. Thus it was that Michell Bearings Ltd.** was incorporated on 2 1 May 1920 with a share capital of 100,000 € 1 shares half of which were allocated without payment to Michell (29,500) and Newbigin (20,500) in consideration of the existing patents and partnership activity. The remaining 50,000 shares were taken up by four shipbuilding companies as follows: Vickers Ltd, 15,000; Came11 Laird and Co. Ltd, 15,000; John Brown and Co. Ltd, 10,000; and The Fairfield Shipbuilding and Engineering Co. Ltd., 10,000. A site was purchased in Newcastle and a substantial factory established (figure 9) with Newbigin, as joint Managing Director, in charge of manufacturing. The commercial offices were in London managed by
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*
From "Modest Man of Genius"; an Account of A.G.M. Michell and the Crankless Engine" produced privately by S.E.A. Walker, 1972, and lodged in the library of the Institute of Engineers, Australia.
**
We are indebted to the directors of Michell Bearings for Vickers PLC much of the following information.
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Col. L.E. Becher, Newbigin's fellow joint Managing Director. The remaining, nonexecutive directors were nominees of the respective shareholding companies and of Michell himself. Gibson was retained as a consultant to the company and later became its London representative. The early years in the new company's life were far from easy. It soon became evident that the available cash resources, the initial €50,000 subscribed by the shipbuilders, were 1922 insufficient and in the period 1921 additional debentures of €10,000 were taken up by the four shareholding companies. By 1925 the company had still paid no dividends and was considerably in debt. A thorough investigation of the company's affairs was undertaken by Vickers Ltd's internal auditing department and this led to a major reconstitution of the board and management. The London office was closed and a new General Manager, Mr. H.B. Scott, appointed in place of the previous joint Managing Directors who resigned. Newbigin, although no longer in an executive role, remained a director of the company until his death in 1928. The shares initially owned by Michell and Newbigin were later bought in by the other owners gradually between 1938 and 1946. The 1925 restructuring proved successful and led to an extended, profitable period of operation lasting right up to the early 1960's and far outlasting any residual income from the early manufacturing licences. It is an interesting reflection of the stability of the company in these years to note that there were only two chief executives appointed between 1925 and 1964, and they were brothers. H.B. Scott, appointed Local Director and General Manager in 1925, became Managing Director in 1939, and was succeeded by his brother W.S. Scott who was Managing Director from 1954 until 1964. Although there was always a strong minority interest in bearings for land-based industrial use, the company's principal business throughout this time remained the supply of bearings for marine applications and there was no shortage of demand. When the "Queen Mary'' was launched in 1934 and commissioned in 1936 she was then the greatest ship ever built. Engines of 200,000 horse power transmitted their driving force to four propellers each weighting 35 tons and drove the vessel at a speed in excess of 30 knots. The thrust was absorbed by four Michell thrust blocks and the intermediate lines of shafting were supported by no less than 46 Michell self lubricating journal bearings, (20).
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5.
RECENT YEARS
During 1962 the company moved to larger premises on the Scotswood Road in Newcastle and unfortunately commenced a period of much reduced profits lasting into the early 1970's. This was partly caused by the costs incurred in moving to a new site but also by worsening trading conditions as the decline in Britain's position as a shipbuilding nation became increasingly evident. Successful efforts were made to increase the industrial component of the company's workload but nevertheless
53
profitability remained low. In 1969 Vickers Ltd., purchased the shareholdings of the other owners and for the first time Michell Bearings Ltd., came under single ownership. Following Vickers' takeover the company entered a healthier period helped initially by the supertanker boom of tke early 1970's and by the increased contribution from the industrial business. When the o i l crisis of 1974 brought the market for supertankers to an abrupt halt the company had enough renewed strength to survive that particular setback and carry on to build a new, purpose built factory officially opened in 1978. Today Michell Bearings is a member of the Vickers PLC group of companies as a specialist supplier, to numerous industries, of bearings the great majority of which are tilting pad thrusts and journals based on A.G.M. Michell's original principles. However, although Michell's ideas are still valid, the company, like other bearing companies, has seen in its 65 year life technological changes affecting its products. Some of these changes are highlighted in the next section. 6.
MODERN TECHNICAL CONSIDERATIONS
Michell's work led to a simple statement of Reynolds' equation for the hydrodynamic film thickness in a tilting pad bearing assuming a constant lubricant viscosity throughout the system. The success of early tilting pad bearings owes much to the good agreement found between their performance and the predictions of Michell's theory. The chances of success were enhanced by many of the applications, especially those in the marine field, having relatively slow rotational speeds and hence generating small amounts of heat. The simple expression of Reynolds' equation for a tilting pad bearing remains valid for many small to medium size applications and for moderate speeds. However market pressures have led to a search for ever more efficient bearing designs in terms of increased loadings and higher speeds. Gradually the limits of tilting pad bearings in terms of performance and size have been explored and extended. The great importance of the considerable changes in temperature which do occur across the face of a thrust pad has been recognised and given rise to a widespread series of experimental investigations much of which is reviewed in a recent paper (11). Temperature variations affect not only the lubricant viscosity but also the surface profile of the pad which distorts under thermal stress and can thus cause a decrease in the load carrying capacity of the bearing. The development in the past two decades of numerical methods and computational power has allowed non-isoviscous solutions to Reynolds' equation to be found simultaneously with solutions to the equations for energy, thermal distortion and elastic bending of the pad. In analysing a bearing it is now possible to take account of a full range of parameters listed in Table 1. It is common for the results of such analysis to be presented graphically in the form of contours of pressure, temperature and film thickness
across the pad surface such as shown in figure 10. In parallel with the increased understanding of bearing behaviour which experimental and theoretical work has brought about, there have been numerous technological advances in bearing design designed to improve performance in particular applications. Some design features have been introduced specifically to counteract the effects of pad distortion referred to earlier, others, such as directed lubrication, have the purpose of reducing energy consumption in the bearing. Table 2 lists some of the alternatives a modern designer can utilise which were not available in Michell's day. 8.
CONCLUSION
The unique contribution of A.G.M. Michell in making practical use of Osborne Reynolds' analysis of lubrication has long been recognised. Michell himself was only involved directly in the industrial exploitation of his invention for a short time in Australia. Nevertheless, Michell bearings, after overcoming initial resistance, helped to bring about a worldwide revolution in lubrication practise. A specialist bearing company, carrying the inventor's name was founded in Newcastle upon Tyne in 1920. This company, Michell Bearings, now a member of the Vickers PLC grcup of companies, has remained a supplier of tilting pad bearings and continues to play a full part in their technical development. 9.
ACKNOWLEDGEMENTS
The authors are grateful to directors of Michell Bearings - Vickers PLC for permission to publish this paper and to their colleagues for generous assistance. References ARCHIBALD, F.R. 'Men of lubrication Anthony G.M. Michell', Lubrication Engineer, September - October 1955, 304
-
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305, 346.
CHERRY, T.M. 'Anthony George Maldon Michell', Biographical Memoirs of Fellows of the Royal Society, Vol. 8, 1962, 9 1 103.
-
.
INSTITUTION OF MECHANICAL ENGINEERS, LONDON 'Award of the James Watt International Medal to Mr. A.G.M. Michell, FRS', 22 January 1943.
DOWSON, D. 'History of Tribology'. Longman : London, New York, 1979. DOWSON, D. 'Men of tribology, 18 : Anthony George Maldon Michell', Journal of Lubrication Technology, Vol. 102, January 1980. MICHELL, A.G.M. 'The lubrication of plane surfaces', Zeitschrift fur Mathematik u. Physik, Bd. 52, Heft 2, 1905,
123
-
137.
WOODRUFF, G.B. 'Lecture on thrust bearings', Proc. Institution of Marine
54 1920's
Engineers, Vol. 20, 12 October, 1908, 21 (8) (9)
-
39.
GIBSON, J.H. 'The Michell thrust block', Trans. Institution of Naval Architects, Vol 61. 1919, 248 - 259. NEWBIGIN, H.T. 'The problem of the thrust bearing', Proc. I.C.E., Vol. 196, 1914, 223
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Simple isoviscous expression of Reynolds' equation for a tilting pad 1980's
265.
(10) GIBSON, J.H.
'The Michell thrust block and journal bearing', Liverpool Engineering Society, Vol. 38, 1917, 330
Following factors all taken into account:
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362.
Heat loss by conduction through thrust collar and tilting pads
(11) HORNER, D., SIMMONS, J.E.L.,
and ADVANI, S., 'Measurements of maximum temperature in tilting-pad thrust bearings', Proc. American Society of Lubrication Engineers, Annual Meeting, Toronto, Canada, 12 - 15 May 1986. (12) NEWBIGIN, H.T. 'Pressure oil film lubrication', Rep. British Association Adv. Sci., 1916. (13) WARD, J. 'Some notes on the theory of lubrication with particular application to Michell thrust and journal Bearings' Proc. Institution of Marine Engineers, Vol. 36, pp. 141 - 185, 1924. (14) "H.M. Battle Cruisers 'Repulse' and 'Renown'", Engineering, Vol. 107, 1919,
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461 464. (15) "The war development of the torpedo boat destroyer", .Engineering,Vol. 107, 1919, 364. 362 (16) "H.M. Battle Cruiser 'Hood'", 399. Engineering, Vol. 109, 1920, 397 (17) GIBSON, J.H. 'Thrust and journal
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Heat loss by convection in the lubricant Thermal distortion of the pad Mechanical bending of the pad due to applied load Mixing of hot and cold oil in the groove between pads prior to entrainment in the hydrodynamic film Turbulence in the hydrodynamic film Lubricant viscosity variable in three dimensions, (x, y and 2). TABLE 1
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bearings', Proc. Institution of Marine Engineers, Vol. 45, part 3 . pp 75 - 97, 1933. (18) KINGSBURY, A.
'Development of the Kingsbury thrust bearing', Mechanical Engineering, December 1950, pp 957-962. (19) Illustrated Official Journal (Patents), Vol. 36, Supplement, 16 July 1919. (20) Journal of Commerce and Shipping Telegraph, "Queen Mary" commemoration issue, 15 May, 1936.
Some Design Innovations since 1920 Alternative pad support systems to minimise distortion such as button and spring mattress types. Methods of lubricant supply to minimise energy losses; directed/low loss lubrication or oil supplied directly to a groove in the pad surface near the leading edge. Development of alternative surface materials like Copper-Lead, Aluminium-Tin and nonmetals. Bearings which use a hybrid hydrodynamic hydrostatic mode of operation.
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Sophisticated means of internal oil circulation for self-contained norizontal and vertical bearings; viscosity pumps and high speed oil rings in which the lubricant is collected from inside the rim. More efficient oil cooling systems like finned and wire-wound tubing and heat pipes for selfcontained bearings. Initial "crowning" of pads to prevent them "ringing" to a collar surface. TABLE 2
55
A.G.M. MICHELL Figure 1
a
w
FACE ELEVATION
MICHELL VERTICAL THRUST BEARING FOR A CENTRIFUGAL PUMP. INSTALLED AT COHUNA ON THE MURRAY RIVER, VICTORIA, 1907 Figure 2
L
I3
HORIZONTAL SECTION
MICHELL BEARING FOR A STEAM TURBINE DESIGNED BY H.T. NEWBIGIN AND TESTED BY HIM AT MESSRS. BELLIS AND MORCOM LIMITED, BIRMINGHAM, JUNE 1910.
e DETAIL OF BEARINQ SPLINTS 6.
Figure 3
i SECTION T H R O U Q H ~ L FELEVATION OIL PABSAQE
VERTICAL SELF CONTAINED JOURNAL BEARING SUPPLIED FOR CIRCULATING PUMPS AT THE NEWPORT POWER STATION, MELBOURNE, AUSTRALIA BEFORE 1916. Figure 4
MICHELL THRUST BEARING SUCH AS USED IN THE EARLY MERCHANT MARINE INSTALLATIONS. NOTE THE FLAT BACKED PADS AND INDIVIDUALLY ADJUSTABLE PIVOTS. Figure 5
56
SCALE MODELS OF THE MICHELL THRUST BLOCK INSTALLED ON H.M.S. HOOD AND THE PLAIN MULTICOLLAR VERSION ORIGINALLY PROPOSED. Figure 6
MICHELL COMBINED THRUST AND JOURNAL BEARING FITTED TO H.M.S. MACKAY LAUNCHED IN 1918. NOTE THE THRUST PADS WITH OFFSET SPHERICAL PIVOTS AND JOURNAL PADS ON THE PERIPHERY OF THE COLLAR. Figure 8
ONE OF THE THRUST BLOCKS FOR H.M.S. HOOD WITH THE TOP COVER REMOVED.
_ , I
. ,
:
... ...
I!!,
.. . . .. :..... . ,.. ..
Figure 7
.... ..,
..
.I:!,,
:. CONTOURS OF TEMPERATURE. DEG C
.1_
...
. . . . . . . . . . . . . . ........ .... .......
THE ORIGINAL MICHELL FACTORY, SCOTSWOOD ROAD, NEWCASTLE UPON TYNE.
. . . . . . . . . . . . . . ........ .... ....... .. s .......
I / . . . _ _ .
.....rj
I.
.
..
I
.
.
,
.
.. . .. ..
.
.
. . .. . I
. %
.
I
,
I
/
: . . . . . . ,. . . . . . . . . . . . . . . . . .. .. .. .. ....... .
I
. I
...... ..
$ 8 , 1 1 1
! .
.. . .
.. ..
111
..I
.I,
,
I
I
.I,._ .,.I
I. % I
..t
Figure 9
. .....................
.. <. ... . . . . . . . . . . ., -
CONTOURS OF FILM THICKNESS, MICRON
CONTOURS OF TEMPERATURE, PRESSURE AND OIL FILM THICKNESS ACROSS A TILTING THRUST PAD SURFACE. Figure 10
SESSION 111 JOURNAL BEARINGS Chairman: Professor J.F. Booker PAPER Ill(i)
An approximate global thermal analysis of journal bearings
PAPER Ill(ii) Negative pressures in statically and dynamically loaded journal bearings PAPER Ill(iii) Mixing inlet temperatures in starved journal bearings PAPER Ill(iv) Starvation effects in two high speed bearing types
This Page Intentionally Left Blank
59
Paper Ill(i)
An approximate global thermal analysis of journal bearings Donald F. Wilcock
The thermal state of a jmrnal bearing has been reccgnized for many WECS as important to its operating characteristics and tx~its ability to survive. Characteristics such as power loss, minimum film thickmess and lubricant flow are directly affected. The ability survive depends upcn lubricant t h e m 1 stability and bearing mterial softening or degradation, bath of which are influenced by the maximm temperature occurring in the film. ?his paper reports cn a full thermal treafment, with appropriate assumptions, of the t h e m a d flaw behavior of a two-pad cylindrical jaunal bearinclding the effect of heat transfer from the lubricant and the heat balance in the feed growes. 'Ihe analysis is -1mmted in a carrprter rode on an IBM PC that is capable of prOduci.ng rapid solutions. Bath inprt and cutput are in tern of ncn-dimensional parameters defined in tern of designerstrolled axditim such as lubricant feed t e m p e r a m and pressure, bearing Efeanetry, loads and speeds. A tearing cooling parameter permits apprcocimte treatment of the influence of beardiameter and type of mxmting. ~
i
INTRODUCTION
Film temperatures in journal bearings may run considerably above the feed lubricant temperature and the ambient temperature. At the time 111 was written, isothermal analyses were becoming available, but in limited availability because of costs and the then current speed and memory limits in mainframe computers. Nevertheless fairly good approximations to measured power consumption and lubricant flow rates were ob-tained by computing bearing performance at a viscosity corresponding to an elevated temperature representing the crossover point between the lubricant viscosity-temperature curve and the curve of bearing exit lubricant temperature vs assumed isothermal viscosity based on AT=H/(FC Q). Witg the recent expansion of memory and speed capability of micro-computers, PC machines can now complete rather complex bearing computaAn tions in a few seconds to a minute or so. example is C21 which presents some results from a finite difference solution that includes variable temperature in the film of a two-pad cylindrical journal bearing, and determines the film inlet temperature based on mixing of hot carryover lubricant with cold feed lubricant in the feed groove. In this paper, the flow and thermal behavior in a two-groove cylindrical bearing are incorporated into a rapid finite difference type PC code. The analysis includes effects of end flow from the feed grooves and a first approximation for heat conduction from the bearing to the environment.
1.1 Notation a Viscosity correction coeff., N-D. b Viscosity correction exponent, N-D. C Radial clearance, m Heat capacity, J/kg/K Diameter, m e Shaft eccentricity, m h Film thickness, m H Power, watts j Power parameter, ratio to "Petroff" power J Joules, Nm Pressure coefficient, 6 wc)2 Bearing length, m
D".
3
N
P
P 9 €2
R t T
f
T W
W Z 2
(1
I'
r E
Shaft speed, fi/min Pressure, N/mL Non-dim. pressure, p/Kp Lubricant flow coeffic ent, 2Q/1.CR2 Lubricant flow rate, m 3 /s Radius, m Effective brg./housing radial thickness, m Temperature, OC Temperature rise above ambient, C' Non-dim. temperature, n(T-To) Load parameter, W/%R2 Load, N Axial coordinate, m Non-dim. axial coordinate, z / R Temperature-viscosity coefficient, 1 / T Conductivity, J/m/s/K Bearing cooling coefficient, 21/(Ctl.>pCp) Eccentricity ratio, e/C Heat transfer rate Angle from TDC2 rad Viscosity,Ns/m Attitude angl Density, kg/mE3 Shaft speed, rad/s
Suhsrripts
e f g i
L o p
U x
entrance to film following pad feed groove inlet, bottom pad =lower pad lubricant feed condition preceding pad =upper pad exit from film
2 FEED GROOVE FLOW AND MIXING RELATIONS The basic function of a feed groove is to replace lubricant forced from the bearing edges by the internal pressures. This key subsystem is shown schematically in Figure 1. A typical feed groove has a depth an order of magnitude or more greater than the bordering entrance and exit film thicknesses, subtends an arc length of loo or more, and extend nearly the axial length of the bearing. As has been pointed out by C11 and others, there must be a flow balance on this volume:
60 QP + Qo = Qg + Qf
(1)
where is flow from the ends of the groove d i r e c t 1 9 to the exit drain resulting from the feed pressure Po. Flow from the film of the preceding pad Q may involve both Couette and Poiseulle elements depending upon the groove location and attitude angle. The same may be said for flow into the following pad. These flows significantly influence thermal behavior in the vicinity of the feed groove. An e n e r g y ION h a l a n c e for the feed groove is conveniently expressed as: Zf.Q. 1 1 = 0
tQg,
FIG. 1.
FEFD GROOVE FLOWS AND TEMPERATURES
(2)
where Ti = T.1 - T 0 (3) For the usual situation where Q > O , and assuming that P =O so that Qg=O, ( 1 ) an8 ( 2 ) are solved to yiel8: (4)
Studies of groove flow, C21, C31, suggest that hot flow f+, continues across the groove as a boundary layer to enter the following film. Nevertheless, if Qf>Qp, some cool feed flow must be added as indicated by ( 2 ) so that the a c e r a g e temperature across the film is less than Tp as indicated by ( 4 ) . If Po>O and if the resulting Qg is significant, say 5% or more of Qf, then further consideration must be given to internal mixing in the groove. In many practical design situations the groove conditions will involve vortex circulation and be turbulent because of the relatively large groove depth. Finally, groove end flow will generally occur across the adjacent film thickness cross-section. Thermocouple measurements C41 of groove end flow temperatures have shown them to be close to average exit flow temperatures. For these reasons, a straightforward assumption of complete thermal mixing in the feed groove has been made and Equation ( 4 ) then becomes:
Tg
Qo,? o = O
3 BEARING FLOW/TEMPERATURE CYCLE
The two-groove cylindrical bearing as a total entity is shown in Figure 2(a). Of the power H required to rotate the shaft in the bearing, the greater part usually is seen as an increase in temperature of the lubricant from inlet to outlet, Tx-To, while the balance, 3 ~ .is removed by heat transfer. The flow pattern within the bearing is diagrammed in Figure 2(b). The pattern of Figure 1 can be recognized at the lower pad and upper pad feed grooves. The total exit flow, combines the side leakage flow from the k e r (loaded) pad, Qs, with the groove leakage flows, 2Qu.
Hshaft
1 Qx, Px
(a) Total B e u i m System.
UPPER PAD
If heat transfer from the bearing is considered (see section 4 . 3 ) , the additional energy flow, 3 ~ ,gives: P
(6)
LOWER PAD
4€4In multi-groove designs, the situation may occur where If &p>(Qf+Qg), reverse flow may occur in he feed groove-at a temperature of Tg, and mixing in the feed system will raise the feed temperature to other grooves above To. This situation does not usually occur in the two-groove cylindrical bearing.
$>+.
'Qout
(b) Internal Bearing Flows, (CCW Rotation).
FIG. 2.
FLOWS AND TEMPERATURES IN A 2-GROOVE
BEARING 2.1 Axial groove flow The groove behavior analysis above has treated the groove as a completely mixed volume with defined inputs and outputs, and with a constant temperature. A design with numerous small inlet ports along the length of the groove would closely simulate this assumption. However, a single inlet port located at the centerline is normally used. In this case groove temperature could be expected to rise from the inlet port to the end of the groove assuming complete mixing at each station along the groove. This behavior has not been included in the analysis in this paper to be consistent with other simplifying assumptions, but is being included in an updated model.
The 'thermal cycle may be considered as beginning at the entrance into the lower pad with lubricant at an intermediate temperature between that of the feed lubricant and that of the fluid exiting from the top pad. Temperature rises through the lower pad. Temperatur-e drops in the upper pad feed groove after mixing with feed fluid, and rises again through the upper pad.
4 ANALYSIS AND CODE STRUCTURE 4.1 Assumptions. Several assumptions have been made in order to extend the scope of thermal treatment of the journal bearing while retaining simplicity for use by designers who may not have ready access to detailed thermal solutions. They are:
61
(a) Temperature, and hence viscosity, is stant across the film.
con-
fb) Exponential viscosity-temperature behavior, to permit predictions independent of the feed inlet temperature. (c) Heat transfer from the film to the bearing is a function of local film temperature and lubricant flow. Heat transfer to and from the shaft is nearly balanced per C31 and C51, and is assumed to be included in thm estimate of transfer through the bearing. (d) Feed groove pressure is assumed to be zero in setting the boundary conditions for solving Reynolds equation. (e) Cavitation occurs in the expanding regions of the film. The film is assumed to break up into a number of filmlets whose total width when combined with the local film thickness and half the shaft velocity equals the constant flow rate established at the entry to the cavitation zone. (f) Locally calculated Reynolds numbers determine the magnitude of a viscosity-increase ratio used in the solution of Reynolds equation, and in calculating power and temperature rise. An exponential curve-fit to the tesults of C61 is used of the form iiT/ii=aRe where a and b depend upon the use as follows: a b In dir. of motion Across dir.motion Viscous drag
0.017 0.055
0.634
0.0138
0.69
0.44
4.2 Film temEerature .. calculation. The method of C71 is used in calculating the local film temperatures. Film temperature is An assumed constant in the axial direction. energy balance between the local energy generation rate and the local circumferential Couette flow is used to establish the local temperature rise. In non-dimensional parameters,
(8)
Starting at a known or assumed temperature, e.g. at the entrance to a film, ( 7 ) is calculated along the film to establish the temperature profile. Kad is a function of inlet conditions, lubricant properties, and bearing geometry. It is also a function of shaft speed so that computed data must be prepared as a function of Kad to provide coverage of a range of speeds.
correction,. 9.,3.H!xG-C.ransfer Correction for heat transfer is made using a very simple assumption: that energy is removed from the lubricant film by conduction through a fixed radial distance to a heat sink at lubricant feed temperature. No allowance for boundary coefficients is made. The radial distance may be varied with the shaft diameter in a family of machines, and may be estimated from test results. Per unit of length, the heat removed by conduction is: 1T €Ic = Ra6 (9) at and the change in heat content of the flowing in the film is: HL = t,’RChf C AT P 2n
__
lubricant (10)
Equating (9) and
(lo), (11)
This correction is smaller at higher speeds, larger clearances (i.e. larger diameters), and thicker bearing/housing structures. It is included in calculation of the temperature profile. In cavitated regions, heat conduction is assumed to occur over the full axial length, the while flow and power loss are limited to filmlets. Correction for conduction loss is also made in adjusting the temperature of the outflow from the grooves, using the total groove area and total flow into the groove. 4 . 4 Reynolds equation
In non-dimensional parameters, Reynolds equation is solved in terms of the local effective viscosity and film thickness:
A finite difference procedure using the marching technique is used to determine the pressure field, i;i .. Boundary pressures of zero at the bearing ends and along the grooves are assumed. Both the pressure and the pressure gradient are set to zero at the end of the full film. Flows and loads are calculated from the pressure field. 4 . 5 Computer code structure The code is structured to find the values of C and + that will carry a given load, G, at given while values of L/D, 0 , Kad, Re , r and attaining an internal therma? balance. ‘$he elements of the code are as follows: JPADTM - computes basic performance of the bottom pad, based on a given value of to establish the temperature TCf field. JBTEM - calls JPADTM, calculatks the groove and usper pad thermal performance to obtain the temperature closure error on the cycle, and then perturbs E and 0 and repeats the calculation to provide the influence coefficients for WH and wv as a basis for a new estimate of E and JBTBIS - calls JPADTM and uses influence coefficients to adjust E and 0. JPADTM is repeated, and w and wv are tested against the desire2 values of w and zero. The process is repeated until the error is less than 0.001, following which the stiffness and damping coefficients are computed, together with flows, power coefficient and energy rate. JBGLTS - calls JBTBIS, tests for the temperature -closure error, corrects the assumed TL if the error is too large or too smafl, and repeats JBTBIS. If the closure error is not within the desired limit, the two values are used to predict a third estimate, and so forth. A closure error limit of 0.01 is used in this paper, refleatinu an accuracy in closure of 0.3C for most oils. Input and output are printed, and the final pressure and temperature fields are stored for later plotting if desired.
+.
62
5 TYPICAL PERFORMANCE
TEMPERATURE PROFILES
FIG. 3.
A
design for which experimental results are available serves as a vehicle for examining the influence of key variables, particularly the new heat transfer variable r and the feed groove end flow variable qg. The bearing is circular with two 150° pads, two 30’ feed grooves, and the load vector directed at the midpoint of the shaft diameter lower pad. The L/D is 1.0, 6.000 in (0.152 m), and radial clearance 0.004 in (0.102 nun). Behavior about a common operating 3690 r/min, unit load 180 ,psi point ( 1 . 0 3 ~ 1 0N/m ) , Q =0.5 gpm (3.79~10- m /min), and n=O. 0306 K-I.gThe corresponding input nondimensional parameters are Kad=O.1-92, and w=O.128. The tests, as described in C11, housed the test bearing in a steel cylinder approximately 15 in (0.38 m) in diameter. Assuming that the heat energy was largely conducted radially, the effective conduction distance t in eq.(11) becomes t=R ln(RZ/R) where R is the shaft is the outer radius of the radius and mounting cylinjer, exposed to lubricant at feed temperature. For R=3, 1 becomes 0.025 for the reference condition.
L/D = 1 . 0
12
w =
0.128
I
= 0.00
05
06
1
7
:
:
o
Im
m
KO
403
e
WE~PITDC-
8
5.1 Temperature Profiles Increasing decreases the bearing film temperature over the entire bearing surface. This is clearly seen in Figure 3 where the circumferential temperature profiles are plotted for values of r ranging from 0 to 0.24 for the reference case. If the bearing radius and clearance are small, if the bearing conductivity is increased from that of steel to copper for example , and if the path is shortened with a thin shell, higher values of 1 are obtained. In such cases bearing temperatures are lowered significantly, and the temperature profile flattens out, particularly beyond the point of minimum film thickness. Bearing cooling parameter r exerts a strong influence on the key operating parameters as rise shown in Figure 4. Pad inlet temperature drops rapidly, as does outlet temperature rise and maximum film temperature rise at the end of the bottom pad. The result is a decrease in eccentricity ratio, an increase in power consumption (j), and a decrease in lubricant flow (q). Non-dimensional temperature rise to the inlet to the bottom pad, and outlet nondimensional temperature rise of the lubricant above feed temperature are shown in Figure 5 as a function of load parameter w f o r the reference case. Also shown are parameters E , j and q. The range shown cover? unit loads from 20 to 500 psi (0.14to 3.44 N/m ) . -__.
FIG. 5. I2
LOAD
-
FIG. 4.
EFFECT OF HEAT TRANSFER
I .I
.-
I
n
01
:I
I
,
,
,
,
,
,
04
001
0
O M
0 12
0 2
0 4
W
a
4
r
5-Z__sp_e_e_d.-a.~d-~l~.a.r~-cRatl~-.I_n_f_?~e~ce-
Q
LOAD PARAMETER
0 2
Effects of varying speed and clearance ratio around the reference case are shown in Figures 6 and 7. In varying speed from 1800 to 9000 rpm, the non-dimensional input parameters w; q , Kad6 r and Reo are adjusted appropriately. Fisure also shows the power measure-ments made on a bearing of this size for comparison with the calculated results. Notice that the input temperature to the loaded pad rises rapidly with speed, while the lubricant outlet temperature pad rises rapidly also, s o that the rise from inlet to outlet is relatively small, varying at 9000 rpm. from IOC at 1800 rpm to ~ O C 7, Increased clearance ratio, Figure proportionately raises flow rate, and correspondingly decreases inlet, outlet and maximum temperatures. The power consumption, however, is almost unchanged by this factor.
Tf
0
0 11
BEARING COOLING COEFFICIENT
L/D R = c = P =
-
,
0
0.0306
-
0.0306
0.7
FIG. 6. (I
=
08
I -
OB -
au
Reo = 101
0.0
= 1.0 0.076 m 0 . 10x1g-3m 155x10 N / - 0 . 0 0 1 9 m’/min
40 C
SPEED
63
5.3 Groove ~. end___ flow. Increasing flow from the ends of the feed groove by enlarging the chamfer if the bearing is split at that point, or by cutting a drain groove, is perhaps the most common adjustment a designer makes if a bearing is running hot. The effect of increasing groove end flow on the same reference bearing from qg=O to q -0.4 is shown in Figure 8. Since both pad inlee-temperature and lubicant exit temperature drop with end flow, the change is more than the cosmetic one of FIG. 8.
FIG. 7.
CLEARANCE RATIO
40
L/D = 1.0 0.076 m 3600 r g m 155x10 N/m2
nil
= 40 C
'T No=
0.0306
GROOVE FLOW
0 0001
,
,
00012
I
1
.
00014
T
I
I
I
00018
00011
I
I
0002
I
9
00022
00024
0
b26
C/R
04
-
-
L
I
0 0.1
0
0.3
0.2
GROOVE LEAKAGE FLOW
0.4
qg
diluting the exit f1o.w with cold feed lubricant. The effect on power consumption is only a modest increase.
The operating line results are compared with variable temperature results for I = O and r=0.025 in Table 1 and Figure 10. The temperature profiles are shown in Figure 10.. Only in the region of high load on the lower pad do the JBGLTS temperatures exceed the isothermal operating line method temperatures. In all other areas they are lower. The two methods give remarkably similar overall results. Variable temperature JBGLTS calculations indicate a somewhat higher E and 9 . The power parameter j, which is based on feed viscosity Petroff power consumption, is high for the operating line method since a high inlet temperature has been assumed. FIG. 10.
OPERATING LINE AND GLOBAL TEW'FXUTIIRE
PROFILES
?,.4__9pe_Eat~~~-__i.n.e-.m.e.t-hod~.
For many decades, a simple operating line method C11 has been used by designers to estimate the thermal performance of a journal bearing. The method is based on the availability of isothermal solutions to Reynolds equation as a function of Somerfeld number, or its reciprocal, w. On a plot of viscosity vs temperature for the lubricant, a second curve termed the operating line of the bearing is drawn. Points on this curve are determined by calculating the temperature rise above feed temperature based on isothermal power loss and lubricant flow for several assumed viscosities. The intersection with the lubricant curve is taken as the operating temperature and viscosity for the bearing. The JBGLTS code provides a convenient way to check the operating line method. .Isothermal operasion is generated by assigning Kad=O, r = O , and Tin=O, and computing results for several feed temperatures. The results are shown in Figure 9 for the reference case, the solution being a feed viscosity of 10.0 CP at a temperature of 30.6OC
12
-
1 1 1 -
09
0.8
-
OP'G LINE
05 04 03 01 07 08
0.1
R = 0.076 m N = 3600 r / in ! P = 515 N/m ! Q~ = 0.0019 m3/min T =4OC NO= 0.0306
-
0 0
103
FIG. 9.
Ecc. ratio,^ Attitude Angle, 9 Power ratio, j Power, kw Flow, lov3 m3/min Pad Ti, OC Pad Te, OC Lubr.out, Tout,OC
OPERATING LINE METHOD.
L/D = 1 . 0 R = 0.076 N = 3600 ramin P= 515 N/m Q% = 0.0019 m3/min To = 40 C N = 0.0306
Im 90
BEARING
m
LUBRICANT
X
C
0
4
I
I2
I8
LUBRICANT VISCOSITY
20
CP
4 I
3w
- om.
OPERATING LINE 0.602 40.6 0.443 6.7 9.16 64.2 64.2 64.2
JBGLTS
.-=0 ___-
0.652 40.9 0.431 6.5 8.00 61.4 85.8 64.7
r=o. - 025 -
...
0.631 41.8 0.459 6.9 7.88 58.8 80.7 61.7
Since the operating line method assumes an adiabatic condition (all the power is converted to heating the lubricant), the dimensional output is best compared to the I = O JBGLTS solution. Here the operating line method is slightly high on power, 15% high on flow, and slightly low on lubricant exit temperature.
70
60
A
T.AE-LE2 COMPARISON OF OPERATING LINE AND VARIABLE TEMPERATURE SOLUTIONS. P&tAME_TE_H
I10
200 M
24
64 6 COMPARISON TO EXPERIMENT
8
A number of carefully made experimental results on cylindrical bearings with two 30' feed
The assistance of Mechanical Technology Inc. in the preparation of this paper is appreciated.
grooves are reported in Cj 1. A few runs have been selected for comparison with the JBGLTS analysis. The bearing dimensions are given in Table 2 . The input conditions and the key out-
References (1)
TABLE 2 BEARINGDIMENSIONS CODE
L/D
4x2 4x4 6 x6 8x4
0.5 1.0 1.0 0.5
(2) FAD. CLEARANCE in mm
DIAMETER in mm 4.000 4.000 6.000 8.000
ACKNOWLEDGEMENT
0.0045 0.00185 0.0040 0.0063
101.6 101.6 152.4 203.2
Wilcock,D.F. and Rosenblatt,Fl. 'Oil Flow, Key Factor i n Sleeve-Bearing Performance', Trans. ASME, '& 849-866 (1952). Knight,J.D. and Bamtt,L.E.,'The Effects of Supply F'ressure on the Operating Characteristics of Two-Axial-Groove Journal Bearings', Trans. ASN, 2&,3, 336-342
(1985).
0.11 0.047 0.10 0. 16
(3) (4-1
put conditions, both experimental and calculated, are summarized in Table 3. TheEe runs ~ ) unit were all made with 150 psi ( 1 . 0 3 ~ 1 0N/m loag and with a feed pressure of 10 psi ( 6 9 ~ 1 0 ~ N/m ) . The lubricant had a viscosity of 2 5 . 5 cp 1 . at 4O.O0C, and a coefficient ~ ~ = 0 . 0 3CThe agreement between calculated and experimental performance is reasonably good in most cases, particularly on power. However, the higher speed runs with the 4x2 bearing show flow lower than calculated, and outlet temperature higher than calculated. A trial of a smaller clearance (row 3 ) which might have been induced by differential thermal expansion at the higher operating temperature reduced the discrepancy somewhat. The 4 x 4 bearing shows a measured flow nearly double the calculated and a temperature rise half that calculated. It seems likely in the case of this very low-flow, lowclearance bearing that lubricant was leaking around the bearing at a rate about equal to the bearing flow rate, serving to reduce the measured outflow temperature.
(5)
(6)
Rhonsari,li.M., ' A Review of Thermal Effects in Hydrodynamic Bearings. W 11: Journal Dearings' , ASLE Preprint 86-AN-24. Private communication. Dowson,D., Hunter,J., and Warch,C., 'An Experimental Investigation of the Thermal Equilibrium of St adily Loaded Journal Eearinzs'. h c . Inst. llech. &em., 101, $3 (19g6-67). Ng,C.W. and Pan,C.H.T., ' A Linearized Turbulent Lubrication Theory, J. Basic Ehg'g., Trans. ASI*lE, Series D, 3, 675-688(1965) Pinkus,@. and Bupara,S.S., 'Adiabatic Solutions for Finite Journal Pearings, J. Lubrication Technology, Trans. ASME October (1979).
a,
(7)
___ TABLE 3 CALCULATED BFARINGSPEED r/min 03DE 4x2 4x2 4x2* 4x4 6x6 8x4 8x4
3000 7000 7000 3000 3600 2700 5000
FEEDTEMP m V E
OC 54.7 53.8 53.8 40.6 40.0 40.2 40.5
vs EXPERIMENTAL mwcE
FLOW,-
r
0.36 0.36 0.36 0.02 0.5 0.5 0.5
.022 .0093
tl CalaAatel. with a reducd
HP Cal'd Kxpt'l
FLOW Cal'd Ekpt'l
0.76 0.75 2.84 3.06 .0105 2.92 3.06 .053 1.69 1.72 .025 7.45 7.56 .018 8.7 8.9 .0098 20.3 22.9
clearance of O.Oo40 in.
7 CONCLUSIONS
The introduction of a term approximating the heat flow from the bearing film through the shell permits a more adequate treatment of the bearing thermal performance. The net effect is moderate for large diameter bearings, but can become appreciable for small bearings to the extent that their operation is more nearly isothermal than adiabatic. This is entirely consistent with our experience in applying adiabatic solutions to fluid film bearing design. In a test of the operating line method with the global thermal method (JBGLTS), the operating line method was found to be a rather good approximation for the adiabatic case. It cannot of course predict a maximum film temperature. Based on the assumption of equilibrium mixing in the feed grooves, increasing end flow from the feed grooves not only decreases the lubricant outlet temperature, but also significantly decreases the bearing film temperatures.
0.88 1.52
0.80 1.22 1.40 1.22 0.18 0.32 1.62 1.74 2.96 3.06 4.89 5.26
TEMP. RISE Cal'd 4.7 11.5 12.4 31.0 23.3 16.7 25.1
m'l 4.6 15.3 15.3 16.4 24.7 17.1 28.4
65
Paper Ill(ii)
Negative pressures in statically and dynamically loaded journal bearings S. Natsumeda and T. Someya
The circumferential pressure distribution in a journal bearing is measured under static and dynamic loading. Compared with conventional pressure transducers a pressure pickup mounted in the journal has given more accurate results and yielded negative pressure o r tension as much as -1.2MPa (absolute) in the oil film. Two patterns are found in the pressure distribution: the one similar to the pressure curves calculated under separation boundary condition, and the other similar to the Sommerfeld's one. In order to expound the negative pressure o r tension, a hydrodynamic theory is developed regarding the lubricant as two-phase liquid of oil and small bubbles. Equation of motion for small spherical bubbles is solved taking into account surfacedilational viscosity. 1
INTRODUCTION
r
journal radius period of dynamic load (=l/f) t time w load Cartesian coordinates X ? y, z void fraction 0 Y angular position of minimum film thickness e angular coordinate (=x/r) K surface-dilational viscosity dynamic viscosity of fluid lJ density of fluid P surface tension 0 w angular velocity of journal
T
A number of numerical 'techniques f o r calculating
In the journal loci has been developed (1). order to obtain reliable results, it is necessary to know the appropriate boundary condition for pressure distribution in solving the Reynolds' equation. However, the lack of appropriate boundary condition has led to the use of boundary condition for static loading a l s o in the case of dynamic loading. In order to solve this problem, some workers have investigated the oil film pressure distribution in journal bearings under dynamic load (2-51, and one of the authors has also presented two papers (4,5)which had dealt with this problem. Nevertheless, the boundary condition for dynamic loading has remained a big question, because there is infinite combinations of experimental parameters and the measurement of an oil film pressure under dynamic condition is a difficult task. In the present paper the circumferential pressure distributions in the statically and dynamically loaded journal bearing are measured. A pressure pickup mounted in the journal has given more accurate results than pickups of conventional type. And in order to explain the negative pressure or tension which is detected in the present experiment, a hydrodynamic lubrication theory is developed regarding the lubricant as two-phase liquid of oil and small bubbles. Notation Fig.1 shows the coordinates system used in the present paper. Notation is listed in the .following. c radial clearance e eccentricity f frequency of load h oil film thickness (=c-e-cos(8-y)) L length of each land N rotational speed of journal (=w/(2~r)) n polytropic exponent p oil film pressure (gauge pressure) R bubble radius
Definitions nondimensional angular velocity B=- 20
c=-"1
nondimensional surface tension
fi
Rapa P1
nondimensional dynamic viscosity of liquid phase
D=-fi 4K
nondimensional viscosity
surface-dilational
R3Ja 1'
by
1.1
IL_I Fig.1. Geometry and coordinates
66
Gap sensor
Load cell
Servocylinder
4
1166 Fig.2. Test apparatus nondimensional squeeze speed
7 inner
surface
nondimensional wedge speed nondimensional film thickness nondimensional viscosity nondimensional density eccentricity ratio
pressure nondimensional axial coordinate nondimensional pressure nondimensional time nondimensional bubble radius clearance ratio a 1 s
Subscripts atmospheric liquid phase (i.e. lubricant without bubbles) supply
2 EXPERIMENTAL APPARATUS The test apparatus used in this study is the same as reported in ref.5 except for pressure pickup mounted in journal (Fig.2). The outline of the test apparatus is described in the following, details are given in ref.5. A journal is supported by two two-lobed bearings, and is driven by an electric motor with speed controller. The test bearing, a circumferentially grooved journal bearing with 100mm diameter, 25mm each land length and 108.5~m radial clearance, is mounted at the end
Fig.3. Pressure pickup mounted in bearing of the journal, and loaded by two electrohydraulic servo-cylinders arranged perpendicularly each other. The load applied to the bearing is measured by means of load cells inserted between the test bearing and the servo-cylinder. The journal center locus is measured by four gap sensors of eddy-current-inductive type. They are fixed in two perpendicular directions at both ends of the test bearing. By eleven pickups placed along the center line of land of the test bearing (Fig.3) the pressure distribution is also measured and compared with the data obtained by the pressure pickup in the journal. In experiments under dynamic condition, the pressure distribution is obtained using a pressure pickup in the journal and using the same method as reported in refs. 2, 4 and 5 under the assumption that the phenomena are exactly periodical. To this end the ratio of the frequency of the load to that of the journal rotation has to be so adjusted, that the least common multiple of both frequencies is sufficiently large. Then, for any given loading
67
914
H
Semiconductor /strain gauge
7
-Journal nal
To the slip ring
t- -7 journal Fig.4. Pressure pickup mounted in journal phase the circumferential position of pickup mounted in rotating journal moves by small amount from cycle to cycle giving the circumferential distribution of the pressure for the given load phase. In the present experiment sixty load cycles have yielded a complete pressure distribution (see dots in Fig.7). 2.1 Pressure pickup mounted in journal Pressure measurement of hydrodynamically lubricated oil film is a difficult task. Especially in the case,of journal bearing, the pressure ranges from low pressure (ambient o r sometimes negative o r tensile) to high pressure (10MPa in the order of magnitude in practice). Some workers have reported on interesting pickups(6-8). In the present study pressure pickup of Nakai type(7) is used. It seems that Nakai et. al. has measured only low pressure (the highest measured pressure is about 0.2MPa). The pickup in the present study has wider measuring range, namely, it can detect up to 5MPa. The pressure pickup mounted in the journal is shown in Fig.4. The membrane to the back of which the semiconductor strain gauge is bonded is positioned as close as possible to the journal surface. This is to avoid the interference to flow in oil film, and also to get good response of pressure pickup. In addition, the diameter of membrane is made as small as possible in order to obtain good spatial resolution. Experimental parameters are chosen as following: journal speed and supply pressure are kept constant at 8rps and 0.2MPa respectively. Frequency of load is set D.C. (i.e. constant load), 2Hz, 4 H z , and BHz. The amplitude and static component of load are varied.
3 EXPERIMENTAL RESULTS AND DISCUSSION 3.1
Static load Examples of the pressure distribution along the center line of bearing land under static load is shown in Fig.5. The theoretical curve is calculated by finite difference technique under Reynolds boundary condition. The agreement of these three kind of pressure data is rather good. It is noteworthy in Fig.5 that tensile stress is detected in the oil film region a little downstream of the minimum film thickness position. and this Dressure curve is similar to
E =0.851
2.0
a! r Q
1.0
0
-1
thmin
.o,
I
I
ll
2Tr 0 rad
g =0.933
Ihrnin
-1.01
I
I
TI
2Tr
0 rad o Measured (from the bearing 1 Measured(from the journal 1 Calculated (Reynolds condition)
---
Fig.5. Circumferential pressure distribution on the center line of bearing land under static load N=8rps; ps=0.2MPa
the one calculated under separation boundary condition. According to the past experiments (9) the
68
bearing characteristics can be calculated under Sommerfeld condition for light load, and under Reynolds condition for heavy load; for medium load separation condition is reasonable. However, in the present experiments sharp negative pressure dent is always observed except under lightly loaded conditions, and the heavier the load is, the greater the negative (tensile) pressure is. The maximum tensile stress detected under static condition is -0.6MPa (absolute), and such large value has scarcely been reported. Dyer et. al. observed tensile stress with maximum value of -0.64MF'a (absolute) (lo), however, their results showed that the magnitude of tensile stress decreased as eccentricity ratio increased contrary to the present results. It is also well known that the position of upstream boundary of cavitated region observed experimentally is always a little more downstream than the Reynolds condition predicts (11-13). In the present experiments this was also confirmed. 3.2 Dynamic load Under dynamic conditions negative pressure or tension is also observed, for which two types are found: the one is the sharp negative dent a little downstream of the minimum film thickness position as observed under static conditions, and the other is the pressure curve similar to the one obtained'under Sommerfeld or full film condition. Figs.6 and 7 present an example of the experimental results under dynamic conditions; namely, Fig.6 shows the load vector diagram, the journal center locus, and nondimensional squeeze and wedge speed, and Fig.7 shows the pressure distributions corresponding to Fig.6. There is a sharp dent of oil film pressure in Nos.19, 20 and 21 of Fig.7; Fig.8, enlargement of No.19 of Fig.7, shows tension of -1.OMPa. On the other hand, the pressure curve is similar to that of full film condition in Nos.15 and 16; Fig.9 is the enlargement of No.15 of Fig.7, except that the theoretical curve in Fig.9 is calculated under full film condition. In the present experiments the magnitude of negative pressure of Sommerfeld type is rather small, the maximum value is -0.3MPa (absolute). In contrast, the magnitude of negative pressure o r tension of separation type sometimes exceed 1.OMPa. The maximum is -1.2MPa (absolute). In contrast to the experiments under static condition, in which negative pressure is always found, the pressure distributions without negative pressures can be also found (for example, Nos.1--12, 24 and 25 in Fig.7) under dynamic condition. There is a --though very scarce-- possibility that the sharp dent of negative pressure is not detected by a pickup mounted in journal if the width of sharp dent is less than the spatial resolution of the pickup in the journal. Mori et. al. (14)have concluded from their experiment with constant G* and E=O: in the case of pure rotating loading, no dent of negative pressure can be observed in the range of G* above -1.0 and below 1.0, so that the Reynolds condition is appropriate, and in the excepted range of G*, the separation boundary condition is appropriate. In the present experiment, however, no such unique relation can be found between the generation of negative pressure and the value of G*. This discrepancy could be attributed at
a. Load vector diagram
b. Journal center locus
3.0
.
f=256ms
p=-$;u
0.0 \
-1.0 \-JE -2.0 3.0;
-
0.6 0.4
I
I
I
114
112
314
.2 -0.4 !-0.6
tlT c. nondimensional wedge and squeeze speed Fig.6. Typical bearing load, orbit and speed of journal center N=8rps; f=3.9Hz; ps=0.2MPa least partly to the difference that in the present experiment of dynamic loading velocities G* and E vary continuously during one load cycle, whereas in the experiment of Mori et. al. the condition is quasi-static.
4
THEORETICAL APPROACH
In order to explain the negative pressure o r tension generated in the oil film of journal bearings, theories have been developed in which
69
1.o
a 1.0
I'
;;
2n
hmin
-
- 1.0
hmln
2n
No. 7(t/T= 6/25) E = 0.524 G*= 1.936 E =-0.054
-1.0
- 1.0
hmin
0
~ T T
N0.11 (t/T=10/25) E = 0.693 G*= 1.051 E = 0.205
- 1.00
No.l6(t/T=l5/25) E = 0.859 G*= 0.652 E = 0.097
-
1 O
.
hmin
o
hmin
TI
No.l2(t/T= 11/25) E = 0.735 G*= 0.908 E = 0.175
~ T T-"OO
hmln
2~
hmin
No. 17(t/T=16/25) E = 0.884 G*= 0.644 E = 0.079
L 2TT
1 0
.
hmin
o
-1.0
0
2n
hmin
- 1.00
0
E
hmin
hmln
~ T T
No. 14(t/T=13/25) E = 0.805 G*= 0.730 E = 0.121
0.0
0.0
0
hmin
2~
- 1.00
I:
I
2V
0 hmin
No.l5(t/T=l4/25) E = 0.832 G*= 0.675 E = 0.107
- 1.0O
hmin
2TT
~ T T
hmin
2TT
No.l9(11T=18/25) E = 0.900 G*= 0.667 E = 0.037
No.20(t/T=19/25) E = 0.900 G*= 0.670 E = 0.01 1
-
-1.01 i J 0 hmin 2TT N0.25 (t /T=24/25) E = 0.877 G*= 0.795 E =-0.090
No.l8(t/T=17/25) E = 0.895 G*= 0.651 E = 0.059
hmin
2n
- 1.0
0
~ T T
N0,13(t/T=12125) E = 0.773 G*= 0.808 E = 0.146
- 1.0
hmin
- 1.0
G*= 1.809 E = 0.129
0
~ T T
No.lO(t/T= 9/25) E = 0.644 G*= 1.244 E = 0.230
= 0.536
- 1.0
hmin
No. 9(t/T= 8125) E = 0.585 G*= 1.508 E = 0.220
NO. 8(t/T= 7/25)
L -1.01 2TT 0
NO.22 (t/T=21 I 2 5) E = 0.885 G*= 0.686 E =-0.014
No. 2 1 (t IT= 201 25) E = 0.890 G*= 0.672 E =-0.006
- 1.0
1.o
+
No. 6(t/T= 5/25] E = 0.569 G*= 1.719 E =-0.230
0
NO. 5(t/T= 4/25) E = 0.650 G*= 1.407 E =-0.290
aQ
d k
0.o
- 1.00
NO. 4(t/T= 3/25) E = 0.723 G'= 1.198 E =-0.271
NO. 3(t/T= 2/25) E = 0.778 G*= 1.062 E =-0.23 1
NO. 2(t/T= 1/25) E = 0.822 G*= 0.947 E 50.185
No. 1 (t/T= 0125) E = 0.858 G*= 0.853
NO.23 (t /T=22/ 25) E = 0.887 G*= 0.712 E =-0.030
1 0
.
hmin
o
L
2TI
N0.24 (t/T=23125) E = 0.884 G*= 0.753 E =-0.055
1.1 I
I.
0.o '"0
0.0 hmin
8
2~
rad 0
1.0 0
0.0 hmin
8
2~ rad
1.o
0
hmin
8
~ T T
rad
1.o
0
hmin
~ T T
8 rad
1.o 0
hmin
0
~ T T
rad
M e a s u r e d ( f r o m t h e bearing) ....... M e a s u r e d ( f r o m t h e j o u r n a l ) Calculated(Reyno1ds condition)
.-__
Fig.7. Circumferential pressure distribution on the center line of bearing land under load during one cycle (Corresponding to Fig.6)
dynamic
70
t T
18
=25
-
.oo
-1
hmin TT
2TT €3 rad o Measured(from the bearing) * * * * - Measured(from the journal 1 ---Calculated (Reynolds condition
bearings, the bubbles flowing with the oil stream experience the pressure variation even under static conditions, and in the vicinity of minimum film thickness position, they experience a large pressure variation especially under heavier load conditions. So, if the surfacedilation effect is not negligible, bubble growth in the region a little down stream of minimum film thickness position is delayed and the bubble radius in this region remains small even though the oil film pressure is very low, namely negative or tensile. With this background, a theory is developed in which the lubricant is regarded as two phase liquid consisting of oil and small bubbles, and surface-dilation effect is taken into account in the motion of bubbles. Five basic equations are given below.
4.1 Reynolds' equation for compressible fluid The bubhly oil must be regarded as compressible the Reynolds' equation in the fluid, so nondimensional form is given by:
Fig.8. Enlargement of No.19 in Fig.7
4.2 The equation of motion for spherical small
Q
bubble In this analysis it is assumed as in ref.5 that the spherical gaseous bubbles are distributed uniformly in the lubricant, and the dynamic effect is more dominant than the thermal one. In addition, diffusion of gas is neglected. Under these assumptions the equation of motion for small spherical bubbles in nondimensional form is given by?
1
I
-1
.oo
, 'J
%i;
,
,
hmin TT
€3 rad
2TT
o
Measured(from the bearing) Measured (f rom the journal 1 --- Calculated (full film condition)
-....
Fig.9. Enlargement of No.15 in Fig.7.
the lubricant is regarded as the emulsion consisting of oil and small bubbles (5,15). However, only the surface tension has been taken into account for the motion of small bubbles. It follows that very small bubble radius must be assumed to obtain the tensile stresses of the order of -1.OMPa observed in the present experiments. For example, bubble radius of 0.05 pm can sustain only -0.57MPa (16). Ida et. al. (17)have postulated that the effect of surface-dilational viscosity (18,191 is not negligible in the motion of a small bubble when anti-foaming additive is added to base oil, and this effect is more dominant when initial bubble radius is small. This effect restrains the natural vibration of bubbles, and delays the transient response under sudden decompression. In the case of oil film in journal
= 1+B-(l+E)X Pa X
. -,
The fourth term on the right hand side represents the term of surface-dilation effect. The bubble is assumed to flow along the path line of lubricant. The path vector is assumed to be equal to the mean velocity for simplicity, namely: the mean circumferential velocity component is equal to the half of the journal surface velocity, and the axial one is equal to zero. In the present study the inertia force is negligibly small compared to viscous force, pressure and surface tension. So, the left hand side of Eq.(2) is omitted.
4 . 3 Other equations The other three equations, namely the relation between void fraction and bubble radius, the relation between specific density and void fraction, and the relation between specific viscosity and void fraction are the same as in ref.5:
QaX3
cr= 1-cr a+CYax3
(3)
6=1-a
(4)
71
E
K=l .O+O. 5062a+9.0LLa2-L6.83a3 +60.13a4-23.85a5
The five equations (1)-(5) are solved simultaneously using finite difference method for Eq. ( 1 ) and Runge-Kutta method for Eq. (2). The data used in this calculation are chosen according to the experimental conditions except for few unknown parameters. For surfacedilational viscosity K the same value as Ida et. al. estimated for mineral oil is used ( 7.85~10-~ kg/s). Void fraction at atmospheric pressure a,, bubble radius at atmospheric pressure Ra, and polytropic exponent n are assumed to be equal to 0.01, 1.0um, and 1.0, respectively. 5
1
6.0
2.0
0
n
I: 1.0
a
4.0
'
.
%
0.0-2.0
---- --_-1.ooE
---
#'
o hmin .
1.0 2Tr 0
e = 0.7
NUMERICAL RESULTS AND DISCUSSION
Fig.10 shows the distribution of the pressure and the nondimensional bubble radius calculated numerically. For large value of eccentricity ratio a sharp dent of negative pressure appears a little downstream of the minimum film thickness position, and the dent is deeper for larger eccentricity ratio, i.e. heavier load. These features correspond well to the above mentioned experimental results of the present study The region of flat pressure distribution after the sharp dent of negative pressure corresponds to the region of film rupture. However, the level of this flat pressure region is nearly equal to that of absolute vacuum, though it is widely accepted that the pressure in the ruptured region is nearly ambient or somewhat less. One can mention two reasons for this discrepancy. Firstly, full film is assumed over both 360; circumference and across the bearing in this analysis. So even in the divergent region film rupture does not occur, but only the void fraction increases. The large value of void fraction is obtained only when the pressure is lruch lower than ambient, and remains so. Secondly, the ambient pressure at the bearing ends is prescribed as the boundary condition, namely air suction is ignored for brevity in the present analysis. To be exact, this condition should be used only for submerged bearing, but not for usual bearing running in the atmosphere. In fact, negative and constant pressure region has been measured for submerged condition (12). The solution cannot be obtained for eccentricity ratio exceeding 0.9, because bubble radius diverges infinitely in solving Eq.(2). The numerical results under dynamic condition will be presented in other opportunity.
=0.6
7
(5)
2.0 -
0
a I:
- 6.0
1.0 -
a
::
- 4.0
%
l q
/
0.0
'
.--- - --_- -
- 1 \
y - 1 , . /'
2.0
---- 1 .o
I
-
o& .-o.l
0.0
hmin &
=0.9
6 CONCLUSIONS The experiments and numerical analysis presented here lead to the following conclusions. In the pressure distribution under static conditions, a sharp dent of negative pressure as known for the separation boundary condition has been found experimentally a little downstream of minimum film thickness position. Under dynamic condition, another type of pressure distribution similar to the Sommerfeld's one is also observed experimentally.
e4Y-n) rad Fig.10. Calculated circumferential distribution of pressure and nondimensional bubble radius on the center line of bearing land under static condition N=8rps; ps=0.2MPa; ~=7.85xIO-~kg/s;aa=O.O1 ; Ra=I .OUm; n=l .O
72
The authors are deeply indebted to Director Dr. R. Wada, Toyoda Machine Works Ltd. for making special pressure pickups mounted in the journal. Thanks are also due to Mr. K. Yoshioka, a post graduate student at that time, and Mr. C. Oikawa and Mr. M. Yamauchi, assistant and technical official in the University of Tokyo, for their contribution and assistance in carrying out the experiment.
cavitation regions in journal bearings', Trans. Chalmers Univ. Technol., 1961, Nr.238. (13) ISHII, A. 'On the vaporization of the oil film in the short journal bearing with a circumferential oil groove', Trans. Jpn. SOC. Mech. Engrs., 1962, 28, 331-338. (in Japanese) (14) MORI, A., KIMURA, S. and MORI, H. 'An experimental investigation of cavitation in a circumferentially grooved journal bearing subjected to rotating loads', Trans. Jpn. SOC. Engrs. (ser.C), 1986, 52, 1426-1434. (in Japanese) (15) SMITH, E.H. 'The influence of surface tension on bearings lubricated with bubbly liquids', Trans. Am. SOC. Mech. Engrs., 1980, J. Lubr. Technol., 102, 91-96. (16) FLOBERG, L. 'On the tensile strength of liquids', Trans. Machine Element Division,
References
1 973 * (17) IDA, T.
(3) Maximum tensile stress detected in the present experiment is -1.2MPa (absolute). (4) A sharp dent of negative pressure can be expounded theoretically by regarding the lubricant as two phase liquid consisting of oil and small bubbles, and by taking into account the surface-dilation effect in the equation of motion of spherical bubbles.
7 ACKNOWLEDGEMENTS
MARTIN, F.A. 'Developments in engine bearings', Proc. 9th Leeds-Lyon Symposium on Tribology, 1982, 9-28. CARL, T.E. 'An experimental investigation of a cylindrical journal bearing under constant and sinusoidal loading', Proc. Instn. Mech.,Engrs., 1964, Lubrication and wear, second convention, 100-119. PATRICK, J.K. 'An experimental investigation into the performance of sleeve bearings subjected to a range of alternating loads', Proc. Instn. Mech. Engrs., 1968, Tribology convention, 77-88. TANAKA, T. and SOMEYA, T. 'Investigation into oil film pressure distribution in journal bearings', Proc. Instn. Mech. Engrs., 1972, Tribology convention, 20-25. KAWASE, T. and SOMEYA, T. 'An investigation into the oil film pressure distribution in dynamically loaded journal bearing', 4th European Tribology Congress, 1985, vol. 11, 5.2.3. BROWN, S.R. and HAMILTON, G.M. 'Pressure measurements between the rings and cylinder liner of an engine', Instn. Mech. Engrs., 1975, Piston ring scuffing, 99-106. NAKAI, M., KAZAMAKI, T. and HATAKE, T. 'Study on pressure distribution in the range of negative pressure in sliding bearings (1st report) --the relation between the hehavior of cavities and the pressure distribution curve--', J. Jpn. SOC. Lubr. Engrs., 1982, 27, 837-844. (in Japanese) RIGHTMIRE, K. 'An improved method for experimental determination of pressure distributions in hydrodynamic bearings', Proc. 10th Leeds-Lyon Symposium on Tribology, 1983, 225-227. MORI, H., YABE, H. and FUJITA, Y. 'On the separation boundary condition for fluid lubrication theories of journal bearings', ASLE Trans., 1968, 11,196-203 DYER, D. and REASON, B.R. 'A study of tensile stresses in a journal-bearing oil film' J. Mech. Engrg. Sci., 1976, 18,46-
52. COLE, J.A. and HUGHES, C.J. 'Oil flow and film extent in complete journal bearings', Proc. Instn. Mech. Engrs., 1956, 170, 499510. (12) FLOBERG, L. 'Experimental Investigation of
and SUGIYA, T. 'Motion of air bubbles in mineral oils subject to sudden change in chamber pressure ( 1 s t report, experimental analysis of single spherical bubbles)', Bull. Jpn. SOC. Mech. Engrs.,
1980, 23, 1132-1139. (18) SCRIVEN, L.E. 'Dynamics
of a fluid interface, equation of motion for Newtonian surface fluids', Chem. Engrg. Sci., 1960,
12, 98-108. (19) =RIVEN, L.E.
'On the dynamics of phase growth', Chem. Engrg. Sci., 1962, l7, 55.
73
Paper Ill(iii)
Mixing inlet temperatures in starved journal bearings H. Heshmat and P. Gorski
Cold and hot oil mixing at the inlet to a hydrodynamic bearing has been an unresolved and critical problem in determining bearing performance; this is aggravated in the case of starved bearings by the fact that the ratio of recirculating hot oil to cold supply oil is higher than in fully lubricated bearings. The paper provides an experimental investigation of the levels of inlet temperatures in journal bearings for the case of insufficient oil supply. Aside from the customary functional dependence on operating conditions, the additional parameter considered here is the degree of starvation which ranges from full lubrication to only some 10% of the required supply. A correlation is offered that provides an estimate of the expected mixing temperature in terms of known geometric and operational parameters.
1
c
INTRODUCTION
The solution of any hydrodynamic bearing problem, whether one uses a thermohydrodynamic analysis or a simple isoviscous approach, hinges on a knowledge of the inlet temperature to the bearing pad. The essence of the problem is portrayed in Figure 1. Most bearings consist of a number of pads arranged in tandem. A quantity of hot oil emerges from an upstream pad and mixes with the cold oil admitted to the oil groove; together, this determines the temperature at the inlet to the downstream pad. Clearly, this inlet temperature is higher than the cold o i l temperature. This subject has been treated in detail in In that work, expressions were Reference 1. derived for full-film bearings based both on theoretical guidelines and on an extensive experimental program which permitted a correlation of the mixing temperature in terms of linear speed and supply temperature. There are, however, many practical applications when the oil delivered to the bearing is only a fraction of that required for a full hydrodynamic film. This occurs when restrictions are placed either on the width of the axial groove, on the supply pressure, or when pump capacity is below that required for full lubrication. It also occurs routinely when wick or splash lubrication is used and, as was shown in Reference 2 , it i s an inherent feature of oil ring bearings. The latter, as the tests of Reference 3 have shown, is due to the fact that only a fraction of the oil scooped up by the ring is delivered to the bearing clearance. A general representation of the hydrodynamics of a starved journal bearing is shown in Figure 2. The results of such a reduced extent of oil film are manifold; film thickness decreases, temperatures in general rise, stiffness and damping decrease, and, oftentimes, a reversal of customary hydrodynamic behavior is observed, such as an increase in E with a rise in speed (due to the strong thermal effects). Likewise, the process of mixing the exiting hot oil and incoming cold oil differs from a full-film bearing because, aside from the different flows and different temperatures at the relevant bearing locations, the cold and hot oil streams must traverse the starvation
Oil Groove
-U 842230
Fig. 1 Groove Mixing in Hydrodynamic Bearings
MTI P284
W
-Cavitation
/ 842416-2
Fig. 2
Mixing Inlet Temperature in a Starved Bearing
74
zone before they can traverse from the inlet stream. The present work reports on an experimental program conducted with metered oil delivery, ranging from flooded conditions (full film) to states of extreme oil starvation. Eccentricity, attitude angle and film extent were measured. In particular, temperature readings were recorded at appropriate locations to provide the basis for correlating the mixing temperatures in terms of the relevant bearing parameters. The equations derived in Reference 1 for full-film bearings were then expanded in terms of the index o f starvation and other relevant parameters to yield the corresponding values of the mixing inlet temperature. 1.1
Notation
C D L N -P P
Bearing radial clearance Bearing diameter Length of bearing in direction normal to U Shaft rotational frequency Unit loading (P, Pa/6.8 x lo5; (P, psi/lOO) Q Volumetric oil flow 9 Q/ITNRLC (Q/Qi 1 Q Cold oil supply (=Qs) Qo Inlet flow Qi 42 Exit flow QR Recirculating flow Bearing side leakage QS 9 s Index of starvation 42 (Qz/Qs) Temperature T Bulk temperature of side leakage Ta TO Cold (supply) oil temperature (9/5 To, OC + 32/120); (To, 'F/120) TO Tav Average - temperature Tmax Maximum temperature (equal to T2) TR Reference temperature Ti Inlet mixing temperature T2 Exit temperature ( = Tmax) U Linear velocity 0 (u, m/sec/2.54); (u. ft/min/5OO) Load on bearing -W W (P/MN) (C/Rl2 Specific heat of oil CP h Film thickness h (h/C) 6 Kronecker delta E Eccentricity ratio e Coordinate in plane of rotat ion Start of hydrodynamic i lm End of hydrodynamic fi m 0: Start of-bearing pad 0 Angular velocity of shaft A Mixing function for full film A S Mixing function for starved conditions p Viscosity '
the mixing process as consisting of the layer of hot oil exiting from the end of an upstream pad with the cold oil delivered to the groove, namely:
The above implies that the journal or thrust runner carries the hot oil as a thermal boundary across the oil groove into the next pad. Of course, some of the enthalpy of that hot oil is lost either by heat conduction or by physical loss due to centrifugal and other mixing forces. To account for these and other effects, the function A was formulated for the departure of the value of the mixing temperatures, Ti, from the simple relationship given by Equation 1. The expression for the mixing temperature is then given by:
where the reference temperature, TR, has been set to 17.8OC (OOF). It was found that the function 1 depends primarily on the linear speed and supply temperature, To. For journal bearings, the function A is given by:
A
[I
= -9 +
0.017
Ci21
[5
1.07 ij
-
3Tol
Figure 2 plots the function versus with To as a parameter. With the flows properly normalized and 1, as given in Figure 2, the value of Ti for full-film bearings is then giveh by:
where 6 = 0 for OF and 6 = 1 for 'C. 3
MIXING TEMPERATURES UNDER STARVED CONDITIONS
The process of flow mixing at the inlet to a starved hydrodynamic bearing differs in a number of respects from that of a full-film bearing. A schematic of flow mixing in a journal bearing is shown in Figure 3 . The major differences for starved conditions are: No flooding of the oil groove. The flow pattern, groove power loss, and convection from the hot oil layer, Q2, and Tmax will all be a1tered.
Subscripts Full-film conditions G Groove max Maximum min Minimum 1 Start of fluid film 2 End of fluid film F
2 GENERAL EXPRESSIONS FOR MIXING TEMPERATURES The approach taken to the derivation for the inlet temperature in hydrodynamic bearings is described fully in Reference 1. Essentially, it formulates
No loss of oil through the groove chamfers and no cold oil backup into the upstream pad. Together with diminished thermal convection, there would be less reason to expect loss of heat from the 42 x Tmaxlayer. Heating of the cold oil QoTo. Since the film is not formed until at some angle, 1. O S , the temperature, To, is likely to absorb heat from the bearing surface so that it will reach el at a temperature Tol 2 To. Higher ratio of hot to cold flows. As shown in Figure 4, with an increase in starvation, the
inflow, Q1, consists progressively of higher percentages of hot flow, 42, and lower percentages of cold flow, Qo (or Qs). At Qs = 0.1, 82% of the inflow consists of the hot flow, 42, and only 18% of the cold inflow, Qo. While Figure 4 plots the flows for a constant load, Figure 5 shows the flow variations as a function of E f o r a roughly similar load variation. (No exact duplicate is possible as the inlet viscosity by which W is normalized will always differ between a starved bearing and a full-film bearing.) Whereas q 2 drops with E in a full-film bearing, this quantity rises with E in a starved bearing.
No loss of recirculating flow. In full-film bearings, the backflow is dumped back into the groove and is lost. Here, as shown in Figure 6 , the lackflow, QR, i s discharged into the clearance space of the starvation region and returned = to the mixing zone. Therefore, ( Q R T ) ~ ~ (QRT)out and the recirculating flow drops out of Equation 1.
The combined effect of the above items is that while Equation 1 tends to overestimate Ti in the case of full-film bearings, ( A 2 01, the overestimate is reduced or perhaps even turned into an underestimation ( A 5 0 ) for starved bearings. The mixing inlet temperature in starved bearings is given by:
The next task, therefore, is to determine the dependence of A on the degree of starvation experienced by the bearing.
-0 1
0
2
6
4
8
10
12
14
16
18
20
22 24
26 28
30
32
U
85667
Fig. 3
Function of
0.3
A for Journal Bearings
0.4
0.5
0.6
0.7
0.8
-
0.9
1.0
c 853082
Fig. 5
Variation of Q2 in Starved and Full-Film Bearings
Mlxing Lone
0' 0
I I
I
0.2
I
I
0.4
I
0.6
I
I
-u
I
I
1.o
0.8
Index of Starvation, Q,
4% 842373-1
Fig. 4
\
Component Flows in Starved Bearings
Fig. 6
842232-1
Recirculating Flow in Starved Bearings
76
4
5
EXPERIMENTAL SETUP
Experiments were conducted on two journal bearing sizes: a moderate one 138 mm (5-7/16 in.) in diameter and a large one 229 mm ( 9 in.) in diameter. The journal bearings had a 150' lower arc with an L/D = 0.93 and two axial grooves at the horizontal split. The experimental rig used in conducting the tests both with full-film oil supply and under starved conditions has been described in Reference 1. Three eccentricity probes were used which, in addition to the journal locus, also provided a check on thermal distortions of the clearance circle. Particular attention was, of course, paid to placing thermocouples at the crucial stations. A sample of such instrumentation for the case of the 229-mm (9-in.) bearing is shown in Figure 7. In the correlations that follow, the flows Qs and Q2 were obtained as follows. Given that the bearing contained careful instrumentation, particularly with regard to thermocouples and clearance probes, these were used to obtain the viscosity map in the bearing, as well as the eccentricity and attitude angle. Based on these experimentally determined parameters, the two flows (Qs and 4 2 ) were calculated from a corresponding computer program based on the accurate solution of the Reynolds equation [41. The required temperatures, To and Tmax, were taken directly from thermocouple readings. However, Ti had to be obtained from the extrapolation of two temperature profiles. Since the start of the film, 8 varied with the operating conditions W), as well as with the amount of (e.g., supplied oil, Qo, its location was obtained by an extension of the measured temperatures in the fluid film, as shown in Figure 8. This, of course, rests on the fact that the temperature gradients in the fluid film differ noticeably from those prevailing in the starvation region, @ s @ I ' In addition, the location of e l (i.e, the start of the film) was determined from the measured values of E , W, and Qs and provided a check on the accuracy of the locus of as determined by the intersection of the temperature gradients.
CORRELATION FOR JOURNAL BEARINGS
Test results are given for both bearing sizes, 138 mm (5-7/16 in.) and 2 2 9 mm (9 in.) in diameter. For the smaller size, data for two different bearings are cited; namely, bearing A, located on the motor side of the test stand, and bearing B, located at the outer end and carrying an adjacent thrust bearing. Because o f the proximity of the thrust bearing, the two journal bearings operated at significantly different temperature levels and provided different sets of data. Figure 9 provides a composite plot o f the as a function o f speed. This plot does value of not serve our purpose of correlating A with the degree of starvation prevailing in the bearings because, as functions of load and speed, these points are at different values of the index of starvation, defined here as:
Tmax
h:
p;;r*-L--I
Fluid Film
4
0s
I
02
.
8 Measured Temperatures
BE
Thermocouples 842404
Fig. 8 Determination of Inlet Temperatures
W
O
1
D
I
I
Bearing A D x L x C = 5 h x 5'h# x 0.0055 in. Load. psi
I
I
Thermocouples 0 Pressure Taps
1 W
m
I I
I I
15
Q 0
I
L
I
9
10.8
- p - Metered Oil Supply 11.6 I 2.4 I 3.2 I
0
1
2
011 &ng
3
4
5
rpm Wmin
spwd x l(r'
842425
Fig. 9
A at High and Low Levels of Starvation
77
Qs = Qo is the external amount of oil delivered to the bearing and QSF = QOF is the delivery required for full-film lubrication. The plots are given here to show in a qualitative manner that, as discussed in the previous section, there is, in fact, a decrease in the value of h at high levels of starvation. This can be seen from the significantly lower values of h for oil ring lubrication than for the metered flows, this being due to the fact that oil rings operate at extremely starved conditions, with 0.1 < Qs 5 0 . 3 (see Reference 2 ) . Figures 10 through 13 give sample experimental results for the value of h-as a function of the index of starvation. Since Qs = 1 denotes full film operation, i t can be seen from all of these plots that there is a strong trend for 1 to the vaLue decrease with a drop in the value of of h becoming negative at very low values of Qs. This trend is retained in the numerous test data obtained, in addition to those shown in Figures 10 through 13. Pulling together the results of the various plots, the following characteristics are noted:
4
-
-2 L 12
DxLxC = 57/10
The h curve intersects the Q s axis at heavy loads earlier than at light loads.
51/10
x
0.0055 in.
P = 90 psi
A
1800 rprn
os,
There is no appreciable influence of starvation up to 50% starvation (i.e., the full film h applies to all Qs 3 0.5). This insensitivity to starvation is consistent with the fact that, as shown in Figure 1 4 , the eccentricity ratio in starved bearings is not affected until Qs falls below 0.5.
x
3200 rprn 0 0
4
1
0
I
I
I
I
‘
I
#
‘
Index of Starvation.
kis 842379-1
Fig. 11 The h Function in the 5-7116-in. 11 11 B Bearing
The h = 0 point seems not to be too sensitive to speed and depends primarily on load.
DXLXC=9X7%xO.O068in. U
5 (900 and 1200 rprn)
0 50 psi
300 m i
DxLxC = S7Ao x 51/1s x 0.0055 in.
1800 rprn 2400 rprn 3200 r m
!-o---
e l
&----
L
P = 290 psi
-4
-8
rp L
4
?=
180 psi
84-
0 P = 90 psi
I
0.1
1
I
0.2 0.3 0.4
0.5
0.6 0.7 0.8
0.9 1.0
% ,
0
0.1 0 2
0.3 0.4
05
Index of Starvation.
Index of Starvation. Q,
0.6
07
08
6,
842380 852695
Fig. 10 The 1 Function in the 5-7116-in. 11 11 A Bearing
I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 12 The h Function in the 9-in. Bearing
78
A s = A - AXs.
7
At standard condition? (U = 5, To = 1, P = 6.8 lo5, 100 psi), we have Qso = 0.15 (zero intercept). Values o f A X s for different P and U are given in Table 1. Thus, Ahs depends on Q s with P and U as parameters. Expressing A X s = f ( Q s ) in
6
x
5 4
quadratic form:
3 2
The boundary conditions in the above equation are :
Qs
= 0.5
AXs = 0
Qs
= 0.5
(dAhs/dQ,) = 0
Qs
= 0.15
-1 -2
(As =
A)
Ahs = -0.025 f o r the reference conditions U = 5, To = 1, and P = 6 . 8 x 10 Pa (100 psi).
05
-3
The above yields: I
-5
AXs
-
U = 5 (900and 1200 rpm ) o 100 psi Q 200 psi
'
= -0.05
+ 0.2
Qs (1 - Qs)
Since there is a linear variation in A X s with both U and P, then:
AXs
as
Index of Starvation,
852696
Fig. 13 The A Function in the 9-in. Bearing f o r Moderate Loads
= [-0.05 + 0.2 Qs (1
and the function
As = A
-
[0.05
-
Qs)]
(P/Po)(U/5)
for starved conditions becomes
-
0.2 Qs (1
-
Qs)lP(U/5) Full Film
I 0 7
o o o
I I
."Q
I
0
0
0
O 0 0
I I
0
P = 290 psi
I
0.6
0.9 o.8
0.7
0.8 0.9
t
I I
[ 1
1.01
0
I
O 0 -
e oo
0 0
P = 180 psi
l
Decrease in A Due to Starvation at a Partic
I I
I
I I
0
I
0
'
'
0.2
'
'
0.4
!
P = 90 ' ' I 0.6 0.8
Index of Starvation,
psi
' 1.0
6, 842423
Fig. 14
Eccentricity Ratio as a Function of Starvation
generic curve of the A function can thus be simulated by the graphs of Figure 15. The portion of A which varies with Qs will now be denoted by BAS so that A s for starved journal bearings is given by:
A
Fig. 15 Generic Curve f o r Values of A s
79
Table 1 Values of A X s at AQs = (0.5 P
10-5 pa(psi)
-
0.15) = 0.35'?
I D mm
(in.) AS
0.150
0 10 0.06
19.7
(290)
I
138
(5-7/16)
I
5.1
I
7.7
I
9.5 0
*Values in percent
3
6
12
15
-
18
21
24
30
27
U
861493
where P = (P/6.8 x lo5) and as derived in Reference 1 is given by Equation 1. Figures 16 and 17 plot the function A s in terms of speed and index of starvation Q s , down to Qs = 0.1.
Fig. 16 Variations of A s with Velocity
0 24
6 ACKNOWLEDGMENT
020
-
The work reported on in this paper was supported by Reliance Electric Company. The author expresses his appreciation to the organization for their support and permission to publish this paper. The helpful suggestions and efforts of Oscar Pinkus and Dave Hosterman of HTI are appreciated. References 1.
-0 16
Heshmat, H. and Pinkus, O . , "Mixing Inlet Temperatures in Hydrodynamic Bearings," Journal of Tribology, Trans. ASME, Vol. 108, 1986.
t'
0
I
I
1
I
I
01
0 2
03
04
05
h
0 s 861492
2.
Heshmat, H. and Pinkus, O., "Performance of Starved Journal Bearings With Oil Ring Lubrication," Journal of Tribology, Vol. 107, No. 1, Trans. ASME, January 1985.
3.
Heshmat, H., Pinkus, O . , "Experimental Study of Stable High-speed Oil Rings," Journal of Tribology, Trans. ASME, Vol. 107, No. 1, January 1985.
Fig. 17
4.
Variation of A s with Index of Starvation
Heshmat, H. and Artiles, A., "Analysis of Starved Journal Bearings Including Temperature and Cavitation Ef€ects," Journal of Trlbology, 'Trans. ASME, Vol. 107, No. 1 , 1985.
This Page Intentionally Left Blank
81
Paper Ill(iv)
Starvation effects in t w o high speed bearing types D T Gethin and J 0. M e d w e l l
t h e o r e t i c a l i n v e s t i g a t i o n i n t o s t a r v a t i o n e f f e c t s i n two b e a r i n g t y p e s i s p r e s e n t e d . The a n a l y s i s shows t h a t f o r a n i s o t h e r m a l model a c o n s i d e r a b l e s h o r t f a l l i n l u b r i c a n t does n o t r e d u c e b e a r i n g load c a r r y i n g a b i l i t y s i g n i f i c a n t l y . When t h e r m a l e f f e c t s , i n c l u d i n g g r o o v e mixing are i n t r o d u c e d , s t a r v a t i o n h a a profound e f f e c t o n a l l b e a r i n g parameters. A
temperature at i n l e t t o loaded f i l m
NOMENrrpITuRE
relative sliding velocity
C
radial c l e a r a n c e
Cf
lubricant specific heat capacity
H
p o w e r loss
H
d i m e n s i o n l e s s p o w e r loss H
i-
-
L
-- -___-
film termination coordinate
.
4pN2LR2
c o n s t a n t i n Walther e q u a t i o n eccentricity
E]
local f i l m t h i c k n e s 8
R
f i l m t h i c k n e s s at t h e i n l e t
bearing length
L,
axial l e n g t h of feed i n d u c e d f i l m
LS
axial l e n g t h of f i l m a t s e c t i o n ' s '
N
r o t a t i o n a l speed ( r e v / s e c )
P
load c a p a c i t y
P
f i l m conrmencement c o o r d i n a t e
film thickness at location ' s ' lubricant thermal c o n d u c t i v i t y
constant i n Walther equati-on pressure
[:Iz]
[;
v e l o c i t y component
d i m e n s i o n l e s s load c a p a c i t y P ZLRW
streamwise velocity component a t f i l m inlet
R
Q
leakage
Qco
l u b r i c a n t carried t h r o u g h c a v i t a t e d film
Qgs* Qgz
lubricant s u p p l i e d a t g r o o v e s 1 and 2
Qs
dimensionless s i d e leakage [=
c r o s s f i l m i n t e g r a t e d v e l o c i t y component
streamwise velocity component a t sect i o n ' s ' coordinate d i r e c t i o n s a n g u l a r l o c a t i o n from f i l m i n l e t a t t i t u d e angle
Q
e c c e n t r i c i t y ratio m o l e c u l a r viscosity R
journal r a d i u s
WCl
c r i t i c a l Reynolds number for o n s e t of nonlaminar f l o w
WCZ
c r i t i c a l Reynolds number for w h o l l y t u r b u l e n t flow
kinematic v i s c o s i t y kinematic v i s c o s i t y - x d i r e c t i o n kinematic v i s c o s i t y - z d i r e c t i o n lubricant density
El
Wf
f i l m Reynolds number
W T
t u r b u l e n c e model Reynolds number
T
temperature
Tcav
cavitated f i l m temperature
T
mean f i l m t e m p e r a t u r e
+
maximum f i l m t e m p e r a t u r e
Tg1 .Tg2
lubricant t e m p e r a t u r e a t g r o o v e s 1, 2
d i m e n s i o n l e s s t u r b u l e n t C o u e t t e stress r o t a t i o n a l speed ( r e v / m i n ) 1
INTRODUCTION
I n the d e s i g n of c y l i n d r i c a l bore hydrodynamic j o u r n a l b e a r i n g s it is u s u a l t o adopt one o f two l u b r i c a n t feed c o n f i g u r a t i o n s as shown i n F i g u r e 1. The a r r a n g e m e n t s sham are a s i n g l e port at t h e maximum f i l m t h i c k n e s s and
82
a twin arrangement where feed is provided at points orthogonal to the load line. BY virtue of its design, the former has optimum load carrying ability for one direction of shaft rotation (clockwise as shown), while the latter has more widespread application due to its suitability to cope with bidirectional shaft rotation. In the design of such bearing types, it is important that there is an adequate lubricant supply to the film. This will ensure that the bearing runs completely flooded otherwise its load-carry capacity will be derated and furthermore, the film temperatures may become excessively high111 which may result in lubricant oxidation in oil fed bearings. Adequacy of oil supply to a bearing is expected to be determined mainly by the design of the feed arrangement and by lubricant delivery pressure in pressure fed bearings. For example, the supply port may comprise a simple hole or, more commonly, a groove where for the latter, depth, circumferential and axial extents are all variables which have been shown by visualisation experiments[2] to influence film formation over the entire bearing width, particularly at high speed. In predicting the lubricant needed by a bearing, it is usual to consider two separate components, that obtained from hydrodynamic action and that induced by having a positive gauge pressure as described in [3] for a large variety of lubricant supply geometries. Here the authors compared a simple flux plotting technique with a more complex numerical model of the groove and correlated the results in a format which could be used readily for estimating the pressure induced flow for various groove designs. This information Can then be combined with the flow predicted from hydrodynamic considerations to determine lubricant supply pump duty. More recently, the influence of feed pressure on journal bearing performance has been investigated both theoretically and experimentally as reported in 14, 51. The model developed accounted for the effect of a pressurised lubricant supply by computing the extent of the wetted film and this was shown to have a significant influence on bearing load capacity and attitude angle at l o w eccentricity ratios. This aspect was not confirmed in the experimental study[5], however, the latter verified fully the ability of the model to describe accurately the film extent and lubricant flowrate. Little is known about the operation of high speed bearings operating with incomplete (or starved) lubricant films. Intuitively, it is expected that they will operate at higher temperatures since the diminished lubricant supply will not provide cooling either by virtue of low feed temperature or by its ability to convect heat away. However, in bearings which operate at comparatively low speeds, experimental results[6] obtained from studies conducted on a bearing supplied by either an oil ring or controlled lubricant flow, confirmed that maximum film temperature could set the operating limt. From this rigorous investigation it was deduced that under the more severe conditions the lubricant film occupied about 25% of the bearing width at cornmencernent of the load-carrying region and that in modelling the bearing behaviour under these conditions it was essential to
incorporate the lubricant mixing process atthe supply grooves. There are few theoretical studies of starved bearing behaviour, however, some of the more recent studies include those described in 171, [8] and [s]. In [7], the authors describe an approach to approximate starved bearing behaviour by modelling the film as a partial arc bearing such that the arc extent determined the degree of starvation, a small arc representing a more severe condition. More recently Bayada presented a predominantly mathematical model of starvation in a journal bearing and illustrated the developnent of a variational approach to model the free boundary corresponding to the film extremity in the convergent gap. The combined effect of starvation and thermal effects has been investigated numerically using the finite element method as described in [s]. In this study the finite element mesh was modified during the iterative solution so that it mapped the wetted surface. the extent of which was determined from flow continuity. However, the analysis did not include any mixing model at the grooves which has since been shown [6] to be important. It is intended to incorporate this in the present study for the bearing types shown in Figure 1.
2
THEORETICAL E(DDEL
2.1 Governing Equations for Hydrodynamic Lubrication The fundamental equations of thermohydrodynamic lubrication are the Navier-Stokes, continuity and energy equations[lO]. However, fqr hydrodynamic lubrication it is possible to adopt a boundary layer type approach where assumptiom are made concerning the form of the velocity and temperature distribution through the film[ 111. For journal bearing lubrication, the crossfilm velocity and temperature fields may be expressed in terms of local mean values viz
... 2.la u(y)
=
:-;[ hZ
and
T(y) =
'i
w f l ~ y - y 2 ] +h
... 2 . a ...
2.1c
These equations may be substituted into the mamentun continuity and energy equations where, after integration, they become
83
[s
h aP
+
hw,
P az d'h wzw-+-
12vz v h
dx'
(axial mmentum)
h ap hv,
P ax 12v,i
; +
:[
6vxu
h
+
avdh
z ] - 2 v z
vh ( =
---
however, gives
a x dx dh
6vz;
av,
aV
az
az
+h [g]- 2 h - - -
,]a; dh
[-I
dh
ReT
-
6wxV
__
4vxV
- --
h
h
2.2a
'
av
h
-+
a;
h
az
-+
dh
u
ax
-= 0
... 2 . 2 ~
(continuity)
dx
and
2.2
... 2 . 2 d
Equations 2.2 i n c l u d e also q u a n t i t i e s which account for nonlaminar flow i n the f i l m , f o r the p r e s e n t analysis, t h e r e s u l t s from 1121 and 1133 were u s e d , i . e .
and
... ~... ~ 2). 3 b ... 2.32
v x = v (1
+
0.000375 R ~ T ~ * ~ ~ 2.3a )
v z = v (1
+
0.000175 R
+
0.0012 ReTQsS4
=
1
~
T
~
*
(141
f o r a w a t e r lubri-
S t a r v a t i o n kAel
Where the lubricant f i l m o c c u p i e s o n l y part of t h e b e a r i n g width a t f i l m i n l e t , as it i s drawn i n t o the convergent gap, t h e n its axial e x t e n t w i l l i n c r e a s e so as t o s a t i s f y flow continuity This w i l l c o n t i n u e u n t i l it reaches t h e b e a r i n g edge at w h i c h p o i n t s i d e f l o w from t h e f i l m w i l l o c c u r (see F i g u r e 2). A t any s e c t i o n 's' i n the load c a r r y i n g f i l m (see F i g u r e 2 ) a flow balance may be written viz
The Reynolds number i n e q u a t i o n s 2 . 3 w a s determined from t h e n a t u r e of t h e flow i n t h e film. C l e a r l y under laminar c o n d i t i o n s i t assumed a v a l u e of z e r o w h i l e f o r t u r b u l e n t a c t i o n it is given by
us dz =
h,
-
1,"
ho uo dz
...
2.4
-
where Ls is l i m i t e d t o the axial e x t e n t o f the b e a r i n g and L, a f i l m w i d t h o b t a i n e d from a c o n s i d e r a t i o n o f l u b r i c a n t supply rate and carry o v e r . S o l u t i o n o f e q u a t i o n 2.4 is not s t r a i g h t f o r w a r d s i n c e the v e l o c i t i e s u and uo are dependent on the hydrodynamic a c t i o n i n the f i l m w h i c h is determined u l t i m a t e l y by the b e a r i n g load. However, the prbblem may be overcome by i n c o r p o r a t i n g its s o l u t i o n i n t o t h e o v e r a l l i t e r a t i v e procedure as explained i n d e t a i l i n 191. 2.3
(energy)
Ref Rec,]
.
+
... 2 . -
momentum)
- ReCl]
=
r e s u l t s presented i n cated bearing.
IF
( streamwise
interpolation
w a s assumed t o h o l d , which is i n agreement w i t h
dxz
dh
linear
The critical Reynolds number ( Recl ) w a s obt a i n e d from T a y l o r s c r i t e r i o n w h i l e Recz=2Rec,
d'h
[-I
appropriate
("cz-
2 wx - - - w x u - +
a x dx
an
("f
...
azu
I n t h e superlaminar regime
Ref).
S o l u t i o n Procedure
In the solution method, the governing e q u a t i o n s w e r e s o l v e d u s i n g t h e f i n i t e element method by a d o p t i n g the Galerkin weighted r e s i d u a l approach t o y i e l d g r a d i e n t s as the n a t u r a l boundary c o n d i t i o n [ 151. However, boundary v a l u e s r e q u i r e s p e c i f i c a t i o n and, t o minimise computational e f f o r t , advantage w a s t a k e n o f c e n t r e l i n e symnetry so that o n l y h a l f the bearing required consideration. of T h e r e f o r e , a boundary c o n d i t i o n f o r t h e -axial momentum e q u a t i o n ( 2 . 2 a ) may be w r i t t e n w = 0 on the b e a r i n g c e n t r e l i n e ( a t PO). The remaining g r a d i e n t v a l u e s at the f i l m inflow, and downstream boundaries were sideflow d i f f i c u l t t o p r e s c r i b e beforehand. However, t h i s w a s overcome by assuming v a l u e s at t h e beginning o f the c a l c u l a t i o n and on each them using the iteration recalculating v e l o c i t y f i e l d and shape f u n c t i o n d e r i v a t i v e s a t the a p p r o p r i a t e b e a r i n g boundaries. This w a s t h e n used as a p r e s c r i p t i o n €or t h e next iteration. Then f o r e q u a t i o n 2.2a
aw
- at
L
aw
- and --
at x = X, and X, w e r e u p az 2 ax d a t e d d u r i n g t h e i t e r a t i v e procedure. z =
84
a; Sunilarly for equation 2.2b
3
az represents centreline symmetry remaining gradient conditions
au L au - a t z = - , - at x az 2 ax
=
RESULTS AND THEIR DISCUSSION
-- = O a t z = O while
the
X, and X, were updated.
When the equations are solved by the finite element method [15], pressure is obtained implicitly from the continuity equation and thus it requires the statement of pressure boundary conditions. For the present study, zero pressure was assumed at the upstream (x = X,), sideflow and downstream boundaries where the latter was adjusted so that the pressure gradient was zero at the centreline also. The energy equation (2.2d) requires boundary condition specifications also. For the present investigation, zero gradient values were applied on the centreline (symmetry condition), downstream extremity and film edge, which are compatible with experimental evidence presented in [2]. At the film inlet ( x =XI) the prescribed temperature depends on the mixing process in the groove( s ), the calculation of the effective film inlet temperature accounting for this will be discussed more fully in Section 3 . 2 . In the solution, the film was divided into 50, @-node isoparametric elements (a 10x5 grid - see Figure 2). Subsequent to the first iteration, the solution procedure for a thermal analysis m y be summarised by the following milestones: (1) Update gradient boundary conditions and solve the hydrodynamic equations.
Before proceeding to a full thermal analysis, it is useful to consider the effect of starvation in isolation. Therefore the following section will be divided into two parts - an isothermal analysis followed by a consideration of thermal effects. 3.1 Isothermal Analysis Initially, numerical experiments were carried out to ascertain the effect of lubricant feed and carry-over on journal bearing performance. To illustrate the analysis, the following bearing geometry was assumed journal radius (R) bearing length (L) radial clearance (C)
50 mm 50 m 0.2mn
The results of calculation for the two bearing types considered are shown in Figure 3 where the film width ( b ) generatd by the lubricant supply amounts to 25% of the bearing length (L). Figures 3(a) and (b) illustrate the extent of the wetted surface for a single groove bearing highlighting the case where lubricant carry-over is included (3(b)). This can be seen to contribute to an increase in film width at the film inlet ( 0 = 0). Similarly, Figures 3(c) to 3(e) depict the situation for the bearing fed by two axial grooves. The latter includes the contribution from the second groove at 0 = 180° which is also assumed to flood 25% of the film width. The result of including the various flow contributions is similar to that for a single feed arrangement. The form of the film boundary also compares favourably with observations described in [ 4.1. Global performance parameters associated with the film conditions illustrated in Figure 3 are listed in Table 1.
(2 ) Determine
the inlet temperature to the load-carrying film from lubricant mixing considerations and solve the energy equation.
(3)
Calculate bearing parameters. 0.157 27.9O 0.348 32.1O 1.127 3 6 . 6 O 0 . 0 3 6 23.10 0.210 29.20 0 . 3 4 0 32.5O 0.604 33.7O
(4)Update
the extent of the wetted film accounting for lubricant supply and carry over.
This sequence was repeated until solution convergence was obtained. However, to do so required the embodiment of the appropriate film thickness statement and an expression for the dependence of lubricant viscosity on temperature. For the present investigation where thermal aspects were included, the Walther equation was used, i.e. log,,
(v
+
0.6)= m log,,
T
+ b
...
Table 1
:
Bearing Global Performance for the Different Film W e l s =
2.5
where m and b are constants determined by the oil viscosity-temperature relationship.
film 3(a) film 3(b) fully flooded film 3(c) film 3(d) film 3(e) fully flooded
Ref
0.8; L/R = 1.0; C/R = 1400; b / L = 0.25
0.004
=
It can be seen that all parameters are affected. For both bearing types it may be noted that load carrying ability is reduced, the effect of starvation being marginally more significant for the single axial groove feed arrangement. In view of the film extents shown in Figure 3 , initially it is surprising that the changes in load-carrying capacity are
so small. However, it may be n o t e d t h a t where s i g n i f i c a n t f i l m pressures are g e n e r a t e d , the f i l m i s flooded f u l l y and t h e r e f o r e t h e a b i l i t y of t h e b e a r i n g t o carry l o a d i s retained. Table 1 shows c l e a r l y t h e change i n l u b r i c a n t leakage f r o m t h e f i l m for t h e d i f f e r e n t n u m e r i c a l models. From t h e r e s u l t s it may be n o t e d t h a t t h e greatest change takes place i n t h e s i n g l e axial g r o o v e b e a r i n g . This arises s i n c e there i s n o l e a k a g e from t h e f i l m i n a r e g i o n w h e r e t h e g a p between the j o u r n a l and bush is greatest. The table s h o w s also t h e effect of i n c l u d i n g t h e v a r i o u s w e t t e d s u r f a c e s o n power loss for t h e n u m e r i c a l model, o b v i o u s l y the r e s u l t s corresponding t o the f i l m e x t e n t s i l l u s t a t e d i n F i g u r e s 3 ( b ) and ( 3 e ) b e i n g m o s t physically realistic. As expected,the smallest w e t t e d s u r f a c e g i v e s rise t o the least power loss. F i n a l l y , the a t t i t u d e a n g l e data p r e s e n t e d i n T a b l e 1 shows h o w its v a l u e d e c r e a s e s when t h e f i l m r u n s starved. This arises s i n c e t h e p r e s s u r e force c o n t r i b u t e d i n t h e r e g i o n where the f i l m i s incomplete i s reduced. This rep o s i t i o n i n g of the j o u r n a l i n the b u s h w i l l have t h e most s i g n i f i c a n t effect o n l u b r i c a n t supply rate at t h e second groove of a t w i n axial groove b e a r i n g d u e t o f i l m t h i c k n e s s I t is f u r t h e r reflected i n t h e changes. o v e r a l l p e r f o r m a n c e of t h i s b e a r i n g type s i n c e it e f f e c t i v e l y d e t e r m i n e s the volumetric supply of lubricant at t h e second g r o o v e . F i g u r e 4 i l l u s t r a t e s a comparison w i t h some d a t a o b t a i n e d f r o m e x p e r i m e n t s u s i n g the procedure d e s c r i b e d i n [I). Both w e t t e d f i l m e x t e n t and b e a r i n g c e n t r e l i n e p r e s s u r e are p r e s e n t e d where the e x p e r i m e n t a l d a t a is p e r t i n e n t t o t h a t for a b e a r i n g f e d b y a s i n g l e h o l e o n t h e load l i n e . I t c a n be s e e n that good q u a l i t a t i v e agreement has b e e n o b t a i n e d f o r t h e e x t e n t of t h e w e t t e d f i l m along w i t h r e a s o n a b l e agreement f o r predicted f i l m p r e s s u r e which is shown for t h e l o a d e d part of the f i l m o n l y . To i n v e s t i g a t e global b e a r i n g performance mre f u l l y , c a l c u l a t i o n s w e r e r u n o f f f o r a range of s p e e d s and e c c e n t r i c i t y ratios w h e r e t h e l u b r i c a n t f e e d a t the b e a r i n g groove amounts t o 25% f l o o d i n g of i t s l e n g t h . The r e s u l t s of t h e s e c a l c u l a t i o n s are i l l u s t r a t e d i n F i g u r e s 5 t o 8 for t h e b e a r i n g geometries c o n s i d e r e d . From the c u r v e s p r e s e n t e d , it c a n be s e e n that l u b r i c a n t leakage from the f i l m is affected most s e v e r e l y for the two b e a r i n g types c o n s i d e r e d . N e v e r t h e l e s s , the r e m a i n i n g bearing parameters (load capacity and loss) a r c ? n o t affected a s s o c i a t e d power significantly. However i t is anticipated t h a t the effect of l u b r i c a n t s h o r t f a l l w i l l be more d r a m a t i c when thermal behaviour is i n c l u d e d . This w i l l be c o n s i d e r e d n e x t . 3.2 Thermal Aspects
I n s t u d y i n g the thermal r e s p o n s e of the bearing t o d i f f e r e n t operating conditions, towards preliminary analysis w a s directed incorporating i n v e s t i g a t i n g the effect of groove mixing models i n a manner similar t o t h a t d e s c r i b e d i n 116). s i n c e t h i s d e t e r m i n e s t h e l u b r i c a n t s u p p l y t e m p e r a t u r e t o the l o a d carrying film. C o n s i d e r i n g first the s i n g l e g r o o v e
bearing. W i t h r e f e r e n c e t o F i g u r e 1, at the downstream e x t r e m i t y the l u b r i c a n t w i l l be a t its maximum t e m p e r a t u r e T ( 2 1 . Then for a l u b r i c a n t h a v i n g constant d e n s i t y and specific heat, t h e e n e r g y transport rate at thi? p o i n t By i n the f i l m is p r o p o r t i o n a l t o Q-T. assuming that i n the cavitated f i l m the v i s c o u s g e n e r a t i o n equates heat removal- by c o n d u c t i o n f r o m the f i l m , the t e m p e r a t u r e T i s s u s t a i n e d t o the feed groove w h e r e mixing takes place w i t h t h e fresh l u b r i c a n t s u p p l y . Although t h e e x p e r i m e n t a l e v i d e n c e i n [2] s u g g e s t s that heat is removed f r o m the cavitated f i l m for t h e p a r t i c u l a r e x p e r i m e n t a l c o n f i g u r a t i o n , the adoption o f the p r e s e n t will give a worst-case bounding model analysis. Then w r i t i n g an e n e r g y balance at the g r o o v e g i v e s a l u b r i c a n t t e m p e r a t u r e at the i n l e t t o t h e l o a d - c a r r y i n g f i l m v i z
To =
[Qco T + Qgr T g r ] [Qco + Q ~ L ]
... 3.1
A s i m i l a r argument may be applied t o the t w i n axial g r o o v e b e a r i n g , w h e r e now lubricant s u p p l y a t the second g r o o v e must be a c c o u n t e d for. Again w i t h r e f e r e n c e t o F i g u r e 1, between the end of t h e load c a r r y i n g f i l m and the s e c o n d g r o o v e , t h e f i l m t e m p e r a t u r e w i l l be a b o u t T while i n t h e cavitated f i l m it w i l l be c o n s t a n t a t a v a l u e g i v e n by
[Qco
+
Qaz Tazl 3.2
Then at t h e i n l e t t o t h e l o a d c a r r y i n g f i l m at the upstream groove g i v e s a mixing temperature [Qco T To
=
+
Qgz Tgz
[Qco+ Qgz
+
+
Qgi Tgr]
Qgr]
... 3 . 3
where for a commn s u p p l y TgL and Tgz are equal w h i l e Qgl and Qqz depend o n t h e f i l m t h i c k n e s s at the g r o o v e l o c a l i t y and the e f f e c t i v e axial e x t e n t of the f i l m due t o l u b r i c a n t s u p p l y conditions. To d e m o n s t r a t e the effect of i n c l u d i n g the mixing model, analyses w e r e carried o u t for the two b e a r i n g types where i n one case, the t e m p e r a t u r e at t h e i n l e t t o the loaded f i l m ( T o ) w a s held at the feed t e m p e r a t u r e (Tgi or Tgz) w h i l e i n the s e c o n d case the e n e r g y b a l a n c e s a t the g r o o v e ( s ) (equations 3.1 t o 3.3) were applied during the iterative procedure. For these calculations, the f o l l a w i n g g e o m e t r i c and p h y s i c a l properties were assumed.
-
journal r a d i u s (R) bearing length ( L ) = radial clearance ( C ) = lubricant d e n s i t y (p) = lubricant specific heat
5Omm 5Omm 0.2m 860kg/m3 (Cf)
2000 J/kg deqK l u b r i c a n t thermal c o n d u c t i v i t y ( k f ) = 0.156 W/m d e g K =
For the Walther e q u a t i o n , the constants m and b assumed v a l u e s of -3.79 and 9.64 which are p e r t i n e n t t o a n o i l of 1.50 v i s c o s i t y grade 32.
The r e s u l t s o f t h e s e a n a l y s e s are shown i n F i g u r e 9 which i l l u s t r a t e s plots of t h e c i r c u m f e r e n t i a l temperature p r o f i l e on t h e bearing centreline. It is apparent imnediately that when groove mixing is incorporated, the temperatures a t the f i l m i n l e t are i n c r e a s e d s i n c e now t h e r e i s i n s u f f i c i e n t fresh l u b r i c a n t t o cool down t h e The combined effect of carry-over component. t h e h i g h temperatures and incomplete f i l m on g l o b a l b e a r i n g performance is l i s t e d in Table 2.
Rttitude Cornnent [ degrees )
can be s e e n t h a t a t the more extreme o p e r a t i n g conditions, t h e starved b e a r i n g runs a t dangerously h i g h temperatures. The consequent e f f e c t on b e a r i n g behaviour is also shown i n F i g u r e s 10 t o 13. As expected, load c a r r y i n g a b i l i t y , s h a f t torque r e a c t i o n , leakage and a t t i t u d e a n g l e are a l l reduced when s t a r v a t i o n t a k e s place. With regard t o o v e r a l l behaviour, it can be seen that t h e t w i n axial groove b e a r i n g is not affected q u i t e as d r a m a t i c a l l y as its s i n g l e This i s due p r i m a r i l y t o groove c o u n t e r p a r t . the f a c t t h a t the combined supply of l u b r i c a n t by these grooves exceeds t h a t s u p p l i e d by the s i n g l e groove geometry and hence t h e twin groove d e s i g n r u n s somewhat cooler or closer t o t h e flooded f i l m temperature.
I -
4
coNcLu.51oNs
no mixing
i n v e s t i g a t i o n of s t a r v a t i o n e f f e c t s i n two has been cumonly used bearing types Both i s o t h e r m a l and thermal investigated. models have been d e s c r i b e d w h e r e the latter i n c l u d e s a mixing model for t h e b e a r i n g From t h i s investigation, the grooves. following conclusions may be drawn. An
mixing
no mixing
Table
2
: ~ l a b a Bearing l
Performance f o r a
S t a r v e d Bearing w i t h Groove Mixing Effects Present C / R = 0.004; W L = 0.25;
I+/R = 1.0; c = 0.8; w = 10000 rev/min.
table shows c l e a r l y that b o t h load capacity and shaft torque reaction are a f f e c t e d markedly. The r e d u c t i o n i n both of these arises f r o m the r e d u c t i o n i n l u b r i c a n t v i s c o s i t y associated w i t h t h e h i g h e r f i l m temperatures. A series o f c a l c u l a t i o n s w e r e carried o u t to compare starved and flooded bearing behaviour under t h e m h y d r o d y n a m i c c o n d i t i o n s w h e r e groove mixing was i n c o r p o r a t e d . The r e s u l t s of t h e s e a n a l y s e s are p r e s e n t e d i n F i g u r e s 10 t o 1 3 f o r a range of speeds and eccentricity ratios. From these f i g u r e s i t can be Been that s t a r v a t i o n r e s u l t s i n h i g h e r inlet and maximum film temperatures p a r t i c u l a r l y at the upper o p e r a t i n g speeds (see F i g u r e s 10 and 11). Under these circumstances, the high film temperatures r e s u l t i n lover l u b r i c a n t v i s c o s i t y and hence a higher film Reynolds number for i n c o r p o r a t i o n i n t o the t u r b u l e n c e model. T h i s r e s u l t s i n higher v i s c o u s g e n e r a t i o n (see equation 2.2d) and ultimately increased o p e r a t i n g temperatures. These t r e n d s are i n q u a l i t a t i v e agreement w i t h experimental data presented i n 123. However, w h e r e j o u r n a l e c c e n t r i c i t y effects have been i n v e s t i g a t e d , t h e maximum temperatures i n the flooded and s t a r v e d s o l u t i o n s show some convergence. This is p m b a b l y due t o the fact that at the mre extreme e c c e n t r i c i t y ratio, the combined effect of reduced v i s c o 8 i t y ( a r i s i n g from high temperature) and nonlaminar f l o w , r e s u l t i n a lower v i s c o u s g e n e r a t i o n when compared w i t h high viscosity and less turbulent the hydxodynamic a c t i o n i n the cooler f i l m . I t
-
The
For the degree of
s t a r v a t i o n considered, under isothernal c o n d i t i o n s , load c a r r y i n g a b i l i t y is affected o n l y marginally while associated power loss w a s reduced. More s i g n i f i c a n t l y , leakage from t h e f i l m w a s diminished v e r y considerably. Under thermal conditions, the i n c o r p o r a t i o n o f a mixing model at he grooves increases film temperature c o n s i d e r a b l y . This becomes i m p o r t a n t when s t a r v a t i o n is accounted for s i n c e now f i l m are increased which is temperatures reflected adversely i n global bearing behaviour
.
-
The o v e r a l l e f f e c t of s t a r v a t i o n is marginally less s i g n i f i c a n t for the twin axial groove b e a r i n g when compared with t h e s i n g l e axial grove c o u n t e r p a r t .
References J.O. and BOWEN, E.R. 'Journal bearings operating in the turbulent Leeds/Lyon Tribology regime',Proc. 2nd Symposium - Superlaminar Flow i n Bearings;
(1) MEDWELL,
Y.E.P.
1976, p.189-193.
( 2 ) MEWIIELL, J.O. and BUNCE, J.R. 'The i n f l u e n c e of b e a r i n g i n l e t c o n d i t i o n s on bush temperature fields' Proc. 6th Leede/Lyon Symposium on Tribology The& Effects in Tribology, 1979, (edited by Dowson, Taylor, Godet and Berthe )
.
( 3 ) MARTIN, F.A.
and LEE, C.S. 'Feed p r e s s u r e i n p l a i n j o u r n a l b e a r i n g s ' Trans. A S L E , V01.26, 1983, p.381-392.
flow
a7
( 4 ) DOWSON, D., TAYLOR, C.M. and MIRANDA, A.A.S. 'The p r e d i c t i o n of liquid film journal bearing performance with a consideration of lubricant film reformation P a r t I ' Theoretical Results, P r o c . I .Mech.E., V o l . 199, 1985, p. 95-102.
D., TAYLOR, C.M. and KIRANDA, ( 5 ) DCUSON, A.A.S., 'The prediction of l i q u i d f i l m journal bearing performance with a c o n s i d e r a ti o n of lubricant film reformation P a r t 2 ' Experimental Results, Proc.1.Mech.E.. V01.199, 1985, p.103-111. ( 6 ) HES€JMAT,H. and PINKUS, O . , 'Performance Of starved j o u r n a l b e a r i n g s w i t h o i l r i n g Trans. ASME, Journal of lubrication' Tribology, V01.107, 1985, p.23-31. ' S o l u t i o n s of ( 7 ) ETSION, I . and PINCUS, O., incomplete f i l m s ' f i n i t e bearings w i t h Trans. ASME, J. of Lub.TeCh., V01.97, 1975, p.89-93.
' V a r i a t i o n a l f o r m u l a t i o n and ( 8 ) BAYADA, G . , a s s o c i a t e d a l g o r i t h m for starved f i n i t e Trans. AsME, J.of journal b e a r i n g s ' , Lub.Tech., VO1.105, 1983, p.453-457. D.T. and MECWELL, J.O., 'An ( 9 ) C;FLIIIN, a n a l y s i s of h i g h speed b e a r i n g s o p e r a t i n g with incomplete films', Tribology I n t e r n a t i o n a l , Vol.18, 1985, p.340-346. FIGURE 1
( 1 0 ) SCMICHTING, H., 'Boundary l a y e r thoery', published by Pergamon P r e s s .
BEARING TYPES AND ASSOCIATED GROOVE MIXING MODELS
(11) LAUNDER, B.E. and LESCHZINER, M.A., 'An efficient numerical scheme for the prediction of t u r b u l e n t f l o w in thrust bearings' Proc. of 2nd Iaeds/Lyon Sympoeium on T r i b o l o g y , 1975 ( e d i t e d by Davson, Godet and T a y l o r ) .
( 1 2 ) NG C.W.
and PAN,
C.H.T.,
'A
t u r b u l e n t l u b r i c a t i o n theory', (D), VO1.95, 1973, p.137-1%.
linearised
Trans ASME
V.N. 'Basic relationships in turbulent lubrication and their extension t o i n c l u d e thermal effects * Trans. ASME J. of L u b r i c a t i o n Tech.,
( 1 3 ) CONSTANTINESCU,
V01.95,
1975, p.147-154.
/ z.0
and PULLEFt, D.D. 'JOUrMl ( 1 4 ) SMITH, M.I. b e a r i n g operation a t superlaminar speeds', T r a n s . A S M E , April 1956, p.469-474. ( 1 5 ) HWD,
P.
and TAYLOR,
C.
'Navier
Stokes
equations using mixed interpolation' Conf . P r o c . on The F i n i t e Element Method i n Flaw Prablema 1974 (edited by Men, Zienkiewicz, G a l l a g h e r and T a y l o r ) . ( 1 6 ) HESHMAT, H. and PINKUS, 0. 'Mixing i n l e t temperatures in hydrodynamic b e a r i n g s ' AsI(E Paper 85-Trib-26.
FIGURE 2
A TYPICAL STARVE0 FILM GEOMETRY AND ASSOCIATED FINITE ELEMENT MESH
m to
200
100 8
0
THEORY
12 N
E
10 8
6 4 2
O O
100
200
8 FIGURE 3
THE EXTENT OF THE WETTED FILM FOR
VARIOUS MODELS Re,=1400
;E
=0.8,L/,=l.O.C/,=O.O04,
FIGURE 4
COMPARISON OF OBSERVED AND PREDICTED DATA FOR A WATER LUBRICATED BEARING ; $=O 004,L/,=l.O. E =O 7,Re=2980,R=114mm
FLOODED - - -STARVED
LO/,
L
=0.25
-FLOODED --- S T AR V ED Lo/L= 0.25
180
lo
10 -
180
in _. v)
0 v)
J
v)
l x
5
0
14C
a
a
? 0
0 J
v) W
z 5 v) z W
a
J
v)
m
v) v) W J
v) v)
1
0
0
a
v)
140
0
LLI
=
W
0
0
W
J
z
5
0
v)
zz
v)
z
W
z 0
z
0
i
0
0
1500
0
6C
3000
0 0
1500
3000
0
3 0.9 0
J
0 I/)
W
0 v) z w
0 3
0.4
C
4 --0 t - r I
0.7
FIGURE 5
1500
3000
0 1500 FILM REYNOLDS NUMBER (Re,)
FLOODED AND STARVED BEHAVIOUR OF A TWIN AXIAL GROOVE BEARING ; c/=O.004,L/,=1.0,~=0.?
0
50"
1500
3000
L
W 1
a
W
0
I n
3
LL!
b
0.5
% L E 0.2 0
3000
u z
W
v, v)
1500
1-
I
U
v)
100
3000
Il-
a
300L 20"
&
0
1500
3000
FILM REYNOLDS NUMBER (Re,) FIGURE 6
FLOODED AND STARVED PERFORMANCE OF A SINGLE GROOVE BEARING ; c/=0.004,L/,=1.0,~ =0.7,R=50rnrn
FLOODED --- STARVED L t d =0.25
CD
0
L 000E0 --- FSTARVED
I L
40
160
1401
a
0
0
a
120
J
W
ga 120
;20
20 100
0 wl z
v,
v,
1 Z
W
I n
52
2 100
0 0
W
g
1
J
Z
10
0
8ot u
0.5
0.7
0.3
0.9
0.5
0.7
z
10
W
0
60
0.3
v,
Z
g -
0
80
60
0
0.9
r
(1:
0
v,
E J
160 140
t
30
30 0
=0.25
0.3
0.5
0.7
0.9
0.3
0.5
0.7
0.9
0.7
1.1 3
0
3
0.6
v,
a
v,
z
0.5
0
Z
2z 0.4 W
0.3
------ -
3
z
Il-
v,
k
a
\
FIGURE 7
0.5
0.7
0" 0.9 0.3 0.5 ECCENTRICITY RATIO
0.7
0.9
60"
a
0.i
W
0
40"
z
TWIN AXIAL GROOVE BEARING BEHAVIOUR, 0.0 0 4 ,'/R= 1.0,Re, = I 4 0 0 ,R= 50mm
c/=
g
I-
c
W
' u
0.3
J
E
20" -
c
W
v,
W
W J
z 0
g 0.5
C J
J
v,
lA
J LL
W
0 w
80"
0
J LL W
0.5 0
a
0.: 0.3
FIGURE 8
0.5
0.7 0.9 0.3 0.5 ECCENTRICITY RATIO
0.7
SINGLE AXIAL GROOVE BEARING BEHAVIOUR 9, 0.004 =1.0,Ref = 140 0, R=5 0 mm I
'/R
0.9
;
91
*
FIGURE 9
THE EFFECT OF INCLUDING GROOVE MIXING MODELS ; $ 0.004 ,PR= 1.0, E: = 0.8,b/L 0.2 5,R= 5 0.0 m m,w = 10,O00rev / mi n (a) SINGLE GROOVE BEARING (b) TWIN GROOVE BEARING
--- STARVED
-FLOODED
L”/L=O 25
60
W Y
0 0
0
0
10
20
30’
0
10
20
40 0
10
ROTATIONAL SPEED 40’ (rev/min)
FIGURE 10
FLOODEO AND STARVED BEHAVIOUR OF A TWIN GROOVE BEARING 004,L/,=l O.E :O 7,R=50mm
g0 .
20
92
-FLOODED
---- STARVE0 L”/LO
24
-
25
24,-
16
0
a 2
0
8
0
10
0
20 ROTATIONAL SPEED x103 Irev/min) COMPARISON OF FLOODED AND STARVED BEARING PERFORMANCE $4 OOL,L/=lO.~=0.7.R=50mm
FIGURE 11
-
SINGLE GROOVE,
--- STARVED LvL=O 25
-FLOODED
30
___---
O
p:::b;:! J
40
30
<- 20
---_-_ 3
W Y I
a
5 10
280
40’
W 2
20‘
0
03
05
FIGURE 12
6o
07
03
07
09
6oob u
-0
a
E 3
___---
n
II
c W
I
J
c LO Ic
Y
\
a 20’ 0 3
05
07
\ 0 $9
60
- 0 03
05
- 180 r
80’
z W
4
03
FLOODED AND STARVED BEHAVIOUR OF A TWIN AXIAL GROOVE BEARING VR=O004,L/=1O.R=50mm.w=10000rev/min
Y 2
w
I
05
07 09 ECCENTRICITY RATIO
09
r
u
---/
05
07
09
03
05
07
09
ECCENTRICITY RATIO FIGURE 13
FLOODED AND STARVED PERFORMANCE OF A SINGLE GROOVE BEARING O.R=50mm.w=10000rev/min
VR=OO O L , L / = l
SESSION IV THRUST BEARINGS (1) Chairman: Professor Y . Hori
PAPER IV(i)
Three dimensional computation of thrust bearings
PAPER IV(ii)
Parametric study and optimization of starved thrust bearings
PAPER IV(iii) Tilting pad thrust bearing tests - Influence of three design variables PAPER IV(iv) An experimental study of sector-pad thrust bearings and evaluation of their thermal characteristics PAPER IV(v) Hard-on-hard water lubricated bearings for marine applications
This Page Intentionally Left Blank
95
Paper IV(i)
Three dimensional computation of thrust bearings C.M.McC. Ettles
The historical development of 2D and 3D bearing analyses is described and discussed. Data for the convection coefficients at the back face of thrust pads is given and correlated against standard variables. From the use of thesedata in a simple analysis it is suggested that the "double layer" pad could operate at substantially lower temperatures then standard pads. A 3D analysis program is used to show the benefits of relieving parts of the support circle in button insert pads. 1.
Definition of the entering temperature profile is required. (The hot oil carry over effect.) The coupling between heat flow in the pad and film was sometimes treated in an awkward way that caused difficulty. The convection coefficients around the free surfaces of the pad were unknown. (A range of measured coefficients is given in this present paper.) If the temperature of the moving surface can be taken as uniform, how should the value be assigned? Non-reversing flow in the film (as always occurred in planwise solutions) allowed single pass (initial value) techniques to determine the temperature field. However these techniques must be radically revised if locally reversed flow occurs in the film. Partly reversed flow in the inlet region does occur in infinitely broad bearings for film convergences greater than about 2.5:l (dependent on the profile between inlet and outlet) and at larger convergences in finite breadth bearings. Reversed flow occurs widely in journal bearings (10).
INTRODUCTION
The generation of heat within a bearing oil film is an important effect that has been widely researched. The first comprehensive treatments include those of Cope ( 1 ) and Christopherson ( 2 ) . Cope's model assumed that there was a negligible temperature variation through the thickness of the film. Effectively the film was viewed in plan. Sternlicht (3) was among the first to demonstrate coupled solutions of the Reynolds and energy equations based on Cope's model. Conduction was assumed to be negligible within the film, the primary mechanisms being convection, flow work and dissipation, although much later treatments (e.g. 4,5) allowed for a proportion of the generated heat to be lost to the bearing components. Two dimensional planwise solutions were developed to a high degree of sophistication by, for example, Castelli and Malanoski (5). However it has been shown by analysis (e.g. 6) and by experiment (7) that large temperature differences develop across a bearing film. The thermal inertia of the moving component is such that temperature fluctuations on the moving surface are negligible. In contrast, the temperature rise along the static component can be of the order 100°C. The large variations in viscosity across a bearing film must be ignored in a 2D planwise solution. In many publications the satisfactory agreement of pad temperatures with the "film temperature" is (incorrectly) considered to be evidence of successful modelling. An alternative two dimensional solution is to view the film in elevation rather than in plan. This dictates that the pads must be of infinite breadth, but gives a far better model of mass and heat transfer within the film and heat transfer within the components. The first THD analysis of the film in elevation was given by Zienkiewicz (8) at the same 1957 conference at which Sternlicht (3) gave THD solutions for the film in plan. Soon afterwards in 1963, Dowson and Hudson (9) gave 2D elevation solutions for the film and components. Although 2D elevation solutions are much more representative of heat flow in a bearing it was the planwise solutions that were intensively developed during the following two decades. This occurred due to several difficulties or unknowns that arise with the elevation solution. These are:
Most of the difficulties 1-5 above have been addressed-andatleast partially solved (these are later discussed). This allowed the development of some 3D solutions which are described in this paper. The only further difficulty that occurs in expanding a 2D elevation solution to 3D is some extra complexity in programming and longer execution times. 1.1
Notation
B
Arc length of pad on mean radius
c
Specific heat
F
Film friction force
h
Film thickness
ho Minimum film thickness H
Surface heat transfer coefficient
k
Thermal conductivity
96 Nu
Nusselt number (HB/k)
P P
Local film pressure
Pr
Prandtl number, (cn/k)
Q
Heat flux
r
Radial coordinate
R
Radial position (fora temperaturemeasurement)
Re
Reynolds number pUB/rl
t
Pad thickness
U uf
velocity, u: u = g[Jfdr
Average film pressure
Rotor speed, ur
-
'I
Aj-ddz
n
+
El-dz 1
[21
where A,B are integral terms of the viscosity across the film. Mass continuity can then be applied to give a generalised form of the Reynolds equation which can be solved (using finite difference equations) for the pressure distribution. Details of the derivation are given by Dowson (11) and Heubner (12). 3.2
The solution for film temperature
Flow velocity over back of pad
W
Deflection due to shear
W
Pad load
rl
Viscosity
t)
Circumferential coordinate
P
Density
2.
THE ADVANTAGES OF 3D COMPUTATION
The principal advantages are accuracy and realism. The locally varying dissipation can be more correctly modelled and .the heat transfer to and from the components and the oil film can be correctly coupled and treated in detail. 'The thermal bending moments within the pad can be accurately computed from the 3D temperature distribution. This allows a realistic determination of thermal deformation, which is particularly important in large bearings. A 3D treatment allows several methods of improving a bearing's operation to be investigated (at least on a comparative basis) without carrying out field tests. Such methods include hydrostatic assistance (jacking), leading edge feed grooves (to partly deflect the carryover of hot oil), the inclusion of pockets (to reduce bearing friction), the inclusion of cooling galleries in the pad (to cool the film and control thermal distortion) and variation of the pad support system (to obtain more efficient film shapes). Although an analysis in 3D allows several important effects to be accurately treated, all of these effects must be actually included if the end result is to be of value. For example, a full treatment of thermo-elastic deformation must be included if the advantage of 3D computation is not to be lost. 3.
METHODS USED IN THIS 3D ANALYSIS
3.1
The solution for film pressure
A Newtonian lubricant and steady, laminar conditions were assumed. The solution for pressure requires the film shape and 3D distribution of viscosity to be completely defined. The method begins with the stress equations, for example:
iiF!
ax
=
a ($2) az a2
[ll
Integrating twice and applying the boundary conditions for u at z = 0,h gives the local
This is the core problem in a 3D analysis. It is usual to neglect conduction in the plane of the film (r,O directions). This apparently sets the problem in initial value form. Treatments such as (6,8,9,10,13) used a finite difference r e p r e s e n t a t i o n , g e n e r a l l y with backwards differencing of the (aT/afJ) term. Finite element treatments have been given by Tieu ( 1 4 1 , Fust ( 1 5 ) and Gero and Ettles (16). Gero (17) recently gave a full discussion of the use of FE methods to solve the energy equation. The main difficulty in obtaining a solution using initial value techniques is the occurrence of reverse flow in the inlet region, because the propagation process must always proceed in the direction of the local flow. In (10) a method is described that identifies separate zones of forward and reverse flow, together with an internal boundary from which propagation must initiate to treat the reverse flow zone. The solution as an elliptic equation is not satisfactory for two reasons. Firstly, the downstream conditions must be specified. Secondly, the band width of an equation set in a 3D treatment is so large as to force a solution of the computing equations by iterative and not by direct means. However the elliptic set is not suitable for iterative solution, since the magnitude of the convection terms compared to the conduction terms results in the equation set not being diagonally dominant, which causes difficulty in obtaining a solution that, as observed, must be obtained by iterative means. This problem has long been realised in computational fluid mechanics, and "solved" by the use of control volume techniques together with upwinding, as described by (for example) Patankar (18). The basis of the method is to divide the film into 3D cells with a node at the centre of each cell. The flow across the cell boundaries is found. A heat balance is then formed considering internal dissipation and the heat flux convected and conducted across the boundaries. The flow is convection dominated and an essential feature of the method is to "upwind" the convected flux. It is assumed that the temperature of the flux crossing a boundary and "flowing" from node to node is defined by the temperature at the node which is upstream, and not by (say) an average value of the temperature at the two nodes. This gives excellent stability to the iterative solution of the equations for temperature, at some expense of increased truncation error. The method is robust and can accommodate reverse flow at any position in the film (which might occur from, say, recesses or hydrostatic supply ports). The domain can be set up and swept in the same way as an elliptic problem and, as observed, is strongly convergent.
91 An advantage of the cell method is that the pad and film can be treated as a unified system without separately applying the interface boundary condition (31 Local sinks can be arranged in the pad to simulate cooling passages. An estimate is needed of the convection coefficient on the free surfaces of the shoe. Some measured values are given later in this paper.
even though only the deflection is required. A finite difference solution was used in the examples that follow. An FD solution gives only the deformation (and consequently requires much less memory) and is faster in execution, although the free edge boundary conditions are tedious to apply. An approximate solution for the deformation is shear w is given by
where D = t 3/[12(1-v 2 ))
3.3
The solution for rotor temperature
Since, as discussed, it is justifiable to assume aT/ao = 0 , the rotor can be modelled as a 2D axisyumetric domain. A non uniform grid is quite easy to arrange if the control volume method is used. Sinks can be specified to model rotor cooling, and irregular shapes can be used if necessary to allow (for example) for cooling at the inner radius of a rotor plate as well as at the outer radius.
3.4
Hot oil carry over
The temperature profile of the incoming oil must be specified. HeshmaE and Pinkus (19) have recently described a method based on inlet mixing that will give a uniform entry value. An inlet profile T(r,z) can be estimated from closed form solutions of the thermal boundary equations (Ettles, discussion to (19)) or by numerically solving the thermal boundary layer equations, as described by Vohr (discussion to (191, see also Vohr (20)). A suitable treatment for turbulent boundary layers on the surface of the rotor has not yet been given. In flat plate flow the onset position of a turbulent boundary layer is usually assumed to occur when Re(x) = lo5, where Re(x) contains the distance from the upstream pad. In high speed bearings or those with a large gap between pads, the boundary layer may become turbulent before reaching the leading edge of a pad. This would give an advantage since turbulent mixing will reduce the temperature of oil in the entry steamtube, even if some of this is in the laminar sub layer. If this is the case, the sudden reduction of pad temperature with increasing high speed (as observed in several laboratory test machines) could be due in part to turbulence and cooling outside the film, rather than viscosity enhancement from super-laminar effects within the film. 3.5
and aw / a r = O
on the inner and outer radii
aw /rao=O at the leadingand trailing edges In the overall program, the shear and bending deflections were determined separately and superposed. A non uniform grid was used to allow the exact position and length of a line pivot to be accommodated. (The effect of infinitely varying the pivot dimensions could therefore be found). A non uniform grid was also used for disc supports, in which the support nodes were uniformly distributed around the support ring. In some examples at the end of the paper, some of the support points are removed to leave two support arcs (the "eyebrow" support). These gave an improved oil film shape in these particular examples.
4. CONVECTION FROM THE BACK SURFACE OF PADS The resistance to heat transfer at the free faces of the shoe is in series with the resistance due to conduction. In some grevious analyses a uniform value of 100 BTU/ft hr OF ( 5 6 8 W/m2 "C) has been used for all bearing sizes and running conditions, and in some case studies in the literature the assumed value has been arbitrarily varied over three or more orders of magnitude. A convection coefficient H may be estimated from measurements of the thermal gradient at the surface. In Fig. 1 below
H(T~-T,)
=
-
kaT/azsurface
Thermo-elastic deformation of the pad
The 3D temperature distribution in the pad allows the thermal bending moments to be accurately determined. These can be incorporated in the biharmonic equation for the bending deflection of an arbitrarily supported plate, as described by Robinson (21,22). Several finite element solutions are available for plate deformation, but these have the disadvantage (implicit in the FE method) of treating the deformation gradients aw/ar and aw/r.aO as unknowns as well as the These gradients must be determined deflection w.
Rotor
U
Fig. 1 Determination of the convection coefficient at the back of the pad.
98 Members of the Tribology Community have been most helpful in contributing to a data base from which surface heat transfer coefficients may be found. The data are for seven bearing assemblies, in the size range 149 mm to 2946 mm outer diameter, a s summarised in Table I. The estimate of H from 151 is inevitably subject to error, due to the requirement to find the flux k(aT/az) at the actual surface. In bearings 1,5,6,7 in Table I, the temperature was measured at only two points, so that a uniform gradient must be assumed. In bearings 2,3,4 the temperature was measured at three points, allowing a parabolic curve fit to obtain the surface gradient. For these particular bearings, the "straight line" coefficient (using points nearest the film surface and the back surface) could be compared with the "parabolic" coefficient using three points. The average of the ratio: A =
arabolic coefficient straight line coefficient
Parameters likely to affect the convection coefficient
The straight line coefficients from the seven bearings were subject to regression analysis, but before this is presented, some classical results from the literature will be discussed for convection from round or flat bodies. Convection from a cylinder in cross flow has been widely investigated. The results for many liquids over a wide size range can be summarised as:
0.26
=
-HD= k
H
UD Oe6 cn k n
0.26
or
0.3 Pr
ReOe6
Or
171
0.3
[81
p-
u0.6 -0.3D-.4
n
-
191
Here D is the cylinder diameter and H is the average coefficient around the surface. For laminar flow over a hot plate, the result can be reduced to: H
-
U
0.5 -0.17x-.5
n
I101
where H is the local coefficient at distance X from the leading edge. For turbulent flow over a hot plate at high Prandtl number, the approximate result is
161
was 0.51. The parabolic measurements are the most representative, but are subject to ill conditioning from experimental error. For example a variation of 0.5OC at the centre could strongly affect the calculated value of H. To give a common basis for the data, & w d u u 06 conv e c t i o n coeddicient given in 6 i g ~ r uand 6omnutke atre "6.ttlaigkt fine1' w d u e ~ . 4.1
Nu
H
=
0.9 -0.9 x-o.l U n
[Ill
when H and X have the same meanings as in 1101. Considering results [9,10,11],a regression test of the form
could be suitable, where U is the free stream velocity over the back of [he pad and the exponent a is less than unity. The exponent b for viscosity can be expected to be negative, with c being negative also, however X in [12] could be a pad size (arc length at the mean radius) or distance from the leading edge. The quantification of Uf is particularly difficult, since the flow over the back of the pad can be expected to be sluggish and affected by the bearing design. Theapproach taken was to assume that Uf was directly proportional to the local sliding velocity U = wR, where R is the radius at which the measurement was taken: Uf = a w R = . < U
[ 131
TABLE I
THRUST BEARINGS
No.
00
-
-
1.
149
steel
31
4000
2.
267
s t e e l ,Cu
30,90
2000 12000
3.
572
steel
46
300- 1000
1.72- 3.45
47- 104
4.
786
steel
90
82- 308
0.31-3.49
37 46
-
26
5.
1664
cast-iron
68
200-428
2.14
31-44
27
6.
1727
steel
65
200
3.03-3.91
58
28
7.
2946
steel
131
86
1.67-4.03
48
29
~
Material
mm
V i sc. CS@4O0C
Pressure
RPM -
-
MPa
Tbath "C = -
Ref.
-
1.79
67
23
0.69-3.45
46
24 25
99 wherea is common to all the bearings designs, and much less than unity. In fact a is likely to vary considerably between designs and to vary locally within a given design, giving large scatter to the results. Other effects-known to give difficulty in the measurement of convection coefficients are surface roughness and deposited films (dirt or lacquering). The relationship subjected to regression was
Speed 25
40
lo4
'
Ido'I$
m/s
1
3 0 c S a11 9OcS oil
Loboratory bearing
v
N
267mmdion
E
2 I
H =
C
Ua q b Xc
[ 141
0 3OcSoll
where C is a constant that emerges from the analysis and U = wR.
I
/
4.2
Results for convection coefficient
All regression analyses will give f confidence limits for the constant C and the exponents. The BMDP:lR linear multiple regression package was used in a series of tests in which the viscosity was either the bulk value in the housing or at the local surface temperature T (in Fig. 1). The value of X was tested as thz distance of the measurement point from the leading edge or the arc length of the pad on the mean radius. For data on individual bearings and for the whole set, the results which consistently gave the least scatter were given by rl = rich (the bulk viscosity) and X = B, the arc length of the pad. The overall result was: ' 0.699k.092 H=25U
2 x103
I 0'
v N
E
2
I
-0.196k.071 B-0.376f.131 I151 'lch
The following rounded version was found to be satisfactory
H
=
25.5
U
0.7
-0.2 'lch
-0.4
10' Io2
lo3
25.5 Uo.7/?:f
1161
Bo4
The units in [15,16] are U m / s ; IIch Pa-s; B m ; H, W/m2 'C. In non dimensional form, the result was: Nu = 0.018 Re0.676f .063 pr0.503f .0861171 The exponents in [16] are fairly similar to those for flow over a cylinder 191. Comparing the non dimensional results 171 and [171 gives an order of magnitude value of the flow velocity over the pads as O.O15U, i.e. a 0.015 in 1131. Itwould seem possible to substantially raise the flow velocity by quite minor modifications in design. The possible benefits of increasing the convection coefficient are considered in the next section. The results for H are compared against Eqn. I161 in Figs. 2,3,4. The dashed diagonal lines show error bounds for the experimental result being twice or half the formula result 1161. A S expected the scatter is quite high, even for individual bearings. In the marine thrust block, two shoes were instrumented identically, withmeasurements of thermal gradient at the 64-79 (r,Q) position (shown by closed points) and at the 93-93 position, where the pad was undercut to about half the average thickness (open circles). The 93-93 values are much less than those at 64-79, possibly due to conduction effects at the corner or to flow stagnation in the undercut region. The 93-93 results were not included in the regression analysis. The results for all bearings are shown in Fig. 5. The scatter is surprisingly low considering the variations in design and the possible
2 10)
103
v
N
$ I
10'
Figs. 2,3,4 "Straight line" convection coefficients for bearing No.Z(upper figure), the marine thrust block (bearing No. 4, centre figure> and the hydro-electric generators (bearing Nos. 5,6,7 lower figure).
100 If Tf-Ta = AT, then .to allow for hot oil carryoverset T -T = % AT. The basis of the f e heat balance is that the dissipation FU is given by AQv and Qc
I o4
Y
N
E
FU = AQV
2
I
The pad breadth L cancels when the heat balance is made and is not shown below. Assuming:
L
a
0"
,
+ Qc
3
10
U/h
(shear stress)
T
= rl
u
q
=
0
AQv = pcq*AT/Z
(convected)
= AT-BkH/(k+tH) *. 2 Ph /UqB = W = 0.07
(conducted)
t
0
c
PC
(volumetric flow)
Uh/2
0
Qc
I o2 lo2
h lo3
lo4
=
(for L = B)
1.5 ho
Gives the solution for the overall rise AT:
2 5 5 U0j/rl:h2604
Fig. 5 Convection coefficients for all bearings plotted against the regression result [161. Closed symbols are for point pivots. Open symbols are for line pivots.
*
The load capacity W is taken for a square pad with a 2:l convergence. The mean film thickness h is used to define the flow and shear stress. Equation [16] is used to define 11. (The experimental error. In Fig. 5 the point pivot "straight line" result may be used since the mesh designs are shown as closed points and lie conlength is "unity"). The model is intended for sistently above the line pivot designs (open comparison studies only. points). The difference could be due to stagTwo of the cases below are for a "doublenated flow behind the pivot. layer" or "sandwich" pad in which a thin plate (of thickness tj7.5) is mounted on pad of regular thickness, as shown in Fig. 7 below. Oil 4 . 3 The beneficial effects of reducing pad thermal flow through the cooling passages is induced from the rotor motion. The chamfer 6n the plate takes resistance advantage of the velocity ram effect to boost the flow* What benefits are likely to result if the thermal resistance of the pad is reduced? Previous studies The double layer pad has been tested by Bahr ( 1 3 , 3 0 ) have shown a small decrease of film tem( 3 1 ) and Kawaike et al. ( 3 2 ) and found to be very perature with increased convection coefficient. effective. An attractive feature is that the However there is a size effect, as the following cooling flow is passively induced,without a pump. case studv will show. Kawaike found that the flow velocity in the channels was 0.15 the rotor velocity, which compares well with the approximate velocity of 0.015 over the back of standard pads. Higher convection coefficients will result. In addition, the conduction resistance (k/t) is substantially reduced, particularly if the plate is of copper alloy. The four cases in Table I1 have the common basis:
I
0 =
P = Fig. 6
20 CP
all cases
4 MPa.all cases
Schematic €or study of pad resistance
The intention in this highly simplified analysis is to obtain a closed form result that allows the convection coefficient H and the pad conductivity k to he varied. Only three temperatures are defined: Ta
ambient temperature (hulk)
Tf
film temperature
Te
entry temperature to film
BABBIlTED LAYER BACKING I E T A L LOWER BLO OIL GUIDE COOLING DUCT
Fig. 7 The double layer pad from Kawaike et al. ( 3 2 ) . From ASLE Transactions, with permission.
101
with
U = 10 m/s, 50 m/s B = 50 mm, 450 mm
The superior performance of this type of shoe has received little attention in the literature.
two speeds two sizes
H = zero, lx, 5x proportions of Eqn. [61
For case l(co1. 1) t = B/4, k = 50 standard steel pads B/4, k
t
case 3
t = B/30, k = 50 steel plate
case 4
t = B/30, k =350Cu alloy plate
=
CASE STUDIES USING 3D COMPUTATION
The computer program has these features
350 copper alloy
case 2
=
5.
A 3D solution for the film and pad temperature using the control cell technique, with upwinding of the film flows. A solution for pad thermo-elastic deformation for any pad support system (including springs). An allowance for hot oil carryover. The convection coefficients for the rotor are taken from standard texts. The coefficients at the backs of the pads are taken as 0.50 of the "straight line" value, since a grid of five intervals is usually used through the thickness. (The straight line results would be satisfactory for only one grid length.) The coefficients at the sides are taken as twice those on the back.
TABLE I1 CASE STUDY OF TEMPERATURE RISE (Case 1 standard)
P = 4 MPA (all cases) Size B = 50 mm U m/s
H ii/mZ°C
10 10 10
0 926 (1x1 4631 (5x)
Temp. rise, deg C Case 1 2 3
4
61 42 26
61 40 18
(61) (61) (40) (39) 18 16
61 56 52
61 54 41
(61) (54) 40
5.1 50
0
50 2858 (lx) 50 14289 (5x1
50
50 50
0 1187 (1x1 5933 (5x1
61 58 58
61 54 46
(61) (54) 46
(61) (53) 36
(61) (52) 35
Effect of varying the convection coefficient
Some results from Neal (34) were used as a basis to find the effect of varying H in the 3D program. The bearing was of 149 mm diameter, with eight centrally pivoted pads in a flooded chamber. Figure 8 shows the reading from a thermocouple (fairly close to the maximum position) a s a function of load at 3000 RPM. The chamber temperature is also shown. Three results from the model are shown, using a chamber temperature of 66OC. too
150
90
130
80 -: 0
110
a E 70
90
60
70
!?
c
The results in Table I1 indicate Increase of the convection coefficient is more effective for small pad sizes at low speeds then large pad sizes at high speeds. (2) High conductivity pad materials are more effective in large sizes. Even s o , Gardner (33) has reported worthwhile reduction in the maximum temperature in aluminum pads of 50 mm length. (3) The double layer pads give a substanadvantage for all sizes and speeds. A value of 5x the standard convection coefficient is likely, consequently the results for zero and lx are shown in parentheses.
50
0
0.5
I 1.5 2.0 Pressure M P o
2.5
50
0
1
2
3
4
Pressure MPo
(1)
From Table I1 it appears that the use of a high condricti.vity plate (compared to steel) gives a greater advantage in larger size bearings. The
Fig. 8 Left: Experimental and model results Right: The effect of varying H I k. The right hand figure shows the model result for the "standard" convection coefficient H=907. (The straight line value would be 1814). The two other curves show the effect of completely insulating the pad on all free surfaces ( H = O ) and of increasing the coefficient on all surfaces by a factor of five. The result for copper alloy pads with H = 907 is shown as filled circles. The results for the full program agree quite well with the simplified analysis. For P = 4 MPa and U = 18 m/s (3000 RPM), the temperature rises from the 3D program and the simple model are:
102
Steel
Copper
5.2
H=O
3D Program Rise, degC
Eqn. [191 Deg C
55
61
H = 907
34
40
H = 907 x5
26
27
H=907
31
37
Hydroelectric bearing, 3048 mm
150 RPM, IS0 VG 150
The effect of pad support extent in a large bearing
For a pad with any given support system, the design can be scaled up directly without affecting the crowning ratio 6,/h0, where 6, is the deflection from elastic effects. However the crowning ratio from thermal effects increases directly with bearing size, and consequently the support system in large bearings must be carefully designed to avoid excessive thermal distortion. Baudry (35) and Kawaike (32) have considered the support problem in detail. In (36) Ettles showed that a deleterious size effect existed in bearing assemblies above about 1 metre diameter. In this present study an example hydroelectric generator bearing is analysed using the 3D program. The bearing is disc supported, with an offset pivot of 0.55. Ths size and running conditions are Outer diameter
3048 mm
Inner diameter
1524 mm
Pads
12
Pad angle
25.5'
Pad thickness
170 mm
Support diameter
varied
Speed
150 RPM
Viscosity (3 40'C
150 mPa
400 Hydroelectric bearing, 3048mm
I
A,B
~
0
2
4 6 Specific lood,P MPa
8
10
0
2
4 6 Specific lood, P MPo
8
Figs. 9,lO The effect of disc support size in a hydro-electric bearing. Figures 9,lO show the effect of disc size on minimum film thickness and maximum local face temperature. The disc size has a negligible effect on performance below 4 MPa, but sudden collapse occurs above 4 MPa with the smaller disc supports. This is due to excessive thermal crowning. The "failure" occurs by a ratcheting mechanism in which the inefficient film shape causes excessive heating, which causes further crowning. The thermal and elastic deformations can be made to cancel using larger diameter disc supports (curves c,d in Figs. 9,101. Figure 11 shows isobars of face temperature, pressure and film thickness for a small disc ( A / B = 0.08) at P = 4.34 MPa and for a large disc ( A / B = 0.91) at P = 6.42 MPa. For the case of the small disc, the crowning is excessive, giving a most inefficient film shape as shown by the isobars for pressure. A very small increase in load will cause complete collapse. With the larger disc, the crowning is moderate and the pressure distribution is more efficient Pads supported on two (inner and outer) discs connected by a pivoting beam are preferred to single disc support (32,351. Figure 12 shows isobars for a disc support that has been relieved (or cutaway) between one and five o'clock and 7 and 11 o'clock approximately. The two remaining arcs of support perform the same function as a two disc support system. From Figs. 9-12 it can be seen that this inexpensive, simple modification to a full disc support gives a substantial improvement. Finally, the effect of water cooled galleries close to the working face is shown in Figs. 9,10,12. The temperature in the centre of the pad face is reduced by about 2OoC, but in addition, the bulk of the pad is at a nearly uniform temperature, which almost eliminates thermal distortion. The relatively large distance between the support arcs causes the centre of the pad to deflect, which gives a "dog-bone" effect to the pressure field.
103 Full Support A/B = 0.08
Full Support A/B = 0.91
Cutaway Support A/B = 0.91
T, OC
Cutaway Support Water Cooling
T, "C
04
62
16 1 6-
\
I
P, MPa
g50 P=4.34 MPa
P=6.42 MPa
I
P = 8.29 MPa
P = 8.49 MPa
14.5
h
mici
I.
h, micron
200
Fig. 11
Fig. 12
Isobars of temperature, pressure and film thickness for example hydro-electric generator bearing.
5
CONCLUDING REMARKS (1)
The 3D analysis of thrust bearings offers far better accuracy and realism than 2D planwise solutions, however the extra complexity can be considerable. (2) Data are given of convection coefficients at the backs of pads. A simple analysis shows that ''double layer" pads could give a worthwhile performance advantage for small as well as large bearings. ( 3 ) The two disc or "eyebrow" support gave (for the cases considered) better pad support than is possible with a single disc. 7 ACKNOWLEDGEMENTS This project was carried out at Rensselaer Polytechnic Institute and funded by the National Science Foundation under grant number MSM-8212511. The author records his thanks for this support and for the test data supplied by S. Advani, K. Brockwell, P. Neal, D. Nelson, A. Mikula and J. Simmons.
References COPE, W.F. 'A hydrodynamical theory of film lubrication', Proc. Roy. SOC., London, Ser. A, 197, 201-217, (1949). CHRISTOPHERSON, D.G. 'A new mathematical model for the solution of film lubrication problems', Proc. I.M.E, 146,126-135, (1942). STERNLICHT, B. 'Energy and Reynolds considerations in thrust bearing analysis', Proc. I.M.E., Conf. on Lubrication and Wear, 1957, Paper 21, 28-38. ETTLES, C.M.McC. 'The development of a generalized computer analysis for sectorshaped tilting pad thrust bearings', Trans 2, 152, (1975). ASLE, 9, CASTELLI, V. and MALANOSKI, S.B. 'Method of solution of lubrication problems with temperature and elasticity effects: application to sector, tilting-pad bearings', Trans ASME, Jrnl. of Lub. Tech., Oct 1969, 634-640. HAHN, E.J. and KETTLEBOROUGH. 'Solution for the pressure and temperature in an infinite slider bearing of arbitrary profile', Trans ASME, Jrnl. of Lubn. Tech., October 1967, 445-452.
DONSON, D., HUDSON, J.D., HUN'I'ER,B.and MARCH, C.N. 'An experimental investigation of the thermal equilibrium of steadily loaded journal bearings', Proc. I.M.E., Symposium on Journal Bearings, 181, 3B, 70-80, 1966-67. ZIENKIEWICZ, O.C. 'Temperature distribution within lubricating films between parallel bearing surfaces and its effect on the pressures developed', Proc. I.M.E.,Conf. on Lubrication and Wear, 1957, Paper 81, 135141. DOWSON, D. and HUDSON, J.D. 'Thermohydrodynamic analysis of the infinite slider bearing, Parts I and 11', I.M.E. Lubn. and Wear Convention, 1963, Papers 4,5. BONCOMPAIN, R., FILLON, M. and FRENE, J. 'Analysis of thermal effects in hydrodynamic bearings', Trans ASME, J r n l . of Tribology, 108, 219-224. DOWSON, D. 'A generalised Reynolds equation for fluid film lubrication', Int. J . Mech. Sci., 4 , 159-170, (1962). HUEBNER, K.H. 'Three-dimensional thermohydrodynamic analysis of sector thrust bearings', ASLE Trans., 17, 1, 62-73, (1974). EZZAT, H.A. and RHODE, 'A study of the thermohydrodynamic performance of finite slider bearings', Trans. ASME, Jrnl. of Lubn. Tech., 298-307, (July 1973). TIEU, A.K. 'A numerical simulation of finite-width thrust bearings, taking into account viscosity variation with temperature and pressure', J. Mech. Eng. Sci., 17,p . 1, (1975). FUST, A. 'Dreidimensionale thermohydrodynamische berechnung von aFialgleitlagern mit punktformig abgestutzten segmenten', Thesis ETH, Zurich, Switzerland (1981). GERO, L.R. and ETTLES, C.M.McC. ' , A three dimensional thermohydrodynamic finite element scheme for fluid film bearings', Submitted, ASLE Trans. GERO, L.R. 'A finite element method for the solution of three-dimensional thermohydrodynamic lubrication problems', Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY, Dec. 1986. PATANKAR, S.V. 'Numerical heat transfer and fluid flow', McGraw Hill (1980). HESHMAT, H. AND PINKUS, 0. ''Mixing inlet temperatures in hydrodynamic bearings', Trans. ASME, Jrnl. of Trib., 108, 231-248, (1986). VOHR, J.H. 'Prediction of the operating temperature of thrust bearings', Trans. ASME, Jrnl. of Lubr. Tech., 103, 97-106, (1981). ROBINSON, C.L. and CAMERON, A. 'Studies in hydrodynamic thrust bearings', Phil. Trans. Roy. SOC., London, A, Math. and Phys. Sciences, 278, No. 1283, 351-395, April 1975. ROBINSON, C.L. 'Thermal and elastic distortions in thrust bearings', Ph.D. Thesis, London University and Imperial College, 1971. NEAL, P.B., Sheffield University, Private Communication, 1981. MIKULA, A. Kingsbury Bearing Co. Private Communication, 1984. SIMMONS, J.L. and ADVANI, G. Michell Bearings PLC, Private Communication , 1983. ELWELL, R.C., GUSTAFSON, R.E. and REID, J.C. 'Performance of centrally pivoted sector sea trials aboard thrust bearing pads USS Barry DD993', Trans ASME, Jrnl. of Basic Eng., 483-497, Sept. 1964.
Sx.
-
32)
33)
341
(35)
(36)
BROCKWELL, K.R. and ADVANI, S . Michell Bearings PLC, Private Communication, 1982. NELSON, D.V., PLUMMER, M.C. and McCULLOCH, R.C. 'Overload performance testing and evaluation of a hydrogenerator tilting pad thrust bearing', Proc. Amer. Power Conf., 2, 1057-1062, (1984). Anon. KIM, K.W., TANAKA, M. and HORI, Y. 'A three-dimensional analysis of thermohydrodynamic performance of sector-shaped, tilting pad thrust bearing', Trans ASME Jrnl. of Trib., 105,406-413, July 1983. BAHR, H.C. 'Recent improvements in load capacity of large steam turbine thrust bearings', Trans ASME Jrn. of Engr. for Power,E, 130-136, Jan. 1961. KAWAIKE, K., OKANO, K. and FURUKAWA, Y. 'Performance of a large thrust bearing with minimized thermal distortion', ASLE Trans, 22, No.2, 125-134, (1979). GARDNER, W.W. 'Performance tests on sixinch tilting pad thrust bearings', Trans. ASME, J r n l . Lubn. Tech., 97, No.3, 430-438, (1975). NEAL, P.B. 'Some factors influening the operating temperature of pad thrust bearings', Proc. of 6th Leeds-Lyon Symp. on Tribology, MEP Ltd., 137-142, (1979). BAUDRY, R.A., KUHN, E.C. and WISE, W.W. 'Influence of load and thermal distoriton on the design of large thrust bearings', Trans ASME, 807-818, (1957). ETTLES, C.M.McC. 'Size effects in Tilting pad thrust bearings', Wear, 2 , 231-245 (1980).
105
Paper IV(ii)
Parametric study and optimization of starved thrust bearings Hooshang Heshmat, Antonio Artiles and Oscar Pinkus
A parametric study was conducted on the characteristics of thrust bearings with insufficient lubricant supply. An analysis was performed for tapered land bearings with a flat at the end of the pad equal to
20% of the pad arc, subject to the assumption that the incomplete fluid film starts along a radial line. Variation in bearing performance was obtained by changing the number of pads, the (R2 - Rl)/R2 ratio, and the degree of starvation. The parametric study yielded an optimized bearing geometry under starved conditions, namely thrust bearings with 12 pads (arcs of 27’) and (R2 - R1)/R2 = 1/2. The performance of this optimized configuration is tabulated herein for the full range of loadings and levels of starvation. Results of the parametric study are compared with experimental data showing excellent agreement between the theory and the test results.
1 INTRODUCTION In several previous papers, the subject of starved bearings has been ivestigated both from theoretical and experimental viewpoints [l, 2, 3 , 4]*. A starved thrust bearing is one where a lubricant is supplied at a rate below the expected side leakage, Q s ~ ,that is required for a full hydrodynamic film. A s a consequence, the fluid film in a starved bearing begins downstream of the physical start of the bearing pads, yielding smaller film thicknesses and higher maximum temperatures. In addition to its numerous conventional parameters, the performance of a starved thrust bearing depends on the degree of starvation. Since lubricant supply has to be equal to the side leakage, we can define an index of starvation as:
Flat at RI+
Qs = (Qs/Qs~). Thus, in terms of dimensionless quantities, a starved thrust bearing would depend on L/R2, B , b, 6r, E2, E, and &, where E makes its appearance when thermal effects are included. This paper is concerned with a study of the effects of some of the above parameters on starved bearings. Three values of L/R2 and B and four values of E2 are considered, with the starvation index ranging from 1.0 (full film) to a value as low as 0.2. From these values, an optimized geometry is obtained for severely starved bearings from the standpoint of highest load capacity. All calculations are for a tapered land bearing with zero radial taper and a flat at the end of the pad equal to 20% of the pad arc (b = 0.81, as shown in Figure la. The analytical results are compared with some experimental results obtained on a thrust bearing with a starvation index between 0.125 and 0.25.
a) Pad with Circvmferentlal Taper Only
b) Pad with Clrcumferential and Radial Tapers 861498
“Numbers in brackets indicate references, which can be found at the end o f the paper.
MTI P283
Fig. 1 Geometry of Thrust Bearing Pads
106 1.1
No ta t ion
A b B
Bearing area Fraction of tapered portion of bearing Circumferential extent of pad at midspan, B(R1 + R2)/2 C Specific heat E Thermal constant, 2poUW/cw (R2/h2l2 h Film thickness Film Thickness at start of pad h0 Film thickness at start of film hl Minimum film thickness _h2 (h2/6) h2 L Radial extent of pad, (R2 - R1) M Torque M l16M/powL2R22 N Shaft rotational frequency, w/2n Pressure P Volumetric flow normal to pad boundary 9 per unit length Volumetric oil flow Q Oil flow at inlet to bearing film Qi Side leakage Qs Side leakage for full film 9sF Q/llNR2L6 Q Index of starvation, (Qs/Qs~) BS r Radial coordinate S Coordinate along pad boundary Inner radius of bearing R1 Outer radius of bearing R2 T Temperature , T (T T1)/T1 Tmax Maximum temperature Average side leakage temperature Ta W Specific weight of lubricant Load WT WT 6 2 / $ ~ pN L ~ ) WT wT* 2 WT 6 /(tL1~R24) a Temperature-viscosity coefficient Bearing arc B Fluid film arc (8 - el) 6
-
BS
6
er
e P w
(B
If we assume that Q1 (the limited quantity of admitted lubricant) is distributed uniformly along the leading groove, then by ignoring the effects of the pressure gradients at 8 we can derive the locus of the starting line. Afnce the amount of available lubricant per unit width is given by (Ql/L), this amount will fill the clearance space wherever it satisfies the Couette flow requiremen t : (2) By solving for 8, we obtain the locus of the start of the fluid film: el(r) = bB(1 + ii2 - QlR2/r) ; R i Ir 5 R2
Depending on the magnitudes of E2 and q1, Equation 3 yields a family of concave lines for el, as shown in Figure 2a. However, if a radial taper exists wherein h varies inversely with r, Equation 3 would yield a constant value for 8 1' The above is all valid if we assume a uniform distribution of Q1 with r. If the lubricant is admitted at the inner periphery of the bearing at R1, the lower portion of the pad may receive the bulk of Q1 at the expense of the upper portions. As for journal bearings, if insufficient lubricant is fed via a central hole in the groove, the shape of the start of the film in thrust bearings may resemble that shown in Figure 2b [5, 6 1 . Since the exact shape of the starting line is a function of bearing shape, the manner of lubricant delivery, and the oil groove geometry, it is not considered here and the fluid film is assumed to start at a constant circumferential position.
Wr
- e )/B
CircumJerential taper, (ho - h2) Radial taper (h01 - h02) Coordinate in direction of rotation Start of fluid film (816) Viscosity of lubricant Angular velocity of shaft
Subscripts a) Unlforrn Lubricant Dlstrlbutlon: h
1 2 s
2
(3)
Inlet to the fluid film Exit from the fluid film Side leakage
----
b) Central Feedlng Hole: h
(1)
-.8 85828
Fig. 2 e/b6); 0 5 8 5 b6
Starting Line
w-
One of the ambiguous aspects of starved thrust bearings is the locus of the start of the fluid film. In journal bearings, the film starts at a constant circumferential position, which was observed experimentally via a transparent bearing, as discussed in Reference 2. Given the uniform conditions along the axial coordinate in a journal bearing, one may expect this. Generally, in a thrust bearing neither the film shape nor the linear velocities are constant along r, and one may expect a nonlinear shape for the starting line. For example, if we consider a bearing with a circumferential taper only, the film thickness in the tapered region is given by:
-
8
Straight Line Approximation
START OF THE FLUID FILM
h = h2 + 6 ( 1
-
Possible Shapes of the Start of the Fluid Film
107
described in Reference 6 , the exact shape of the starting line has no great impact on the load capacity, which is our main focus here. In effect, by assuming 8 independent of r, the film i s made to begin at a+ocation that is the average of whatever the actual starting line is, because the total amount of fluid at the start of the film must be the same in all cases, regardless of the shape of e l ( r ) . As
3
EFFECT OF DESIGN PARAMETERS
The parametric study was conducted for tapered land bearings of the shape shown in Figure la, Three L/R2 ratios and three pad with b = 0 .8 . angles were evaluated as follows:
0
02
06 O4
L/R2 = 113, 112, 2 1 3 0 = 2 l 0 , 42', 51'.
10 85831 1
Fig. 4 Effect of Starvation on Load Capacity: ( L / R 2 ) = 113
With the oil grooves extending over an arc of 3 ' each, the above angles correspond to thrust bearings consisting of 1 2 , 8, or 6 pads, respectively. All comparisons were made with the parameter, 6, kept constant, or, in dimensionless form, the same K2. This corresponds to having similar slopes for the different L/R2 ratios, and different slopes for the different 6 values, as shown in Figure 3.
a
08
n
0,
I
t
h2
85832
Fig. 5 85829
Fig. 3
Effect of Starvation on Load Capacity: ( L / R ~ )= 112
Geometry of Bearing Pads for Various Values of 6
If constant slopes t o r the various values of 6 were maintained, then the values of 62 could not have remained the same, and, as is known from the thrust bearing theory, 62 is the more crucial parameter in determining bearing performance. Moreover, the performance parameters such as i ~ 01, and & 2 , could not (in the case of a variable 6) have served as a valid index of performance, since they are all normalized with respect to 6. The data were obtained by a proper solution of the Reynolds equation in finite difference form, as described in Reference 3, with the thermal effects accounted for via the Couette approxiBy ignoring the effect of the mation [ 7 ] . pressure gradients on temperature variation, the latter permits a decoupling of the energy equation from the Reynolds equation. Some of the major parametric effects on bearing performance are as follows: Effect of Gs. Figures 4, 5, and 6 show the load capacity as a function of the index of starvation, Q s , for L/R2 = 1 1 3 , 112, and 213, respectively. The load capacity decreases with increased starvation, a 50% drop being reached when 0.25 5 Q s 2 0 . 5 (see Table 1). The higher the L/R2 ratio and the larger the 62, the more sensitive the load capacity is to the starvation
,
6, Fig. 6
85840
Effect of Starvation on Load Capacity: (L/R2) = 2/3
as:
index, A rapid decline in WT occurs in all cases as Qs approaches 0 . 2 . This is due to the fact that, at this high level of starvation, the film is forming near the flat portion of the pad. The above qualitative description is quantified in Table 1, where it is shown that the higher the L / R 2 ratio and the more lightly loaded the bearing, the more pronounced is the effect of 0,.
108 Table 1 Effect of Starvation on Load Capacity
Value of ij, h2
Crossover Occurs from Low to High 6
0.125
---
(L/R2)
1/3
0.25 0.69 1 .o 1/2
0.69 1.0 2/3
0.125 0.25 0.69 1 .o
Drops By 50%
--
0.25 0.30 0.35 0.35
0.25 0.30 0.45 0.52
0.25 0.30 0.45 0.50
0.40 0.45 0.75
0.35 0.35 0.45 0.55
0.15
0.125 0.25
WT
1 .o
0,
Fig. 7
05839
Effect of Starvation on Bulk Temperature at Constant Load
1.2
1 .o
0.8
Effect of 6. Since UT is synonymous with unit load capacity, it is also a measure of total bearing load capacity regardless of the value of 6. For all L/R2 ratios, 6 = 2 7 ' ( 1 2 pads) seems to be the kest geometry. It is only at very low values of Qs that the higher 8 becomes superior. The-crossover point (when 6 = 57' yields a higher WT than 6 = 2 7 ' ) depends on both the L/R2 ratio and ii2. The higher these two parameters are, the-earlier the crossover. Thus, at L/R2 = 2 / 3 and h2 = l , A 8 = 5 7 ' is preferable almostfrom the start of Qs = 1. At L/R2 = 113 and h2 = 0 . 1 ? 5 , the 6 = 27' preference is maintained down to Qs = 0.1.
I;"
0.6
0.4
0.2
"
01
Qs
=
=$
$
q(T
-
0.6
0.4
1.o
0.8
85 Fig. 8
Temperature Effects. Figures 7, 8, and 9 show the nvariations in temperature rise as a function of Qs. The average side leakage temperature rise, Ta - Ti, is calculated by summing the local energy convected by the local side leakage and dividing by the side leakage:
AT,
0.2
85848
Effect of Starvation on Bulk Temperature at Constant hp
1.2 1.6
1.o
Tl)ds/Qs
1.4
ds
where the integration over s is performed over the inner and outer radii (r = R1 and R2) of the pad. For the practical case of constant UT, ATa rises with increased starvation. However, in terms of constant film thickness, AT, first rises wit! increased starvation, then approaches zero as Qs + 0. Likewise, the maximum temperature rises with increased starvation. Figure 10 illustrates the interdependence of the various relevant quantities. As seen for a constant WT, the side leakage, qs, drops faster than the drop in torque, H, yielding a rise in bulk and maximum temperatures with an increase in starvation. The minimum film thickness naturally drops with a drop in Qs. Thus, the two usual criteria of hydrodynamic performance, h2 and Tmax,also remain the two operative limitations with starved bearings, with Tmax being the more sensitive parameter with increased levels of starvation.
1.2
0.8
2
1.2
I@
2
I -4
4
15' 0.6
1.o
Id 0.4
0.8 0.2
0.6 0
0
0.2
0.4
6, Fig. 9
Effect of Starvation on Maximum Temperature at Constant Load
109 P
2.2
=
Constant
1 .B
m I+!
a
1.4
1.o
I
nfil
0
I
I
I
I
I
1
-1-
I
I
I
I
I
I
I
I 0.6
I
I
I
0.2
0.4
0.8
8, Fig. 10
1 .o 85838
Interaction of Various Parameters as a Function of Degree of Starvation
Extent of Oil Film.-When the fractional extent of the fluid film, 6, = BS/0, is plotted as a function of the various parameters involved, it retains its value independent of either L/R2 or 6. Moreover, there is only a small variation with changes in i~ (or its equivalent, a change in h2). The variation of 6, with an eightfold variation in fi2 is shown in Figure 11. Consequently, the extent of the fluid film,ABs, is a function of the index of starvation, Q s , only. Using a single curve inserted into the band of solutions of Figure 11 and extended to the point 6, = 0 . 2 at Q s = 0 ; one can write a simple equation for 6, in the form of:
85827
Fig. 12
Shape of Starved Fluid Film at Two Different Values of L
and for the second pad at a film thickness given by : 4 Q,
h2a
-
w [R',
-
(R2
- L b)']
Neglecting the difference between the contributions due to the pressure gradients at the start of the film, it follows that since Lb > La, we must have:
0
Fig. 11
0.2
0.4
0.6
0.8
1 .o
The Extent of 8, as a Function o f Starvation Index
4 OPTIMUM GEOMETRY In full bearings, the larger the radial extent of the pad, L, the higher its load capacity. However, in starved bearings, this is no longer true. The situation is schematically portrayed in Figure 12, where a pad of a given 6 and R 2 i s given two different widths, La and Lb. For a restricted quantity of available lubricant Q1 5 Q ~ F , the film will start for the first pad at a film thickness given by:
and the film in the pad with the larger width, Lb, starts later than that with the smaller width, La. For the two cases, we then obtain two different film shapes in which the first is long in the e direction but narrow radially and the second has a short arc but is wide radially. Now, since the ability of a hydrodynamic film to develop pressures depends on both the arc length and its width, an optimum film configuration in terms of the best L/R2 ratio should result. In order to plot the load versus L/R2 ratio, we must first make
independent of the L dimension. To accomplish this, we define a new dimensionless load capacity: WT;~= 2 WT 62 /(Bp1NR2 4 which is now independent of L and, when plotted against L/R2, represents the true variation in total load capacity with a change in L/R2. Figures 13 aFd 14 show the parametric variation of WT* and Qs for optimizingthe L/R2 ratio. Figure 13 shows the variation of h2 for different levels of loading at a constant Qs, while Figure 14 shows the variation of h2 for different levels of starvation at a constant lo_ad. As shown, the highest minimum film thickness h2 is obtained for an L/R2 = 1/2 in all cases. There is a slight
110
.,0.2
0.6 WR2
0.8 05031
85835
tendency for 6 = 27' to lean toward lower ratios and for 6 = 57O to lean toward higher ones, but the deviations are less than 5 % and L/R2 = 1 / 2 is a clear optimum. For the three 6 ' s considered here, namely, 27', 42O, and 57', L/R2 = 112 yields B/L ratios of 0 . 7 , 1.1, and 1.5, respectively. Table 2, together with some of the previously plotted relationships, provides a complete set of computer-obtained data for the entire spectrum of loads and levels of starvation of the optimized (L/R2 = 1/21 thrust bearing.
Fig. 14
Tests were performed on a tapered land thrust bearing with dimensions of R 1 = 74.9 mm ( 2 . 9 5 in.), R2 = 119 mm ( 4 . 7 in;), 6 = 0.0381 mm (1.5 x in.), and B = 2 7 with a 20% flat at the end. The bearing was instrumented with proper flowmeters, thermocouples, and distance probes to measure film thickness. The tests were conducted under severe starvation conditions that gave a values from 0.125 to 0 . 2 5 . The test range of results are compared with the theoretical values in Figure 15. As shown, there is excellent agreement between theory and experiment, both qualitatively and quantitatively. Where the test values of RT are plotted versus h 2 , they fall precisely between the theoretical curves that delimit the constant starvation lines of 0.125 and 0 . 2 5 . The experimental values of B,, which are of the order of 10 to 14O, fall consistently within the corresponding theoretical envelope and do not vary with ii2.
Optimum Geometry for Different Levels of Starvation
This confirms the theoretical r'esults plotted in Figure 11. The reason why BS does not vary with the load can be seen from Figure 16, which shows that: The end leakage, 0 2 , is independent of the_level of starvation and is proportional to h2, as shown on Table 3 . The side leakage for full-film lubrication is nearly constant at all values of ii2 (or UT) and, therefore, i t is an absolute indicator of the level of starvation in starved bearings
COMPARISON WITH EXPERIMENTS
as
0.6
0.8
Fig. 13 Optimum Geometry for Different Levels of starvation
5
0.4
0.2 0.4
This excellent agreement with test results seems to justify the constant circumferential position for the start of the film assumed in the analysis. This may be because the lubricant distributes itself along the groove in such a way as to provide each radial segment with a proportionate amount of lubricant. This, together with the presence of a radial taper, would tend to bend the curve of the starting film more closely to the radial line postulated in the analysis. 6
ACKNOWLEDGMENT
The authors wish to express their appreciation t o MTI for its support in the presentation of this paper. Acknowledgment i s also due to Reliance Electric Company, whose facilities were used to conduct the experiment described herein, and to Mr. Paul Gorski for his support and interest in this project.
111
Table 2 Performance of Starved Thrust Bearings (L/R2) = 1/2
B 27'
42
57O
-
-
Qi
QS
QS
ATa
n
7.807 7.637 6.373 3.966
0.125 0.125 0.125 0.125
1.00 0.96 0.75 0.50
0.43 0.40 0.30 0.20
0.29 0.27 0.17 0.08
1.00 0.92 0.58 0.27
0.61 0.62 0.68 0.72
1.69 1.68 1.62 1.59
1.07 1.05 0.91 0.69
4.325 4.104 3.156 1.769 0.0
0.250 0.250 0.250 0.250 0.250
1.00 0.94 0.72 0.47 0.20
0.54 0.50 0.40 0.30 0.20
0.30 0.26 0.16 0.07 0.01
1 .oo 0.87 0.53 0.24 0.04
0.45 0.46 0.49 0.48 0.11
1.17 1.16 1.11 1.05 0.92
0.83 0.80 0.67 0.49 0.21
1.116 0.825 0.502 0.173
0.690 0.690 0.690 0.690
1.00 0.81 0.60 0.36
0.91 0.80 0.70 0.60
0.30 0.20 0.11 0.04
1 .oo 0.66 0.37 0.14
0.21 0.21 0.20 0.15
0.53 0.50 0.45 0.37
0.52 0.45 0.36 0.25
0.571 0.333 0.032
1.000 1.000 1 .ooo
1 .oo 0.72 0.28
1.15 1 .oo 0.80
0.31 0.16 0.03
1.00 0.53 0.08
0.14 0.13 0.07
0.36 0.31 0.19
0.42 0.33 0.16
7.113 6.343 5.284 3.392
0.125 0.125 0.125 0.125
1 .oo 0.79 0.62 0.42
0.55 0.40 0.30 0.20
0.42 0.27 0.18 0.08
1.00 0.65 0.42 0.20
0.75 0.82 0.86 0.88
1.89 1.84 1.81 1.77
1.17 1.04 0.91 0.71
3.925 3.344 2.668 1.572 0.0
0.250 0.250 0.250 0.250 0.250
1.00 0.78 0.61 0.41 0.20
0.65 0.50 0.40 0.30 0.20
0.42 0.27 0.17 0.08 0.01
1.00 0.63 0.40 0.18 0.03
0.57 0.61 0.63 0.61 0.16
1.36 1.31 1.27 1.21 1.10
0.96 0.83 0.71 0.54 0.26
1.062 0.704 0.459 0.179
0.690 0.690 0.690 0.690
1.00 0.70 0.52 0.33
1.01 0.80 0.70 0.60
0.43 0.22 0.13 0.05
1.00 0.51 0.30 0.12
0.29 0.29 0.27 0.20
0.68 0.62 0.56 0.47
0.68 0.53 0.44 0.32
0.556 0.298 0.037
1.000 1 .ooo 1 .ooo
1.00 0.64 0.26
1.25 1.00 0.30
0.43 0.18 0.03
1.00 0.43 0.07
0.20 0.19 0.10
0.48 0.30 0.26
0.57 0.42 0.22
5.862 5.126 4.378 2.895
0.125 0.125 0.125 0.125
1 .oo 0.70 0.55 0.38
0.63 0.40 0.30 0.20
0.51 0.28 0.18 0.09
1 .oo 0.36 0.17
0.86 0.95 0.98 1.. 00
2.03 1.97 1.93 1.91
1.19 1.01 0.90 0.70
3.256 2.714 2.234 1.397 0.0
0.250 0.250 0.250 0.250 0.250
1 .oo 0.70 0.54 0.37 0.20
0.73 0.50 0.40 0.30 0.20
0.51 0.28 0.18 0.08 0.01
1 .oo 0.54 0.35 0.16 0.02
0.67 0.72 0.73 0.70 0.21
1.49 1.42 1.38 1.33 1.23
1.02 0.84 0.73 0.58 0.29
0.903 0.594 0.408 0.174
0.690 0.690 0.690 0.690
1 .oo 0.64 0.48 0.31
1.07 0.80 0.70 0.60
0.51 0.23 0.14 0.06
1.00 0.46 0.27 0.11
0.36 0.35 0.32 0.25
0.80 0.71 0.65 0.55
0.79 0.60 0.50 0.38
0.482 0.348 0.260
1 .ooo 1 .ooo 1 .ooo
1 .oo 0.73 0.58
1.31 1.10 1.00
0.51 0.30 0.20
1.00 0.58 0.39
0.26 0.25 0.23
0.58 0.52 0.47
0.70 0.57 0.49
0.55
112
Table 3 Values of 4 2 as Function of 6 2 and 6s
6, =
m
-h2
U
4"
&
0.25
62
= 1.0
Ratio
02
Ratio
1%
1.0
0.870
1 .o
0.882
1.0
0.4
0.355
0.41
0.364
0.41
0.2
0.175
0.20
0.182
0.21
0.1
0.087
0.1
0.089
0.10
2 0 0.1
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
h2
Fig. 15
842312
Comparison of Experimental Results with Theory 6 = 27'; L/R2 = 0.37; b = 0.8 References 1.
Artiles, A. and Heshmat, H. "Analysis of Starved Journal Bearings Including Temperature and Cavitation Effects." Journal of Tribology, Trans. ASME, Vol. 107, No. 1, January 1985.
2.
Heshmat, H. and Pinkus, 0 . "Performacne of Starved Journal Bearings with Oil Ring Lubrication." Journal of Tribology, Vol. 107, No. 1, Trans. ASME, January 1985.
3.
Artiles, A. and Heshmat, H. "Analysis of Starved Thrust Bearings Including Temperature Effects." Submitted for the ASME/ASLE Tribology Conference, October 1985.
4. Heshmat, H. and Pinkus, 0. "Performance of Starved Thrust Bearings." International Scientific Conference on Friction, Wear, and Lubricants, Tashkent, USSR, May 1985. 5.
6. Etsion, I. and Finite Journal Films." Journal Trans. ASME; Vol.
h2
85834
7. Fig. 16
Etsion, I. and Pinkus, 0 . "Analysis of Short Journal Bearings with New Upstream Conditions." Journal of Lubrication Technology, Trans. ASME, Vol. 96, No. 3 , July 1974.
Flow Components in a Starved Bearing
Pinkus, 0 . "Solutions of Bearings with Incomplete of Lubrication Technology, 97, No. 1, January 1975.
"Adiabatic Pinkus, 0. and Bupara, S . Solutions for Finite Journal Bearings." Journal of Lubrication Technology, Trans. ASME, Vol. 101, No. 4, October 1979.
113
Paper IV(iii)
Tilting pad thrust bearing tests - Influence of three design variables W.W. Gardner
The purpose of this paper is to report the results of an extensive series of tests on a special tilting pad thrust bearing. The primary interest was to determine the effects on bearing performance of using circumferential pivot locations outside the normal range of 50 to 60% of the pad arc, from the leading edge. Secondary results were also obtained from tests with pad SUPport disks of various diameters, and with pads of different thicknesses. 1 INTRODUCTION
The test work reported here was prompted by results of design analyses of large tilting pad thrust bearings for vertical shaft hydro-electric turbine-generator applications. Although the primary purpose of this paper is to present results from laboratory tests on a special tilting pad thrust bearing, some background with respect to the analytical work leading to these tests is in order.
bearings by the company with which the author is associated and by others, and has been the subject of previous publications, including ( l ) , (2) and ( 3 ) . The closed form solutions used in the design program for the pad distortions resulting from pressure and thermal loads are similar to For this specific those developed in (1). construction (Fig. 1) and witn paa aspect ratios of approximately 1 . 0 , this method of calculating pad distortions (and ultimately film shape) appears reasonable.
The program used for this design work incorporates finite difference forms of the Reynolds and energy equations for the oil film, plus closed-form equations for the pressure and thermal distortions of the pad. These are all iteratively related, including factors to compensate for the carryover of heat from pad to pad. This program was first written in the mid 60's and has been revised numerous times since then to incorporate advances in bearing analysis and to include empirical correction factors from test results.
The influence of the carryover of heat from pad to pad has been found to be significant in obtaining analytical solutions representative of actual bearing operation. The method used in this design program to account for heat carryover is based on the work of Ettles in ( 4 ) with modifications indicated by test data.
The specific pad and pad support configuration of interest here is shown in Figure 1. The back face of the thrust pad is simply a flat plane finished by grinding. This surface is then supported by a disk of high strength steel with a spherical pivoting surface on one face to transmit the thrust load to the machine base, and a raised annular ring (also finished flat by grinding) at the outer edge of the opposite face in contact with the flat back face of the Dad. This construction has been used for large thrust
Through the use of this design program, it is convenient to study the effects on bearing performance of pad and/or support geometry variations. For machines that rotate primarily in one direction; circumferentially offset p-ivots are often used t o improve bearing performance. Historical theoretical analysis ((5), for example) indicated that a pivot placed circumferentially about 60% of the pad arc length from the leading edge provided maximum
Numerous large tilting pad thrust bearings have been designed with the aid o f this program and have been successfully applied in the field.
I
Section "A-A"
Fig. 1 Thrust pad and support configuration.
114
1I 1 0
I
d/b= d/b= d/b= d/b=
36 45 54 63
v
1 d/b= 36 2 d/b=45 3 d/b= 54 4 dlb= 63
I 50
Fig. 2
0
I 1 2 3 4
0
Z
C a l c u l a t e d minimum o i l f i l m t h i c k n e s s vs. c i r c u m f e r e n t i a l p i v o t p o s i t i o n .
60 70 Pivot Position, %of Pad Arc
I0
Fig. 3
C a l c u l a t e d maximum .pad t e m p e r a t u r e vs. circumferential pivot position.
Fig. 4
C a l c u l a t e d maximum o i l f i l m p r e s s u r e vs. c i r c u m f e r e n t i a1 p i v o t p o s i t i o n .
l o a d c a p a c i t y , and t h a t , i n f a c t , a p i v o t a t t h e c e n t e r (50% l o c a t i o n ) r e s u l t e d i n z e r o l o a d capacity. This s i t u a t i o n i s t r u e f o r p e r f e c t l y f l a t pads and c o n s t a n t v i s c o s i t y f l u i d s , b u t i s n o t t r u e i n g e n e r a l s i n c e t h e r e a r e numberless machines o p e r a t i n g s a t i s f a c t o r i l y w i t h c e n t e r p i v o t e d t i l t i n g pad t h r u s t b e a r i n g s . The a b i l i t y o f such b e a r i n g s t o , i n f a c t , provide s i g n i f i c a n t load capacity i s explained when pad d e f o r m a t i o n s and v i s c o s i t y v a r i a t i o n s are included i n t h e analysis. This i s w e l l accepted today, and t h e s e f a c t o r s a r e i n c l u d e d i n d e s i g n a n a l y s i s programs, ( 6 ) and ( 7 ) f o r examp l e , f o r t i l t i n g pad t h r u s t b e a r i n g s . Early r e c o g n i t i o n o f t h e i n f l u e n c e o f pad d e f o r m a t i o n s on t h e l o a d c a p a c i t y o f c e n t e r p i v o t pads was p r o v i d e d by Raimondi ( 8 ) . I n t h e d e s i g n a n a l y s i s work which prompted t h e t e s t work r e p o r t e d here, t h e e f f e c t o f c i r c u m f e r e n t i a l p i v o t p o s i t i o n on b e a r i n g p e r f o r mance was s t u d i e d . T h i s r e s u l t e d i n d a t a as g i v e n i n F i g u r e s 2 , 3 and 4 . A second i t e m o f i n t e r e s t was t h e e f f e c t o f s u p p o r t d i s k diamet e r , and t h i s i s a l s o r e f l e c t e d i n t h e s e f i g ures. Pad t h i c k n e s s i s t h e t h i r d d e s i g n v a r i a b l e s t u d i e d , and F i g . 5 g i v e s r e p r e s e n t a t i v e c a l c u l a t e d temperature d a t a f r o m t h e d e s i g n work. I n these p l o t s o f c a l c u l a t e d bearing p e r formance, as w i t h t h e l a t e r p l o t s o f measured b e a r i n g performance, d i m e n s i o n l e s s r a t i o s (d/b and t / b ) a r e used f o r convenience. T h i s i s n o t t o imply t h a t the data presented i s v a l i d f o r a l l d e s i g n s where such r a t i o s a r e o b t a i n e d ,
115
The c a l c u l a t e d d a t a i n F i g u r e s 2, 3 and 4 i n d i c a t e improved performance f o r p i v o t s beyond 60% f o r a l l t h r e e i t e m s ; f i l m t h i c k n e s s , temp e r a t u r e , and p r e s s u r e . I n the tests reported here, pad t e m p e r a t u r e was t h e o n l y i t e m o f t h e s e t h r e e performance c h a r a c t e r i s t i c s t h a t was measured, and i t i s t h u s used as t h e b a s i s f o r c o n c l u s i o n s r e g a r d i n g performance. For b a b b i t t e d b e a r i n g s , a c o m b i n a t i o n o f l o c a l temperat u r e s and p r e s s u r e s may a l s o be o f v a l u e i n judging bearing capabilities due to the strength-temperature r e l a t i o n s h i p o f babbitt. 2
TEST BEARING AND SETUP
A s p e c i a l 381 mm (15 i n c h ) d i a m e t e r t i l t i n g pad t h r u s t b e a r i n g was designed and manufactured f o r these t e s t s . I t i s shown i n F i g u r e s 6 and 7.
t/b
Fig. 5
C a l c u l a t e d maximum pad t e m p e r a t u r e (62% p i v o t ) v s . pad t h i c k n e s s / p a d r a d i a l length, t/b.
r e g a r d l e s s o f b e a r i n g s i z e , geometry o r o p e r a t ing conditions. For the l a r g e t h r u s t bearing ( c a l c u l a t e d data presented i n Figures 2 through 5), t h e b a s i c dimension, b, i s 561 mm. For t h e t e s t The b e a r i n g , t h i s same dimension i s 95 mm. t e s t s were conducted t o d e t e r m i n e v a r i a t i o n s i n performance due t o changes i n t h e t h r e e d e s i g n v a r i a b l e s , and t h e r e was no a t t e m p t o r i n t e n t t o model t h e f i l m c o n d i t i o n s c a l c u l a t e d f o r t h e 1arge b e a r i n g .
instrumented
Fig* 6
T e s t b e a r i n g with
Fig. 7
T e s t b e a r i n g components.
One
O f p r i m a r y i n t e r e s t was t h e a n a l y t i c a l i n d i c a t i o n t h a t b e a r i n g performance c o u l d be improved (reduced f i l m temperatures and p r e s sures, and i n c r e a s e d f i l m t h i c k n e s s e s ) by moving t h e p i v o t beyond t h e 60% p o s i t i o n t o somewhere i n t h e area o f 70 t o 80%.
Because t h e use o f a p i v o t i n t h i s area (75%) d e v i a t e s f r o m e x p e r i e n c e ( a t l e a s t t h a t o f t h e a u t h o r ) and because t h e l i t e r a t u r e o f f e r e d l i m i t e d d a t a , i t was concluded t h a t l a b o r a t o r y t e s t s s p e c i f i c a l l y d i r e c t e d a t t h i s p o i n t were needed t o p r o v i d e a d d i t i o n a l guidance. Reference ( 9 ) i n c l u d e s t e s t d a t a on e l a s tomer f a c e d t h r u s t pads o v e r a range o f c i r c u m f e r e n t i a l p i v o t p o s i t i o n s f r o m 55 t o 85%. These t e s t s showed an i n c r e a s e i n f i l m t h i c k n e s s as t h e p i v o t p o s i t i o n was i n c r e a s e d o v e r t h i s range, and a l s o an i n c r e a s e i n t h e peak f i l m pressure. The t h r u s t pads o f i n t e r e s t i n t h e work r e p o r t e d here a r e s i g n i f i c a n t l y d i f f e r e n t i n b o t h m a t e r i a l s and s u p p o r t geometry f r o m those i n ( 9 ) .
pad'
116
It is a two pad bearing in which the pad and support construction described earlier, and shown in Fig. 1, is used. The two pad configuration was chosen to allow the arrangement shown for testing various support disk diameters. The disk retainer plate could be rotated within a circumferential slot in the base ring. This allowed the insertion of the disks to be tested into their corresponding holes in the retainer plate, which was then rotated to place these disks into position for support of the pads. In addition, a series of thirteen notches associated with each pair of support disk holes was used to position the support disk at the desired circumferential pivot position. A key was used to align and maintain this relationship between the retainer plate and the base ring. The thirteen notch positions correspond to pivot positions from 0.20 to 0 . 8 0 in 0.05 steps. Seven support disk diameters were used, varying in 9.5 mm (0.375 inch) steps from 19.1 mm (0.75 inch) to 76.2 mm (3.00 inch). This provided d/b ratios of 0.2 to 0.8 in 0.1 steps. Pads of three thicknesses were made for test, 25.4 mm ( 1 . 0 0 inch), 19.1 mm (0.75 inch), and 12.7 mm (0.50 inch). In terms of ratios to the pad radial length (or arc length at mid-radius since the pad aspect ratio was 1.0) the values are 0.27, 0.20, and 0.13. The base ring of the test bearing was pivoted on its back face at 90 degrees to the pad locations. This provided for equal loading of the two pads. Copper-constantan thermocouples were embedded in the babbitt facing in all six of the pads tested, at the following locations: 85-85 75-75 75-25
65-65 50-85
50-65
The seven support disk diameters, the three pad thicknesses, and the various pivot location possibilities translated into a large number of possible geometry combinations. In order to keep the test program within manageable proportions, several limits were initially set. These were : a) Tests would be made only at pivot locations at which the support disk was fully within the pad trailing (or leading) edge. This condition limited the largest support disk to a range of pivots from 40% to 60%. Only the three smallest disks could encompass the full range from 20% to 80%. b) Tests would be run at speeds up to 4000 RPM. Higher speeds were avoided to prevent effects from non-laminar film conditions. This was consistent with the field applications of the particular bearings that prompted this work. c) Thrust bearing loadings would be limited to 4.14 MPa (600 psi). The goal was to collect data, not to fail bearings. d) In this same respect, a maximum pad temperature of 121C (250F) was set. e) IS0 VG32 turbine oil was used for all tests at an inlet temperature held between 48.6C (119.5F) and 49.2C (120.5F). f) A constant flow of 57 liters/min. (15 GPM) was used for all tests reported here. The test procedure was simply to set the load and speed conditions desired and maintain this for a minimum o f ten minutes. A reading was taken at that time if the oil supply temperature was within limits. If not, adjustments were made to bring it within limits, and data was then recorded. The test data collected consisted of the temperatures from the eight thermocouples embedded in each of the two pads plus the oil inlet and drain, plus the oil flow, bearing load, and shaft speed. 4
50-50
50-25
These thermocouples were located approximately 0.75 mm (0.03 inch) below the pad face in the 1.5 mm (0.06 inch) thick babbitt facing. The tests on this bearing were made in the thrust bearing test stand described in ( 1 0 ) . Briefly, this is a hydraulically loaded, D.C. motor driven facility capable of accommodating thrust bearings in the 250 mm (10 inch) to 500 mm (20 inch) range of outside diameters. The 750 kw (1000 HP) geared drive has a maximum test stand speed of about 10,000 RPM. The test and slave thrust bearings are enclosed in separate housings, and each thus operates in conjunction with a separate thrust collar. 3
range, also.
TEST CONDITIONS AND PROCEDURE
As noted previously, the area of primary interest was the influence of circumferential pivot position on bearing performance. The test bearing a1 lowed thirteen distinct 1 ocations for this variable. Although the locations of most interest were from 50% to 80%, the bearing design readily permitted tests in the 20% to 50%
TEST RESULTS
Even with the limits noted above imposed, a large amount of data resulted from these tests. The use of a data base program to sort and compare the information was invaluable. The data acquisition program averaged the two temperatures at a specific location on each of the two pads, and also recorded the difference. This difference was typically less than three degrees centigrade and often less than one degree. This gave confidence in the load equalization between pads and in the consistency of the thermocouple instal 1 ations. The temperature data used in all of the plots presented here is the average value of the two readings from corresponding locations on the two pads. The term "maximum pad temperature" is used, and it is, of course, the maximum recorded pad temperature. The thermocouple locations were chosen to cover the area where the highest pad temperatures have normally been found, both by analysis and test. However, only a limited number of distinct locations on the pad face can be sensed. In the majority of cases, the highest temperature was recorded at 85-85. However, depending on load, support disk diameter, and pivot location, the maximum temperature moved to
117
Pivot Position. ?6 of Pad Arc
Fig. 8
Measured maximum pad t e m p e r a t u r e p i v o t p o s i t i o n , 2000 RPM, 4.14 l o a d i n g , d = 3 8 . 1 mm.
vs. MPa
other l o c a t i o n s . There was even one t e s t run, w i t h t h e p i v o t a t t h e 20% l o c a t i o n , where t h e maximum temperature was r e c o r d e d a t 50-25. T e s t s w i t h p i v o t s i n t h e 20 t o 50% range were r u n p r i m a r i l y t o d e t e r m i n e what performance could be expected from an " o f f s e t " p i v o t b e a r i n g r u n n i n g backwards. Reference (11) p r o v i d e s a d d i t i o n a l r e c e n t d a t a i n t h i s area. F i g u r e s 8 t h r o u g h 16 p r o v i d e a r e p r e s e n t a t i v e view o f t h e t e s t r e s u l t s o b t a i n e d . 5 DISCUSSION
A two pad b e a r i n g was used f o r t h e s e t e s t s p r i m a r i l y because i t a l l o w e d t h e s p e c i a l d e s i g n arrangement f o r changing s u p p o r t d i s k s i z e s and pivot position. It i s recognized t h a t the performance o f a t h r u s t b e a r i n g w i t h l e s s t h a n a f u l l complement o f pads (which would be e i g h t here) i s n o t reduced i n p r o p o r t i o n t o t h e r e d u c t i o n i n t o t a l b e a r i n g area. However, f o r t h e purpose here o f i n v e s t i g a t i n g r e l a t i v e p e r f o r mance due t o o t h e r g e o m e t r i c a l changes, t h e two pad d e s i g n was deemed s a t i s f a c t o r y . A s i z e e f f e c t was a l s o r e c o g n i z e d (Ref. 3 ) , b u t t h e pad t h i c k n e s s v a l u e s chosen f o r t e s t were b e l i e v e d t o c o v e r a range s u f f i c i e n t t o r e s u l t i n measurable changes i n b e a r i n g p e r f o r mance. F i g u r e 8 shows t h e e f f e c t o f p i v o t p o s i t i o n on maximum pad temperature f o r t h e t h r e e pad thicknesses a t a s p e c i f i c o p e r a t i n g c o n d i t i o n . This p l o t c o v e r s t h e f u l l range o f p i v o t p o s i -
Fig. 9
Measured maximum pad t e m p e r a t u r e p i v o t p o s i t i o n , 3000 RPM, 4.14 l o a d i n g , t = 25.4 mm.
vs. MPa
tions. I t s h o u l d be n o t e d t h a t f o r p i v o t s n e a r t h e l e a d i n g edge (20%-30%), t h e h o t s p o t may w e l l be i n t h e l e a d i n g edge h a l f ' o f t h e pad. T h i s p o r t i o n o f t h e pad had a l i m i t e d number o f thermocouples so t h e p o s s i b i l i t y o f t h e maximum recorded temperature being representative o f the maximum b e a r i n g t e m p e r a t u r e i s somewhat l e s s t h a n f o r o t h e r p i v o t l o c a t i o n s (40% and h i g h e r ) . A c l e a r trend w i t h respect t o the influence o f p i v o t p o s i t i o n i s shown. T h i s i s n o t t h e case f o r t h e pad t h i c k n e s s . F i g u r e 9 shows t h e e f f e c t o f p i v o t p o s i t i o n on maximum pad temperature f o r t h r e e s u p p o r t d i s k diameters (d) a t a s p e c i f i c operating condition. Here, also, t h e e f f e c t o f p i v o t p o s i t i o n i s c l e a r , b u t t h e same i s n o t t r u e f o r support d i s k diameter. T h i s p l o t i s d a t a from t e s t s on t h e t h i c k e s t pad, however, where supp o r t d i s k d i a m e t e r would be expected t o have 1 e a s t in f l uence. F i g u r e 10, then, i s temperature d a t a p l o t t e d a g a i n s t pad t h i c k n e s s f o r t h r e e s u p p o r t d i s k diameters. For t h e t h i n pad, t h e advantage o f increasing t h e support d i s k diameter i s evident. I t became e v i d e n t d u r i n g t h e a n a l y s i s o f the t e s t data that the e f f e c t o f the pivot p o s i t i o n was c l e a r and c o n s i s t e n t . The l o w e s t temperatures were found i n t h e t e s t s w i t h t h e p i v o t a t 75%. Evidence w i t h r e s p e c t t o t h e e f f e c t o f s u p p o r t d i s k d i a m e t e r and pad t h i c k ness was n o t as c l e a r , b u t i t appeared t h a t some f o r m o f t h r e e dimensional p l o t m i g h t be h e l p f u l .
Contour p l o t s o f measured maximum tempera-
CONTOUR VALUES DEGREES C
120
1 - 80 2 - 84 3 - 88 4 - 92 5 - 96 6 - 100 7 - 104 8-108 9 -112 10- 116
110
v 100
f P
O
?!
Ea
j
90
80
50.00
60.00
70.00
80.00
70 0 Pivot Position.% of Pad Arc
F i g . 10 Measured maximum pad t e m p e r a t u r e vs. pad t h i c k n e s s , 3000 RPM, 4.14 MPa l o a d i n g , 70% p i v o t .
CONTOUR VALUES DEGREES C 1 - 80 2 - 84 3 - 88 4 - 92
F i g . 12 Measured maximum pad temperature, 3000 RPM, 2.76 MPa l o a d i n g , t = 12.7 mm.
t u r e v s . d/b r a t i o s and p i v o t p o s i t i o n a r e shown i n F i g u r e s 11 t h r o u g h 16. III a l l o f these, t h e i n f l u e n c e o f t h e p i v o t p o s i t i o n i s t o be seen, and t h e i n f l u e n c e o f s u p p o r t d i s k d i a m e t e r and i t s r e l a t i o n s h i p t o pad t h i c k n e s s i s more e v i dent. F i g u r e s 11 and 12 a r e f o r t h e t h i n pad, 12.7 mm t h i c k ; F i g u r e s 13 and 14 a r e f o r t h e medium t h i c k n e s s pad, 19.1 mm; and F i g u r e s 15 and 16 a r e f o r t h e t h i c k pads, 25.4 mm. Note t h a t t h e l a c k o f d a t a i n t h e upper, r i g h t hand p o r t i o n s o f t h e s e p l o t s i s due t o t h e f a c t t h a t t e s t s were l i m i t e d , as n o t e d e a r l i e r , t o combin a t i o n s o f s u p p o r t d i s k d i a m e t e r s and p i v o t l o c a t i o n s s u c h . t h a t t h e support d i s k did not e x t e n d beyond t h e t r a i l i n g edge o f t h e pad. I n F i g u r e s 11 and 12 ( t h e t h i n pad), a small d i a m e t e r s u p p o r t d i s k p r o v i d e s reduced pad temperatures f o r c e n t e r p i v o t pads, b u t a t t h e 75% p i v o t l o c a t i o n , a l a r g e d i a m e t e r support disk i s better. I n F i g u r e s 15 and 16 ( t h e t h i c k pad), t h e 28.6 mm d i a m e t e r s u p p o r t d i s k (d/b = 0.3) gave t h e l o w e s t temperatures f o r c e n t e r p i v o t pads, w h i l e an even s m a l l e r d i s k i s i n d i c a t e d f o r t h e 75% p i v o t p o s i t i o n .
50.00
60.00
70.00
80.00
Pivot Postllon. % of Pad Arc
F i g . 11 Measured maximum pad temperature, 2000 RPM, 4.14 MPa l o a d i n g , t = 12.7 mm.
F i g u r e s 13 and 14 a r e f o r t h e pads o f medium t h i c k n e s s , and t h e i n d i c a t i o n s a t t h e 75% p i v o t a r e somewhat between t h o s e f o r t h e o t h e r pads (as m i g h t be expected). Best r e s u l t s appear t o be w i t h d/b r a t i o s o f about 0.4 t o 0.5, a l t h o u g h t h e use o f t h e s m a l l e s t support d i s k gave temperatures almost as low.
119 CONTOUR VALUES DEGREES C
CONTOUR VALUES DEGREES C
1-
1-78 2.82 3 - 86 4.90 5 - 94 6.98
I
I
1
50 00
I
I
1
I
70 00
60 00
234567-
1
50.00
80 00
60.00
CONTOUR VALUES
I
50.00
I 6000
1
1 70.00
80.00
Fig. 15 Measured maximum pad temperature, 2000 RPM, 4.14 MPa loading, t = 25.4 mm.
Fig. 13 Measured maximum pad temperature, 2000 RPM, 4.14 MPa loading, t = 19.1 mm.
I
70.00
Pivot Position. % of Pad Arc
Pivot Position. %of Pad Arc
I
78 82 86 90 94 98 102
1
I 80.00
Pivot Position. %of Pad Arc
Fig. 14 Measured maximum pad temperature, 3000 RPM, 2.76 MPa loading, t = 19.1 mm.
It can be argued that the variations in temperature are small regardless of the disk diameter (d/b ratio) for any given pivot position. This is true, and apparentqy the result of too small a variation in the pad thicknesses tested. Trends are indicated, however, particularly for the thin pads (Figures 11 and 1 2 ) , that support theoretical results which require pad crowning for load capacity with center pivots (thus a small disk is better), but less crowning for offset pivots (thus a larger disk gives better results). Pad crowning is also believed to be the basis for the improved performance found with the pivot at the 75% location rather than 60%. Crowning, due to pressure and thermal loads, commonly results in a diverging film at the trailing edge. The extent of this divergence directly affects bearing performance. As the pivot is moved further downstream, this divergence decreases and is transferred to the leading edge as an increase in the convergence angle. The decrease in divergence on the trailing edge improves the bearing performance, while the increase on the leading edge somewhat reduces performance. The net result is, however, an increase in performance until the pivot is SO far downstream that very high local pressures (and reduced film thicknesses) have to be deve1 oped for pad bal ance.
1
6 CONCLUSIONS 1- For the bearings and conditions tested,
the lowest maximum recorded Dad temoeratures were consistently found with the pivot at 75%.
120
7
ACKNOWLEDGMENTS
The a u t h o r wishes t o express a p p r e c i a t i o n t o t h e l a b t e c h n i c i a n , John Grigg, f o r h i s p a t i e n c e i n r u n n i n g t h e many t e s t s i n v o l v e d i n t h i s p r o j e c t , and a l s o t o my w i f e , Joyce, f o r h e r a s s i s t a n c e i n t a b u l a t i n g the extensive data collected. References ETTLES, C . and CAMERON, A . 'Thermal and e l a s t i c distortions i n thrust-bearings', Proc. I M E , 1963, L u b r i c a t i o n and Wear Convention. 58-69.
ETTLES, C . and A D V A N I , S . 'The c o n t r o l o f thermal and e l a s t i c e f f e c t s i n t h r u s t b e a r i n g s ' , Proc. 6 t h Leeds-Lyon Symposium, 1979, 105-116. ' S i z e e f f e c t s i n t i l t i n g pad ETTLES C . t h r u s t b e a r i n g s ' , Wear, 59, 1, 1980, 231246.
I 50 00
I
I 60 00
ETTLES, C . bearings', I
I 70 00
I
1
1
80 00
Pivot Rosition. % 01 Pad Arc
F i g . 16 Measured maximum pad temperature, 3000 RPM, 2.76 MPa l o a d i n g , t = 25.4 mm.
2 - The t e s t b e a r i n g s o p e r a t e d w i t h p i v o t p o s i t i o n s l e s s t h a n 50%, t o as l o w as 20%. Decreased performance was found, as compared t o p i v o t s a t 50% o r g r e a t e r , b u t d e f i n i t e l o a d c a p a c i t y was p r e s e n t . 3 - The i n f l u e n c e o f s u p p o r t d i s k d i a m e t e r on b e a r i n g performance was most e v i d e n t w i t h t h e t h i n n e s t o f t h e t h r e e pad t h i c k n e s s e s t e s t e d , as a n t i c i p a t e d . The c o n t o u r p l o t s show t h e need f o r a small s u p p o r t d i s k d i a m e t e r f o r c e n t e r p i v o t pads and t h e advantage o f a l a r g e r d i s k f o r t h e 75% pivot location. These p l o t s ( t h i n pad) i n d i c a t e t h a t an optimum c o n d i t i o n e x i s t s w i t h t h e 75% p i v o t l o c a t i o n and t h e l a r g e s t support d i s k possible f o r t h a t l o c a t i o n (d/b = 0 . 5 ) . 4 - As t h e pad t h i c k n e s s i n c r e a s e s , t h e 75% p i v o t l o c a t i o n remains optimum ( a t l e a s t as determined by pad t e m p e r a t u r e ) , b u t t h e optimum s u p p o r t d i s k d i a m e t e r decreases. 5 - F u r t h e r s t u d i e s need t o be pursued i n two areas: a) A n a l y t i c a l e f f o r t s t o c o r r e l a t e these findings with the "size" effect o f l a r g e r bearings. b) A d d i t i o n a l t e s t work t o d e t e r m i n e i f t h e 75% p i v o t l o c a t i o n produces s i m i l a r r e s u l t s w i t h o t h e r pad support configurations.
'Hot o i l c a r r y - o v e r i n t h r u s t Proc. I M E , 184, 3L, 1970, 75-
0 1
01.
RAIMONDI, A. and BOYD, J. ' A p p l y i n g b e a r i n g t h e o r y t o t h e a n a l y s i s and d e s i g n o f p a d - t y p e b e a r i n g s ' , Trans. ASME, 77, 1955, 287-309. ETTLES, C . 'The development o f a g e n e r a l i z e d computer a n a l y s i s f o r s e c t o r shaped t i l t i n g pad t h r u s t b e a r i n g s ' , Trans. ASLE, 19, 2, 1976, 153-163. 'Prediction o f the operating VOHR, J. temperature o f t h r u s t b e a r i n g s ' , Trans. ASME, JOLT, 103, 1, 1981,'97-106. RAIMONDI, A. 'The i n f l u e n c e o f s u r f a c e p r o f i l e on t h e l o a d c a p a c i t y o f t h r u s t b e a r i n g s w i t h c e n t r a l l y p i v o t e d pads', Trans. ASME, 77, 1955, 321-330. R I G H T M I R E , G . , C A S T E L L I , V . and FULLER, D. 'An e x p e r i m e n t a l i n v e s t i g a t i o n o f a t i l t ing-pad, compliant surface t h r u s t beari n g ' , Trans. ASME, JOLT, 98, 1, 1976, 95110. GARDNER, W : 'Performance c h a r a c t e r i s t i c s o f two t i l t i n g pad t h r u s t b e a r i n g des i g n s ' , Proc. JSLE I n t e r n a t i o n a l T r i b o l o g y Conf., 1985, 61-66.
HORNER, D . , SIMMONS, J . and ADVANI, S . 'Measurements o f maximum temperature i n t i l t i n g pad t h r u s t b e a r i n g s ' , ASLE paper 86-AM-3A-1, 1986.
121
Paper IV(iv)
An experimental study of sector-pad thrust bearings and evaluation of their thermal characteristics T.G. Rajaswamy, T. Muralidhara Rao and B.S. Prabhu
Experiments were condlicted on a t h r u s t b e a r i n g o f 0.8 m o u t s i d e d i a m e t e r and 0.4 m i n s i d e d i a m e t e r w i th 3.25 ,'b Fa u n i t p r e s s u m s . E x p er i men t al r e s u l t s were o b t a i n e d f o r d i f f e r e n t l o a d and spe e d combinations and t h e p ar amet er s measured were p r e s s u r e d i s t r i b u t i o n , teffiperature d i s t r i b u t i o n and f i l m t h i c k n e s s a t l e a d i n u and trailiw e d g e s o f t h e b e a r i n g pad. The e x p e r i m e n t a l r e s u l t s were compared w i t h t h e analytical t e m p e r a t u r e s o b t a i n e d by s o l v i n g t h e e s t a b l i s h e d s t e a d y - s t a t e Reynolds e q u a t i o n for i n c omp r es - i b l e l u b r i c a n t a n d t h e e ne rgy e q u a t i o n . Heat balance was a l s o made between t h e h e a t g e n e r a t e d i n t h e l u b r i c a n t f i l m and t h e h e a t d i s , i p a t e d t o t h e s u r r o u n d i n g s of t h e t h r u s t b e a r i w pad- i n t h e b e a r i n p housing. 1
INTRODUCTION
I n t h e e n d e a v o u r of c o n t i n u e d r e s e a r c h i n t h e a r e a o f hydrodynamic f l u i d f i l m t h r u s t b e a r i n g s , it h a s become n r c e s s a r y t o s u b s t a n t i a t e t h e anal .yt ical c a7 c u l a t i o n s w i t h t h e c o rre sponding e x p c r i m e n t a l r e s u l t s i n o r d e r t o make improvements i n t h e Aesiqn o f s ect o r - p ad t h r u s t b ea rings. I n a d d i t i o n t o t h i s , t h e r e i s also a need on t h e p a r t o f t h e m a n u f a c t u r e r s o f e l e c t r i c a l r o t a t i n p machinery such as h y d r o g e n e r a t o r s t o prove t h e p r e d i c t e d v a l u e s of o p e r a t i n g temperatures of t h r u s t b e a r i n g s by e x p e r i m e n t a t i o n t o s a t i s f y t h e c u s t o m e r s of t h e s e p r o d u ct s . The a u t h o r s of t h e p r e s e n t p a p e r co n d u ct ed e x p e r i ments on s e c t o r - p a d t h r u s t b e a r i n g s w i t h d i f f e r e n t l o a d and speed co mb i n at i o n s t o accur a t e l y measure t h e o p e r a t i n g t e m p e r a t u r e o f t h e t h r u s t b e a r i n g s . These e x p e r i m e n t s were conduct e d on a T h r u - t Bearing T e s t Kachine which was developed by t h e a u t h o r s and i n s t a l l e d i n t h e Corporate r e s e a r c h and Development D i v i s i o n o f S h a r a t Heavy E l e c t r i c a l s L i m i t e d , Hyderabad, India. The f e a t u r e s o f t h i s machine are g i v e n s e p a r a t e l y i n t h e f o l l o v i n c p ar ag r ap h s . In t h e theoretical side, there i s considerable literature available f o r predicting t h e operating t e r r p e r a t u r e o f t h r u s t b e a r i n g s . The ap p r o ach for t h e p r e s e n t t h e o r e t i c a l work w a s adopted from t h e ncetbo4s p u b l i s h e d by S t e r n l i c h t , B . (1.2). E t t l e s , C . ( j , d ) and Vohr,J.H.(5). A computer prommmc= was developed t o s o l v e t h e Reynolds and EherFy E r u a t i o n s t o g e t h e r w i t h e l a s t i c and t h e r m a l d i s t o r s i o n n . The h e a t b a la n c e between t h e h e a t g e n e r a t e d i n t h e o i l f i l m and t h e h e a t d i s s i p a t e d t o t h e s u r r o u n d i n g s of t h a t pads was a l s o o b t a i n e d by t h e method qiven h y Vohr,J.H.( 5). F i n i t e d i f f e r e n c e method was a m l i e d i n s o l v i n g t h e Reynolds and Fhergy E m a t i o n s and a n i t e r a t i v e scheme i s o u t l i n e d under t h e o r e t i c a l a n a l y s i s . To l i m i t t h e complicacy i n t h e p r e s e n t work, a l l t h e u e u a l assumptions were ad o p t ed for s o l v i n g t h e Reynolds and Energy Erluations.
1.1
Notation
b
R a d i a l l e n g t h o f t h e pad
C
Ta pe r in 8
c
F
direction
Specific heat of o i l
D
O u t e r d i a m e t e r o f t h e sector
h
B e a r i n g film t h i c k n e s s
HU
Heat t r a n s f e r c o e f f i c i e n t f o r - upper surface of bearing film
H1
Heat t r a n s f e r c o e f f i c i e n t f o r lower s u r f a c e of b e a r i n g f i l m
m n
Number of mesh s u b d i v i s i o n s i n 8 direct ion Number o f mesh s u b d i v i s i o n s i n direction
r
N
R o t a t i o n a l speed
n
Number o f pa ds .
F
Fressum
r
Inside radius
qr
Radial lubricant f l u x
9
C i r c u m f e r e n t i a l l u b r i c a n t flux
T
Te m pe ra ture o f l u b r i c a n t
T1
Temperature of bottom s u r f a c e o f pad
Tu
Temperature on t h e u p p e r s i d e of the film
8
Angular c o o r d i n a t e
122
0
Angular v e l o c i t y
7
Absolute v i s c o s i t y
4
Conztant u s e d i n v i s c o s i t y e i u a t i o n
j'
Density of o i l
S u b s c r i p tP i
Index X defi n i n g v al u e of r i n t h e t h r u s t b e a r i n r mesh r u n n i n g from 1 t o n
j
Index X d e f i n i n c v a l u e of B i n t h e t h r u s t b e a r i n g mesh r u n n i n E from 1 t o m
1
Inlet cuantities
2
Outlet quantities
2
~ E 3 2 R I ~ Or I O THE ~ THRU=T i3.lARING TEST MACHINE
The T h r u s t B e a r i n p T e s t Y a c h i n e on which t h e p r e s e n t e x p e r i m e n t a l work o f s e c t o r pad b e a r i n g s was c a r r i e d o u t i n shown i n F i g s . 1 and 2. F l e x i b i l i t y i n t h e a p p l i c a t i o n o f load i n a r a n g e o f r-1.3 NN and c o n t r o l o v e r speed between 0-1000 RFW and i t s h i g h l y s e n s i t i v e m o n i t o r i n g system t o measure t h e l o a d a n d t e m p e r a t u r e , P r e s s u r e and f i l m t h i c k n e s s d u r i n g c o n t i n u o u s o p e r a t i o n a r e t h e f e a t u r e s o f t h e t h r u s t beari n g t e s t machine. "hP machine m a i n l y c o n s i s t s o f a v e r t i c a l r o t o r c o u p l e d t o a 350 *W D.C. N o t o r and s u p p o r t e d on a s e c t o r - p a d t h r u s t bearinp. In addition t o the test thrust bearing which s u p p o r t s t h e r o t o r , t h e r e i s a n o t h e r t h r u s t b e a r i n K which e n a b l e s t o t r a n s m i t hydraul i c load t o t h e systerr.. The r e q u i r e d l o a d i s a p p l i e d on t h i s b e a r i n g t h r o u g h e i K h t h y d r a u l i c a c t u a t o r s . Both t h e t e s t bearinp: and t h e facil i t y b e a r i n g are mounted on sprinp: mattress. The t e s t b e a r i n g i s i r a e r s e d i n a n o i l h o u s i n g l o c a t e d on t h e t o p o f t h e Irachine w h i l e t h e l u b r i c a t i o n f o r t h e f a c i l i t y b e a r i n g i s provided t h r o u e h s n e x t e r n a l l u b r i c a t i o n system. The speed o f t h e motor i s c o n t r o l l e d t h r o u g h a t h y r i s t o r c o n t r o l l e r . T h e r e a r e e i g h t plug-int y p e o i l c o o l e r s f i x e d t o t h e t o p bsarinp-. h o u s i n g t o c o o l t h e o i l i n t h e housing. The r e a u i r e d water i s made a v a i l a b l e t o flow t h r o u g h t h e s e c o o l e r s from a w a t e r c o o l i n r - t o w e r i n p t a l l e d i n t h e open a i r . The t e s t machine i s s u i t a b l e for continuous operation.
3
Fig.1:
G e n e r a l view o f T h r u s t B e a r i n g T e n t Machine
Fig.2:
Skematic s k e t c h of T h r u s t Hearing Test E;achine.
THEORLTICAL ANALYS1.i
Computer programmes have been developed f o r d e t e r m i n i n g ( i) t h e p r e s s u r e an.) tenaperature d i s t r i b u t i o n s on t h e b e a r i n g o i l s u r f a c e , ( i i ) e l a s t i c and t h e r m a l d i s t o r t i o n s o f t h e pad and ( i i i ) t h e h e a t b a l a n c e between t h e h e a t g e n e r a t e d i n t h e o i l f i b an? t h e h e a t d i s s i pated t h r o u g h d i f f e r e n t modes. V i s c o s i t y was allowed t o v a r y w i t h t e s r e r a t u r e a c c o r d i n g t o r e l a t e i o n s h i p g i v e n i n Eq. 1.
-d T
123 3.1
Pressure Distribution
F r e ss u r e d i s t r i b u t i o n i n t h e l u b r i c a n t f i l m is o b ta i n e d by so l v i n R t h e e s t a b l i s h e d steadys t a t e Reynolds Eq.2 f o r i n c o m p r e s s i b l e l u b r i c a n t by f i n i t e d i f f e r e n c e method. The g o v e r n i n g Reynolds e q u a t i o n w i t h a l l t h e u z u a l as s u mptions i n P o l a r c o o r d i n a t e s i s g i v e n as
T h i s e q u a t i o n is f i r s t non-dimensionalised by i n t r o d u c i n g t h e d i m e n s i o n l e s s p a r a m e t e r s ( E 0 . j ) . Then it is solver! n u m e r i c a l l y u s i n g a 7 x 7 mesh s i z e and u s i n g t h e c e n t r a l d i f f e r e n c e e x p r e s s i o n s for t h e f i r s t and second d e r i v a t i v e s Fig.3 shows i n t h e Reynolds e q u a t i o n (Eq.2). t h e i n d i c e s used f o r t h e p r e s s u r e a t each mesh p o in t o f t h e g r i d selected for d e t e r m i n i n g t h e presstire d i s t r i b u t i o n i n t h e l u b r i c a n t f i l m .
I
0
Dimensionless- rarametzrs ~
h
=
hJ c
8
=t3 Fig.,:
Pad Geometry d e t a i l s and d e f i n i t i o n
of I n d i c e s used i n t h e Computer Prograwe 3.2
(3) The boundary c o n d i t i o n s c o n s i d e r e d i n s o l v i n g t h e 4 . 2 are t h a t t h e p r e s s u r e s f a l l t o mero around t h e pad p e r i m e t e r an d f o r a l l t h e fictitious image p o i n t s as s een in Fig.3, t h e p r e s s u r e s are t r e a t e d t o be e o u a l in magnitude b e t o p p o s i t e i n s i g n t o t h e v a l u e s of t h e pressures at t h e corresponding p o i n t s j u s t i n s i d e t h e boundary. These are g i v e n i n Eq.4.
(4)
B a r above t h e symbols i n d i c a t e s t h e dimensionless q u a n t i t i e s .
Temperature D i s t r i b u t i o n
The e n e r g y e q u a t i o n (Eq.5) i s s o l v e d numeric a l l y for o b t a i n i n g t h e temperature d i s t r i b u t i o n o f t h e l u b r i c a n t film
(5)
where
The e n e r g y e q u a t i o n (dq.5) i s s o l v e d by num e ric a l method, w i t h an a ssum ption t h a t t h e t e m p e r a t u r e i s c o n s t a n t across t h e t h i c k n e s s o f t h e l u b r i c a n t film b u t i t v a r i e s in t h e The boundary c o n d i t i o n p l a n e of the bearing. c o n s i d e r e d i s t h a t , t h e t e m p e r a t u r e is c o n s t a n t a t t h e l e a d i n g edge o f t h e pad and a known v a l u e o f t h e o i l t e m p e r a t u r e i s a ssigne d.
124
3.3
E l a s t i c and Thermal D i s t o r t i o n s
An attempt was made i n t h e p r e s e n t work t o include t h e e f f e c t of e l a s t i c and thermal d i s t o r t i o n s on t h e l u b r i c a n t film thickness. But because of t h e low speeds and low l o a d s chosen i n t h e p r e s e n t experimental work it is considered t h a t the e f f e c t of e l a s t i c and thermal d i s t o r t i o n s on t h e p r e d i c t i o n of o p e r a t i n g temperature is reasonably small and hence t h e s e effects a r e neglected.
t h e o p e r a t i n g v a r i a b l e s are p r i n t e d a f t e r the load and Tin converge w i t h i n t h e prescribed limit o f e r r o r .
Y I
READ I N P U T D A T A R,r, h 2 . d h w ,CD,9 , PITin.
3.4
Heat Balance SET INITIAL PRESSURES TEMPERATURES
I t was r e p o r t e d i n t h e e a r l i e r works of Vohr,J.H.( 5) t h a t t h e true o p e r a t i n g temperat u r e can be determined o n l y by c o n s i d e r i n g t h e h e a t balance between t h e h e a t generated i n t h e o i l film due t o viscous s h e a r i n g o f t h e o i l and t h e amount of h e a t d i s s i p a t e d through d i f f e r e n t modes. These modes a r e swnmarised mainly as,
I1
SOLVE FILM THICKNESS D I S TRlB UTIO N I
1) Heat l o s t due t o s i d e leakage of t h e o i l from t h e i n n e r and o u t e r c i r c u m f e r e n t i a l edges o f t h e pad Q1
SOLVE REYNOLDS EQUATION
-
I M 0 D I F Y V I SCOS I T I E S
Heat d i s s i p a t e d t o t h e runner and thence from runner t o t h e o i l i n t h e b a t h Qr
2)
I
-
I
Heat conducted through t h e pad from t h e f i l m t o the tub o i l &P Heat l o s t by convection i n t h e groove 4) between t h e pads Qg The h e a t l o s t through t h e s e modes is evaluated t a k i n g i n t o account t h e h e a t t r a n s f e r c o e f f i c i e n t s as g i v e n i n Ref.(5).
3)
I
-
SOLVE
DISTORTION
I
I TER
= ITER + 1
-
3.5
TEMP. CHECK €OR CONVERGENCE C R I T ER IA P R I N T THE OPERATING VARIABLE6
I t e r a t i o n Scheme
P r e ~ s u r ed i s t r i b u t i o n , temperature d i s t r i b u t i o n and t h e h e a t balance were a n a l y t i c a l l y obtained with t h e following i t e r a t i o n procedure. Fig.4 shows t h e flow diagram used i n t h e computer propramne. With a n assumed f i l m thickn e s s a t the i n l e t ewe o f t h e pad, t h e f i l m t h i c k n e s s e s a t every mesh p o i n t is determined. The first set of non-dimensionalised p r e s s u r e s are obtained f o r m x n mesh points. Then with a known inlet temperature assigned at t h e leadi n g edge o f t h e pad t h e f i r s t set of temperature d i s t r i b u t i o n is determined. Basing on t h e s e temperatures, t h e v i s c o s i t i e s are modified. Then with t h e f i r s t set of p r e s s u r e s and temperatures the e l a s t i c and thermal d i s t o r t i o n s are c a l c u l a t e d . Then t h e second s e t of p r e s s u r e s are determined and t h i s process of i t e r a t i o n c o n t i n u e s till t h e p r e s s u r e s converge t o a limit o f e r r o r of 0.001. Now, f i n a l p r e s s u r e s , temperatures and e l a s t i c and thermal d i s t o r t i o n s are determined.
DlSTO
Fig.4:
4.
Flow D i a g r a m used i n t h e Computer Frogramme DESCRIPTION OF THE MPERIMENTAL WORK
A s p r i n g mounted t h r u s t b e a r i n g of c o n s t a n t
Basing on t h e above p r o m a m e t h e h e a t transfer q u a n t i t i e s Ql, Qrr $ and Qg are evalu-
pad geometry was t e s t e d f o r s e v e r a l load and speed combinations on t h e t h r u s t bearing test machine. The pads were faced with white metal about 5.0 mm thick. Miniature thermocouples made of copper constantan type were mounted as shown i n Fig.5, at 1 5 p o i n t s on each pad f l u s h i n g with t h e b a b b i t t l i n i n g i n about 2 nun below t h e pad surface. The temperature d i s t r i bution from l e a d i n g edge t o t r a i l i n g edge on t h e pad could be rconitored with t h e h e l p of 1 5 thermocouples covering some o f t h e i n t e r i o r l o c a t i o n s . Thermocouples were a l s o mounted i n between t h e pads t o measure t h e temperature of t h e o i l i n t h e bearing housing. 5 Pressure t r a n s d u c e r s w e r e i n s e r t e d on each pad as shown i n F i g . 6 t o measure t h e p r e s s u r e s a t d i f f e r e n t l o c a t i o n s on t h e pad surface. These p r e s s u r e s were measured a t 2 l o c a t i o n p o i n t s near t h e l e a d i n g edge, at J l o c a t i o n p o i n t s n e a r t h e t r a i l i n g edge. O i l f i l m t h i c k n e s s is measured a t both l e a d i n g and t r a i l i n g edges with t h e h e l p of non-contact type eddy c u r r e n t probes as shown i n Fig.7.
a t e d as p e r t h e flow d i a p a m Riven by Vohr,J.H (5) f o r heat balance and l o a d i t e r a t i o n . A l l
I n a d d i t i o n t o t h e above, t h e flow r a t e of water t o t h e plug-in-type o i l c o o l e r s and the
I n t h e Vohr's a n a l y s i s t h e s o l u t i o n s for t h e thermal boundary l a y e r across t h e groove were used t o c a l c u l a t e t h e h e a t t r a n s f e r r e d t o t h e cold o i l i n t h e groove.
125 i n l e t and o u t l e t t e n p e r a t u r e s o f tb.e water were also recorded. 4.1
D e t a i l s of pad georcetry and 1ubricar.t p r o p e r t i e s
Outside d i a of t h e pad
0 . 8 rt
I n s i d e d i a o f t h e p:id
0.4 m
Pad included a n y k
38 d e g r e e s
liumber o f pads
R
Cap between pads
C.05 m
Thickness o f t h e pad
0.05 m
Tropc’rt
ips of
lubricant 2
Absolute v i 3 c o s i t . v a t 3n°C
@.@go9 E;s/m
g e n s i t y of t h e o i l
889 e / m J
Specifi.? h e a t of t h e lubricant
1 . 7 6 ~ 05J/kg. 1 k)
Fig. 6:
Arrangercent of P r e s s u r e T r a n s d u c e r s i n t h e T h r u s t Jearing Pad.
The t h r u s t b e a r i n # t e s t machine was r u n c o n t i n u o w l y round t h e c l o c k f o r s u f f i c i e n t l y lonu: time i n o r d e r t o r e c o r d t h e t e l r p a r a t u r e s and p r e s s u r e f i f o r + i f f e r e n t sets of loeds and speeds. The t e m p e r a t u r e s , p r e s s u r e s and film t h i c k n e s s e s were r e c o r d e d f o r f i v e s t e p s o f loads r a n g i n 6 from 1 Y. Fa t o 3.25 K € a arid e i g h t d i f f e r e n t speeds. The r e s u l t s o f maximum t e m p e r a t u r e s r e c o r d e d are shown i n Fig.!? t h r o u g h Fig.12 and i n Ta’lle 1.
MOUNTING PROBE
Fig.7:
Fig.5:
Arrangement of Thermocouples i n t h e ThruPt Bearing Pad.
v
Arrangerrent oT Eddy C u r r e n t Frobes i n t h e T h r u a t R e a r i n g Fad.
Fig.8 and 9 show t h e t e m p e r a t u r e c o n t o u r s for t h e s p e e d s of 300 and 40C rpm r e s p e c t i v e l y . These t e m p e r a t u r e c o n t o u r s show a n i n c r e a s e d zone of teriiperature for t h e i n c r e a s e f i s p e e d s w i t h a typical s p e c i f i c l o a d o f 3.25 1. Fa. T h i s s i g n i f i e s n o t o n l y the u x i m u m tercperature r i s e b u t a l s o i n c r e a s e d zone o f maximum t e n p r a t u r e s f o r t h e i n c r e a s i n g speeds. S i m i l a r l y Figs.12 t h r o u g h 1 5 a l s o show a marked i n c r e a s e i n temperature with t h e increasing s t e p s o f speeds. However e x p e r i m e n t a l r e s u l t s as s e e n i n T a b l e 1 d i d n o t show, YO much of i n c r e a s e i n maximum t e e p e r a t u r e w i t h t h e increasing loads on t h e p a d s b u t t h e measured temperatures
1.
2.
3. A.
5. 6.
350 35c 350 350 350 359
0.24
50.5
55.6
11-73
0.32
52.0
57.2
16.76
0.48
55- 6
58.8
18.48
0. 56
59.4
59.2
20.26
0.72
63.0
62.4
22.66
0.92
65.5
64.8
25.00
1 =
3 * 5500 KN/tA2, 4 = 6500 KN/M2. Fig.10:
Contours o f Pressure for Speed = 300 RW., Sp.Load = 3.25 M Fa.
Pig.11: Contours of Pressure f o r Speed = 400 R l X , Sp.Load = 3.25 M Pa.
127
t r a n s f e r c o e f f i c i e n t s considered i n t h e calcul a t i o n o f h e a t l o s s e s u n d e r d i f f e r e n t modes. However, t h e a n a l y t i c a l p r e d i c t i o n s are i n good agreement w i t h t h e p r e s e n t e x p e r i m e n t a l results, in most of t h e o t h e r cases. So, t h e p r o c e d u r e adopted i n t h e a n a l y s i f i of h e a t b a l a n c e i s found t o b e s a t i s f a c t o r y . Figs.10 and 11 show t h e p r e s r i i r e c o n t o u r s a t 300 and 400 rpn respectively. I t was also n o t i c e d for t h e speed and loads c o n s i d e r e d i n t h e p r e s e n t work, t h e e f f e c t o f t h e r m a l and e l a s t i c d i s t o r t i o n s on t h e film p o m e t r y is found t o be much low.
35 r
t OV
w
sp. L O A D = 3 . 2 5 M p a
3025-
K
3
20K ul
a L
-
15
ul t0-
a
10ANALYTICAL
5 01 0
SPEED = 3 5 0 R P M
70
I
50
100
150
M A X . F I L M TEHPERATUR
200
250
300
350
4
S P E ED-RPM-
Fig. 14 : Comparison o f Temperature r i s e V e r s u s Speed
K
w
30
n
t 0 1
I
I
1.o
I
I
I
I
2 .o
I
I
I
4 .O
3.0
S P E C I F I C LOAD-
Fig.12:
Comparison o f Faximum Temperature Versus Sp.Load on t h e b e a r i n g
t
22
-
20
-
18-
U
I
a0
I
I
r 3 . 2 5 MDE
\
2 . 5 Mpa,
1
14
-
12
-
K
w
3
*
U
6o
K YI
I
LO
2U
40
a 10t
'
w
c
a Ly a 30
5 c
16-
!!
a
U
w K
ul
I
c Q
-
6 -
2ol
4 2 -
10
0 1 0
8 -
I
SO
I
100
I
I50
1
200
1
250
I
300
I
350
SPEED-RPM
Fig. 13: Comparison of maximum T e n p e r a t u r e Versus Speed
I
400
01
0
I
I
I
I
1
I
I
50
100
150
200
250
300
350
400
SPEED R P M t
Fig. 15: Comparison of 'l'emperature Versus Speed showing E x p e r i m e n t a l and A n a l y t i c a l h d i ctions
128
7
ACKNOH LkEUtWI
The a i i t h o r s w i s h t o e x t e n a t h e i r s i n c e r e t h a n k s t o Fir F C L a h i r i , G e n e r a l h a n a g e r , C o r p o r a t e R&C b i v i s i o n , BILL, Hyderabad, I n d i a for t h e encouragement and p e m i s s i o n g i v e n i n p u b l i s h i n g t h i s p a p e r . The a u t h o r s also wish t o t h a n k hr F Y w i a n John Cy.Genera1 k a n a g e r , and hr 1 K John, S e n i o r Fanager of C o r p o r a t e R&D D i v i s i o n , BHbL, Myderabad f o r t h e u s e f u l suqgestions given i n t h e course of present study.
i(/ YTQ RUNNER
10
References
,
GROOVE
300
200
100
100
3hRNLICHT, 3 . . 'Xnergy a n d Reynolds consi6erations i n Thrust Bearing Analysis', IF:E C o n f e r e n c e on L u b r i c a t i o n and d e a r , London, Lngland, O c t o b e r , 1957, pp 28-38.
SPEED-RPM
Fig.16:
C o e p a r i s o n of p e r c e n t a g e Heat L o s s e s Versus Speed
bTERNLICHT, 5.. CAHTNR, G.K. and AFWAS, 'Adiabatic Analysis of E l a s t i c C e n t r a l l y F i v o t e d , S e c t o r , T h r u s t Bearing P a d s ' , A3.E J o u r n a l of Applied kechs., Vol.28, June, 1961, pp 179-187.
LB.,
1 70
-
60
-
50
-
40
-
S P E E D = 3 5 0 RPM
LEAKAGE
fiTTLX.3, C., and CAT..WON, A., 'Thermal and E l a s t i c D i s t o r t i o n s i n T h r u s t Bearings', I n s t i t u t i o n o f Nech. E n g i n e e r s , Lubricat i o n and wear Convention, 1963, F a p e r 7, pp 60-71.
-Y
In v)
0
dl'TLG,, C., 'The Ijevelopment of a Gener a l i z e d Computer A n a l y s i s f o r S e c t o r shaped T i l t i n g - P a d T h r u s t Bearings', ASLd T r a n s . Vol.19, 2 , 1975, pp 153-163,
-I
c
U y1
30 -
r
s
20
-
10
L
d
VOHR, J . H . , 'Prediction o f the operating Temperature o f T h r u s t Bearings', A P E T r a n s . J o u r n a l of Lub. Tech., Jan. 1981, Vol.lOj, pp 97-106.
PAD
-GROOVE I I I
-
I
I
I
I
l
l
I
1
Fig. 17: Comparison o f p e r c e n t a g e Heat L o s s e s V e r s u s Sp.Load
6
CONCLUSIONS
d i t h t h e p r e s e n t developnent o f e x p e r h e n t a l f a c i l i t y , f o r s p r i n p mounted s e c t o r - p a d t h r u s t b e a r i w s t h e a u t h o r s are i n a p o s i t i o n t o p r e - i c t t h e t h e n r a l behaviour o f t h e s e c t o r p a d s and correlate t h e s e a n a l y t i c a l v a l u e s w i t h t h e e x p e r i m e n t a l r e s u l t s . The r e s u l t s o b t a i n e d i n t h e p r e s e n t work are s a t i s f a c t o r y . I t is now p o s s i b l e f o r t h e m a n u f a c t u r e r s o f e l e c t r i c a l r o t a t i n g machinery such a s hydrog e n e r a t o r s t o b r i n g down t h e g a p between t h e p r e d i c t e d and t h e a c t u a l f i e l d t e m p e r a t u r e s measured.
129
Paper IV(v)
Hard-on-hard water lubricated bearings for marine applications P.J. Lidgitt, D.W.F. Goslin, C. Rodwell, G.S. Ritchie
SYNOPSIS Current trends in marine engineering are leading to the development of water lubricated bearings using hard materials for the bearing surfaces. The design and manufacture of such bearings is at the limit of established technology. The most promising designs at present are believed to lie in the area of ceramic/ceramic material combinations although in the less demanding situations hard coatings on metallic substrates may be satisfactory. The paper identifies areas for further research leading to improvements in the technology which will be necessary before such bearings can be consistently designed and manufactured with confidence. 1.
operating against a hard material. Such an approach implicitly accepts a rate of wear which, for reasons that have been described earlier, is unacceptable.
INTRODUCTION '1.1 Trends in marine technolory require equipment that has low first cost, high reliabili-tj.,low maintenance and is durable. i'n order to meet these requireiiients increasing attention is being paid to the design of wearinc components, particularly bearing;s. 'This paper addresses the issues raised in designin;, water lubricated bearings and indicates zoi,:e oi' the problems that have been encountered in practice.
1.5 A development presently in its infancy but gaining increasing acceptance is the use of hard bearing surface materials. These are harder than the abrasive particles likely to be drawn into the contact and as such able to withstand the associated wear. However, the engineering of these surfaces is poorly developed and there are many problems to be resolved.
1.2 As far is possible the general design Cuide that a machine should use its working fluid as a lubricant is followed in marine engineering practice. This has obvious advantages. Water lubricated bearings are therefore commonly found in circulating water pumps, high pressure feed pumps, extraction pumps, fire and bilge pumps and in the modern glandless pumps. Propulsion shaft bearings in the ' A ' frame and stern tube are also water lubricated. 1.3 Whilst there are many attractions in using the machine's working fluid as a lubricant, water and in particular seawater bring in their wake certain problems. Fluid film thicknesses are very much smaller than would be obtained using a more viscous fluid. More significantly, seawater carries abrasives eg sand and silt which accelerate wear of the bearing surfaces. In the cases of taper-land designs this can be potentially serious leading to loss of load carrying capacity. 1.4 The traditional response to these problems has been to use bearing surface materials which are tolerant of abrasives and which are capable of operating under mixed film or boundary lubrication conditions eg ferrobestos or cutless rubber
This paper considers various aspects bearing development for sea water lubricated thrust and journal bearings. Section 2 discusses theoretical aspects of bearing design with low viscosity lubricants and makes reference to experimental and practical evidence. Section 3 considers aspects of bearing materials and indicates an approach which has been found to'be helpful in material selection. Section 4 illustrates some of the problems to be overcome in the manufacture of bearing components to the required accuracy. 1.6
Of
2.
BEARING DESIGN CONCEPTS 2.1 The ideal bearing system should be self-feeding, maintenance-free and have infinite life. Additionally, when water (or sea water with possible particulate contamination) is to be used as the lubricating medium, the bearing, as well as exhibiting corrosion resistance and material stability, must, due to the low viscosity of the medium, operate with very small lubricant film thickness unless very conservative and often impractical designs are adopted.
130 difference are set at near optimum values for maximising minimum film thickness at the most heavily loaded conditions. The situation in terms of carrying the loads is therefore satisfactory, in that, with near optimum hydrodynamic designs, contact free operation exists except during starting and stopping transients, where rubbing speeds are low enough to give insignificant wear or distress. This latter feature has been confirmed by long term testing of various equipments in both rigs and prototype machines. These tests involve multiple start/stop cycling and continuous running in both clean sea-water,and in sea-water containing measured quantities of silica powder (representing sand contamination).
The above requirements are being met in many cases by using bearing surfaces of extreme hardness, in mutally compatible pairs. Predominately hydrodynamic (ie contact-free) operation may be guaranteed by using extremely accurate stable bearing surfaces both in terms of siirface finish and form. Since by the nature of such hard materials no significant running-in capability exists the alignment of stationary and moving bearing surfaces necessary to promote hydrodynamic operation at the small film thicknesses anticipated (
-
2.2 For horizontal journal bearings the route chosen is to use bearingsofthe tilting-pad type. The pads are then free to tilt,in pitch, roll and yaw on spherical pivots, and can then adapt to any equilibrium position for a given lubricant film thickness at the pivot point. A further significant advantage of such bearings is that they are stable to self-excited (whirl) vibrations since each pad can only supply a reaction normal to the pivot. This is in distinct contrast to fixed pad bearings, which are prone to whirl.
LIFT (MICRONS)
&or 4
35 -THEORY O EXPERIMENT
30
25 20 15 10
. 0
2.3 For thrust bearings, particularly in horizontal machines with a single rotatior. direction, fixed taper-land bearings are most appropriate. The alignment between collar and bearing is assisted by resiliently mountink the latter component. The dimensions of the bearings are dictated to an extent by the overall dimensions of the machines, eg shaft and hence journal diameters are generally determined with little reference to the bearing requirements. Exploratory design studies, have been carried out using the MELBA suite of bearing analysis programs (refs 1,2 and 3 ) where the various parameters (eg bearing length, pad dimensions, pivot position, pad/journal radius difference etc in the case of journal bearings) are explored over the expected ranges of load, speed and temperature. If the bearings operate in the laminar regime, with modest power losses,the predictions have been found to be accurate to a high level of confidence even with isothermal theory. Experimental checks have been made, where journal/pad eccentricity and attitude measurements confirm the accuracy of the mathematical model, particularly at the more heavily loaded conditions (see Fig 1). The predicted values of minimum film thickness, generally of the order 5-10pm, set the standards to be met for the hard bearing surfaces, both in terms of form (cylindricity OF flatness) and surface finish. In the case ofjournal bearings, with diameters and lengths of the order of lOOmm, cylindricity of 2 - 5 m is required while surface finish Ra-values of 0.2pm are held. Values of pad/journal radius 2.4
3
5 6 7 8 9 10 11 12 DUTY PARAMETER FIGURE 1 VARIATION OF LIFT WITH DUTY PARAMETER 1
2.5
2
4
One feature of journal bearing operation using tilting pads which is not predicted well theoretically and which has causedpractical problems particularly in vertical shaft machinks, is that at large film thicknesses, tilting pads d o not behave as predicted. The bearing clearance ratio is determined at the design stage to cope firstly with thermal transients and secondly with pivot setting tolerances of the order of 1/1000 ie in the range where oil lubricated bearings normally operate. The low viscosity of water dictates that with any significant load on the bearing, pads opposite those taking the load operate at large mean film thicknesses, and therefore with nlinimal loads generated by viscous entrainment. Under such conditions, pad gravity loading, viscous friction effects and fluid inertia effects may be expected to become at least of equal significance as the viscous wedge effects. Although in principle, and at considerable expense computationally, fluid inertia effects in the film can be calculated, they cannot be expected to give the complete picture, since pressure effects in the pad inlets are undoubtedly of significance - and may even predominate. Measurements of pad inlet pressure in a tilting pad bearing (made via a tapping connected to a manometer) suggest that excess pressures of the order 0.3-0.5 times the surface dynamic head pressure exist. Such
131
pressures are much larger than the specific loading expected onpurely viscous grounds so that the normal theory, even if fluid inertia effects are included,would appear insufficiently comprehensive. A separate and more recent investigation using the tilting pad version of MELBA which allows the effect of offsetting moments to be calculated, has indicated that a tilting pad, due to its own gravity offsetting moments (roll and yaw) does not exhibit a stable loaded equilibrium beyond a certain quite small film thickness at the pivot, and as a result will tend to 'sprag'. This has been found to happen in practice. 3.
MATERIAL SELECTION 3.1
Background
The following materials have been selected and extensively tested for both sea water lubricated journal and thrust bearings :
extensive detachment of the coating from the backing. It is not known whether these failures were due to poor coating quality or to a poor performance of chromium oxide coatings during nonhydrodynamic rubbing. Further consideration has been given to material selection, probably the greatest challenge in hard on hard bearing design. Two semi-quantitative investigations are briefly described below.
3.2
The theory was taken fr,oaBlok's work of 1937. Ref 4, to addresF. the temperature rise due to frictional heating at the interface between two bodies rubbing together as shown in Figure 2.
-
Phosphor bronze
STATIONARY BODY
Reinforced phenolic resin
MOVING BODY
-
Plasma sprayed chromium oxide
-
Selection Based on Flash Temperature Calculations
-
t
OB/
Lot pressed silicon nitride (HPSN)
Detonation gun applied, tungsten carbide + cobalt and chromium.
%a -I F I G U R E 2. HEAT PARTITION BETWEEN TWO BODIES V = Rubbing Velocity Q = Q, + Qc
Where: ;oft Aaterials: The soft materials, phosphor bronze and reinforced phenolic resin, work well as one half of a bearing pair but wear too quickly in the presence 3f fine sand and corrosion debris. '-iardMaterials: EPSIV has performed well against chromium oxide and tungsten carbide. It has been used successfully in journal bearings in the form of 6mm thick liners lightly clamped into inconel tilting pad backings and running against 3 chromium oxide coating on the shaft. The tungsten carbide coating has been found to show signs of deterioration in sea water due to cobalt leaching. In journal bearings chromium oxide has been found to perform well, both against itself and against HPSN. During deliberate failure tests when bearings were allowed to run dry and seize, some crazing of the coating was observed inthe seizure band. However the bearing would run hydrodynamically under full load after the seizure tests albeit with some spalling. This ehcouraging result has not been repeated in the case of thrust bearings. Several failureshave been experienced due to
The heat generation at the interface = The part of Q flowing into B = The part of Q flowing into C =
Q Q, Q,
4aKB em
Where :
The mean temperature rise over the area of contact = Radius of contact area = Thermal conductivity of stational body =
a KB m
Where
- - equ 2
= 0.31 QC
e
also:
a KCPCCPcaV
K,
=
The thermal conductivity)
p C
= =
Density) of the moving Specific heat) body
To determine the 'heat partition' or division ofheat flowing into the two bodies, use: 1 1 - + - =- -
eBm Where:
m'
----
1
equ 3
e
m
-
the mean temperature rise Bm = of the area of contact if
e
all the heat was flowing into the stationary body.
132
Where:
-e
=
Cm
similarly, if all the heat was flowing into the moving body.
If very small contact areas were considered, this assumption may not be valid.
-
An arbitrary 'rub' was assumed: Load = 2500N v = 1.73m/sec z = 1.5mm p = 0.2, co-efficient of friction The above theory is applicable to 'high speed' contacts and a check was made, again following Blok, to ensure that all the contacts considered where in fact high speed contacts. The rtb was partly based on known conditions during one LC4/LC4 failure situation. For such a 'rub' the heat generation is: Q
=
865 watts
The material properties used are given in Table 1. TABLE 1
e
Kc oat
MATERIAL
P
K coat = Conductivity of Coating
Kine =
thus e Bm
Conductivity of Inconel
=
9712 + 14711
=
24,423OC
For the moving body an allowance for the coating was made fairly arbitrarily by taking ;cm as the average of that obtained using data for solid Inconel and that of iisine data for solid chromium oxide. Using the relevant data in equation 2 gives:
e
-
C J/Eg°C
kg/m3
4aKinc
Where: 1 = length of conduction path through coating (typically 150pm)
for solid Inconel = 610.1OC
Cm
for solid Cr~oj
'Cm K W/mbC
Q
Q 1 +
=
Bm
Thus
8
= 972OC
average
Cm
= 1333.37OC
Now using equation 3: Crlo3 coating Sic HPSN Inconel(Sub-strate)
1.89
5000
733
200/50
3100
800
20
3200
750
9.8
8440
400
This data is representative but not authoritative. Various sources quote difference values. The thermal conductivity of silicon carbide varies greatly with temperature. An approximate allowance was made for this by taking the high figure at low mean interface temperatures and vice versa. For the homogeneous materials the theory was used is outlineti. For chromium oxide on Inconel it was modified as follows by ray of a worked example. Consider the case N h e r e both bodies are made from chromit3.m oxide coated Inconel. For the stationary body the theory is modified by assuming that the heat flows directly through the coating in the contact area before diffusing into tlie Inconel, F1zi:rc 3.
INCONEL
LCL COATING
1 + 1 24,423 e m
em
= 935OC
QB
=
+L 972
the flash temperature
865 x 935 = 33 .watts 24423 Qc = 865 x 935 = 832 watts 972 Table 2 shows the results of similar calculations for a number of material combinations. (Ref.5)
TABLE 2 FLASH HEAT PARTITION MOVING TEMP WATTS SURFACE RISE,
MATERIAL PAIR
SiC/SiC
EITHER
129.3
710
155
HPSN/HPSN
EITHER
474
808
57
Cr, 0, /Cr20, EITHER (on Inconel)
935
83%
33
I
SIC/Cr2 0 3
HPSN/Cr203
I
Sic HPSN Sic LC4
I
154 431
1
I 1 156.5 727
496 HPSN ~ c 4 857
It can be concluded that FIGURE 3 HEAT F L O W INTO COATED COMPONENT
MOVING STATIC
OC
846 736
I
19 129
I
5.5 218
847.5 763
17.5 102
859.5 647
1
1
133
1. The flash temperatures can be very high, as expected.
f = tensile stress in coating, a function f(x)
2.
T'= shear stress at bond,
Silicon carbide on itself produces the lowest flash temperatures by a large margin and this, together with its tolerance of high temperatures, probably explains its good performance as a bearing under extreme load conditions as reported by Kamelmacher in Ref 6.
a function
7 (XI Consider the bond as elastic developing a shear stress,T, proportional to the relative movement between the coating and the Inconel. ie T = K x relative movement
3.
Tungsten carbide coatings wer; also considered in addition to chromium oxide There is little to choose between t l i e m in respect of calculated flash temperatures. 4. In all cases the majority of the heat generated flows into the moving body which, if' a choice is possible, should have the highest thermal conductivity. Unfortunatc'y either side of a bearing pair couldbe t h e movingbody in most bearings and it is difficult to derivt.ary advantage from this conclusion. Thus even when using Sic, with i t s high thermal conductivity, against chromium oxjdc, the flash temperatures may be almost as high as with chromium oxide against chromium oxide.
5. The SiC/HPSN combination is probably acceptable in respect of flash temperature as the higher tem'peratures are not likely to dairiage ceramics.
Ignore lateral stresses, ie consider a one dimensional case. Assume that the thick Inconel is infinitely stiff relative to the thin coating. At a distance x from g, coating element moves relative to g by: sc =
Where: ac = AT =
E
=
The hard coatings mentioned above are applied about 15pmthick on Inconel. If raised in temperature uniformly the Inconel expands more than the coatirig and away from the edges a perfect coating is put in tension but the bond is not stressedin shear. At about400OC temperature rise from an unstressed state, a chromium oxide coating can be expected to Ci'dCi; as its tensile strength is exceeded. At the edges, or local to cracks in the coating, a mope comple, stress situation is found with potentially hikh shear stresses at the bond.
+/
f(x) dx
JO Ec Co efficient of expansion of coating
Uniform temperature rise Youngs Modulus of coating
The Inconel moves relative to the s. =
Where a . =
3.3 Thermal Stress in Hard Coatings
xacAT
by:
x a i A ~ Co-efficient of expansion of Inconel
Thus at the element:
JO
Thus: 7 ( x ) = K [ xhT (ai -ac) X
Consider a piece of coated Inconel, length 2 L , coating thickness t, and of unit width
:Figure 4 ) .
This equation has been modelled in a step by step iterative.computer programme, and typical results are shown on Figures 5 and 6 for the following case:
f(LI-07 f=flOl#O
f-f(X1
I
Y-
t
I"-r=no).o
I
I\. , -I .I I
I
1
\\IY
DUE TO SYMMETRY
I
FIGURE 4 STRESS MODEL FOR A COATING ON A SUBSTRATE
I
Coating: Coating thickness: AT: K:
LC4 150m 350O C 500 x 106 MN/m3
134
r
25r
TABLE 3
MAX COATING TENSILE STRESS, f (0)
I
I K MN/m3
M P X COATING TENSILE STRESS f ( 0 ),MN/mz
I
PEAK BOND SHEAR STRESS T max,MN/mZ 12.84
I
1
0.5
1
x -mm
1.5
50 x 10
188.9
500 x 10
217.1
178.4
57.54
5000 x 10
217.3
514.2
The variation of coating stress and bond shear stress with coating thickness is shown in Table 4 with K constant at 500 x 1Q6 MN/m’
.
TABLE 4
FIGURE 5 COATING TENSILE STRESS DISTRIBUTION
COATING THICKNESS t,pm
t
N
PEAK BOND SHEAR
MAX COATING TENSILE STRESS f( 0 ),MN/mz
50
217.1
150
217.1
450
217.1
PEAK BOND SHEAR STRESS T max,MN/mz 99.8
178.4 315.0 ~~
K = 500 x 1Q6 W / m 3 . References 7 and 8 address the same problem in greater depth. 4. DESIGN, MANUFACTURE AND MEASUREMENT 4.1 Journal Bearings
Tilting pad journal bearings are used in pumps to eliminate noise and vibration due to half speed whirl. However the selfaligning feature is also essential because it is not possible to make a complete pump to the accuracy demanded by water lubricated bearings. Two essential design requirements have to be addressed, viz:
-2 00
4
I
FIGURE 6. B O N D SHEAR S T R E S S D I S T R I B U T I O N
a. Establishing and maintaining the necessary small radius difference,A R shaft to pad, for the loaded pads.
b.
The value for K was chosen on the assumption that a layer of Inconel 150pm thick (ie same as coating) is distortedin shear due to the bond shear stresses. This is obviously a very questionable assumption, so the variation of naximum coating tensile stress and peak bond shear stress with K is shown in Table 3.
Stability of unloaded pads.
4.1.1 . The performance of a loaded pad is generally as shown on Figure 7. A minimum allowable film thickness must be chosen in light of the accuracy of form and surface finish achievable, ard thus the allowable tolerance on AR can be extracted for the design load. In more demanding applications the best manufacturing accuracy that can presently be achieved is: Surface finish: Roundness: Cylindricity:
.2 2.0 3.0
um um um
Such journal bearings have been shown to rub as expected when the load is increased to give a minimum film thickness of 2 um. Thus, design film
135 thicknesses as low as 4pr. are currently thought to embody an adequate safety factor. Such bearings require a degree of tolerance to asperity contact. However, hard on hard coated sea water bearings may not be sc: tolerant of rubbing, and film thickness of 10 to 12pm are preferred. Silicon carbide against silicon carbide may be very tolerant of rubbing but this has yet to be proied and a limit of 8pm film thickness is presently being adopted for design purposes.
generated by them is small. However, as previously discuseed,a minimum load generation is essential to maintain them in a stable position against the pivot pinz ard prevent random niovements which resu;t in leading edge and corner rubbing damage. In particular the mall hydrodynamic forces generated prcovide little stabilisation in yaw, ie rotation about the pivot axis. For this recson, offset pivotted pads are not satisfactory in vertical axis journal bearings unless they can tte mass balanced about the pivot; a difficult design feature to ac:hieve in practice.
TOTAL AVAILABLEAR
kpiiZE-l
Unloaded pad stability requires much further theoreticul and practical investigation. Ramps at the leading and trailing edges of pads produce a greater incrc:ase in hydrodynamic loed than expected from current thecry applied to large film thicknesses. Ramping is clearly beneficial providing the pad geometry is such that the remaining pad arc length effective under load, is sufficient when the pad is loaded. 4.1.3
~~~
~
RADIUS DIFFERENCE, PAD TO SHAFT,AR
-
FIGURE 7 VARIATION OF MINIMUM. FILM THICKNESS WITH A R FOR VARIOUS BEARING LOADS
The space avtiilable for bearings is normally restricted. The bigger the bearing the mere difficult it is to achieve tight tolerances of form. Thus, whatever minimum film thickness limit is adopted, the allowable tolerance o n A R is usually small and places extreme demands on manufacturing accuracy and measurement capability. The A R tolerance has to cover:
A Typical Design
A journal bearing configuration found to perform satisfactorily employs four pads as shown in Figure 8. Hot
1 I
INCONEL 625 BACKING PAD
EPOXY BEDDING COMPOUND7
HPSNJ LIGHT CIRCUMFERENTIAL CLAMPING
(1) Shaft diar:eter tolerance.
(2) Pad radius tolerances as manufactured
.
( 3 ) The dimensional stability of the pad in service. ( 4 ) Any char:ges in the zhaft size and pad form due to temperature Effects in service.
4.1.2
Unloaded Pad Stability
With bearing diar:eters in the region of 90 to 150 mni the diametral running clearar.ce (set by shimming the pivot pins and often limited by thermal expansion effects) is typically in the region of 75 to 150pm. Thus the 'unloaded' pads run at large film thickness and the hydro-dynamic load
7-
FIGURE 8 JOURNAL PAD ASSEMBLY
pressed silicon nitride pad facings are lightly clamped circumferentially in to the Inconel pad backings. Any bedding cmpound is used on assembly, but a release agent is employed to prevent adhesion. Thus the HPSN can expand independently of the Inconel during temperature transients to minimise the changes in pad radius. After assembly the working face of the HPSN is hand lapped to final radius.
136 require approximately one tenth the stiffness of the water film s o that the loaded taper-land disc Can follow the runner. The mass of the taper-land discs is minimised to reduce dynamic loading and the discs themselves are relatively flexible to assist load sharing between the six active taper-land areas.
Although coatings have been successfully used on the shaft, improvements may be possible in light of the work discussed in Section 5, by replacing the HPSN with silicon carbide and emploling a silicon carbide sleeve on the &.aft. However the technical difficulties of fitting ceramic sleeves to shafts are well known and such problems have not been definitively resolved. 4.2
Stress relieving heat treatments of the runner and discs must be specified at stages during manufacture. It is interesting to note that even the fine grit blasting operation to prepare the taper-land disc blanks for coating, can, unless carefully controlled, result in unacceptable distortion. To obtain the requisite tolerances runner surfaces are Ciamond ground and lapped flat to within 10 helium light bands and checked using a large o~~tical flat. The land areas of the discs are finishing likewise. Taper areas, only 25 to 39am depth, are ground and checked on a Leitz three-dimensional co-ordinate measuring machine.
Thrust Bearings 4.2.1
Background
Dcuble acting tilting pad thrust bearings on horizontal shafts present a number of problems concerning load sharing between pads and unloaded pad instability. The wall known beam 'matircss' system of load sharing cannot be used with its axis horizontal and so each pad requires a flexible pivotted scpport. Such a system has been operated successfully or. the loaded side but it is difficult to to2erance a double acting bearing to ensure a very small axial. clearance. With the excessively large clearance achieveable there is no load generated at the unlcaded pad assemblies which rattle about allowing pad edges to touch the runner leading to damage. 4.2.2
Current Design Trends
For these reasons, design and developn8er;t of large thrust tearings is now looking towards double ac.ting 'taper-land' configurations. Again the overall ac.curacy required of the zssembly is demanding, as are the f1at;nc:ssand surface fi nS sh of the working surfaces. Spacer rings and resilient mounts require to be m5.tc.hfitted so that the taper- land disc:iare just nipped against the resilient mcnunts when unloaded. Thus the unloaded disc is stabilised. Great care is necessary with the runner to shaft fit to limit the swash of the r'lnner to 13/an total indicated reading. Bearings operate with a calculated minimum film thickness of 10 to lpm. Thus the swash is significant, and to accommodate it the resilient rings
5
DISCUSSION 5.1 The requirement for 'hard' materials in water lubricated bearings is established and experience to date indicates that the concept is practicable. However, the technology is imperfectly understood, and is very much a matter of development fcr particular'applications: 5.2 Tiltin& pad designs have been found satisfactory in horizontal journal bearing applications. However, in applications where pads operate vertically in the unloaded condition serious instability has occurred. It is believed that the pads can tilt to present a divergent film s.hape. This generates hydrodynamic forces tending to pull the pad into contact with the moving bearing component and leads,quickly tc catastrophic failure. Variations to the pad shape, by way of
137
'ramps' to the pad leading and trailing edges,have been attempted with some success to try and modify the film shape and so prevent the divergent film from becoming established. This is clearly an area where further research is required. In particular, conditions at the pad inlet and the influence of fluid inertia effects merit investigation. Practically, the problems experienced with unloaded pad instability have led to the more conservative taper-land designs being adopted for thrust bearing applications. 5.3 From the data presented in the text it will be apparent that water lubricated bearings using hard materials are indeed precision engineering components. Fluid film thicknesses of the order 5-10pm, surface finishes of O.2pm Ra value, flatness of runners (26 cm dia) of the order 10 helium light bands and swash of 13pm present design problems which are difficult to resolve in practice. Quality assurance during the manufacturing process is of paramount importance. Particular care must be taken in designing adequate stiffness into the bearing yet enabling sufficient compliance to accommodate misalignments which are inevitable given an accumulation of,manufacturingtolerances. 5.4 Perhaps the most significant problems in designing 'hard-on-hard' water lubricated bearings lie in the area of material selection and design for the use of such materials, particularly ceramics. The design problem can be seen in two parts. Firstly to provide adequate structural strength and rigidity to withstand high transient loads eg shock and stiffness to maintain the bearing form. Secondly to accommodate situations where the bearing operates non-hydrodynamically and touchdown occurs.
5.5 Without taking a particular design for detailed study and presentation it is only possible to make general observations for the first case. Where monolithic ceramic components are used, given that the component sizes are within the size limit that is possible (presently about 26cm dia disc for Sic components) then the method of attachment of a thrust runner to its shaft is often the area where the greatest problems arise. It is important to avoid stess concentrations. Securing the runner by bolts through holes in the ceramic is not likely to succeed. More complex designs using friction devices may be necessary. 5.6 Particular care also has to be paid to the design of the bearing flexible support system. Under load the bearing will deflect. Given the small film thicknesses that exist only minor deflections can be tolerated. Resolution of such problems in practice is as likely to be a matter for test rig development as of accurate calculation and will therefore be costly.
5.7 It is possible to give more quantitative, general guidelines for the second case ie: non-hydrodynamic operation. Essentially we are trying to design to resist wear and potential failure as a result of the high flash temperatures that can be generated. There are three materials approached that are available.
a. Hard coatings (plasma sprayed or applied by detonation gun). b. Ceramics (either monolithic or fabricated structures). c. Hard thin films (ion deposition process 1. 5.8 A useful material screening method has been developed by applying Blok's Flash Temperature theory and by theoretical analysis of the thermal stresses between a coating and its substrate. It has been shown that coatings in contact (Chrome oxide and Tungsten carbide) generate substantially higher flash temperatures than a coating against a ceramic or ceramic against a ceramic. Indeed temperatures sufficiently high to melt the coating can occur. Theoretical work also indicates that the temperatures generated are certainly sufficient to cause coating cracking and bond failure. This would explain catastrophic failures that have been observed in practice, and which resulted in almost total destruction of the bearing surface. Where such damaging contact conditions can be avoided a coating vs coating combination has been found to operate effectively with mi?imal wear. 5.9 It is believed that a more promising way forward lies in the use of ceramics in both thrust and journal bearing designs. For tilting-pad journal bearings,ceramic inserts in a metal backing have been used successfully,albeit not without some difficulty in manufacture. Ceramic sleeves on shafts have not been so successful and it still remains for a wholly reliable attachment method to be devised.
5.10 It is not thought that ceramic facings can presently.be successfully engineered into large thrust bearings given the difficulties of manufacture and the close tolerances that are required. The most promising way forward is seen in monolithic designs paying particular regard to the problems of runner attachment to the shaft. 5.11 Work has only recently started to investigate the likely advantages of hard thin films formed by the ion deposition process. This route offers attractive prospects. The coatings promise to be less prone to catastrophic failure and might even be tolerant to penetrating abrasives in much the same way as traditional white metal bearings. Components would be relatively cheap to manufacture and in the structural sense would certainly be more
138 attractive than ceramics. 5.12 Finally, an area which would particularly benefit from further research is that Of non destructive examination of coatings. At present there is no proven method which can measure the quality of attachment of a coating to its substrate. Equally, the ability to identify coating delamination or inclusions in monolithic ceramic components would be extremely valuable. The most promising way forward probably lies in the direction of ultrasonic scanning techniques. 6.
CONCLUSIONS 6.1 The design and manufacture of water lubricated bearings with hard bearing surface materials is possible but is at the limit of established technology. 6.2 Combinations of ceramic materials are presently believed to other better prospects of success than ceramic/coating or coating/coating combinations.
6.3 Hard coatings require a highly corrosion resistant substrate. Nothing less than IncQnel has been found to be satisfactory in sea water. 7.4 Further research is required in the areas of:
-
lightly loaded tilting pad bearings.
-
NDE methods to assess the adhesive quality between a coating and substrate.
-
the use of ion deposition methods to produce hard coatings for such applications.
References
- Clearance considerations jn Pivotted Pad Journal Bearings. Trans. ASLE, Part 5, pp. 418-426,
(1) BOYD, J., RAIMONDI, A. A.
1962
(2) COWKING, E.W. - 'Thermal Hydrodynamic Analysis of Multi-Arc Journal Bearings' Tribology International August 1981 pp 217-223. (3) HEATH, H.H. COWKING, E.W. - 'MELBA A Suite of Computer Programmes for Analysis of Self-Acting Plain Bearing', GEC Journal of Science and Technology Vo1.45, No.2 1975. (4) BLOK, H - Proc General Discussion of Lubricants, Vol 2, Page 14, Institution of Mechanical Engineers, London, 1937.
(5) ROBINOWICZ, E. Materials.165.
-
Friction and Wear of
- Desigii and Performance of Silicon Carbide Product Lubricated Bearings. Institution of Mechanical Engineers Proceedings, Vol.l9'7A, October 1983.
(6) KAMELMACHER, E.
(7) TING, B-Y, WINER, W.0. RAMALINGHAM, S - A Semi-Quantitative Method for Thin Film
Adhesion Measurement, Trans ASME, Vol 107, Page 472, October 1985. (8) TING, 8-Y, WINER, W.O. RAMALINGHAM, S.
-
An Experimental Investigation of the Film to Substrate Bond Strength of Sputtered Thin Film Using a Semi-Quantitative Test Method. Trans ASME, Vol 107, Page 478, October 1985.
SESSION V THRUST BEARINGS (2) Chairman: Professor C.H.T. Pan
PAPER V(i)
Inlet boundary condition for submerged multi-pad bearings relative to fluid inertia forces
PAPER V(ii)
Pressure boundary conditions at inlet edge of turbulent thrust bearings
PAPER V(iii)
Dynamic analysis of tilting pad thrust bearings
PAPER V(iv) Hydrodynamically lubricated plane slider bearings using elastic surfaces
This Page Intentionally Left Blank
141
Paper V(i)
Inlet boundary condition for submerged multi-pad bearings relative to fluid inertia forces A. Mori and H. Mori
Based on the assumption of conservation of mechanical energy of the lubricant flow through the leading edge of the pad, a simple theoretical model is presented to analyse the inlet pressure jump in submerged multi-pad bearings. To calculate the mechanical energy just upstream the leading edge of the subject pad, the velocity profile of the lubricant flow there is estimated by applying the concept of momentum integral method to a control volume settled in the space between the subject and preceding pads. Using the inlet pressure obtained, the generation of pressure in the lubricating film is numerically calculated and compared with experimental measurements. 1
INTRODUCTION
In hydrodynamic lubrication for high speed sliding with low viscosity.lubricating fluids, the inertia forces of the fluid flow influence the generation of pressure in the lubricating film irrespective of laminar and turbulent conditions. The influence can be classified into two categories. The first is the influence due to the convective inertia forces in the film flow, plus the centrifugal force if polar coordinates are required. This has already been attacked by many workers as found in the Special issue of the Journal of Lubrication Technology, Vol. 96, 1974(1), and in the Proceedings of the'2nd LeedsLyon Symposium on Tribology(2). This type of influence seems to be sinall in a full circular bearing, because of its film thickness profile. In a plane inclined slider pad or a pocketed pad, however, the influence is considerable and pronounced when the maximum to minimum film thickness ratio becomes large. Many workers have concentrated their interests to such pad bearings. Typical results of the analyses can be found in the papers by Smalley & co-workers(3), King & Taylor(4) and Launder & Leschziner(5, 6). Sectorial configuration for an annular thrust bearing requires polar coordinates. In a pad of such configuration, the contribution of the centrifugal force should be included. This force discourages the generation of pressure in the lubricating film, while the convective inertia forces encourage it. Pinkus & Lund(7) analysed the influence of the centrifugal force alone. Mori & co-workers(8) analysed the composite influence of both the centrifugal and convective inertia forces. According to their discussion on the load carrying capacity, the effect of convective inertia forces almost cancels the effect of centrifugal force for any speed, if the circumferential taper of the pad is set for its maximum load carrying capacity to be obtained in the inertialess case. This means that the load carrying capacity of a sector pad properly designed is scarcely influenced by the inertia forces in the film flow in result. All of the above examples are, a priori, based on the assumption that the pressure bound-
ary condition on the periphery of the pad is never influenced by the inertia forces. This assumption may be reasonable only if an adequate amount of the lubricant is fed directly just upstream the pad. Excepting such a case, the pressure boundary condition at the leading edge of the pad is influenced by the inertia forces of the lubricant flov impinging against the pad. The influence of the inertia forces of the second category is such a one through the inlet pressure condition. This problem has been considered by Pan(9), Constantinescu & co-workers (lo), Tipei(ll), Mori & co-workers(l2, 13, 14) If this impinging flow and Tichy & Chen(l5). developes sufficiently over the runner surface, the inlet pressure, which is often called rampressure, can be built up remarkably, and its influence on the film pressure can be very much larger than that of the inertia forces in the film flow. In a submerged system without free surfaces of the lubricant flow, the developement of this impinging flow is not so significant, but its influence on the film pressure seems to be considerable. Such an inlet pressure jump is governed by the developement of the impinging flow over the runner surface. It is an important problem to estimate its developement for a given lubrication condition. We are, however, not aware of any methods of such estimations in the literature. The purpose of the present work is to propose a simple theoretical model to analyse the inlet pressure jump in submerged multi-pad journal and thrust bearings. The modei is based on the assumption of conservation of mechanical energy of the lubricant flow through the leading edge of the subject pad. The velocity profile of the impinging flow developing over the runner surface, which is required for calculation of the mechanical energy of the flow just upstream the leading edge, is estimated by applying the concept of momentum integral method to a control volume settled in the space between the subject and preceding pads. Premising a small value of the inertia parameter, which is defined by the Couette Reynolds number multiplied by the clearance ratio, the influence of the inertia forces of the first category is ignored. Using the
142 inlet pressure obtained in this manner as the inlet boundary condition for the conventional inertialess Reynolds equation, the film pressure profile over the pad is calculated and compared with experimental measurements.
1.1
Not at ion
C
Radial clearance Film thickness Hight of the control volume at the point A Correction factor for warped stream lines Width of the journal bearing pad Pressure in gauge Inertia parameter Re* = pwC2/p ------ journal bearing Re* = pwhzo2/p ---- thrust bearing Radial coordinate Radius of the journal, or outer radius of the thrust bearing Velocity(vectoria1) Velocity component in the sliding direction Velocity component in the lateral or radial direction Load carrying capacity Coordinate along the sliding direction Coordinate across the film thickness Coordinate perpendicular to the 2-y plane Circumferential taper, ( h 1 0 - h 2 o ) / h 2 ~ Radial taper, (h&-h4o)/h20 Small increment Thickness df the impinging flow Eccentricity ratio Angular coordinate Viscosity Density Angular speed of rotation
h h*
k
z
P
Re *
r PO
u U W c)
2
Y
z a 6 A
6 E
e lJ P w
Subscript Reference point Inlet Out let Leading edge 2 Trailing edge 3 Inner periphery of the sector pad 4 Outer periphery of the sector pad
A in out I
2
THEORETICAL MODEL
The lubricant flow between the pads is modeled The point A just upstream as shown in Fig. 1. the leading edge of the subject pad is a reference point where the lubricant flow developing over the runner surface is scarcely influenced by the ram effects. Its distaice from the leading edge of the subject pad, O A , can be several times of the film thickness, and small enough
Over the compared with the pad spacing, &'. static pressure of the flow is assumed to be equal to the ambient pressure. It is also assumed that the lubricant velocity has no component perpendicular to the sheet. The-velocity profile of the impinging flow through the point A is assumed as @ t
(1) where the exponent rn should be given as a function of the Reynolds number. Since, however, its value scarcely influences the final results, it can be set as rn = 2 for any condition. The thickness 6 is estimated by applying the concept of the momentum integral method to the control volume ABQ'QA x Az*, where Az* denotes the elementary width perpendicular to the sheet. The velocity profile of the flow out of the trailing edge of the preceding pad is given by
because the inertia forces in the film flow are ignored. Assuming that the shearing stress is uniform over A& and represented by its value at the point A , the drag force are given by (3)
AX
where is2eplaced by @, the pad spacing, because A& :OQ. Momentum balance in the control volume ABQ' &A x Az* results in
If the film pressure profile' over the preceding pad is known, the thickness 6 can be calculated by using this relation. Thus, we car. estimate the velocity profile of the impinging flow developing over the runner surface. In order to obtain the equations for the inlet pressure jump, the following assumptions are used; ( 1 ) The transitional entrance region from the leading edge to the section where the film flow developes fully across the thickness is negligibly small compared with the length of the pad. ( 2 ) The generation of pressure in the film flow is governed by the conventional inertialess Rey-
Runner
Pad
>
\
+ Fig. 1
Model of the lubricant flow between the pads
Subject Pad
143 nolds equation, i.e., (5)
the calculated results of the inlet pressure, so that it can be supposed freely. The conservation equation of mechanical energy of a fluid flow is given by
for the journal bearing pad, and
for the thrust bearing sector pad. ( 3 ) The mechanical energy of the flow at the reference point A is conserved through the entrance region. ( 4 ) In this region, the stream lines are warped toward the pad corners, and, just downstream this region, the velocity of the flow has the component W i n which corresponds to the lateral Poiseuille flow caused by the gradient of the inlet pressure in the z-direction. This is shown in Fig. 2. Since strict determination of the stream line is very difficult, it is assumed that the effect of its warp can approximately be treated by introducing the correction factor k as Az*= k ( z ) A z ,
or
Ar*= k ( r ) A r
P
t
v
Ed
where p i n represents the inlet pressure jump. The continuity equation of the flow through the control volume is
(7)
The value of k ( z ) or k ( r ) should be equal to unity around the center of the pad width, and off the center it should be decreased from unity to some value. The supposition of the profile of k ( z ) or k ( r ) does not influence significantly Z
where C represents the control volume, the vectorial velocity and d& the volumetric flow rate through dZ. If the film flow has no reverse flow toward the leading edge, the conservation equation is applied to the control volume AOO'A'A shown in Fig. 1, which results in
If the film flow has a reverse flow, the conservation equation should be applied to the shown in Fig. 3. For control volume AOO"SA'A this case, h i in Eqs. ( 9 ) and (10) should be replaced by h i ' . The general procedure of calculation of the inlet pressure jump and film pressure profile is as follows; ( 1 ) Suppose a certain profile of P i n , and solve the Reynolds equation, Eq. (5) or (61, over the pad by using it as the boundary condition of the film pressure at the leading edge. On the other boundaries, the film pressure.is set at zero. (2) Calculate the velocity components, u i n and W i n , by means of the following relations; (11)
Pad
x=o,z
12)
I
x=o, z
(a) Journal bearing Fig. 2
( b ) Thrust bearing
Impinging flow projected on the sliding surface
( 3 ) Calculate the velocity component Uout from Eq- ( 2 ) . ( 4 ) Calculate.the thickness 6 from Eq. ( 4 ) and determine the profile of the impinging flow, U A . (5) Calculate the value of h* from the continuity equation, Eq. (10). (6) Calculate a new profile of P i n from the conservation equation, Eq. ( 9 ) . ( 7 ) Repeat the above process until it becomes equal to the supposed one. 3
Y
Fig. 3
-1
Flow model with a reverse flow
RESULTS OF CALCULATION
Using the above-mentioned theoretical model, the film pressure profile has been calculated over the loaded pads of a non-preloaded, centerpivoted, tilting pad journal bearing shown in Fig. 4(a), and over the pads of a fixed-tapered sector pad thrust bearing shown in Fig. 4(b). Width of the pad for the journal bearing is set = 4P0/3. Their load carrying capacities as have also been calculated by integrating the resultant film pressure over the pad.
144 0.10 E
(a) Journal bearing Fig. 4
= 0.5
(b) Thrust bearing
Schematics of the bearings subjected to calculation of film pressure profiles
tl deg.
Fig. 7
Calculated example of the circumferential film pressure profile for the -journal bearing pad x10-2
1.5 a = 1.0
tl deg.
Fig. 5 Calculated example of the film pressure profile for the journal bearing pad
Fig. 8 Calculated example of the circumferential film pressure profile for the thrust bearing sector pad between the upstream and downstream pads, but the difference is very small. This example is one for the downstream pad. Fig. 6 is the result for the sector pad of the thrust bearing whose operating condition is as follows; The circumferential taper a = (hlo - h2,)/h2, = 1.0 The radial taper B = (hS0 - hg0)/h2, = 0 The inertia parameter Re* = puhpn2/v = 0.2 where hl,, h2,, h,, and h,, are the film thickness at the middle points of the leading edge, of the trailing edge, of the inner periphery and of the outer periphery of the pad respectively. In Fig. 5 for the journal bearin pad, the pressure is normalized with ~~OJP,~/C', while in Fig. 6 for the thrust bearing sector pad, it is normalized with 6 u ~ P o ~ / h 2 ~ ~ . In Figs. 7 and 8, the circumferential pressure profiles along the center lines of the pads are illustrated compared with the results which are obtained without the inlet pressure jump. From these results, it can more clearly recognized that the inlet pressure jump enhances the generation of pressure in the lubricating film through the inlet boundary condition. In Fig. 9, the load carrying capacity of the journal bearing which is normalized by 6pwrO3Z/C2 is plotted against the inertia parameter. In Fig. 10, that of the thrust bearing is plotted, in which the load carrying capacity is Y"
Fig. 6
Calculated example of the film pressure profile for the thrust bearing sector Pad
Fig. 5 is a typical example of the film pressure profile over the loaded pad for the journal bearing. The operation condition is as f01 lows ; The eccentricity ratio E = 0.5 The inertia parameter Re* = pwC2/p = 1.0 where C denotes the radial clearance. Strictly speaking, the results are not the same
145 normalized by 6 ~ w r , ~ / h ~ , ~ .As the inertia parameter increases, the load carrying capacity increases in the early stage. It reaches to its maximum, and decreases if the inertia parameter becomes large. At a certain stage, as found in
Fig. 10, it decreases to the value for the inertialess case, in other words, the inlet pressure jump disappears. After this stage, the inlet pressure is calculated as a negative value, and the load carrying capacity becomes less than the value for the inertialess case. This is shown by the dotted lines in Fig. 10. This notes that the developement of the flow upstream the leading edge is insufficient to produce a fully developed film flow just downstream the leading edge. Under such a condition, the theoretical model proposed above loses its foundation. This can be a subject of further speculation. 4
COMPARISON WITH EXPERIMENTAL MEASUREMENTS
To verify the usability
0
Fig. 9
0.5
1 Re* =
1.5
of
the proposed model,
2
pwC2/p
-
Influence of the inertia parameter on the load carrying capacity of the journa 1 bearing
I -
(a) Non-preloaded, center-pivoted tilting pad journal bearing
0
0.1
0.2 0.3 3.4 Re* = pwh202/p
0.5
Fig. 10 Influence of the inertia parameter on the load carrying capacity of the thrust bearing
(b) Fixed-tapered, sector pad thrust bearing Fig. 11
Test bearing configuration
146 This may be caused by the fault in estimation of the viscosity of the lubricant used. The viscosity was estimated by using the oil bath temperature which could be a little lower than the film temperature. The error could be pronounced under higher speeds. Such details aside, the results of experimental measurements seem to support the theoretical results.
the calculated film pressure profiles are compared with experimental measurements. The test bearings are the same types as those subjected to calculation in the preceding chapter. Their schematics are shown in Figs. 11 (a) and (b). The radial clearance of the journal bearing, C, is 0.60 mm which is equal to the difference between the radii of curvature of the pad arc and journal surface. The clearance ratio is 0.005. As for the thrust bearing, the film thickness at the center of the trailing edge of the pad, hZo, is set at 0.208 mm, and the thickness at the center of the leading edge, h l o , is set at 0.408 mm, s o that the circumferential taper, a, is 0 . 9 6 , and the clearance ratio is 0.004. The radial taper, B , is set at zero. The objective of the measurements is to extract the influence of the inertia forces on the pressure profile. To avoid experimental difficulties such as thermal problems, larger film thickness and slower sliding speed than those of usual bearing systems are used. The results are plotted in Fig. 12 for the journal bearing, and in Fig. 13 for the thrust bearing. Solid lines in both figures represent the results of calculation. The lower plots in Fig. 13 show the radial profiles of the pressure just downstream the leading edge, which are close to the profiles of the inlet pressure jump. In these plots, the data for larger values of Re* tend to decrease slightly, while the calculated results tend to increase slightly.
Re*
CONCLUSIONS
A simple theoretical model has been presented to
analyse the inlet pressure jump in the submerged multi-pad journal and thrust bearings. The model is based on the assumption of conservation of mechanical energy of the lubricant flow through the entrance region of the pad. To calculate the mechanical energy at the reference point just upstream the leading edge, the velocity profile of the flow there is estimated by applying the concept of momentum integral method to a control volume settled between the subject and preceding pads. Data from the experimental measurements of the film pressure profile for both the journal and thrust bearings show that the theoretical model correctly predicts the influence of fluid inertia forces on the generation of pressure in the lubricating film through the inlet pressure jump For higher values of the inertia parameter, this model yields a negative pressure jump,
.
0.8
Re*
0.35
E
=
E =
h
5
1.0
Re*
0.34
E
= =
= =
1.2
0.34
N
0
Re*
=
Re* = 0.15
0.05
h
N
0
3
u l
17.5 0 deg.
35
4-
x
n 0.5
0.75
1
P/PO
Fig. 13 Experimental and calculated, circumferential film pressure profiles at the center of radial extent, and those radial profiles near the leading edge in the thrust bearing sector pad
147 which means that the developement of the impinging flow estimated is insufficient to produce a fully developed film flow in the pad. Though the experimental measurements in the present work could not certify such a phenomenon because of the limitation in the rotational speed of the test rigs, this should be a subject of further speculation in submerged multi-pad bearings, including sertification of its existence. 6 ACKNOWLEDGEMENT The authors are pleased to acknowledge help of Mr. S. Suemori, the former graduate student of Kyoto University, and Mr. K. Ikei, the graduate student of Kyoto University, in the experimental works. REFERENCES "Fluid Lubrication: Turbulent and Related Phenomenal', ASME, Journal of Lubrication Technology, Vol. 96, No. 1, 1974. "Super-Laminar Flow in Bearings", Proc. of the 2-nd Leeds-Lyon Symposium on Tribology, Lyon, 1975, Mechanical Engineering Publication Ltd., 1977. SMALLEY, A.J., VOHR, J.H., CASTELLI, V. and WACHMANN, C., "An Analytical and Experimental Investigation of Turbulent Flow in Bearing Films Including Convective Fluid Inertia Forces", Trans. ASME, Journal of Lubrication Technology, Vol. 96, No. 1, 1974, 151-157. KING, K.F. and TAYLOR, C.M., "An Estimation of the Effect of Fluid Inertia on the Performance of the Plane Inclined Slider Thrust Bearing With Particular Regard to Turbulent Lubrication", Trans. ASME, Journal of Lubrication Technology, Vol, 99, NO. 1, 1977, 129-135. LAUNDER, B.E. and LESCHZINER, M., "Flow in Finite-Width, Thrust Bearings Including Inertial Effects I- Laminar Flow", Trans. ASME, Journal of Lubrication Technology, Vol. 100, NO. 2, 1978, 330-338. LAUNDER, B.E. and LESCHZINER, M. A., "Flow in Finite-Width Thrust Bearings Including Inertial Effects 11-Turbulent Flow", Trans. ASME, Journal of Lubrication Technology, Vol. 100, NO. 2, 1978, 339-345.
(7) PINKUS, 0. and LUND, J.W., "Centrifugal Effects in Thrust Bearings and Seals Under Laminar Conditions", Trans. ASME, Journal of Lubrication Technology, Vol. 103, No. 1, 1981, 126-136. (8) MORI, A., TANAKA, K. and MORI, H., "Effects of Fluid Inertia Forces on the Performance of a Plane Inclined Sector Pad for an Annular Thrust Bearing Under Laminar Condition", Trans. ASME, Journal of Tribology, Vol. 107, NO. 1, 1985, 46-52. (9) PAN, C. H. T., "Calculation of Pressure, Shear, and Flow in Lubricating Films for High Speed Bearings", Trans. ASME, Journal of Lubrication Technology, Vol. 96, No. 1, 1974, 80-94. (10) CONSTANTINESCU, V. N., GALETUSE, S . and KENNEDY, F., "On the Comparison Between Lubrication Theory, Including Turbulence and Inertia Forces, and Some Existing Experimental Data", Trans. ASME, Journal of Lubrication Technology, Vol. 97, No. 3, 1975, 439-449. (11) TIPEI, N., "Flow and Pressure Head at the Inlet of Narrow Passages, Without Upstream Free Surface", Trans. ASME, Journal of Lubrication Technology, Vol. 104, No. 2, 1982, 196-202. (12) MORI, A., IWAMOTO, M. and MORI, H., "Performance Analysis of a Plane Inclined Slider Pad Relative to Fluid Inertia Forces-2nd Report: Solution for Infinite Width in Consideration of Ram-Pressure at the Leading Edge", Journal of JSLE International Edition, No. 5 , 1984, 77-82. (13) MORI, A., IWAMOTO, M. and MORI, H., "Performance Analysis of a Plane Inclined Slider Pad Relative to Fluid Inertia Forces-3rd Report: Numerical Solution for a Finite Width Pad", Journal of JSLE International Edition, No. 5, 1984, 83-88. (14) MORI, A., IWAMOTO, M. and MORI, H., "Performance Analysis of a Plane Inclined Slider Pad Relative t o Fluid Inertia Forces-4th Report: Experimental Verification", Journal of JSLE International Edition, No. 6 , 1985, 89-94. (15) TICHY, J.A. and CHEN, S.H., "Plane Slider Bearing Load Due to Fluid Inertia-Experiment and Theory", Trans. ASME, Journal of Tribology, Vol. 107, No. 1, 1985, 32-38.
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This Page Intentionally Left Blank
149
Paper V(ii)
Pressure boundary conditions at inlet edge of turbulent thrust bearings Hiromu Hashimoto and Sanae Wada
A discussion is presented concerning three kinds of pressure boundary conditions at the inlet edge of hydrodynamic thrust bearings under the turbulent operating conditions including the inertia effects. The one-dimensional turbulent lubrication equation considering the inertia effects is solved for infinitely long slider bearings, in which three kinds of pressure boundary conditions are applied at the inlet edge of the bearings in accordance with three types of lubrication conditions, namely: flooded condition, overflooded condition and starved condition, and an analytical solution for each condition is presented f o r the plane slider bearings. Some results are indicated in graphical form and the relationships between three kinds of pressure boundary conditions and the static characteristics of the bearings are discussed.
1 INTRODUCTION In recent years, with an increase of speeds and sizes in rotating machinery, hydrodynamic thrust bearings are increasingly operating in the turbulent flow regime, and the effects of turbulence on the performance characteristics of the bearings have come to be analyzed by several investigators (1-4). It is often the practice, in the analysis of the turbulent thrust bearings, to assume that the film pressure is equal to the ambient pressure at the inlet edge of the bearings. Such an assumption may be reasonable when the supply flow-rate is a proper quantity. However, when the supply flow-rate is too much or too little, the assumption is not applicable. In the former case, a pressure jump (ram-pressure) may be occur at the inlet edge of the bearings and the pressure distribution will be changed significantly. On the other hand, in the latter case, any pressure does not develop until the film thickness-has decreased sufficiently to establish a full film. Pan (5) and Constantinescu ( 6 ) demonstrated a method to estimate a ram-pressure by considering the conservation of mechanikal energy-of lubrican; flows in the vicinity of the inlet edge. Utilizing the Pan's method, Moii et al. theoretically deter-mined the effects of ram-pressure on the static and dynamic characteristics of plane slider bearings in the laminar flow regime-(7) and they also examined the effects experimentally (8). Moreover, Burton (9) and Tichy et al. (10) simplified Pan's method by approximating the velocity head of incomming iubi-icant flow-with the modified coefficient determined experimentally. Constantinescu (11) and Bonneau et a1.(12) discussed the conditions of film formation when the inlet region Of the bearings is inadequately filled with lubricant. As mentioned above, considerable investigations have been presented in the treatment of inlet boundary conditions. However, there are few papers attempting to discuss the conditions systematically in accordance with a quantity of the supply flowrate. In the present paper, three kinds of pressure boundary conditions at the inlet edge of the high
speed thrust bearings operated in the turbulent flow regime are discussed in accordance with three types of lubrication conditions and an analytical solution for each condition is derived for the infinitely long plane slider bearings. Some results are indicated in graphical form and the relationships between three kinds of pressure boundary conditions and the static characteristics of the bearings are examined f o r a wide range of Reynolds numbers.
1.1 Notation A
Film thickness ratio ( = h o / h z )
GX
Turbulent coefficient
h
Film thickness
H
Normalized film thickness ( = h / h Z )
h,
Mean film thickness ( = ( h , + h z ) / Z . ) Normalized mean film thickness (=h,/hz)
H, 1 '
D
Slider length Film Dressure
P
Normalized film pressure (=h22(p-pa)/(6vu,l))
D, =a
Ambient pressure
qs
Supply flow-rate
qt
Lubricant flow-rate under the flooded condition
Q
Normalized lubricant flow-rate ( = q / ( h 2 U S ) )
Re Rh
Mean number (=phmus/U) Local Reynolds number (=ohus/u)
Re* ua
Modified number ( = p h m 2 u s / ( l J l1) Velocity of inccoming lubricant flow
us
Mean Sliding velocity
Lubricant flow-rate
Of
lubricant
Load carrying capacity k,
Narmalized load carrying capacity
150
(=h2
2 ~ / ( 6 ~ 1~ s l
x
Coordinate in the sliding direction
X
Normalized coordinate in the sliding direction (b) Overflooded condition; when the supply flowrate qs is larger than the theoretical inlet flow( = ( x - x o )/I 1 Location of pressure center Coordinate in the film thickness direction
xc y
Viscosity of lubricant
5
Supply flow-rate coefficient ( = q s / q t ) Effective coefficient for velocity head of incomming lubricant flow
q
Density of lubricant
p
( ) o Refer to the inlet edge of the bearing ( (
Refer to the starting point of fluid film under the starved condition )2
Refer to the outlet edge of the bearing
-
2 THEORY I
2.1 Pressure gradient equation The momentum and continuity equations for steady, incompressible turbulent flows in the infinitely long thrust bearings with an arbitrary film geometry are approximated respectively in terms of mean flow velocity as. follows: d p-(hum2) dx
=- h
d ( u m dxa - 2hGx
W
R-XO
-9)
(1)
E
where the turbulent coefficient Gx is given by the following form (13).
1
- 12( 1 + aRh[J)
4
h
(3)
In equation(3), the coefficients a and f3 are determined in accordance with the ranges of local Reynolds number, Rh, and ~ ~ = 0 . 0 0 0 6 98=0.95 , for Rh 2 104(13). Integrating equation(2) with respect to x , we have : U m = h
I
x=x2
(a) Flooded condition ( q s = q t )
h
G -
S
1
x=xo
usus
I
X'X2
(b) Overflooded condition ( q s > q t )
(4)
where 4 is a lubricant flow-rate to be estimated by the pressure boundary condition as mentioned in the latter section. Substituting um from equation(4) into the momentum equation(l), the pressure gradient equation is obtained as follows: (5)
2.2 Pressure boundary conditions In the analysis of the pressure gradient equation ( 5 ) , the following pressure boundary conditions are considered in accordance with three types of lubrication conditions. (a) Flooded condition; when the supply flow-rate qs is a proper quantity and exactly equal to the theoretical inlet flow-rate q t , the pressure distribution will take the form as shown in Fig.l(a). In such a case, the pressure boundary conditions are given as:
(c) Starved condition ( q s < q t ) Fig.1 Relationships between lubrication conditions and pressure distributions
151 rate under the flooded condition q t , the excess lubricant flows along with the front edge of bearing pad and the pressure jump due to the inertia effects (ram-pressure) may be induced at the inlet edge as shown in Fig.l(b). For a small sliding velocity, the ram-pressure is negligibly small, but for a large sliding velocity the pressure is not negligible as compared with the film pressure developed in the bearing. Applying Bernoulli's theorem to the flows in the vicinity of the inlet edge, we have: p a t ~P~ U S z = p O t P - u m O z; rlSl
2
P + +rlus2
- - pP u s z
= 0
(14)
Moreover, integrating equation(5) under the conditions(9), the pressure distribution for the starved condition is given as follows:
(7)
where p o denotes the ram-pressure and rl the "effective coefficient" for the velocity head of incomming lubricant flow to be determined experimentally relating mainly to a quantity of supply flow-rate and the velocity profile of incomming flow (9,lO). From equation(7), the pressure boundary conditions for the overflooded condition are given as follows: p(x =xo) =pa
where 4 is estimated from ,the following quadratic equation in q .
- umo2)
(15) where q t is the theoretical inlet flow-rate under the flooded condition determined from equation (12) and the location of starting point of fluid film, X I , is estimated numerically from the following equation.
(8) p(x =X
Z )
= Pa
(c) Starved condition; when the supply flow-rate qs is smaller than the theoretical inlet flow-rate q t , any pressure does not develop until the film
thickness has decreased sufficiently to establish a full film as shown in Fig.l(c). The pressure boundary conditions for such a case are given as: p(x=xl) = p a , P(x=xz)=Pa
(9)
2.4 Analysis for infinitely long slider bearings
where the location of starting point of fluid film, The film thickness of infinitely long slider bearx1,is unknown and determined by the following coning as shown in Fig.2 is expressed as follows: tinuity condition. h = ho + v ( x - x o ) (17) q(x=x1)=Sqt ; 561 (10) where 5 is a "supply flow-rate coefficient", which means a ratio of supply flow-rate qs to the theoretical inlet flow-rate under the flooded condition q t . 2.3 Pressure distribution
Substituting h from equation(l7) into equations (11) to (16) and normalizing the results by means of the nondimensional quantities listed in Section 1.1, the normalized pressure distribution for each lubrication condition is obtained. The normalized pressure distribution for the flooded condition (Q,=Q) is given as:
Integrating the pressure gradient equation(5) with the conditions(6), the pressure distribution for the flooded condition ( q s = q t ) is obtained as follows:
where H and y are given respectively as follows: where q t is estimated from the following quadratic equation in q t .
Integrating equation(5) with the conditions (8), the pressure distribution for the overflooded condition ( q s > q t ) is given as follows:
H=A+(l-A)X
X
t?r
I,/ fk h2
Y i f f I f / I
(19)
m / / ,/,, us
XO
dx h2Gx
Fig.2 Geometry of inclined slider bearing
152
3 RESULTS AND DISCUSSION
and Qt is determined from the followingquadratic equation in Qt.
-* fir+(+ * & 1 -m)}Qt
1('
=o
1
+&
1 (1 -;ir=~)} (21)
The normalized pressure distribution for the overflooded condition (Qs>Qt) is: p =-
-+ tI ) (*-&I
+ L { ' +- (* A-1 H
]
}-)*A
Re* + 12Hm2(v
(22)
where Q is determined from the following quadratic equation in Q .
The calculated results of the static characteristics such as film pressure, lubricant flow-rate, load carrying capacity and location of pressure center are presented in the following figures for the case of A=2.0 and h2/1=0.002. Figure 3 shows the variation in the normalized lubricant flow-rate with mean Reynolds number under the flooded condition. The flow-rate neglecting the inertia effects is affected insignificantly by the Reynolds number but the flow-rate considering the inertia effects gradually decreases with an increase of Reynolds number. When the supply flow-rate Qs exceeds the theoretical flow-rate Qt indicated in Fig.3, the lubrication condition changes from "flooded" to "overflooded". On the other hand, when the supply flow-rate becomes less than Q t , the lubrication condition changes to "starved". In Figs.4 to 7, the calculated results for the overflooded condition (Qs>Qt) are presented for various values of effective coefficient Q. Figure 4 shows the normalized pressure distribution for Re=5x103. For Q=l, which is a maximum value of 0 and corresponds to the case that the velocity profile of incomming lubricant flow is uniform in the film thickness direction and nearly
a
t
- With incrtk effects ---
Without inertia effects
And the normalized pressure distribution for the starved condition (Qs
I
1
I
I
I
IIII
I ,,Ill
I
50 I00 X l @ 5 10 Mean Reynolds number R
Fig.3 Variation in the normalized lubricant flow-rate with mean Reynolds number under the flooded condition (A=2.0, h z / l =O. 002) where the normalized location of starting point of fluid film X l , which is implicitly included in equation(24), is determined numerically from the following equation.
t
0 1-8
1
-
=
0
(25)
The normalized load carrying capacity, W, and the location of pressure center, X c , are given respectively by integrating the pressure distribution as follows:
1 W = Ix*Pd X
Ix* 1
,
X, =
0
XPdX'W
(26)
where X*=O for the flooded and overflooded conditions and X*=X1 for the starved condition, respectively.
02
04
06
08
1
Normallzed dlslance X
Fig.4 Normalized pressure distribution under the overflooded condition (Re=5x103, A=2.0, h 2 / 1 =O .002)
153
equal to the sliding velocity u s , the magnitude of the ram-pressure is considerably large compared with that of fluid film pressure. As the coefficient q decreases the ram-pressure becomes smaller and the pressure distribution approaches to the distribution for the flooded condition. Figure 5 shows the variation in the normalized load carrying capacity with mean Reynolds number. As the effective coefficient q approaches to unity the magnitude of the ram-pressure becomes larger, and then the load carrying capacity under the constant Reynolds number increases considerably. Figure 6 indicates the variation in the normalized lubricant flow-rate with mean Reynolds number. The flow-rate under the overflooded condition is larger than that under the flooded condition and the tendency becomes pronounced as the Reynolds number Re and the effective coefficient q increase. The relationships between the location of pressure center and mean Reynolds number are illustrated in Fig.7. The location of pressure center under the flooded condition is affected insignificantly by the Reynolds number. The location under the overflooded condition moves forward in the bearings, and the tendency becomes pronounced as the Reynolds number Re and the effective coefficient q increase. The minimum value of the effective coeffi-
cient, r$,,in, is obtained by putting P(x=Xo)=Pa in equation(8), and the result is given as follows :
Then, it may be considered that the effective coefficient q exists within the following range. Figure 8 shows the variation in the minimum effective coefficient qmin with mean Reynolds number for different values of A . The value of qmin gradually decreases with an increase of Reynolds number and it becomes finally constant when the Reynolds number exceeds 5 ~ 1 0 ~ . The calculated results for the starved condition (&
2.0
3
g 0.8
h
c
'ij 1.5
8
9
t
2 0.6 0,
0
-8 1.0 -P.-
Flooded
a5
9 1
I
I
I
I 1 1 1 1
5 10 50 Mean Reynolds number
100 Re
0
x'
Fig.5 Variation in the normalized load carrying capacity with mean Reynolds number under the overflooded condition (A=2.0, h z / l
I 1 1 1 1 1
1
1 1 1 1 1 1
5 10 50 loo Mean Reyndds number Re
x
13
Fig.7 Variationin the locationof pressurecenter with mean Reynolds number under the overflooded condition (A=2.0, h2/1=0.002) ,
=0.002)
I
I
I
I
5
I
,,,I
I
I
I
I
I
,,,I
10 50 100 Mean Reynolds number Re
I xi03
Fig.6 Variation in the normalized lubricant flow-rate with mean Reynolds number under the overflooded condition iA=2.0, h 2 / 1
=o .002)
Mcan Reynolds nu*
Rc
Fig.8 Variation in the minimum effective coefficient with mean Reynolds number ( h 2 / l
=o. 002 )
154
neglecting the effects is gradually extended with a decrease of 5 because the location of starting point of fluid film slightly moves backward due to the inertia effects. Figure 10 shows the variation in the location of starting point of fluid film with mean Reynolds number. For all values of 5 indicated in the figure, the location gradually moves forward with an increase of Reynolds number and it approaches to constant when the Reynolds number exceeds lo". On the other hand, under the constant Reynolds number, the location rapidly steps backward with a decrease of 5. The inertia effects slightly moves the location backward for a wide range of Reynolds numbers. Figure 11 shows the variation in the normalized load carrying capacity with mean Reynolds number. As the supply flow-rate Qs decreases with a decrease of 5, the load carrying capacity under the constant Reynolds number decreases considerably. For all 5 indicated in the figure, the load carrying capacity increases rapidly with an increase
0.16
1
Without inertia effects
-----Wlth inertia effects
of Reynolds number. In the case ofe=l, which corresponds to the flooded condition, the inertia effects increases the load carrying capacity for a wide range of Reynolds numbers, but the effects becomes smaller with a decrease of 6 . Figure 12 shows the variation in the location of pressure center with mean Reynolds number. The turbulent and inertia effects are insignificant in the location of pressure center. The coefficient 5 , however, affects significantly the location and the location considerably moves backward as 5 decreases. In Fig.13, the theoretical pressure distributions under the overflooded condition are compared with the experimental results by King and Taylor (14). In their experiment, King and Taylor measured the pressure distributions and the film profiles under the turbulent condition for a fixed, 25mm square inclined pad, in which the water was pumped to the leading edge of the bearing pad as the lubricant. In the calculation of pressure distributions, the value of the effective coefficientn=0.55 was determined by utilizing the measured ram-pressure at Re=1750 for A=2.0 and Re=3108 for A=3.0, respectively. The good agreement was obtained between the theoretical and experimental results.
€4.0
0 14
0 12 a golo
Without inertia effects
r
With inertia effects
h008
-E
r
- 006 z 004 002
Mcan Reynolds number Re
o
a2
04
06
08
11
Normalized dislance X
Fig.9 Narmalized pressure distribution under the starved condition (A=2.0, h 2 / l =0.002 , Re=5x103)
Fig.11 Variation in the normalized load carrying capacity with mean Reynolds number under the starved condition (A=2.0. h s / l
=O.OOZ)
._I
J
-----___
L
E*OB
i08 ;
0.85 0.9 0.95 1.0
--_____,
U
u
5 0.6
-
!
6 04C 0
-
7 0.2
-
-
Without inertia effects
--_--With inertia effects
8
-I
:
01 1
I
1
I
I , , , , ,
,
, ,
I
, , , ,I
10 50 100 X I ? Mean Reynolds number Re
5
Fig.10 Variation in the location of starting point of fluid film with mean Reynolds number under the starved condition (Az2.0 h2 /1=O. 002)
0
I
I #
I
I
I
I
. ,,,!
I
Fig.12 Variatio in the location of pressure center with mean Reynolds number under the starved condition (A=2.0, h2/1=0.002)
155
4 CONCLUSIONS
-
0.28
In the present paper, three kinds of inlet pressure boundary conditions for high speed thrust bearings operated in the turbulent flow regime are discussed in accordance with three types of lubrication conditions. The analytical expression of pressure distribution f o r each pressure boundary condition is derived and the static characteristics such as film pressure, load carrying capacity, lubricant flow-rate and the location of pressure center are calculated for a wide range of Reynolds numbers. The following two conclusions are obtained from the calculated results: 1. In the case of overflooded condition, the film pressure, load carrying capacity and lubricant flow-rate are considerably larger than those under the flooded condition, and the location of pressure center moves forward in the bearings. 2. In the case of starved condition, the location of starting point of fluid film and the location of pressure center move backward in the bearings, and the film pressure and the load carrying capacity decreases considerably.
1 Turbulent theory(P055) Laminar theory ('1.0.55)
0.2 0.4 0.6 0.8 Normalized dislance X
1
(a) A=2.0, h2/1=0.003)
References 0.28
,
CAPITA0,J.W. 'Influence of turbulence on - Turbulenl thcory(PQ55) -_Laminar lheory (7455) performance characteristics of tilting pad --0-Experlmenl('4~ thrust bearings', ASME Journal of Lubr. Tech., 1974, 96, No.1, 1'10-117. a 0.20 HUEBNER,K.H. 'Solution for the pressure and 3 temperature in thrust bearings operating in w the thermohydrodynamic turbulent regime', k 0.16 ASME Journal of Lubr. Tech., 1974, 96, No.1, E .58-68. c 0 KING,K.F. and TAYLOR,C.M. 'An estimation of R 0.12 .the effect of fluid inertia on the performance of the plane inclined slider thrust bearing with particular regard to turbulent lubriz" 0.08 cation', 1977, 99, No.1, 129-135. HASHIMOT0.H. and WADA,S. 'Turbulent lubrica0.04 tion of tilting-pad thrust bearings with thermal and elastic deformations', ASME Journal of Tribology, 1985, 107, No.1, 1290.2 0.4 0.6 0.8 I 135. PAN,C.H.T. 'Calculation of pressure, shear, Normalized distance X and flow in lubrication films for high speed (b) Az3.0, h2/1=0.002) bearings', ASME Journal of Lubr. Tech., 1974, 96, No.1, 80-94. CONSTANTINESCU,V.N., GALETUSE,S. and KENNEDY, Fig.13 Comparison of theoretical andexperimental pressure distributionsunderthe overflooded F. 'On the comparison between lubrication condition theory, including turbulence and inertia forces, and some existing experimental data', ASME Journal of Lubr. Tech., 1975, 97, No.3, 107, No.1, 32-38. 439-449. CONSTANTINESCU,V.N. 'On some starvation MORI,A., IWAMOT0,M. and MOR1,H. 'Performance phenomena in fluid films', ASME Journal of analysis of a plane slider pad relative to Lubr. Tech., 1977, 99, No.4, 441-448. fluid inertia forces (2nd report, Solution BONNEAU,D. and FRENE,J. 'Film formation and for infinitely width in consideration of flow characteristics at the inlet edge of ram-pressure at the leading edge)', Journal starved contact-theoretical study', ASME of JSLE, 1983, 28, No.9, 677-684 (in JapaJournal of Lubr. Tech., 1983, 105, No.2, nese). 178-186. MORI,A., IWAMOT0,M. and MOR1,H. 'Performance HASHIMOT0,H. and WADA,S. 'An influence of analysis of a plane slider pad relative to inertia forces on stability of turbulent fluid inertia forces (4th report, Experimental journal bearings', Bulletin of the JSME, verification)', Journal of JSLE, 1984, 29, 1982, 25, N0.202, 653-662. No .6, 435-442 (in Japanese). BURTON,R.A 'Approximations in turbulent film (14) KING,K.F. and TAYLOR,C.M. 'An experimental investigation of a single pad thrust bearanalysis', ASME Journal of Jubr. Tech., 1974, ing capable of operating in the turbulent 96, No.1, 103-109. lubrication regime', Superlaminar flow in TICHY,J.A. and CHEN.S.-H. 'Plane slider bearbearings, Proc. of 2nd Leeds-Lyon Symposium, ing load due to fluid inertia-experimental Lyon, France, Sept. 1975, 149-153. and theory', ASME Journal of Tribology, 1985,
-
-
This Page Intentionally Left Blank
157
Paper V(iii)
Dynamic analysis of tilting pad thrust bearings A. Benali, A. Bonifacie and D. Nicolas
Pivoted-pads thrust bearing are commonly used to support axial load in rotating machinery. Analysis of such bearing is complicated by the ability of each pad to pivot, as the rotor is displaced and to translate axially due to the flexibility of the pad support. This leads to dependance of linearized stiffness and damping coefficients on rotor vibrational frequency, pad inertia and support dynamic characteristics. This paper describes an approach based on the pad assembly technique often used to develop pad journal bearings. A reduced matrix ( 3 , 3 ) can be developped once a vibrational frequency is assumed. Conclusions on the role of pivot support and mass pad are derived. 1
INTRODUCTION
Tilting pad journal and thrust bearings are commonly used in rotating machinery, in particular to avoid oil whirl stabilities and to accomodate misalignment and distortion effects. The first researcher to evaluate the linearized dynamic coefficients of a tilting pad journal bearingwas Lund 11 I . Since many papers on dynamic behavior of tilting pad journal bearings have been puplished. These studies show the good stability of tilting pad journal bearings and the role of some parameters as preload, pad inertia, frequency or pivot flexibility on the stiffness and damping coefficients 12-4 I In the thrust bearingscasethe studies have been carried out aimed to gain knowledge about bearing oil film under running static conditions. The effects of turbulence, thermal and elastic deformation were presented by several investigators 15-7 I More recently Shapiro and all have presented the concept of fluid equalized tilting pad thrust bearing where load augmentation is introduced by external presurization through the pad pivots
.
.
181.
This paper presents a linear dynamic analysis of tilting pad thrust bearing with damped flexible supports. The dynamic behavior is characterized by a set of reduced dynamic coefficients which depend on pad mass, stiffness and damping support and frequency of vibration. 1.1
P
Z direction pressure
m
pad mass
Kij
C
S
damping coefficient in the ith coordinate direction due to perturbation in the jth coordinate direction (Cij = c who/Wo) support damping
(c
=
Cs who/Wo)
Ix’ Iy pad moment inertia e pad thickness at pivot line P h film thickness minimum oil film thickness hmin hok
film thickness at pivot line
‘lk
pad translational degree of freedom in direction
‘2
rotor translational degree of freedom in
Z
2 m ho w /Wo)
stiffness coefficient in the ith coordinate direction due to perturbation in the jthe coordinate direction (Tij= K. .ho/Wo) 1J
KS
r
radius
R.
inner radius of pad
Re R P
pivot radius
outer radius of pad load per pad (Go
=
2
Wo ho /pwR
wO
4 ) P
%+zk
pad coordinate system
X,Y,Z
reference frame
ak’’k
pad inclinations in the xk,ykdirections
y,
6
rotor inclinations in the X,Ydirections
U
frequency of oscillation
P
dynamic viscosity
‘k Ic,
0
‘ij
=
support stiffness (Es = Ks ho/Wo)
$P
Notation
(m
2
(7 =
u/w)
angular pivot location from axis
x
pad arc extent angular pivot location from leading edge rotor angular velocity TRHUST DESCRIPTION AND ANALYSIS
Fig 1 shows a schematic representation of the thrust consists of a runner which rotates at a constant angular speed w and tilting pads (k) with pivots (0 ) which are mounted on a plane througth a combfiant support (ks, cs). It is assumed that the pads and the runner are rigid and plane. The reference frame (X, Y, Z) has its origin located in the center 0 of the pad carrier ring with the Z axis normal toring face. The local pad coordinate systems (xk, yk, z ) with x axis colinear to 00 is also fixed.kThe circumferential coordinateP\ is taken counterclockwise from the axis x :
158 E
(-JIp,
a11 kinematic variables :
9)
and the radial coordinate r from the origin 0 : r E (Ri, Re) The coordinate of the pivot position is denoted by (R , JI ) . Each pad can tilt in boththe radial (a , gk) Pnd circumferential ( B , yk) directions an8 translate along axial direction (L1 2). The runner has its rotation axis inclined to the axis ( 2 ) . y and 6 represent the tilt in the X et Y directions. In normal running conditions y and 6 are nil. The film thickness at any point M 1 on pad number k (1 < k < N) is given by
tl)
( 1 ) hlk(r,e) = L l k + e + (R -rcose) BR+rak sine
P where e is the pad thickness at the pivot line and L the distance between the pivot 0 and the pA8 carrier ring. Pk The film thickness at any point M2 on the runner is : ( 2 ) h2(r,8) = R2 - r 6 cos(8+$k)+rysin (8+Ok)
F=F(R2Y '2, Y, -+ (5) Mo2
.
-+ X = M -+
-+
.Y
oo2
The governing equations are derived under the following assumptions usually adopted in lubrication theory : - incompressible, newtonian fluid - isothermal, laminar flow - body forces and fluid film inertia forces neglected - elastic deformation of surfaces neglected The Reynolds equation is :
AF = F - F
I
-ra sin e - (R - r cos e l . ki k P where h = h2(r, 8 ) - hlk(r, 0 ) is the film thickness at any point on pad number k in the thrust bearing. In the normal steady state running conditions (y = 6 = 0) each pad has the same equilibrium position characterized by a = a 0' B = B o y i l k= R l 0 . Force and moment exerted by lubricant on runner are :
%,
ap
Bk)
(. ..)
=
+ _ aF AR
= X A L 2 O
8112 n + I : t- aF ~
k=i
aa.
all2
2
Alllk+-
lk
+-aF Ay + aY
...
-
aF
Alllk +
... ]
aLlk
If the vibrational frequency can be assumed the degrees of freedom can be reduced by the pad assembly technique commonly used for tilting pad journal bearings. First, pad data are calculated for a fixed pad and then the pad are assembled to form a complete thrust bearing. The degrees freedom of each pad are eliminated in terms of the degrees of freedom of the runner. The stiffness and damping matrix order may be reduced to (3x3). 3
. . +12r t i 2 - ~ l k + r ; s i n ( ~ + $ k ) - r i c o s ( e + g k ) -
& 'ilk, 'lk,
M (. ..) Y Hence a fully calculation of the thrustrunner system would involve all of the runner degrees of freedom (3) as well as the pad degrees of freedom (3n). For small excursion of the runner about its equilibrium position, the non linear functions (5) can be linearized with adequate accuracy for most practical purposes. The force and moment can be represented by their first order Taylor series expansion e.g. Mo2
where R2 is the film thickness on the 2 axis : L2 =
X
6 Y
DYNAMIC COEFFICIENTS OF A SINGLE PAD
The properties of a single pad are calculated in the pad coordinate system aF.d the-rotor motion is characterized by the tilt y and 6 in the xk and yk directions and the translation ll2k along pivot line. Transformation matrix relating local degrees of freedom to global rotor degrees of freedom is :
where $k is the angular pivot location k from X axis. Reynolds equation is given by :
a --ar There is an optimun pivot position where, for a minimal film thickness given, the load capacity is maximum. When the runner and pad carrier ring axis are misaligned, the flexibility of the pad support tend to equalise the load distribution among the pads in the bearing. The more loaded pad will tend to force the support inward and reciprocally the less loaded support will force the pad outward to help pick up the load 191. In dynamic regim, during the motion of the runner about its equilibrium position defined in ( 4 ) , the pad will pivot and translate according to the condition determined by the dynamics of the whole system :
-
Eu1er"s equation for runner and for each pad Reynolds'equation for lubricant ;the lubricant force and moment components are function of
a a IF3 P arIa e IKkI=-6r pr ae (8k-6k)c0s
+ 12 r t izk-Rlk-
2
wl(ak-yk)sin8
+
eI r(&k-fk)sin8-(R
-rcos0Mb-ik) P We note that the geometric parameters are arranged in pairs. At the static equilibrium position the lubricant force on the runner is :
with h = R-2k-R1
k-e-r(ak-Yk)sine-(R
-rcosO) P
159 and for small perturbations about static equilibrium
whi th
is the reduced impedance ; the reduced dyne coefficients are given by :
zi. mid
Where k , , are stiffness coefficients and cij are dampici coefficients of a fixed pad. e,g. 6(!?20+A!?2,0, 0, 0 . ..)- Wo kll = A!?
The dynamic coefficients may be computed by direct numerical differentiation or from a perturbation of Reynolds equation 1101. A s they are derived for a given equilibrium position, the coefficients depend on the bearing geometry and the operating parameters. If all pads are identical, the stiffness and damping matrix areidentical in their local frame. Assuming that the runner execut small harmonic motion with angular frequency v, themotion of the pad which is induced by the motion of the runner, will also be harmonic : (9)
7
=
V
ivt e
,..., ak
I=-
=
,.
ivt ak e
= - Im
ij
4
I
where z represent the impedance : z =k..+iv c ij ij ij i j Eulers equations for the pad are : m Alllk = - AF - ks Balk - c s
1 v
=
-
AM Yk
I
Ixkv2 , - I v 21 Yk the linear system (12) enables us to write system (10) in the form :
1
13
.I
= 1Pl-l
Ik I
/PI
and a similar relation for damping matrix. With the stiffness and dampingcontribution of each pad in terms of global coordinates, the assembled reduced stiffness and damDing global coordinates, coefficients are : n n cij = kgl (cij k kij = k& (kij)k If it is assumed than the pivot location are given by : R = cste, $k+l = $k + 2?/N P then the reduced stiffness terms are
Gll
k 1 2 = k13 = k21 = kjl = 0
where the matrix lMij is defined by : 2
(Z..)
(13)
These equations are written in the relative (L, yk, z) coordinate system located at the pivot point 0 k’ Substiteing (9) into (1 I ) taking into account (lo), we obtain the following linearsystem :
lMij/ = diag { - mv +zs,
1J
The pad stiffness and damping parameters are directional and their contributions to the thrust stiffness and damping matrix are thus a function of the pad location. A transformation matrix relating global degrees of freedom to local degrees of freedom is given by equation ( 6 ) . Applying the transformation from local to global coordinate, the stiffness matrix isgiven by :
k l l= n Iyk ABk
(i. .)
DYNAMIC COEFFICIENTS OF A THRUST BEARING
1J
lziJal [RI(qk-ak) -. Re ( 6k- Bk)
S
Ik.
ALZk - A!?
AGyk/Rp
= Real
These reduced coefficients identical for each pad, depend on frequency v, pad mass m,pad inertia I and I and support characteristics ks and cS. xk Yk
Equations (8) take the form
(101:xk/Rp
Gij
-
k 2 * = k33=:l
=“I k23=-k32 2
1 1 +E22+i33-’13-’311 - +k -k -g I 12 23 21 32
and a similar relations for damping. Relation (13) shows that for a small displacement around normal running conditions the “cross-coupled” terms which describe the change of oil film force due to a tilt displacement of runner are nil. The thrust is isotrope. These results are in good accuracy with static results presented elsewhere 18 I
.
5
RESULTS AND DISCUSSION
Sample results for a typical eight pads thrust bearing are presented. Dimensions and operating conditions are indicated in table 1. The pivot location (I ) corresponds to an optimum position base onpa maximum load capacity for a minimal film thickness given.
(2,
160
R.
= 0.2
1:
= 0.4
1
R P
=
0.304 m
J,
=
38"
J,
=
22.8"
=
0.08 m
P e
N
m
m
=
200tr/mn 0.03 Pa.s
=
25 kg
=
table 1 dimensions and operating conditions. The results are presented in dimensionless form. Table 2 tabulates load capacity W , equiBo and dynamic coefficients librium position a 0 ' defined in (8) for a single pad. We note than for a fixed pad : - the load carrying is affected by a variation of oil film thickness at the pivot line ( A t ) or pad inclination in the circumferential direction (Aa) but not by inclination in the radial ( A B ) . - a variation of the geometrical parameters ( 1 , a , 6) has only small effects on the pressure center position; 2.991 0.846 -0.017
1.622-0.008 0.019
0.007-0.007 0.048
0.019-0.002 0.019
table 2 : dynamic coefficients for a fixed pad The effect of frequency ratio 7 = v/w and dimensionless compliance support (ks, cs) on reduced dynamic coefficients are now examined. Fig. 2 through 4 show the dimensionless reduced coefficients plotted versus the dimensionless support stiffness Fs for various values of pad mass E, support damping cs and frequency ratio V.
- the inertiapad has a negligible effect on direct stiffness and damping. - a flexible support with small or mediumdamping has a significant effect if its stiffness is lower than ten times the oil film thickness, - direct stiffness increase and direct damping decrease when frequence ratio increases, - the dependance of cross coupled terms on inertia and pad frequency is important, - critical pad mass is very higher than this encountered in practice. REFERENCE 11
I
LUND J.W. "Spring and Damping coefficients for the tilting pad journal bearing". ASLE Trans., 7, 4, 1965, pp. 342-352.
12 I ABDUL WAHED "Comportement dynamique des pa-
liers fluides. Etude lin6aireetnon-lin6aire" These de Doctorat Es-Sciences, Lyon 1982. 131 PARSEIL K.K., ALLAIRE P.E. and BARRET L.E. "Frequency effects in tilting pad journal bearing dynamic coefficients". ASLE Trans., 26, 2, 1983, pp. 222-227. 141 ROUCH K.E. "Dynamics of Pivoted-pad journal
bearings including pad translation and rotation effects". ASLE Trans., 26, 1, 1983, pp. 102-109. 151 VOHR J.M. "Prediction of the operating temperature of thrust bearings", ASME Trans, journal of lub. techn., vol. 103, 1981, pp. 97-106. 161 HASHIMOTO H. and WADE S. "Turbulent lubrication of tilting pad thrust bearings with thermal and elastic deformation, ASME Trans, journal of Tribology, Vol. 107, 1985, pp. 82-86.
.
171 KIM K.W., TANAKA M. and HORI Y. "A three dimensional analysis of thermohydrodynamic performance of sector shaped tilting pad thrust bearings". ASME Trans., journal of lub. Techn., Vol. 105, 1983, pp. 406-412.
.
18I SHAPIRO W. , GRAHAM R. and ANDERSON G. "Predicted performance characteristics of hybrid, fluid equalized, tilting pad thrust bearing$' ASME Trans., j. of lub. techn., Vol. 105, 1983, pp. 476-483.
Fig.-2 shows the variation of the axial stiffness k Except in the case of low support stiffness add high frequency, the effect of pad mass is negligible. The stiffness increases with frequency. Similar tendancies are observed for the stiffness k Direcg'damping reduced coefficients c and c are independant of the pad mass (fig. 3j! TiZy decrease when the frequency increases and their behavior versus stiffness support depends on the damping values. The cross coupled terms k23 and c23 are function of the pad mass. If the mass is -3nil, they are also nil. For a pad mass of 10 and a small damping the "cross coupled" terms increase with damping support. The effect of frequency ratio can be quite substantial. For small and medium damping "cross coupled" terms go through a zero point andchangin signs (fig. 4 ) . Pad stability analysis show that the critical pad mass is very higher (100 or l000times) than mass encountered in industrial thrust, The pad are always stable. 6 CONCLUSION A linear analysis to determine the reduced dynamic coefficients of a tilting pad thrust bearing has been presented. This one is then used to study the effects of frequency, pad mass and compliance support on reduced dynamiccoefficients. The eight pads thrust bearing have been examined. The results shown :
19I BENALI A. , BONIFACIE A. and NICOLAS D. "Comportement d'une but6e B patins oscillants montes sur appuis flexibles". to be presented at the seventh world congress an the theory of machines and mechanisms, Sevilla, 1987. 110) LUND J.W. "Review of analytical methods in rotor bearing dynamics". Tribology Int., Vol. 13, no 3, pp. 233-236, 1980.
161
--+
Fig.
--
Y
1 : Pad s c h e m a t i c s
-
c11
-
C22--
10:.
10. .'
+*-------,
v 0.1
1 .O"
1.0
/
LO.
0.1..
1.o
1 0.1
r-1 c22
-
I
Fig.3 : d i n k = n s i o n l e s s damping C,,
-
and CtZ
versus dimensionless
-
s t i f f n e s s s u p p o r t k,
-
frequency r a t i o
LO
0.1
10'
10
--
5=o.1-
0.1-
--
1.0 10.
for three
10'
\
.
3
-c,
= 100
01 0.1
LO
103
162
-v = 10.
c 0.I
I0
-
lo3
*S
K22
4 10
14
a '
1.0"
0.1
1.
/
O.l/
c
0.1
10
103
l-
v = 10
Flg.2:
-
K1l
Dimensionless
-
stiffness
and K Z 2 versus dimensionless
-
-
stiffness support k, frequency ratio
for three
3
14 0.I
10
103
.
163
=23
I
10-2.
I
I I 1 I
I/
0.1
--
K < O
-K>O.
16
c
0. I
LO
0.1
-v
23 z
I
10.
-c,.
10
to3
LO.
103
1
104 0.1
10.
103
c
0.1
~
-
K-47
d
Y
=
cs = 100
10.
-
10-2-
1
= 10.
10
O I .
01.
-cs=loo. 10-4.
---
0.I
-
\ 0.1
\
*
104 0.1
10.
__
,061
lo3
0.1
-
P i g . 4 : Dimensionless cross coupled stiffness K
dimensionless stiffness support ksfor three
10.
frequency ratio 2 32
to3
-
and damping C versus
ts
This Page Intentionally Left Blank
165
Paper V(iv)
Hydrodynamically lubricated plane slider bearings using elastic surfaces C. Giannikos and R.H. Buckholz
Abstract The problem of compliant surface slider bearings with large slenderness ratios (1.e. the axial length/sliding length ratio is very large) is analyzed. The one-dimensional Reynolds lubrication equation is used to treat the hydrodynamic fluid film. The lubricant film is bounded on one side by an elastic layer (i.e. the bearing surface) and on the other by a translating rigid surface. The rigid surface has a uniform velocity in the sliding direction. The elastic surface is rubber; this elastic surface deforms as a consequence of the hydrodynamic pressure. The deformation of the rubber is constrained by incompressibility (Poisson's ratio equal to 1/21. Both the Reynolds equation for the pressure and the elastic field equations for deformation are linear; the nonlinearity is a consequence of the. surface traction boundary conditions. The fluid film gap thickness is unknown due to the bearing surface deformation. A boundary element solution method is used to treat the elastic deformation; this method of solution yields the boundary deformation and surface traction vector directly. An iterative calculation scheme is used to determine both the elastic deformation and the subsequent fluid film pressures. 1.
INTRODUCTION The rotating components of machines all suffer from wear. These rotating components are confined in the stationary world by several different methods; two prominent methods are rolling element bearings and fluid film bearings. Fluid film bearings are used to separate the surfaces in relative motion; typical fluids used as lubricants are water, oils, and grease. Bearing surfaces tend to deform and to disintegrate with use as a consequence of stresses, heat, and abrasive particulates suspended in the lubricant. Excessive sliding contact between the surfaces in relative motion also contributes to the wear. These factors threaten the reliability of the bearing surfaces. Studies have shown that the susceptibility of the bearing to the said factors is much reduced when plastic-coated or rubber-coated bearing surfaces are used. These bearings are often referred to as compliant surface bearings. Some limited theoretical and experimental studies on compliant bearings has been accomplished. However, a complete description of their performance related In this to their potential is not available. paper we are concerned with the mathematical modeling of both the bearing elastic deformation using linear elasticity and the fluid flow using Reynolds lubrication theory. [a] investigated the use of Fogg water-lubricated rubber bearings. In a subsequent study Benjamin [ 2 , 3 ] applied a finite difference method to study the performance of rubber journal bearings; he used a finite difference scheme to solve the coupled hydrodynamic and elastic deformation field equations. Benjamin experienced a number of computational difficulties since the Poisson's ratio is near to one-half. Hori [9] also studied elastic surface bearings; he studied
It is rubber-surf aced squeeze film bearings. worth noting that Hori's study indicated that viscoelastic effects in the rubber. deformation occured at high squeeze frequencies. Rightmire [16,17] showed the usefulness of rubber coated tilting and swing-pad bearings. A number of experimental studies have shown that rubber is a nearly incompressible solid; thus the Poisson's ratio is nearly one-half. Rightmire [15 ] studied the bulk compressibility characteristics of rubber. He designed a Poisson's ratio tester. His results showed Poisson's ratio to be effectively one-half for a number,of different rubber samples tested. Thus, the Poisson's ratio is assumed to be one-half in this analysis. The limiting behavior of the Navier elasticity equation for incompressible materials is given by Sokonikoff [18]. The resulting elasticity equation is identical to the Stokes flow field equations. The elastic deformation vector is analogous to the velocity vector in the Stokes equations. In lubrication applications the fluid inertia is usually not important. Thus the form for both the elastic deformation equations and the fluid velocity equations are the same; however, the meaning of the dependent variables is different. Buckholz [l] applied a singularity superposition method to solve the coupled flow and elastic deformation equations; the method was applied to squeeze film journal bearings. Singularity superposition methods of solution for Stokes flows are discussed by Chwang and Wu [5]. Several numerical methods are available to solve the Stokes flow field equations. In the elastohydrodynamic problem considered here the elastic boundary displacements and tractions are most important. We are less interested in the displacement and stress field within the elastic
.
166
bearing coating. The boundary element method of solution is ideal for this type of problem. The application of the boundary element method to Stokes flow problems has been reported in a number of studies; however, these applictions tend to be in the area of fluid mechanics. Youngren and Acrivos [19] have applied the boundary element method to Stokes flow past an arbitrary particle. Fairweather [7] discusses methods to improve the numerical solution for the boundary element integrals. Kelmanson [lo] presented a boundary element solution to the biharmonic equation; this is a Stokes flow problem. He also applied this solution method to study the effect of arbitrary bearing surface geometries in lubrication (Kelmanson, [ll]). Rallison and Acrivos [14] used the boundary integral method to study viscous drop deformation in extensional flows. They studied both the bounded and unbounded flows. The domain of interest here for the elastic deformation is bounded; this is analogous to an internal flow problem. The solution to the elasticity problem is reduced to the solution of a linear system of equations. The elastic deformation is first calculated by starting from an initial pressure distribution for the fluid film. The fluid film gap thickness is then updated and the hydrodynamic pressure 10 recalculated. A new elastic deformation is calculated. This process of calculation is repeated until both the pressure and elastic deformation converge. 2. ANALYSIS In hydrodynamic lubrication a fluid film pressure change is established when lubricant is swept through a changing gap geometry. This fluid film pressure depends upon the film gap geometry. The prominent component of this study is the change in the hydrodynamic gap geometry due to the deformation of the bearing surfaces. The bearing material is a uniform thickness rubber layer which is attached to a rigid backing. The momentum and mass conservation equations are used to describe the deformation. The boundary element method is used to calculate the elastic deformation. The subsequent fluid film pressure is calculated using integrals which are directly derived from the Reynolds lubrication equation. 2-A. GOVERNING EQUATIONS The elastic deformation equations are governed by the Navier linear elasticity field equations. This conservation equation is expressed in the limit as Poisson's ratio approaches one-half. This asymptotic form for the field equations is written in terms of material displacement and pressure. This resulting field equation for displacement is analogous to the Stokes flow field equations. The elasticity constitutive relationship between the stress and the deformation is given below:
follows:
(4) The occurrence of the pressure P is a direct consequence of the incompressibility constraint imposed on the solid. In general this pressure field is unknown and its determination is part of the solution. The conservation equations are given below: first the mass conservation,
*
= o
(5)
i,i and then the local balance of forces,
*
(6) w;,jj We now consider the equations of motion for the fluid flow. The lubricant is Newtonian viscous. The velocity fields and gap geometry are in a range that permits the use of the lubrication approximation. The dominant effects in the Navier+tokes equations are the pressure gradient and the viscous forces. The lubrication approximation is applied to the Stokes flow equations. The one-dimensional Reynolds lubrication equation is then obtained using mass conservation: a h3 ap ah (7) - ( ( ij- ) - ) = 6Uo ax, 3x1 ax 1 It is computationally inconvenient to use the dimensional equations. We apply a nondimensional scaling to the variables of both the elastic solid and the viscous fluid. We rewrite the elasticity equations using the following nondimensional scales: P = PIPo (8.4
O a -
PS
A
h
w = ywo h
X
=X/d i i o The Po is the hydrodynamic pressure scale and d o is the elastic rubber layer thickness. The local balance of elastic forces in nondimensional form is given below: pswo A2VP=(1o w h
-
doPo
A nondimensional displacement W is selected such that all the coefficients ine!t above equation are unity. POdO W" = US
Nondimensional scales for the lubrication equation are defined below;
(1l.a) *
(1l.b)
x1 = xl/L. h, = h0
-
-
h (L))/L
(11.C)
ax, a = 1
- h(L)/h(O)
(1l-d)
= (h(0)
ax 1s
I.
AA
A
hl = hl/ho = 1
-
The Reynolds equation is given below: A
where
*
=-
aw*i
(2)
ax The relationship betweenjstress and strain for an incompressible solid (1.e. Poisson's ratio equal to one-half) is given as i,j
Here P is a staiic pressure field and is the shear modulus of elasticity. The shear modulus is related to the linear elasticity modulus as
Reynolds
a
ap
ax 1
ax 1
h
ah1) -r (hl A 3 = -r and the nondimensional equations are given as
A
v-
% deformation h2
(12)
field
A
P = V y
(13)
The actual gap thickness is defined as a linear combination of the undeformed linear gap thickness and the resulting rubber displacement in the X2 direction. The fluid film thickness is
167
This gap thickness nondimensionalized below
is
wo
h
A
conveniently
~~
A
h (x) = h +-w3 1 1 h Consequently, the parameter (!governing the gap thickness change relative to the undeformed linear gap thickness is Wo/ho. This deformation parameter is given below:
2-B. SOLUTION METHOD We are to solve the above system of linear partial differential equations which correspond to the elastic rubber displacement; a boundary element method is applied to the Navier equations. The geometry of interest is shown in The elastic rubber layer has uniform figure 1. thickness. The elastic deformation boundary conditions require that either the displacements or the surface tractions be specified on the boundary of the domain. The surface traction vector for the elastic solid is given as: Here nj is the unit outward normal to the surface on which the traction ti acts. The corresponding nondimensional surface traction is: A
A
A
We now introduce the boundary element method as applied to the elasticity equations. The basic displacement field equations are given below: A
is determined provided that either the deformation or surface traction vector is prescribed on the boundary. The fundamental solutions for the elastic deforpation and surface traction are further discussed in Appendix 1. The integral equation for the elastic boundary tractions and displacements is solved by numerical methods. The boundary is divided into In this paper the N finite length elements. surface tractions and displacements are assumed to change linearly over each of the boundary elements. Figure 2 shows a typical boundary; the surface tractions are prescribed on part of the boundary and the displacements are prescribed on the remaining part of the boundary. At each node point on the boundary there are two components of traction and two components of displacement. Thus, for N elements and N nodes there are 2N unknowns in the discretized system. The boundary integral equation for the elasticity problem is rewritten as below:
.
A
'a W i i
h
+ 1 tl(x,y; s
,
A
=
) W.dS =
1 t,(x)
w
(x,y; a ) d s
(24)
S
Unfortunately, all details of operations used to construct a set of linear equations for equation (24) cannot be described. We may however mention some. The vector ai is first set equal to (1,O) and then (0,l). The boundary of the domain on which a solution i s sought is then divided into elements. The displacements and tractions are assumed to change lincarly over each element. The displacement vectorzis defined as, h
h
(19.a) n
(19.b)
= O
The following field equations are given for the fundamental displacement solution to the elasticitv eauations: (20.a) PYi + 2 a 6(r) = 0 i,jj i
L
Here w 1(1) is the displacement in the X direction at the 1 node and w l(N) is the displacement in the x 1 direction at the N node. The corresponding surface traction vector 1. is,
-
= o (20.b) i,i As is well-known these field equations are used in the following integral identity. This identity is used to obtain the boundary element equation.
-
~~
-
The field'equations for w and p are substituted into the above identity. The singular point in the fundamental solution is then placed on the boundary of the solution domain. The following integral equation is found: A
A
aW
i i
A
=
/(t
s
W.
i i
-
A
t'iWi)dS1
The two-dimensional, fundamental solutions to the elasticity equation are given below: the displacement is, '-1 W' = a Zn r + (a.x .) x i/r - 2 (23.a)
.
i
f
and the pressure is,
.
p'
J
= 2
J
(a x. j J
)
/ r
'2
(23.b)
The solution to this type of elasticity problem
When the boundary integral discretization is done the resulting 2N system of equations is given below:
E
=
2:
(27)
Here H and 5; are determined from the integral of the findamental solutions. It is appropriate to mention here that both y and E are singular operators. The uniform displacement vector W is mapped to the zero vector by g and the cons?ant pressure surface traction vector t is mapped to zero by E. This is because both*to and are homogeneous solutions to the Stokes equatqons. In the present work the vectors 1 and t are specified on different parts of the boundary. Complete mathematical details of this method and application are to be reported elsewhere. Results obtained using equation ( 2 7 ) were compared with two exact solutions to the Stokes equation. First, the solution for a simple shearing motion was checked:W = -x2, w2 = 0. Secondly, the Stokes stagnation point f ow field was checked w1 = x x W = -1/2x ?! Both solutions were checaed 'in $he boundgd domain shown in figure ( 2 ) . Agreement between the exact and numerical values was always within 3%.
.
168 3 RESULTS In accord with the basic equations given in section I1 the fluid film pressure and the elastic displacement are shown to depend upon the nondimensional bearing slope and the deformation parameter. A diagram of the basic configuration modeled is shown in figure 1. The elastic layer is uniformly thick. The deformation parameter is a measure of the relative rubber displacement at the bearing surface. The variations in the deformation parameter occur as a consequence of a number of bearing design changes. Larger deformation parameters forewarn the occurrence of larger bearing gap changes during operation. The rubber deformation due to positive hydrodynamic pressure will increase the bearing gap. Factors which individually portend increased bearing surface deformation are: increasing the lubricant shear viscosity, increasing the rubber thickness, increasing the sliding speed, and increasing the bearing length. But a decreased relative rubber deformation is heralded by increasing rubber shear moduli and increasing fluid film thickness To anticipate trends in load, pressure and h eyastic deformation several different deformation parameter values are studied. The solution of these coupled equations is examined for three values of a ; the rubber deformation is calculated for Wo/hoequal to 10,50, and 75. The slope of the undeformed gap was set equal to .1, . 3 , and .5. The pressure and deformation calculations were begun by determining the elastic deformation due to the fluid film pressure in a linear gap. Then the new bearing gap thickness was determined. This new thickness was introduced into the Reynolds lubrication equation and the next pressure iteration was calculated. Again the elastic deformation was calculated. This process was repeated until pressure and deformation convergence was obtained. The elastic deformation was calculated by using nineteen boundary elements on the bearing surface; the pressure at these nodes was calculated using the one dimensional Reynolds equation solution. Figures (3A,3B) show the film gap and the pressure for a dimensionless slope of one-tenth. Results are given for both the pressure and gap thickness for three values of the deformation parameter (10,50,75). As the deformation parameter increases the elastic deformation increases. The predominant effect of bearing surface deformation is to flatten the fluid film gap and to widen the gap. The wide gap is the cause of the decrease in the fluid film pressure. The elastic deformation tends to move the pressure peak toward the bearing trailing edge as the deformation parameter increases. This pressure peak shift is due to the bearing surface deformation. The fluid film gap is shown to increase in thickness due to displacement of the bearing surface. Consequently, the pressure is shown to decrease everywhere in the gap when compared to the undeformed linear gap pressure distribution. The fluid film pressure and elastic deformation results are given for a dimensionless slope of .3. The deformation parameter for these calculations was set equal to 10, 50, and 75. The results are shown in figures 4A and 4B. These results show trends similar to those shown in figure 3. The fluid film gap is more uniform in thickness away from the trailing edge. The gap is everywhere more wide. Fluid film pressure is shown to decrease; the pressure maximum is
.
shifted toward the bearing trailing edge. A final calculation is presented for a dimensionless slope of one-half. These results are shown in figure 5. The pressure profile corresponding to the undeformed gap is larger than the results shown in figures 3 and 4. But tendencies in the bearing surface deformation are similar. The results are briefly summarized. A model was given for the elastic deformation of a uniformly thick elastic coating on a bearing surface. The elastic material is incompressible. The bearing gap is allowed to deform due to hydrodynamic pressures. To forecast actual deformations due to fluid film pressures, a boundary element method was used to solve the elasticity field equations. ReynoIds lubrication equation was used to calculate the fluid film pressure; however, this equation appears as a nonlinear boundary condition for the elastic deformation. The resulting fluid film gap, after deformation of the elastic surface, was regular enough to be consistent with the fluid flow assumptions used to derive Reynolds equation. The elastic surface deformation tends to form a lip near the trailing edge. This sharp change in bearing geometry results in a rapidly converging gap near the trailing edge. The fluid film pressure gradient is large in this region. The results are consistent with what would be anticipated. The increased deformation parameter values do promote larger bearing deformation and consequently decreased values for the pressure. .The results disclose that the pressure peak is shifted more to the trailing edge of the bearing with increasing deformation parameter. In advance of these results it would have been difficult to predict the nearly uniformly thick bearing gap at higher values of the deformation parameter; this is of course for regions away from the bearing trailing edge. 4. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support by the National Science Foundation, Grant NO. MSM-84-14933.
-
Appendix I The system of equations for the fundamental solution to the elasticity equations has been introduced. The nonhomogeneous part of the solution is due to the point force in the deformation field. The following displacement field was found to be the fundamental solution:
where
=
a i Znr
cLi =
a i+a
Wi
1-
-1
-2
+ ajx.x J ir
(1.1)
21
(1.2)
In component form the displacement field is: 2 2 112 1 - a.1 In((x 1- y 1 ) + (x 1 - y 2 ) ) we’=
-
al(xl
+ [ (XI
-
Y,) + a2(x2 2
Y1)
-
+ (x2 - Y,)
(1.3) Y*) 2
1
‘Xi
-
Yi)
The associated surface traction vector depends on the unit outward normal on the surface where the traction is acting. The unit outward normal components are (n1, n2); the surface traction vector components are: (tfl,tf2)-
169
IIIST = n 1 (x1 r2 = (xl
-
Y,)
2
y,)
+
+ n2 (x2
-
-
(x2 y,)
y,)
2
(1.4)
Appendix I1 To obtain an estimate of the deformation parameter we choose a few typical values from slider bearing applictions. Based on these estimates we select a range of study for the deformation parameter. Our aim is to find the behavior of this elastohydrodynamic system for a practical parameter range. We select (11.1) L = 1 in. do = .25 in. (11.2) ho = .002 in. (11.3) Uo = 12 in./sec2 (11.4) = 150 lb /$n (11.54 = 50 x 16lbf -S/IN (11.6) The associated value of the deformation parameter is 90. As the deformation parameter increases the material becomes "softer"
E;
.
References
1. Buckholz, R. H., (1984) "On The Role of a Compliant Surface in Long Squeeze Film Bearings", Journal of Applied Mechanics, Vol. 51, pp. 885-891. 2. Benjamin, M. K., and Castelli, V.,(1971) "A Theoretical Investigation of Compliant Surface Journal Bearings," J. of Lub. Tech., Vol. 93, pp. 191-202. 3. Benjamin, M. K., "Compliant Surface Bearings: An Analytic Investigation," Columbia University Thesis, 1969. 4. Cameron, A., The Principles of Lubrication, 385-386. Wilev, . pp. .. _ . New York, 1966, 5. Chwang, A. T., Wu, T. Y., "Hydromechanics of Low-Reynolds-Number Flow. Part 2. The Singularity Method for Stokes Flows," J. Fluid Mech., Vol. 67, 1975, pp. 787-815. 6 . Elrod, H. C., "Exact and Approximate Theory for Linearly Compliant Layer with Inflexible Adhesive Backing ," Columbia University Lub. Res. Lab., Report No. 2, 1965.
7. Fairweather, G., Rizzo, F., Shippy, D., Wu, Y. (1979) "On The Numerical Solution of Two-Dimensional Potential Problems by an Improved Boundary Integral Equation Method" Journal of Computational Physics, No. 31, pp. 96-112. 8. Fogg, A., Hunwicks, S. A,, "Some Experiments With Water Lubricated Rubber Bearings ,*' General Discussions on Lubrication and Lubricants, Vol. 1, Institution of Mechanical Engineers, London, 1937, pp. 101-106. 9. Hori, Y., Kato, T., and Narumiya, H., "Rubber Surface Squeeze Films," J. of Lub. Tech., Vol. 103, 1981, pp. 398-405. 10. Kelmanson, M. A. (1983)" An Integral Equation Method for the Solution of Singular Slow Flow Problems", Journal of Computational Physics, No. 51, pp. 139-158. 11. Kelmanson, M. A. (1984) "A Boundary Integral Equation Method for the Study of Slow Flow in Bearings with Arbitrary Geometries", Journal of Tribology, Vol. 106, pp. 260-264. 12. Pirvics, J . , Castelli, V., "Elastomer Inertia Effects in Compliant Surface Bearings," Columbia University Lub. Res. Lab., Report No. 19, 1972. Castelli, V., "ElastomFr 13. Pirvics, J., Viscoelasticity Effects in Compliant Surface Bearings Columbia University Lub. Res. Lab., Report No. 20, 1972. 14. Rallison, J. M., Acrivos, A. (1978) "A Numerical Study of the Deformation and Burst of A Viscous Drop In An Extensional Flow", Journal of Fluid Mechanics, Vol. 89, Pt. 1, pp. 191-200. 15. Rightmire, G. K., "An Experimental Method for Determining Poisson's Ratio for Elastomers," J . of Lub. Tech., Vol. 92, July, 1970, pp. 381-388. 16. Rightmire, G. K., Castelli, V., and Fuller, D., '* An Experimental Investigation of a Tilting-Pad, Compliant Surface, Thrust Bearing," J. of Lub. Tech., Vol. 98, Jan., 1976, pp. 95-110. 17. Rightmire, G. K., "The Swing-Pad Bearing: An Experimental Evaluation," Lubrication Research Laboratory, Dept. of Mechanical Engineering, Report No. 29, 1982. 18. Sokonikoff, I. S . , Mathematical Theory of Elasticity, McGraw-Hill, New York, pp. 404-411, 1956. 19. Youngren, G. K., Acrivos, A. (1975) "Stokes Flow Past a Particle of Arbitrary Shape: A Numerical Method, of Solution", Journal of Fluid Mechanics, Vol. 69, pt. 2, pp. 377-403. ,*I
Surface Sliding Direction-
U,
Figure 1. Configuration of the elastic bearing system. Bottom surface moves at a constant speed. Elastic deformation occurs due to the hydrodynamic pressure; Lubricant is swept into the gap. rubber side walls are traction-free. Top surface cannot deform due to the rigid backing.
170
t
x2
- -
Surface 1 ; W * 0 ( 1,O)
(-1,O)
Surface
3
- 1 * L2
Surface 2 ,
Solution Domain
(-1,l 1
(1,l) Surface3
,1 * 1
Figure 2. Configuration used to test the boundary element equation against the exact Stokes solution. Surface one always has zero deformation. Surface tractions are specified on surfaces two, three and four. Tractions are calculated for simple shear and for stagnation point flow. Domain of interest is bounded. 0.01480
i!
ii PI
0.00740
0.00000 0.00
0.20
0.60
0.40
= 10 h0
1.00
0.80
WO
50
75
--
---
LENGTH
0.90 0.00
I
0.20
I
0.4 0
I
0.60
--. \ -. I
0.80
J . -
1-00
Figure 3 . Pressure and total gap thickness results for a = . l , ( A ) fluid film gap thickness for w 0 / h equal to 10, 50 and 100 (B) fluid film pressure results for a linear gap and for w / h equal 0 0 to 10; 50 and 100.
171
PRESSURE 0.06400
,/--
wo = l o h0
50 75
\
----
/
/ /'
\
'\
/
\
\
0.03200
0.00000 0.00
0.20
0.40
0.80
0.60
1.00
LENGTH
1.20
1.10
-
-woh0
-
-10
50 -75 ---
Figure 4 . Pressure and total fluid film gap thickness for a = .3, (A) fluid film gap thickness for the linear gap and for w / h equal to 10,50, and 100, (B) fluid film pressure results for a linear gap and f o r w / h equal Q o 50, and 100. 0 0
Po,
172
PRESSURE 0.18 /---
10 /
50 75 /
/
/
/
/’
\
\
\
\
/
\
\
\
/’
\
/
0.09
0.00 0.00
0.20
0.40 0.60 LENGTH
1.00
0.80
GAP 1.10
0.90
0.70
\
0.50 0.00
I
I
1
0.20
0.40
\
\
\
\
1
0.60
0.80
1.oo
LENGTH
.
Figure 5. Pressure distribution and total film thickness f o r a = 5, ( A ) fluid film gap thickness for the linear gap and for w / h o equal t o 10,50, and 100, (B) fluid film pressure results for a linear gap and for w / h equa? to 1 0 , 50, and 100. 0 0
SESSION VI ELASTOHYDRODYNAMIC LUBRICATION (1) Chairman: Professor B.J. Hamrock
PAPER Vl(i)
Solving Reynolds' equation for E.H.L. line contacts by application of a multigrid method
PAPER Vl(ii)
The use of multi-level adaptive techniques for E. H. L. line contact analysis
PAPER Vl(iii) Solutions for isoviscous line contacts using a closed form elasticity solution
This Page Intentionally Left Blank
175
Paper Vl(i)
Solving Reynolds' equation for E. H.L. line contacts by application of a multigrid method A.A. Lubrecht, G.A.C. Breukink, H. Moes, W.E. ten Napel and R. Bosma
line contacts, for a variety of
Film thickness and pressure profiles have been calculated for E.H.L.
load and rolling speed conditions. Compressibility of the lubricant is taken into account, and the pressure-viscosity relations according to Barus as well as Roelands have been applied. Results of these numerical calculations are compared with well-established asymptotic solutions (Martin-Giimbel, Moes, Grubin). A formula is presented, which incorporates these asymptotic solutions ahd which accurately predicts
the minimum film thickness throughout the entire film thickness plot. A description is given of the difficulties encountered in numerical calculations, in the neighbourhood
of these asymptotes.
1
INTRODUCTION
complementarity
theory
Over the last ten years, an increasing number of
cavitational boundary.
papers
introduced by
has
been
calculations
31,
provided by Kostreva [ 2 ,
dedicated
in
Elasto
Lubricated (E.H.L.)
to
numerical
Hydrodynamically
line and point contacts.
introducing the
to
cope
Among
the
with
the
refinements
various other authors are the
influence of the compressibility of the fluid, and the application of an accurate pressure-
This renewed interest in numerical calculations
viscosity equation (Houpert [ 4 ] ) . This Newton-
can be attributed to two different developments.
Raphson
approach
has
several
important
In the first place, there is the revolution in
advantages over the iterative solution methods,
calculational speed and
storage capacity
of
with or without over-relaxation, like Gauss-
modern
a
mini
Seidel. These include, low calculational times
computers;
even
medium
size
computer is nowadays suited for the numerical
(for small problems),
fast convergence in the
solution of the E.H.L.
neighbourhood
solution and
problem.
of
the
easy
and
The second, perhaps most important, reason is
straightforward to program.
the development of modern solution techniques,
However, this method has two severe drawbacks,
which became available over the last decade.
as pointed out in [ 5 ] . The first one is the
This
treatment of the cavitational boundary, see Oh
paper
is
mainly
based
on
the
second
development, in particular, it deals with one of
[6].
the promising modern numerical techniques; the
calculational time and storage capacity. The
MultiGrid method.
The
second
disadvantage
concerns
required cpu time is proportional to N3 the storage is proportional to N2
Since Okamura [ l ] presented his paper at
,
the
, while
where N is
the number of calculational points.
the 1982 Leeds-Lyon conference, using a Newton-
Both limitations make the approach unsuitable
Raphson based algorithm, a number of papers has
for larger problems (for instance the E.H.L.
been written on the topic of E.H.L.
point 'contact problem).
line contact
calculations, exploring the new possibilities of this method. A sound mathematical basis has been
In order to overcome the difficulties
176
mentioned above, a modern solution technique (MultiGrid) problem.
has
been
MultiGrid
convergence
applied
can
accelerator
solvers. The
to
the E.H.L.
be
regarded
for
slow
as
Moes L
dimensionless material parameter according
a
iterative
natural choice of this solver is
the Gauss-Seidel iterative scheme, in which the cavitational condition can be introduced quite naturally (Brandt [ 7 ] ) . In the following paragraphs an outline is
to Moes pressure
P P
dimensionless pressure P = p/Ph maximum Hertzian pressure
h' R U
reduced radius, R = l/(l/Rl+ l/RZ) mean tangential velocity
U
dimensionless velocity parameter according
given of the MultiGrid method, as well as its application to the E.H.L.
line contact problem.
to Dowson and Higginson W
load per unit length
W
dimensionless load parameter according to
1.1 Notation
Dowson and Higginson coordinate
X
MultiGrid symbols f
= right hand side
f -
=
right hand side vector, approx. f
L
=
differential operator
L
=
with superscript matrix operator approx. L
u
= solution
u
= solution vector', approximating u
r -
=
residual vector
v -
=
error vector
1;
=
restriction operator from grid h to grid H
=
interpolation operator from grid H to h
IH :=
Sub/superscripts fine grid
=
H
= coarse grid
-.
=
A
Roelands pressure viscosity coefficient
a
Barus pressure viscosity coefficient
a
dimensionless pressure
A
a = UP h distance between two neighbouring grid
-
viscosity
-
11
viscosity
-V O
viscosity at atmospheric pressure
11
rl
dimensionless viscosity
P
density
-Po
density at atmospheric conditions
coeff.
= q/qo
P
dimensionless density
h
dimensionless velocity parameter
2
approximation
= FAS
dimensionless coordinate
Z
points
= gets the value of
h
X
= p/po
MULTIGRID METHOD
The concept of Multi Level (MultiGrid)
coarse grid variable
fast
solvers is based on a certain understanding of the
E.H.L. symbols
nature
of
slowly
converging errors
in
iterative schemes. It can be shown that error components, which have wavelengths of the order
b
=
half the contact width
of the mesh size, are fast to converge. Hence,
E'
=
reduced Youngs modulus
these error components will disappear quickly,
G
= dimensionless material parameter according
to Dowson and Higginson
that, whenever an iterative scheme is slow to
h
=
film thickness
converge, the errors in neighbouring points are
H
=
dimensionless film thickness, H = W / b 2
almost equal. This means that the error is a
2/E' = (l-v:)/El
+
(1-vi)/E2
i.e.
in
just
a
few
iterations,
from
the
solution. On the other hand, it can be shown
Hmin= dimensionless minimum film thickness
smooth
H'
wavelengths which are large compared with the
=
dimensionless
minimum
film
thickness
according to Moes Hdh = dimensionless
minimum
over
the
domain,
with
distance between two adjacent grid points. film
thickness
according to Dowson and Higginson M
function
= dimensionless load parameter according to
This means that, whenever the convergence of a numerical
process
is
slow,
the
error
adequately be represented on a coarser grid.
can
177
Solving the error on this coarser grid has two
Instead
advantages:
coarser grid, where this error is defined by:
-
convergence
is
faster,
since
the
of
solving
the
error _vh on a
next
ratio
wavelength to gridsize is smaller
- there is less work involved per iteration as is done in the Correction Scheme, a new
(less points).
coarse grid variable is introduced, in the non When this error is determined on the coarser grid,
it
is
used
to
correct
the
linear case (FAS):
current
approximation to the solution, on the finer grid. where IH is a restriction operator from h grid h to grid H
In case of a linear differential equation, the so-called Correction Scheme is applicable (51. Since the Reynolds equation is non linear,
since the error _vh
the Full Approximation Scheme (FAS) had to be
itself, because of the non linear differential
used in the calculations.
operator.
In
the
following
section,
this
cannot be approximated by
Full
Approximation Scheme is described. For a more
The FAS coarse grid equations are given by:
detailed description of FAS and of MultiGrid in general, the reader is referred to Brandt [ 8 , 91.
For the sake of simplicity only two levels
cH to inis found,
(fine h and coarse H) will be assumed. Consider
When an approximate solution
the differential equation written in a general
it is used to correct the solution fine grid, according to:
form: L u = f
1)
-new
where L = non linear differential operator
ih
-
on the
7)
where Ih is an interpolation operator from H grid H to grid h
u = solution
f = right hand side
i"
The discretized equations (on an equidistant
- IHh Note that error (eq. 5),
grid, with meshsize h) can be written in matrix
directly.
2
-h u is an approximation to the which could not be calculated
notation: 3
E.H.L. THEORY
The isothermal line contact where Lh = matrix operator approximating L
problem can be
described by three equations.
fh = right hand side vector approx. f uh - = solution vector approximating u
The first equation, is the Reynolds equation for a
compressible
fluid p(p)
,
pressure-viscosity relation q(p)
with
a
.
After some iterations on eq. 2 ) an approximation h to 2 , called is found. Subsequently, residuals fh can be calculated according to:
ih
with p>O at the interior (cavitation
-rh = ih - L~ G~
3)
condition) p=O at the boundaries
general
178
Written in a dimensionless form, (see appendix
in a dimensionless and discretized form as:
l ) , this equation reads:
--
N
x )- h d
-H3 dP
el-
(
14)
-
9)
( pH ) = 0
n
I
1 2 q0uR2
---.J----
=
with
4
CALCULATIONAL DETAILS
h' The density is assumed to depend on the pressure
In solving the isothermal line contact problem,
according to Dowson and Higginson.
the Full Multi Grid (FMG) algorithm has been
The pressure-viscositity relation can be written
used. The procedure is started on the coarsest
as :
grid with the Hertzian pressure as an initial value.
The
calculated results on
this grid
provide an accurate starting solution for the next according
to
Barus,
or
more
accurately,
according to Roelands, using S.I. units:
finer grid, and
on. The relaxation
so
scheme used is Gauss-Seidel, using red-black ordening. As is shown by Brandt and Cryer [7], the cavitational
pa= 1 .96E8/Ph
boundary
can
be
treated
in a
straightforward manner. Only the usual fullweighting of residuals of the Reynolds equation,
Using second order central discretization for the
first
reasons,
term, first
and
because
order
of
stability
backwards
(upstream)
going from a fine grid to a coarser grid, has to be
abandoned
in
the
of
neighbourhood
the
cavitational zone, since the Reynolds equation
discretization for the second term, the Reynolds
is not valid in the cavitational region.
equation becomes:
In order to effectively utilize the Gauss-Seidel relaxation, it
should
be
realized
that
the
second term in eq. 9) plays a dominating role in the spike region. The influence of changes in this term is calculated by linearizing this term with regard to the central pressure. Furthermore, it is important to treat the three
13)
12),
equations
14)
and
identically,
calculating residuals and transferring them to coarser grids. The third equation for force balance is updated,
-
) and q
where:
i+h
-
changing H 0 0 , only on the coarsest grid. Since pi+l+pi
= q(-------
2
1
residuals of this equation have been transferred from the finest grid down, equation 1 4 ) will be
Writing
the
film
dimensionless way,
thickness
equation
see
integrating and
[5],
in
a
satisfied
on
the
finest
grid,
through
the
changes on the coarsest grid.
discretizing results in: H(X
x2
i
)-H00- zi
Convergence up to the discretization error
N
+ 7l-1 1 j=1
Pj
of the differential equations, on the different grids, was checked using solutions which had converged to algebraic errors being ten times smaller
than
solutions.
For
the
errors
detailed
of
the
original
information on
this
topic, the reader is referred to Brandt [9]. The equation of force equilibrium can be written
179
5
RESULTS The differences in the calculated values of the
Solutions of the E.H.L.
line contact problem
minimum
have been calculated for different conditions of
film
thickness
based
on
the
two
load and rolling speed. The results of these
pressure-viscosity relations are very small. In fig. 3 a pressure and film thickness profile
calculations are displayed in the well known
is presented for a case of moderate load and
Moes film thickness plot. In fig. 1 the results
rolling speed. WC.h
P
of the calculated minimum film thicknesses are
-
shown in case of the Barus pressure-viscosity
= + 1 . 8 8 E*81 n '2.88 E.81 Hmin- *3.79 €+El8
relation, in fig. 2 for the Roelands pressure-
xn
viscosity
relation,
(see
2
appendix
for
L
44.1
- 4 . 8 8 €+BE
a
description of the parameters).
4 ::
H4 ,
ZOO
I
STRIP - CONTACl
1
h
0 : bwui
m
I
20
fig. 3
Pressure and film thickness profile for
M=20, L=10, using a compressible fluid and the Roelands pressure-viscosity relation and 1217 nodal points. Fig. 4 shows the minimum film thickness and the 0.1
0.1
fig. 1
0.1
1
z
10
1
20
100
10
I00
100
maximum mass flux defect, plotted against the
Film thickness plot with numerically
number of nodal points. From this figure it can
calculated values of the minimum film thickness,
be seen that the minimum film thickness is close
using
to its final value, from s i x levels upwards.
a
compressible
pressure-viscosity calculated
with
fluid
and
the
Barus
relation. Drawn lines are the
minimum
film
thickness
AMF 96.
00.
formula (eq. 15).
60.
0
40. ZOO 100
80
n
STRIP - CONTACT
4
30.
0 : ro.l.nd.ls0.
20.
0
10.
.
O+
7 4.8
10
0
0
I
O
1
Z t
1
0.1
3*6 0.1
0.1
0.1
I
Z
1
LO
20
50
I00
ZOO
500
I
fig. 2
Film thickness plot with numerically
1
2
3
4
calculated values of the minimum film thickness,
fig.
using a compressible fluid and
minimum film thickness (H,)
pressure-viscosity lines are
relation
calculated with
thickness formula (eq. 15).
the Roelands
(Z=0.69).
Drawn
the minimum
film
4
Maximum
mass
6
6
7
LEVEL
flux defect (Amf) and as a function of the
number of nodal points, for M-20, Lp10.
180
LEVEL= 1 2
3 4 5 6
7
N= 20 39 77 153 305 609 1217
A
film
thickness
formula
was
derived,
incorporating all the known asymptotes, using exponential functions for a smooth change from one asymptote to the other. The dimensionless parameters describing the E.H.L.
line contact
problem were first published by Moes
[lo].
This minimum film thickness formula reads (see Most solutions have been calculated with the first order backwards
discretization of
also appendix 2):
the HI= [{(0.99M -1f8L3f4)r+
second term of equation 1 2 ) , which proved to be very
robust.
For
the
lightly
loaded
(2.05M -115 ) r} s/r+
(2.45M -1 ) s ] 1 1 s
case,
15)
however, a second order backward discretization gave better predictions of
where s = 4-exp(-L/Z)-exp(-2/M) r = exp{l-4/(~+5)}
the minimum film
thickness, when using few calculational points. However,
this
second
order
scheme
In this formula, the following asymptotes can be
caused
convergence problems, for high loads.
recognized: -The Grubin asymptote:
A few words are appropriate on the different
types of calculational troubles encountered in the different regions of
the film thickness
plot.
see
Terril
[ll],
taking
the
minimum
In the first place there is the region close to
thickness as 314 of the central value.
the rigidfisoviscous asymptote, the Weber and
-The
Saalfeld
Martin 1121 and GUmbel 1131 :
region,
where
the
inlet
should
film
rigid/isoviscous asymptote according
to
virtually be at minus infinity to avoid starved lubrication. On the other hand, the density of calculational points must be above a certain limit, demanding a large number of calculational
-The elasticfisoviscous asymptote, according to
points for accurate results.
Moes [14]:
For high M instabilities coupling
and
high L
occur,
between
values,
because
equations
of in
the
common
the
strong
H'= Z . O ~ M - ~ / ~
neighbouring
points. A moderate under-relaxation factor ( o f
merging with the Herrebrugh solution [15].
4) was used in the spike region only, to avoid these instabilities. However, to penetrate this S T R I P - CONTACT
region, a new type of relaxation (distributed relaxation) has to be developed. For high values of M a different type of trouble so
was encountered. The problem lies in the large pressure gradients in the inlet zone, where the pressure changes over a very small traject, from almost
zero
to
a
fraction of
the
maximum
Hertzian pressure. To accurately calculate these large pressure gradients, a high density of nodal points is necessary. Most solutions have been calculated, using 1200 points,
taking
an
hour
VAXllf750, see table 1).
of
cpu-time
on
a fig. 5
Minimum film thickness plot with the
predictions according to the Moes formula (eq. 6
FILM THICKNESS FORMULA
15 drawn) and the Dowson and Higginson formula (dotted).
181
table 1
Comparison of the required cpu time for
The two parameters s and r have been used to
Newton-Raphson and MultiGrid, as a function of
ensure a
the number of calculational points (N).
smooth transition, in
the
regions
between the asymptotes. The presented film thickness equation 1 5 ) has
cpu time
the advantage over excisting formulae, that it accurately predicts the minimum film thickness in the entire region of load, speed and material
level
N
N.R.
M.G.
2
39
50 sec
10 sec
3
77
5 min
20 sec
35 min
parameters; it is not restricted to a small part region, see fig. 5 .
of the E.H.L.
The solid lines in fig. 1 and 2 are calculated
4
153
with this formula and it can be seen that the
5
305
4 hours
4 min
predictions agree quite well with the numerical
6
609
33 hours*
1 5 min
calculations.
7 8
1217
11
2433
89 days*
7
1 min
* days
1 hour 4 hours
CONCLUSIONS
From fig. 1 and 2 it can be seen that a large
*
These
part of the E.H.L.
to
o(N~)
region is covered by the
results were extrapolated
according
presented MultiGrid solution routine. The differences in the 'calculated value of the minimum film thickness, caused by application of
8
ACKNOWLEDGEMENTS
the two pressure-viscosity relations, are small. In case of the standard line contact, numerical
The authors would like to thank Prof. Dr. A.
solutions are not necessary, since the minimum
Brandt
film thickness, in all regions of the plot, is
suggestions on the MultiGrid approach.
for the many
helpful discussions and
accurately predicted by the Moes formula. In fig. 5 the Moes formula is compared to the
Part of this work was done during a stay at
Dowson and Higginson formula. It can be seen
the Weizmann Institute of
that the latter formula predicts film thickness
Israel,
only in a part of the film thickness plot. For
different where
geometries
numerical
or
by
a
grant
.
other
solutions
are
Appendix 1
necessary, this new approach provides the user with
the
possibility
to
work
with
many
Starting with equation 8)
calculational points, requiring only moderate cpu times, on medium size computers. In table 1 the required computational time of the MultiGrid method is compared with the time needed by the Newton-Raphson method. The cpu time required by this new approach is obviously
and introducing
much smaller, especially for larger problems. P = p/Ph The problems outlined in the previous section demand a non equidistant grid; the possibility of local grid refinements is currently being studied.
of
the
"Hogeschoolfonds van de Technische Hogeschool Twente"
applications
supported
Science, Rehovot,
,
H = hR/b2
,
a
= aP
h'
182
HOUPERT, L. G., the result reads:
B. J. HAMROCK, "A Fast
Approach for Calculating Film Thicknesses and
Pressures
Lubricated
in Elasto Hydrodynamically
Contacts
at
High
Loads".
presented at the ASMEIASLE Joint Lubrication Conference, Atlanta, October 1985. LUBRECHT, A. A . , W. E. TEN NAPEL, R. BOSMA,
Appendix 2
"Multigrid,
an
Alternative
Calculating
Film
Thickness
Profiles
in
Elasto
this
appendix
parameters
are
the
related
Moes to
dimensionless the
parameters
according to Dowson and Higginson (see also [lOl).
for
Pressure
Hydrodynamically
Lubricated Line Contacts". In
Method and
to appear in
Trans. ASME JOT. OH, K. P. ( 1 9 8 5 ) , "The Numerical Solution of Dynamically Contacts as
Loaded Elastohydrodynamic a Nonlinear Complementarity
Problem". Trans. ASME JOT, 1 0 6 , pp 88-95
H'= Hdh(2U)
-112
BRANDT A., C. W. CRYER ( 1 9 8 3 ) , Algorithms
M = W (2U)
-112
=
G (2U) 114
Higginson .are given by: =
Solution
Linear
of
SIAM. J.
Problems".
Sci.
Stat.
Comput. vol 4 , No. 4 pp 655-684.
Where the parameters according to Dowson and
u
the
Complementarity Problems Arising from Free Boundary
L
for
"Multigrid
BRANDT A. ( 1 9 7 7 ) , "Multi-Level Adaptive Solutions to Boundary-Value Problems". Math. of Comp. 31, No. 138 pp 333-390 BRANDT A. ( 1 9 8 4 ) , "Multigrid Techniques:
7OU
---
1984
E'R
guide
with
applications
to
fluid
dynamics". Monograph. Available as G.M.D.h
studie No. 8 5 ,
Hdh'
1 2 4 0 , D-5205,
w
i
W --E'R
from G.M.D.-FLT,
postfach
St. Augustin 1 W.-Germany.
MOES H. ( 1 9 6 5 ) , "Discussion on a paper by D. Dowson". Proc. Inst. Mech. Engrs, vol 1 8 0 ,
G
=
pp 244-245.
aE'
TERRIL, R. M. ( 1 9 8 3 ) , "On Grubin's Formula in Elastohydrodynamic Lubrication Theory". WEAR, Vol. 9 2 , pp 67-78.
References
MARTIN, H. M. ( 1 9 1 6 ) , "Lubrication of Gear OKAMURA,
Numerical
H. ( 1 9 8 2 ) , "A Contribution to the Analysis
of
Isothermal Elasto
Teeth". Engineering, 1 0 2 , pp 119-121. GUMBEL,
(1916),
L.
"Uber
Geschmierte
Hydrodynamic Lubrication". Proc. "9th Leeds-
Arbeitsrader".
Lyon Symp. on Tribology", Leeds. KOSTREVA, M. M. ( 1 9 8 4 ) , "Pressure Spikes and
MOES
Stability
internal report, Laboratory of Tribology,
Hydrodynamic
Considerations Lubrication
in Models".
Elasto Trans.
2.
ges. Turbinenwesen, 1 3 ,
357.
H.
(1985),
"On
Survey
Diagrams".
Twente University of Technology.
ASME JOLT, vol 106 pp 386-397 KOSTREVA, M. M. ( 1 9 8 4 ) , "Elasto Hydrodynamic
HERREBRUGH
Lubrication: a Non-Linear Complementarity Problem". Int. Jou. Num. Meth. in Fluids,
Elastohydrodynamic Lubrication through an
vol. 4 pp 377-397
9 0 , pp 262-270
K.
Incompressible and
(1968),
"Solving
the
Isothermal Problem in
Integral Equation". Trans. ASME JOLT, Vol.
183
Paper Vl(ii)
The use of multi-level adaptive techniques for E. H.L. line contact analysis R.J. Chittenden, D. Dowson, N.P. Sheldrake and C.M. Taylor
SYNOPSIS
The numerical analysis of the isothermal elastohydrodynamic lubrication of line contacts has made considerable advances in recent years. One of the most recent developments has been the application of Multi-level Adaptive Techniques to such problems, which has allowed solutions to be obtained with much reduced computational times. The application of this new method is outlined, and applied to a model which includes both lubricant compressibility and pressure viscosity effects. Solutions are then presented which cover a wide range of material and lubricant combinations, contact loads and rolling speeds. These results are found to exhibit good agreement with previously reported solutions. The problems, as well as the benefits of the Multi-level method, when applied to elastohydrodynamic line'contacts,are considered, and comparisons are drawn with other solution techniques. INTRODUCTION
to finite element scheme, as well as to finite difference algorithms.
Since the first papers on the Elastohydrodynamic Lubrication of line contacts were published over twenty years ago, the power of the numerical methods employed has been progressively increased. The developments in computer technology that have enabled this progress to be made have, to a large extent, been paralleled by faster more efficient numerical techniques. These advances enabled solutions to be obtained in hundreds of minutes, whereas previously such solutions took many months of hard calculation [l]. The more recent advances of Okamura [2] and the extension of this work by Houpert and Hamrock [3] have reduced solution times still further, typically to 5 minutes CPU time (Table 1). Authors
Time
Dowson 6 Higginson Hamrock 6 Jacobson Houpert 6 Hamrock
[l] [71 [31
Months Hours Minutes I
Table
1
A recent development in numerical methods is that of Multigrids (MGs) or, more generally, Multi-level Adaptive Techniques (MLATs) and this is the subject of this paper.
It is important to realise that MLATs are not 'solvers' in their own right, but are merely convergence accelerators and as such require a solution technique such as Gauss-Seidel or some other iterative procedure. Another important feature, although not one dealt with in this paper is the possible application of multigrids
The underlying solution technique used in this paper is the Newton-Raphson scheme, refined by Houpert and Hamrock 131, which incorporates lubricant compressibility, the Roelands pressure-viscosity relationship, an improved elastic calculation and variable mesh spacing. The Newton-Xaphson technique is used since it offers better convergence than the Gauss-Seidel scheme. There are, however some limitations to this technique, since the matrix inversion procedure requires a CPY time approximately proportional t9 O(N ) and the storage is proportional to N (where N is the total number of nodes in the computational region). These factors combine to make the approach unsuitable for extension to point contact problems where the number of nodes is large. NOTATION ELASTOHYDRODYNAMIC LUBRICATION NOTATION b
Hertzian half width R
m
h
Film thickness
m
W
u1 + u2 Entraining velocity 2 Applied load
E'
Reduced elastic modulus
U
~
+
ml s
184
such a change are of an appropriate wavelength. This may be brought about by the suitable choice of interpolation operators.
G
Materials parameter
H
Dimensionless film thickness
HO
Dimensionless constant used in calculation of H
The use of such techniques brings about two major advantages:
H
Dimensionless film thickness
(i)
R
Effective radius of curvature
-
( aE')
(h/R)
-1 = - 1+ - l
m
R1 R2
U
Dimensionless speed parameter
y
W
Dimensionless load parameter
W -
a
Pressure viscosity coefficient
E'R E'R
Dynamic viscosity at ambient pressure.
[+J f
g1
Viscosity parameter
g3
-
h
Film thickness parameter
HW U
MULTI-LEVEL ADAPTIVE TECHNIQUE NOTATION r
Residual value
u
Current estimate of the solution
v
Error estimate f
Coarser grids involve fewer nodes, hence less computational work is required.
Ic
Coarse to fine interpolation (prolongation) operator
If
Fine to coarse interpolation (restriction) operator.
f
Non-linear system of equations
L
Jacobian of non-linear system
U
Exact solution of non-linear system i.e. (U) = 0.
MULTI-LEVEL TECHNIQUES Multi-level Adaptive Techniques (MLATs) are methods which involve the use of two or more grids which participate co-operatively in the solution process to give a solution to a system of equations on the finest grid. The importance of finishing on the finest grid should not be ignored since some methods may not use the same form of approximation to the system of equations on coarser grids. The existence of these techniques has been known for a number of years but it is only recently that they have gained widespread acceptance in numerical work. The behaviour of Multi-level schemes depends on an understanding of errors and the convergence properties of numerical techniques. It is known that errors of approximately the same wavelength as the grid size converge faster than other wavelength errors, hence changing the grid size may be beneficial to the solution process provided that any errors introduced by
(ii) Errors of the same wavelength as the grid spacing converge faster.
These two features combine to improve computational speed although there is a penalty in both storage and programming terms. The essential difference between Multi-level Adaptive Techniques and Multi-grids (MGg) is that, as the name suggests, MLATs use locally refined grids (a locally refined grid is one in which certain areas of the computational domain have a finer mesh spacing than that used elsewhere) whereas MGs use globally refined grids. Local grid refinement (or adaptive meshing) is used where the solution requires greater accuracy than could be achieved using the coarse grid; usually because the solution changes too rapidly to be accurately modelled on a coarse grid. A two dimensional grid is shown in Figure 1 to illustrate the idea of adaptive meshing. In elastohydrodynamically lubricated line contacts there are two distinct regions which experience rapidly changing pressure solutions. Firstly, the Hertzian pressure distribution, used to initialize many solution procedures, has pressure gradient discontinuities at the ends of the contact region. The second region, is that of the pressure 'spike' where rapidly changing pressure gradients may be encountered. Both these features are illustrated in Figure 2. The solution technique described in this
FINE GRIDS Used with 1. Multi-grid
2. Multi-level Adaptive Technique
Figure 1 Typical 2 Dimensional Grid Structures
185 T h e r e are a number o f v a r i a t i o n s on t h e basic multi-level algorithm (steps i vii). I t I s a c c e p t a b l e , and many improve convergence, t o i t e r a t e t h r u g h s t e p s i - v a number o f times between s t e p s v i and v i i . The work d e s c r i b e d i n t h i s paper embodied t h e s e f e a t u r e s s i n c e c o n v e r g e n c e rates were i n c r e a s e d . It is a l s o p o s s i b l e t o u s e d i f f e r e n t forms of a p p r o x i m a t i o n on t h e coarser g r i d s which i n v o l v e less c o m p u t a t i o n a l work. A d i a g o n a l l y dominant m a t r i x may be approximated by i t s d i a g o n a l terms a l o n e , and i n t h i s case i t i s e v e n more e s s e n t i a l t h a t c y c l e s f i n i s h on t h e f i n e g r i d where t h e f u l l a p p r o x i m a t i o n is used.
-
,-
275
225 200 175 -
250
Oirncnrionlcrr X Coordinate Ix/bl
Figure 2
-
Line Contact P r o f i l e
(U = 1.98x10-''
G=4174 w = 1 . 4 5 ~ 1 0 - ~ )
w.
6
paper t a k e s t h e s e f e a t u r e s i n t o a c c o u n t by u s i n g g r i d s which have h i g h e r n o d a l d e n s i t i e s i n t h e s e regions t hus r e t a i n i n g t h e accuracy of l a r g e r g r i d s . A g r a p h o f n o d a l d e n s i t y as a f u n c t i o n of p o s i t i o n is i n c l u d e d i n F i g u r e 3 where t h e abscissa corresponds t o t h a t i n Figure 2 t o f a c i l i t a t e comparison.
150
-
125
-
75
-
YI
5 100 B
50 25
-
ou -6
-1.
'
A s i n g l e M u l t i - l e v e l c y c l e is composed of v a r i o u s s t e p s and a t y p i c a l two g r i d c y c l e i s d e s c r i b e d below.
I f t h e i n i t i a l estimate of t h e e x a c t s o l u t i o n U i s d e n o t e d by u t h e n a Newton-Raphson s t e p is c a r r i e d o u t and a new estimate ( u ' ) is formed on t h e f i n e g r i d . The r e s i d u a l i s t h e n c a l c u l a t e d a t e v e r y node on t h e f i n e g r i d u s i n g t h e n o n - l i n e a r system.
These r e s i d u a l v a l u e s are t h e n t r a n s f e r r e d t o the coarse g r i d v i a t h e r e s t r i c t i o n operator,
rc
rf
=
Figure 3
:I I
'
-5
-
-3
-1 0 Position of Node IX/b)
-2
1
2
3
Nodal D e n s i t y a s a F u n c t i o n of Posit ion
COMPUTATIONAL DETAILS The i t e r a t i v e p r o c e d u r e is a Newton-Raphson a l g o r i t h m u s i n g a two-level a d a p t i v e meshing t e c h n i q u e , i n c o r p o r a t i n g two non-uniform g r i d s . The scheme u s e s a H e r t z i a n p r e s s u r e p r o f i l e t o i n i t i a l i z e t h e p r o c e d u r e on t h e c o a r s e g r i d . The g r i d s w i t c h i n g is c o n t r o l l e d u s i n g a 'Fixed Sawtooth Algorithm' ( 4 ) u n t i l t h e s p e c i f i e d convergence c r i t e r i a are m e t , t h e f i n a l i t e r a t i o n is t h e n c a r r i e d o u t on t h e f i n e g r i d (see F i g u r e 4).
(if)
The c o a r s e g r i d e q u a t i o n s
Lc ( u ' ~ ) vc = re
(iii)
a r e t h e n s o l v e d f o r t h e e r r o r estimate v which is t h e n t r a n s f e r r e d back t o t h e f i n e g r i i v i a the p r o l o n g a t i o n o p e r a t o r
/
/ /
/
Coarse grid Vf
= Icf vc
(iv)
T h i s e r r o r estimate i s t h e n used t o u p d a t e the c u r r e n t estimate t o t h e s o l u t i o n Ul'
= u'
+
Figure 4 (V)
Vf
A Newton-Raphson s t e p i s t h e n c a r r i e d o u t using t h i s estimate,
L~ (u") vf
5
rf
(Vi)
t o o b t a i n a new e r r o r estimate and hence and new estimate t o u, u
= u"
+ Vf
(vii)
F i x e d Sawtooth Algorithm n1 i t e r a t i o n s on c o a r s e g r i d n2 i t e r a t i o n s on f i n e g r i d n3 f i n i s h i n g i t e r a t i o n s on f i n e g r i d
The u s e o f t h e Newton-Raphson a l g o r i t h m p e r m i t s a l i n e a r m u l t i - g r i d approach t o t h e s o l u t i o n of t h e Reynolds' e q u a t i o n s i n c e t h i s t e c h n i q u e i n v o l v e s l i n e a r i s a t i o n as p a r t of t h e i t e r a t i v e p r o c e d u r e . The a p p l i c a t i o n o f a Newton-Raphson a p p r o a c h u s i n g m u l t i - g r i d s is covered b r i e f l y in IS].
186
The convergence of t h e scheme is examined i n two ways. C o m p a t i b i l i t y w i t h t h e d i f f e r e n t i a l o p e r a t o r (Reynolds' e q u a t i o n ) i s checked by examining t h e r e s i d u a l v a l u e s a t e a c h n o d a l p o i n t and a f u r t h e r check on t h e convergence of t h e e l a s t i c c a l c u l a t i o n s i s a l s o performed. The i n t e r p o l a t i o n between t h e g r i d s is u s u s a l l y c a r r i e d o u t u s i n g polynomial f u n c t i o n s , t h e d e g r e e o f which depends on t h e It is d i s c r e t i z a t i o n and o t h e r f a c t o r s . i m p o r t a n t t o r e a l i z e t h a t e r r o r s may be i n t r o d u c e d by t h e i n t e r p o l a t i o n r o u t i n e s , s i n c e i t i s p o s s i b l e t o i n t r o d u c e v a l u e s t h a t are incompatible w i t h t h e formulation of t h e o r i g i n a l problem. Q u a d r a t i c polynomials were used f o r t h e c o a r s e t o f i n e o p e r a t o r and i n one case a n e g a t i v e p r e s s u r e was g e n e r a t e d by t h i s p r o l o n g a t i o n o p e r a t o r . T h i s was overcome by r e v e r t i n g t o l i n e a r i n t e r p o l a t i o n i f such a case occurred. The f i n e t o c o a r s e o p e r a t o r ( r e s t r i c t i o n ) used was d i r e c t i n j e c t i o n (The c o a r s e g r i d v a l u e is t h e v a l u e a t t h e c o r r e s p o n d i n g node on t h e fine grid).
1 1
Co r c e c t ed
181 181 181
1500 33 20
Nodes
Au t h o r s
CPU Time
~~
Hamrock S J a c o b s o n Houpert S Hamrock P r e s e n t Study I
[7] [3] I
Table 2
I
Comparison of T y p i c a l Run T i m e
For t h e purpose o f comparison i t i s i m p o r t a n t t h a t t h e number of nodes i s comparable, b u t i t should a l s o be noted t h a t t h e a d a p t i v e meshing used i n t h i s paper g i v e s n o d a l d e n s i t i e s up t o 16 times g r e a t e r i n t h e pressure spike region than i n the i n l e t region, where p r e s s u r e g r a d i e n t s a r e changing less r a p i d l y . For a f i n e mesh c o n t a i n i n g 340 nodes t h e e f f e c t i v e n o d a l d e n s i t y i n t h e v i c i n i t y of t h e p r e s s u r e s p i k e i s a p p r o x i m a t e l y 250 nodes p e r H e r t z i a n h a l f - w i d t h and is e q u i v a l e n t t o a u n i € o r m mesh of a p r o x i m a t e l y 1800 nodes. (The c o m p u t a t i o n a l zone i n h i g h l o a d , m o d e r a t e t o low s p e e d cases e x t e n d s from -6.0 t o + 1.5 H e r t z i a n half-widths).
RESULTS The r e s u l t s p r e s e n t e d i n t h i s paper e x t e n d o v e r a c o n s i d e r a b l e ' r a n g e of p a r a m e t e r s and t h i s is i l l u s t r a t e d i n F i g u r e 5 by means of a p l o t o f t h e p a r a m e t e r s s u g g e s t e d by Johnson [81. The a i m of t h i s paper i s t o e s t a b l i s h a b a s i s f o r f a s t e r numerical techniques i n t h e f i e l d of Elastohydrodynamic L u b r i c a t i o n and some comparisons w i t h o t h e r methods are shown i n T a b l e 2. The comparison of run times is based on a t y p i c a l r u n time c a r r i e d o u t u s i n g t h e method of Houpert and Hamrock [ 3 ] on t h e Amdahl 580 computer a t Leeds U n i v e r s i t y and d a t a p u b l i s h e d i n [31.
4
I
I
Theoretical Results -- Elasto-lso/Piezovisrous
I
Boundary
,
_ _ _ _ Rigid ~SO-viscousBoundary 3 -
-
OI
L
-Elasto/Rigid Boundary
/
/
XW
/
I
" "
Figure 5
/
xlr
I
-
I CY
/
x
I
P l o t of R e s u l t s
/ /
The r e s u l t s a r e i n good agreement w i t h previous findings, reported i n references [ l ] , (61 and [71. DISCUSSION
/
'.C".
L
A l a r g e number o f r e s u l t s have been produced i n t h e r e g i o n c o r r e s p o n d i n g t o a steel-steel c o n t a c t l u b r i c a t e d by a m i n e r a l o i l , w i t h a lesser number of r e s u l t s b e i n g o b t a i n e d f o r v a l u e s of t h e m a t e r i a l s parameter (G) p r o g r e s s i v e l y d e c r e a s i n g from a p p r o x i m a t e l y 4500 t o 500 (see T a b l e 3 ) . F i f t y s i x r e s u l t s f o r v a r i o u s arameter v a l u e (U r a n g i n g from 0.198 x lo-'' t o 0.53 x lo-', W from 1.0 x and G r a n g i n g from 3000 t o 4600) t o 1.5 x have been used t o o b t a i n Dowson C Higginson 111 r e l a t i o n s h i p s f o r b o t h c e n t r a l and minimum f i l m t h i c k n e s s . It i s recognized t h a t such a f o r m u l a t i o n h a s l i m i t a t i o n s but f o r t h e s i t u a t i o n s under c o n s i d e r a t i o n such approximations are a c c e p t a b l e .
/-
Piezovisrous
2-
The r e a l b e n e f i t s o € t h i s t e c h n i q u e are t o be found i n t h e a b i l i t y t o h a n d l e non-uniform meshing which r e s u l t s i n a more a c c u r a t e d e f i n i t i o n of t h e p r e s s u r e s p i k e r e g i o n f o r a g i v e n number o f nodes.
-
The powers on t h e d i m e n s i o n l e s s p a r a m e t e r s
(U,G,W,) were g e n e r a t e d u s i n g t h e f i r s t 56 r e s u l t s r e c o r d e d i n T a b l e 3 by m u l t i - v a r i a t e l i n e a r r e g r e s s i o n by c o n v e r t i n g t o l o g a r i t h m i c form and s u b m i t t i n g t h e s e r e s u l t s t o s t a t i s t i c a l a n a l y s i s . The d i f f e r e n c e s i n t h e r e l a t i o n s h i p s €or minimum f i l m t h i c k n e s s a r e i n s i g n i f i c a n t . For c e n t r a l f i l m t h i c k n e s s , however, t h e power on t h e d i m e n s i o n l e s s l o a d p a r a m e t e r is s i g n i f i c a n t l y d i f f e r e n t from p r e v i o u s l y r e p o r t e d v a l u e s [ 6 ] , a s is t h e power on t h e materials p a r a m e t e r (GI. The c e n t r a l f i l m t h i c k n e s s formula g e n e r a t e d from s o l u t i o n s o b t a i n e d by t h e new method g i v e s s l i g h t l y h i g h e r f i l m t h i c k n e s s v a l u e s , b u t i n t h e r a n g e f o r which t h e f i l m t h i c k n e s s p r e d i c t i o n s a r e v a l i d (G 5 25001, such e r r o r s are g e n e r a l l y less t h a n 10%.
187
FILM
M inh u m
THICKNESS x
Central
lo5
%
DIMENSIONLESS PARAMETERS
Speed
Materials
x 10 1 1
Load
x 10
5
DIFFERENCE
Minimum
Central
[ll
(21
~~~~
31.870 28.620 24.630 20.090 15.420 13.080 10.560 8.012 6.488 5.923 5.307 4.666 4.345 3.970 3.620 3.247 3.002 2.621 2.402 2.181 1.935 1.724 1.445 1.195 0.898 0.550 3.018 2.767 2.728 2.686 2.688 2.671 2.655 2.658 2.622 2.621 2.586 2.559 2.525 2.513 2.511 2.462 2.455 2.449 2.439 2.436 0.660 0.641 0.628 0.614 0.602 0.592 0.699 0.656 0.624 0.611 0.818 0.731 0.680 0.681 1.608
33.880 30.210 25.950 21.380 16.290 13.960 11.430 8.605 7.203 6.578 5.913 5.214 4.845 4.464 4.068 3.651 3.380 2.966 2.700 2.449 2.192 1.965 ' 1.638 1.369 1.026 0.625 3.547 3.157 3.073 3.034 2.999 2.968 2.937 2.911 2.885 2.860 2.835 2.792 2.752 2.733 2.714 2.679 2.663 2.647 2.632 2.617 0.794 0.767 0.744 0.724 0.706 0.690 0.854 0.787 0.740 0.725 1.037 0.895 0.820 0.824 2.003
53.300 44.410 35.530 26.650 17.770 14.210 10.660 7.106 5.685 4.974 4.264 3.553 3.198 2.843 2.487 2.132 1.954 1.599 1.421 1.244 1.066 0.888 0.711 0.533 0.355 0.178 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 1.978 0.198 0.198 0.198 0.198 0.198 0.198 0.273 0.273 0.273 0.273 0.409 0.409 0.409 0.546 1.638
4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4647 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4174 4138 4138 4138 4138 4138 4138 3024 3024 3024 3024 2016 2016 2016 1512 1512
Table 3
3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 3.947 2.198 4.836 5.715 6.155 6.594 7.034 7.473 7.91'3 8.353 8.792 9.232 10.110 10.990 11.430 11.870 12.750 13.190 13.630 14.070 14.510 1.099 1.319 1.539 1.758 1.978 2.198 1.213 1.820 2.427 3.033 0.910 1.820 2.730 2.427 2.427
.
4.2 6.1 6.7 6.5 8.2 7.5 6.3 7.0 1.8 2.0 1.8 1.7 2.0 1.2 1.3 1.2 -0.5 -0.1 -0.5 -0.9 -2.0 -0.8 -2.9 -1.7 -1.9 -2.5 -2.7 -1.1 -0.3 -0.9 0.0 0.3 0.4 1.3 0.6 1.3 0.6 0.7 0.4 0.5 0.9 -0.2 0.0 0.2 0.2 0.5 -2.0 -2.7 -2.7 -3.2 -3.6 -4.0 -0.6 -1.7 -2.9 -2.2 4.8 2.6 0.6 3.7 3.6
5.0 6.0 6.2 6.7 7.4 7.4 7.2 6.8 4.6 4.7 4.7 4.7 4.6 4.6 4.5 4.3 2.7 3.4 2.2 1.6 1.2 2.8 0.1 2.0 1.1 -0.6 6.4 2.8 1.8 1.3 0.8 . 0.5 0.0 -0.3 -0.7 -1.0 -1.4 -2.1 -2.7 -3.0 -3.3 -3.9 -4.2 -4.5 -4.7 -5.0 9.0 7.4 6.0 4.7 3.4 2.2 11.6 7.9 4.9 5.1 20.4 14.0 9.9 11.6 16.8
188
Table 3 (continued)
1.506 1.437 1.383 1.935 1.704 1.585 2.567 2.217 1.929 2.492 2.450
1.851 1.752 1.676 2.449 2.127 1.959 3.130 2.819 2.416 3.017 2.968
1.638 1.638 1.638 2.457 2.457 2.457 4.194 4.914 4.914 4.914
3.640 4.835 6.067 1.820 3.640 5.460 1.820 3.640 7.280 1.820 1.820
1512 1512 1512 I008 1008 1008 504 504 504 390 334
2.4 1.5 0.5 11.3 8.0 6.1 25.3 21.0 17.0 33.0 37.3
13.5 11.2 9.2 25.5 20.0 16.6 35.3 33.0 27.1 41.6 45.4
(1) DOWSON AND HIGGINSON [l] (2) DOWSON AND TOYODA [61
Minimum
Central
U G
-0.14
Table 4 (1) (2) (3) (4)
-
-0.13
-0.11
-0.16
-0.10
Comparison of Film Thickness Techniques
Present Study Dowson and Higginson [11 Hamrock and Jacohson [71 Dowson and Toyoda [61
1
It is important to realise that the results presented in this paper demonstrate how consistent and accurate the film thickness relationship of Dowson and Higginson [l] is considering the small number of solutions that were available at the time that it was derived. This applies both to the powers on the independent variables and the overall accuracy of film thickness predictions. This new method is an important advance in elastohydrodynamic lubrication analysis and its application demonstrates some of the possibilities for improving solution times and the accuracy of solutions. REFERENCES
The regression formulae were limited to the 56 cases since residual analysis, even on this limited range, indicated on underlying trend incompatible with the relationship for an extended range. The method proposed in this paper produces results which are in good agreement with previously reported work, whilst substantially reducing the amount of computational work involved. This improves the prospects for obtaining more accurate data over differing ranges of actual working conditions. CONCLUSIONS The method presented in this paper has introduced Mulci-level Adaptive Techniques into the field of Elasto-hydrodynamic lubrication and has demonstrated some of the benefits that such techniques offer. The results presented here cover an extensive range of conditions, and comparisons with the findings of other authors show good agreement, whilst demonstrating a substantial reduction in CPU time. Adaptive meshing permits solutions to be obtained with fewer nodes, whilst still retaining accurate definition of the pressure spike region. No special relaxation procedures are required to cope with the pressure spike or the high loads considered, but the rescaling suggested by Houpert and Hamrock [3] may enable the parameter range to be extended still further
.
DOWSON, D. and HIGGINSON, G.R., "Elasto-hydrodynam4c Lubrication" S.I. Edition, 1977 (Pergamon Press). OKAMURA, H., 'A Contribution to the Numerical Analysis of Isothermal Elasto-hydrodynamic Lubrication', 9th Leeds-Lyon Symposium on Tribology 1982. HOUPERT, L. and HAMROCK, B.J., 'Fast Approach for Calculating Film Thicknesses and Pressures in Elasto-hydrodynamic Lubrication Contacts at High Loads', ASME/ASLE Joint Conference, Atlanta, Georgia, 1985; BRANDT, A., in 'Numerical Methods for Partial Differential Equations', p.69. Editor: Parker, S., 1978 (Academic Press). HACKBUSCH, W., 'Multigrid Methods, 1985 (Springer Verlag.). DOWSON, D. and TOYODA, S., 'A Central Film Thickness Formula for E.H.L. Line Contacts', Proc. 5th Leeds-Lyon Symposium on Tribology, 1979. HAMROCK, B.J. and JACOBSON, B., 'Elasto-hydrodynamic Lubrication of Line Contacts', Trans. ASLE 1984, 24, 275-287. JOHNSONTK.L., 'Regimes of Elasto-hydrodynamic Lubrication', Journal. Mech. Eng. Sci., 1970, 12, 9-16.
189
APPENDIX : Formulation of t h e Newton Raphson Approach The Newton-Raphson t e c h n i q u e is f u l l y d e s c r i b e d i n [ 3 ] , b u t a b r i e f d e s c r i p t i o n is given below. The f i r s t o r d e r Reynolds' e q u a t i o n i n non-dimensional form is g i v e n by:
where
-H
-
H is g i v e n by,
-
- H o + X2 r
P I n IX-s(
ds
(ii)
'min The e q u a t i o n s above (i) and (ii) are d i s c r e t i z e d and a s o l u t i o n g i v i n g fi and P is required a t each nodal point. A system o f N + 1 unknown v a r i a b l e s is formed, t h e s e b e i n g N-1 p r e s s u r e changes (AP ), t h e change i n t h e f i l m and She change i n t h e o u t l e t t h i c k n e s s (AH boundary p o s i ? i o n (AH ) and t h e change i n t h e o u t l e t boundary p o s i t y o n (AH p ). (The p r e s s u r e a t t h e f i r s t node is set t o $ego and n o t included). N e q u a t i o n s a r e g e n e r a t e d u s i n g Reynolds' equation a t t h e N p o i n t s i n s i d e t h e f l u i d r egio n with t h e f i n a l e q u a t i o n b e i n g g e n e r a t e d by a c o n s t a n t l o a d requiement. The m a t r i x e q u a t i o n g e n e r a t e d is:
a f 2 -a f 2 -
a(peHe)
0
aP
c2
af2 .... aPN
....
cN
af2
aHo
0
The v a l u e s o f t h e m a t r i x e l e m e n t s a r e b o t h p r e s s u r e and f i l m t h i c k n e s s dependent and a r e r e q u i r e d t o be r e c a l c u l a t e d f o r e v e r y i t e r a t i o n . Expressions f o r t h e v a r i o u s elements are contained i n 131 and are dependent on t h e p r e s s u r e v i s c o s i t y model used. The s y s t e m o f e q u a t i o n s is s o l v e d u s i n g Gaussian e l i m i n a t i o n w i t h p a r t i a l p i v o t i n g b u t s i n c e t h e Newton-Raphson t e c h n i q u e r e q u i r e s r e l a t i v e l y few i t e r a t i o n s t o a c h i e v e convergence ( t y p i c a l l y 15 i t e r a t i o n s ) t h i s d o e s n o t r e q u i r e l a r g e q u a n t i t i e s of CPU time.
T h i s f o r m u l a t i o n of t h e E.H.L. line c o n t a c t problem a l s o i n c o p o r a t e s a n improved e l a s t i c c a l c u l a t i o n which g e n e r a t e s a s e t o € deformation c o e f f i c i e n t s using a l o c a l quadratic pressure expression using three nodes. The l o a d c o e f f i c i e n t C 2 , , CN a r e produced i n a similar manner and are d e f i n e d i n [31.
...
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191
Paper Vl(iii)
Solutions for isoviscous line contacts using a closed form analytic solution R. Hall and M.D. Savage
This paper i s concerned w i t h o b t a i n i n g f u l l s o l u t i o n s t o t h e i s o v i s c o u s elastohydrodynamic l i n e c o n t a c t problem. The method p r e s e n t e d , d i f f e r s from p r e v i o u s a n a l y s e s i n t h e u s e of a c l o s e d form s o l u t i o n of t h e e l a s t i c i t y problem which r e l a t e s s u r f a c e d i s p l a c e m e n t v ( x ) t o a p p l i e d p r e s s u r e P ( x ) .
1
INTRODUCTION
Figure 1 i l l u s t r a t e s a t y p i c a l elastohydrodynamic c o n t a c t i n which superambient p r e s s u r e s a r e g e n e r a t e d o v e r a domain which i n c l u d e s a n i n l e t r e g i o n and a H e r t z i a n c o n t a c t zone 1x1 2 ah Taking i n l e t and o u t l e t l o c a t i o n s d e f i n e d by
r a t i o and e l a s t i c modulus r e s p e c t i v e l y . The m a t h e m a t i c a l f o r m u l a t i o n i s complete w i t h t h e s p e c i f i c a t i o n of Reynolds boundary c o n d i t i o n s ; p(-a)
.
-x . -
=
-
=
ah-
6
t h e n , a s shown i n f i g u r e 2 , 2L i s a measure of t h e i n l e t sweep and 6 t h e d i s t a n c e of t h e o u t l e t from t h e end of t h e H e r t z i a n c o n t a c t zone. By i n t r o d u c i n g a c o o r d i n a t e system 121
; + ( a , + & )
=
t h e o r i g i n i s a t t h e c e n t r e of t h e p r e s s u r e band which i n t u r n h a s h a l f w i d t h , a , g i v e n by
a
=
ah + '
.
PORITSKY'S EXACT SOLUTION OF THE ELASTICITY PROBLEM
I n a r e c e n t p a p e r , H a l l and Savage (1987) p o i n t o u t t h a t e v e r y E.H.L. problem h a s a n a s s o c i a t e d d r y f r i c t i o n l e s s c o n t a c t problem w i t h i d e n t i c a l s u r f a c e d i s p l a c e m e n t v ( x ) and p r e s s u r e P(x) r e l a t e d by e q u a t i o n [ 5 ] . They p r f v i d e a d e t a i l e d d e r i v a t i o n of t h e c l a s s i c a l P o r i t s k y Solution [2] f o r dry, f r i c t i o n l e s s contacts which c o n s t i t u t e s an e x a c t s o l u t i o n of t h e e l a s t i c i t y problem g i v e n by e q u a t i o n ( 5 ) . Writing
and expanding t h e s u r f a c e d i s p l a c e m e n t a s a F o u r i e r c o s i n e series on t h e h a l f range [O,a] v(;)
together with an i n t e g r a l equation r e l a t i n g s u r f a c e displacement v(x) t o applied pr essu re ;
a
,(
=
o
=
ds + const
P ( s ) loglx-sl
b ( x ) = ho + ( x - i r 6 ) k 2 + v ( x ) i s t h e gap t h i c k n e s s
1 Bn
cos n
u = R
n
2
u2
[lo1
11
F
i s the t o t a l load. I t i s perhaps most c o n v e n i e n t t o i n t r o d u c e Chebyshev polynomials where T n ( x / a ) i s t h e nth Chebyshev polynomial of t h e f i r s t k i n d d e f i n e d by =
cos n
r)
where x
=
a cos!
[ll]
i s a mean s u r f a c e speed 1
-
=
1
1
R1
R2
where v and
E
and hence d i s p l a c e m e n t and p r e s s u r e are now g i v e n by
-+ -
i s v i s c o s i t y ( a s s d t o be constant) E
[91
n c o s n 11 ,+zi F 1
a sin where
Tn(:)
i s a mean r a d i u s g i v e n by
a = - 1 -v2
+ const
B~
=
[6I +
Y
then the pressure d i s t r i b u t i o n i n the contact i s g i v e n by
P(n)
which a r e t o be s o l v e d o v e r t h e c o n t a c t zone 1x1 5 a ; where f o r two c y l i n d e r s of r a d i i R l and R2 and p e r i p h e r a l s p e e d s U1 and U2
=
n
[51
-a
"1
[71
t31
The f o r m u l a t i o n of t h e E.H.L. l i n e c o n t a c t problem i n v o l v e s Reynolds e q u a t i o n
v(x)
(a)
0
=
( 2 ~ .+ ah + 6) 2
x
x
2
p(a)
=
and
represent Poisson's
192
o n l y B2 and B4 which i s symmetrical a b o u t x = 0. . .
C o n d i t i o n s 1161, [ 1 7 ] , [ 2 0 ] and [ 2 1 ] y i e l d
+ 2B2 + 4B4
In addition the surface displacements outside t h e c o n t a c t zone a r e a l s o g i v e n by P o r i t s k y (1950); x
> a ,
v(y') =
-2aF yr -
+
1 B~
8B2 + 64B4
+
e-"y'
c
< -a , v ( y ' )
=
-2aF y'
+
1 ( - 1 1 ~B~
0
[231
+ c
e-"Y'
n
[221
and hence
n x
=
0
=
[I41
Consequent 1
where P(rl)
=
a sin:
3
REYNOLDS BOUNDARY CONDITIONS
With p r e s s u r e now e x p r e s s e d a s a Chebyshev s e r i e s t h e n w e should e x p e c t Reynolds c o n d i t i o n s , i . e . e q u a t i o n s [ 7 ] and [ 8 ] , t o g e n e r a t e c o n d i t i o n s on t h e Chebyshev c o e f f i c i e n t s , Bn , and indeed t h i s i s t h e case. Close examination of e q u a t i o n [ l o ] r e v e a l s t h a t p r e s s u r e i s , - i n g e n e r a l , % i n g u l a r a t t h e ends of t h e c o n t a c t , q = 0 and r) = n , due t o t h e p r e s e n c e of s i n ;i i n t h e denominator. C l e a r l y t h e n p e r a t o r must b e i d e n t i c a l l y z e r o a t rl = 0 and 0 = n i f p r e s s u r e i s t o be f i n i t e in which c a s e i t i s a l s o z e r o ( a t m o s p h e r i c ) . Hence c o n d i t i o n s [ 6 ] reduce t o
-
-2aF+
~=
o
1 (-tin n
B~
y
n
~
f o r which t h e c o r r e s p o n d i n g s u r f a c e d i s p l a c e a i s g i v e n by ment i n 1x1
<
v(x) 4
8aF x2 3 6n (F - 7 )
=
+ const
.
[26 1
SOLUTION OF THE ISOVISCOUS ELASTOHYDRODYNAMIC PROBLEM
W r i t i n g t h e gap t h i c k n e s s e q u a t i o n [ 6 ] i n terms of Chebyshev polynomials y i e l d s
[I61
n and
+ const
2aF
+
=
o
[171
n With z e r o p r e s s u r e a t t h e end p o i n t s it i s r e a d i l y shown t h a t p r e s s u r e g r a d i e n t s are g i v e n by
-
Z a dp -=l dx
n x
+
-
a
(0 +
where t h e series h a s been t r u n c a t e d a f t e r N terms. D i f f e r e n t i a t i n g e q u a t i o n 1131 w i t h respect t o x gives the following expression f o r p r e s s u r e g r a d i e n t i n 1x1 5 a ,
-
n + cos~~(cosnq-l)]
-
sin) 0
a2
and hence a s
-
-
n Bn[n s i n n q s i n
[I81 where
0)
na
cn(x) =
dP dx
Consequently
n
3
~ =
-312 (a2-x2)
X
w i l l be non s i n g u l a r a n d , i n
f a c t , i d e n t i c a l l y zero a t I n
[271
x = a
provided
I n t e g r a t i n g t h e Reynolds e q u a t i o n [ 4 ] and s u b s t i t u t i n g e x p r e s s i o n s [ 2 7 ] and [ 2 8 ] f o r 120 1
0~ .
It i s a l s o noted t h a t the condition
3 dx
(-a) = 0 ,
B~
=
o
respectively yields
dx
N-2 const +
see s e c t i o n 4, would r e d u c e t o
1 ( - I ) ~n3
h ( x ) and
1
B~ T~(:)
=
n= 1
.
n Example :
-az
A s a n i l l u s t r a t i o n of a p r e s s u r e d i s t r i b u t i o n
satisfying x = +a
,
P =
9 = o dx
at both
x = -a
and
t h e s i m p l e s t i s perhaps t h a t i n v o l v i n g
x a ~ ~ ( +2 ,(il+s) )
X
T~);(
-B
~ - , T ~(:>-B~T~(:) - ~
where, f o r c o n v e n i e n c e , we d e f i n e Q(x)
=
h3 (x)
.
"1
193 I n a d d i t i o n t o t h e (N+1) unknown q u a n t i t i e s o c c u r r i n g above, t h e f i l m t h i c k n e s s a t some s p e c i f i e d p o i n t i s r e q u i r e d ( s i n c e B1 t o B and N 6 g i v e o n l y t h e f i l m s h a p e ) and t h i s p o i n t c a n be chosen a s t h e c y l i n d e r c e n t r e l i n e x = II + 6 where h ( x ) = hc the c e n t r a l f i l m thickness. Consequently t h e f i l m t h i c k n e s s a t any p o i n t x i s t h e n g i v e n by
-
The elastohydrodynamic problem i s now reduced t o t h e d e t e r m i n a t i o n of B1 t o BNY and h which i n some s e n s e s a t i s f y e q u a t i o n [ 3 0 ] . T k s c a n b e a c h i e v e d by a p p l y i n g a c o l l o c a t i o n (matching) method as d e s c r i b e d by Fox and P a r k e r (1968) where N-2 c o l l o c a t i o n p o i n t s a r e g i v e n by
and i t i s n o t e d t h a t w i t h a c h o i c e of more p o i n t s t h a n (N-2) t h e method o u t l i n e d h e r e becomes a l e a s t s q u a r e s method. Denoting t h e r i g h t hand s i d e of e q u a t i o n [ 3 0 ] by f ( x ) and t h e c o n s t a n t t e r m on t h e l e f t hand s i d e by Bo y i e l d s
Br =
N-2 1 (N-1) 6nU K=O X
r = 3,
..., N-2
,
K
[381
where
7
X
K
By means of t h e d i s c r e t e o r t h o g o n a l i t y p r o p e r t y f o r Chebyshev polynomials
[34 1
m, n 5 N-2 the
,
Bn's
1
5 n 5 N-2
Bn
=
N- 1
1
K-0
N-2
,
are now g i v e n by
XK f(xK) T n ( a )
.
[351
I n p a r t i c u l a r when f (x,) i s s u b s t i t u t e d using t h e r i g h t hand s i d e of e q u a t i o n [ 3 0 ] t h e followi n g are o b t a i n e d
E q u a t i o n s [36]-[38] p r o v i d e N-2 e q u a t i o n s i n t h e N+2 unknowns. The system of e q u a t i o n s i s complete by i n c l u d i n g e q u a t i o n s [ 1 6 ] , [ 1 7 ] , [ 2 0 ] and [ 2 1 ] , t h e f i r s t t h r e e of which a r i s e from Reynolds boundary c o n d i t i o n s . The i m p o s i t i o n of e q u a t i o n s [ 2 1 ] c o r r e s p o n d s t o t h e g r a d i e n t of p r e s s u r e b e i n g z e r o a t t h e i n l e t x = -a From e q u a t i o n [ 3 2 ] it i s c l e a r t h a t x = -a i s n o t a c o l l o c a t i o n p o i n t and a n i n c o n s i s t e n c y w i t h Reynolds e q u a t i o n i s t h u s a v o i d e d . T h i s a d d i t i o n a l c o n s t r a i n t may b e i n t e r p r e t e d a s requiring t h a t pressure gradients i n the neighbourhood of t h e i n l e t p o i n t x = -a are small and i t should b e n o t e d t h a t t h e u s e of a Chebyshev series e x p a n s i o n n e c e s s i t a t e s t h a t p r e s s u r e g r a d i e n t a t a n end p o i n t i s e i t h e r z e r o o r i n f i n i t e . The c o n s t r a i n t i s t h e r e f o r e a p p r o p r i a t e from b o t h p h y s i c a l and mathematical standpoints A s o l u t i o n of t h e now complete set of nonl i n e a r e q u a t i o n s was a c h i e v e d by d e v e l o p i n g an i t e r a t i v e scheme based on t h e s u c c e s s i v e a p p r o x i m a t i o n of t h e f u n c t i o n Q(x). The method w a s programmed i n F o r t r a n 77 on t h e Amdahl computer a t Leeds U n i v e r s i t y t o g i v e t h e following r e s u l t s .
.
.
5
RESULTS
The program was r u n f o r a r a n g e of l o a d i n g l s p e e d c o n d i t i o n s c o r r e s p o n d i n g t o m o d e r a t e l y deformed ( c l o s e l y H e r t z i a n ) c o n t a c t s and t h e number of terms (N) i n t h e f i l m t h i c k n e s s expansion was v a r i e d between 20 and 50. A f t e r much e x p e r i m e n t a t i o n i t w a s found t h a t a b o u t 40 terms i n t h e series e x p a n s i o n were s u f f i c i e n t t o g i v e
194 a c c u r a t e r e s u l t s w i t h o u t e x c e s s i v e computation a l t h o u g h some a c c u r a t e s o l u t i o n s were o b t a i n e d u s i n g o n l y 30 t e r m s when t h e domain of computat i o n w a s small (under h i g h l y deformed c o n d i t i o n s ) . Also t h e l e a s t s q u a r e s method was t r i e d b u t was c o n s i d e r e d t o be m a r g i n a l l y i n f e r i o r i n some r e s p e c t s t o t h e c o l l o c a t i o n method which gave a Two computed b e t t e r defined o u t l e t constriction. f l u i d p r e s s u r e d i s t r i b u t i o n s and f i l m shapes a r e shown i n f i g u r e s 3 and 4 w i t h t h e c o r r e s p o n d i n g H e r t z i a n c o n d i t i o n s i n d i c a t e d by t h e d o t t e d l i n e s . For comparison, f i g u r e 5 shows t h e r e s u l t s produced u s i n g t h e p r e s e n t method t o g e t h e r w i t h t h o s e of Hooke and O'Donoghue (1972) and Herrebrugh (1968) o b t a i n e d u s i n g a l t e r n a t i v e methods. The computed s o l u t i o n s a r e e x p r e s s e d i n terms of t h e non-dimensional groupings H and WZ/nU where Fig.1 and t h e u s u a l l o a d and speed p a r a m e t e r s U a r e g i v e n by W
E
-
W
and
Fa 2R
x=-a h
'
and
I
u hm
A t y p i c a l elastohydrodynamic c o n t a c t
=
nuu
1
2R
2
I
=
I
i
(2)
(3) (4)
(5)
4
)I
I t
I
!
I
I
r
x =a
x = -a
Extent of fluid pressures
Fig. 2
Co-ordinate Scheme
system f o r t h e Numerical
TABLE 1
FOX, L. and PARKER, B. Chebyshev Polynomials i n Numerical A n a l y s i s , London (1968) O . U . P . HALL, R. and SAVAGE, M.D. J . I n s t . M a t h s Applic ( a w a i t i n g p u b l i c a t i o n ) 1987. HERREBRUGH, K. 262-270.
I I
a(=eta h
References (1)
I
I
0.
One of t h e p r i n c i p l e a d v a n t a g e s of t h e method o u t l i n e d h e r e , i s t h a t t h e u s e of t h e r a p i d l y convergent series expansion p e r m i t s a n a c c u r a t e c a l c u l a t i o n of e l a s t i c d i s p l a c e m e n t . T h i s f e a t u r e of t h e method i s c l e a r l y of g r e a t importance under h i g h l y l o a d e d c o n d i t i o n s when t h e f i l m t h i c k n e s s becomes small compared t o t h e e l a s t i c d e f o r m a t i o n . For comparison, t h e t h i r t y non-dimensionalised B ' s f o r t h e most h i g h l y deformed c a s e i n v e s t i i a t e d a r e shown i n t a b l e 1 ( t h e s e have been non-dimensionalised i n t h e same way as hm so as t o p r o v i d e a comparison w i t h H). A s can be s e e n , t h e c o e f f i c i e n t s d e c r e a s e u n t i l they are about t h r e e o r d e r s of magnitude less t h a n t h e f i l m t h i c k n e s s . This i n d i c a t e s an a c c u r a t e e l a s t i c c a l c u l a t i o n and a small t r u n c a t i o n e r r o r even under s u c h s e v e r e c o n d i t i o n s .
+I
v
I I
I
is t h e v a l u e of f i l m t h i c k n e s s where
;=ah
Hertzian Region
Trans.A.S.M.E.
HOOKE, C . J . and O'DONOGHUE, (1972), 14, No.1, 34-98.
and
(1968), F90,
J.P.
J.M.E.S.
PORITSKY, H. J.App1 .Mech. Trans. Ameri. Soc. Mech.Engrs. (1950), 72, 191.
B .W
n
1
__ 2.U.R 1.713~10~
Bn .W
n
16
2.U.R
2.755
2
-4.606~10~
17
3
-3.469~10'
18
4
2.456~10'
19
2.767
5
- 1.547~10'
20
-1.419
5.94QxlO-' -1.755
6
7.536~10'
21
7.738~10-1
7
-2.217~10'
22
8.928~10-~
8
-1.008~10'
23
-9.320~10-~
9
2.076~10'
24
10
-1.938~10'
25
-2.933x10-'
1.096~10'
26
-6.484~10-~
11
1.217
12
-2.107
27
7.649~10-1
13
-3.844
28
-4.966~10-~
14
6.371
29
7.42Ox lo-'
15
-4.944
30
4.008~10-~
195
-3.0
-2.5
-2.0
-1.5
-0.5
-1.0
0.5
0.0
-3.0
1.0 15 x /ah
-2.5
-1.5
-2.0
-0.5
-1.0
0.0
I
7
"
Hooke and O'Donoghuc
..
+
Hrr rebrugli
0
Prrseir t Metliod
b-.
2 -.
XIO'
1
--
HIGHLY DEFDRHEO
RIGID
+
8 -. 7 .-
+ + +
5
-.
3
..
+
1
2
+
X l d
+: ::
n:::
2
3 4 567891
XlU'
XI00
Fig.5
1.0
x /ah
F i g . 3 & 4 Computed P r e s s u r e D i s t r i b u t i o n s and F i l m Shapes w i t h t h e H e r t z i a n P r o f i l e s i n d i c a t e d by t h e d o t t e d l i n e s .
9 8 .
0.5
2
3 b 567891 X10'
:
+t: :
2
3 b 567891
xld
3 b 567811 w2/,u
Comparison of R e s u l t s w i t h t h o s e of Hooke a n d O'Donoghue ( 1 9 7 2 ) a n d H e r r e s b r u g h ( 1 968).
2 x10'
3 4 5
1.5
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SESSION VII ELAST0HYDR 0DY NA M IC LUBR ICAT10N (2) Chairman: Professor M. Godet
PAPER Vll(i)
Parametric study of performance in elastohydrodynamic lubricated line contacts
PAPER Vll(ii)
Elastohydrodynamic lubrication of point contacts for various Iubricants
PAPER Vll(iii) A numerical solution of the elastohydrodynamic lubrication of elliptical contacts with thermal effects PAPER Vll(iv) A full E.H.L. solution for line contacts under sliding-rolling conditions with a non-Newtonian rheological model
This Page Intentionally Left Blank
199
Paper Vll(i)
Parametric study of performance in elastohydrodynamic lubricated line contacts B.J. Hamrock, R.T. Lee and L.G. Houpert
Abstract
The i n f l u e n c e of o p e r a t i n g p a r a m e t e r s (W, U, a n d C) o n p e r f o r m a n c e p a r a m e t e r s was s t u d i e d f o r e l a s t o h y d r o d y n a m i c l u b r i c a t e d c o n j u n c t i o n s w i t h i d e a l i z e d c o n d i t i o n s of n o s i d e l e a k a g e , f u l l y f l o o d e d c o n d i t i o n s , i s o t h e r m a l b e h a v i o r , smooth s u r f a c e s , and a s s u m i n g a Newtonian f l u i d c o n d i t i o n . Twenty-two cases were i n v e s t i g a t e d c o v e r i n g a c o m p l e t e r a n g e of o p e r a t i n g p a r a m e t e r s n o r m a l l y e x p e r i e n c e d i n e l a s t o h y d r o d y n a m i c l u b r i c a t i o n c o n j u n c t i o n s . From t h e s e s t u d i e s s i m p l e f o r m u l a s were d e v e l o p e d f o r t h e a m p l i t u d e , w i d t h a n d l o c a t i o n of t h e p r e s s u r e s p i k e , t h e c e n t e r o f p r e s s u r e , minimum f i l m t h i c k n e s s a n d t h e mass f l o w r a t e . Introduction E l a s t o h y d r o d y n a m i c l u b r i c a t i o n is a form of f l u i d - f i l m l u b r i c a t i o n where e l a s t i c d e f o r m a t i o n s o f t h e l u b r i c a t e d s u r f a c e s become s i g n i f i c a n t . I t is u s u a l l y associated w i t h h i g h l y stressed m a c h i n e e l e m e n t s s u c h a s r o l l i n g - e l e m e n t b e a r i n g s and g e a r s . S o l u t i o n s t o t h e coupled Reynolds, e l a s t i c i t y and rheology e q u a t i o n s have proved t o be q u i t e d i f f i c u l t . An a t t e m p t t o r e v i e w t h e m e t h o d s of o b t a i n i n g numerical s o l u t i o n s t o t h e e l a s t o h y d r o d y n a m i c l u b r i c a t i o n p r o b l e m is p r e s e n t e d by Hamrock a n d T r i p p ( 1 9 8 4 ) . H i g h l i g h t s o f f o u r main a p p r o a c h e s are p r e s e n t e d , n a m e l y , t h e d i r e c t method, t h e i n v e r s e method, t h e q u a s i - i n v e r s e method, a n d t h e s y s t e m a p p r o a c h method. The a d v a n t a g e s and d i s a d v a n t a g e s o f each method are discussed. A developed approach n o t covered i n Hamrock and T r i p p ' s r e v i e w ( 1 9 8 4 ) i s t h a t of Okamura ( 1 9 8 2 ) . Okamura u s e s a Newton-Raphson method t o s o l v e t h e e l a s t o h y d r o d y n a m i c l u b r i c a t i o n problem i n a system form. I t e r a t i o n s on t h e s y s t e m a p p r o a c h a r e s t i l l r e q u i r e d , a s was t h e case i n t h e d i r e c t method; h o w e v e r , much less c o m p u t e r time i s used. F u r t h e r m o r e , t h e major a d v a n t a g e o f t h e method is t h a t s o l u t i o n s c a n b e o b t a i n e d f o r h i g h e r l o a d s t h a n b y t h e d i r e c t method. (Maximum d i m e n s i o n l e s s l o a d W was o f t h e o r d e r of 8 x ) N e v e r t h e l e s s , these l o a d s a r e much lower t h a n t h o s e i n n o n - c o n f o r m a l c o n t a c t s such as r o l l i n g element b e a r i n g s , where W r e a c h e s t h e o r d e r o f 1 x At h i g h l o a d s t h e e l a s t i c d e f o r m a t i o n s are s e v e r a l o r d e r s of m a g n i t u d e l a r g e r t h a n t h e f i l m t h i c k n e s s . Any i n a c c u r a c y i n p r e s s u r e w i l l cause ( v i a e l a s t i c i t y c a l c u l a t i o n s ) d r a s t i c changes i n t h e f i l m t h i c k n e s s t h a t w i l l , i n t u r n , ( v i a t h e Reynolds e q u a t i o n ) cause l a r g e p r e s s u r e f l u c t u a t i o n s and n u m e r i c a l i n s t a b i l i t i e s , e s p e c i a l l y when t h e v i s c o s i t y is l a r g e , a s is t h e case i n n o r m a l elastohydrodynamic l u b r i c a t i o n c o n t a c t s . A t high loads, t h e v i s c o s i t y of t h e f l u i d can
v a r y by t e n orders of m a g n i t u d e w i t h i n t h e conjunct ion. R e c e n t l y , a n improved v e r s i o n of Okamura's a p p r o a c h was d e v e l o p e d by H o u p e r t a n d Hamrock ( 1 9 8 6 ) t h a n e n a b l e s s o l u t i o n s t o elastohydrodynamic l u b r i c a t e d l i n e contacts t o be made w i t h no l o a d l i m i t a t i o n s . S u c c e s s f u l r u n s were r e p o r t e d t o h a v e b e e n o b t a i n e d a t h i g h p r e s s u r e ( t o 4 . 8 C P a ) w i t h lower CPU times. The new a p p r o a c h p r e s e n t e d by H o u p e r t a n d Hamrock ( 1 9 8 6 ) allows f o r l u b r i c a n t c o m p r e s s i b i l i t y , t h e u s e of R o e l a n d s p r e s s u r e v i s c o s i t y , a g e n e r a l mesh ( n o n c o n s t a n t s t e p ) , and a c c u r a t e c a l c u l a t i o n s o f t h e e l a s t i c deformations. This approach enables elastohydrodynamic l u b r i c a t i o n c a l c u l a t i o n s t o be p e r f o r m e d a t t h e o p e r a t i n g c o n d i t i o n s normally e x p e r i e n c e d i n non-conformal conjunctions such as those t h a t e x i s t i n r o l l i n g element b e a r i n g s and g e a r s . The p r e s e n t p a p e r w i l l u s e t h e v e r y f a s t a n d a c c u r a t e method o f c a l c u l a t i n g t h e p r e s s u r e and f i l m t h i c k n e s s i n elastohydrodynamic l u b r i c a t e d conjunctions d e v e l o p e d by H o u p e r t a n d Hamrock ( 1 9 8 6 ) a n d i n v e s t i g a t e t h e e f f e c t s of a c o m p l e t e o p e r a t i n g r a n g e of l o a d , s p e e d a n d materials p a r a m e t e r s . The p e r f o r m a n c e p a r a m e t e r s t o be s t u d i e d are t h e d e t a i l s of t h e p r e s s u r e s p i k e i n c l u d i n g t h e a m p l i t u d e , w i d t h a n d l o c a t i o n of t h e s p i k e , t h e mass flow r a t e , t h e c e n t e r of p r e s s u r e t h e minimum f i l m t h i c k n e s s . F o r m u l a s w i l l b e developed t o describe t h e performance p a r a m e t e r s a s a f u n c t i o n of t h e o p e r a t i n g p a r a m e t e r s . The r e s u l t s t o b e p r e s e n t e d are o n l y a p p l i c a b l e f o r i d e a l i z e d s i t u a t i o n s of smooth s u r f a c e s , N e w t o n i a n f l u i d b e h a v i o r , i s o t h e r m a l and f u l l y flooded l u b r i c a t i o n conditions. Symbols b
s e m i w i d t h of H e r t z i a n contact,
~ R J Z.m~ Z
200
E
modulus of elasticity, N/m2
El
effective elastic modulus,
dimensionless viscosity,
2
1 -va
2/{-
Ns/m2
II
dIIo
l-vb
+
-1 ,
Ea
N/m2
viscosity at atmospheric pressure,
'lo
Eb
Ns/m2
c
dimensionless material parameter, aE 1
V
Poisson's ratio
H
dimensionless film thickness, h/Rb2
P
lubricant density, kg/m3
Hmi n
dimensionle s minimum film 'min thickness,
P
dimensionless density, p / p o
Hmin
R
dimensionless minimum film thickness
-
density when dp/dx
Pe
=
0
density at atmospheric conditions
PO
obtained from least-square fit of data
Subscripts
h
film thickness, m
a
solid a
he
film thickness when
b
solid b
dp/dx
=
0, m
K
minimum film thickness 2 2 constant. 3n U/(4W
P
dimensionless pressure, p/(PHlmax
pS
dimensionless pressure spike,
P
ps/(pH )max pressure, Pa
('H)max
ma ximum Hertzian
hmin
Pre*ssure Spike
rate, qR/(upobL)
The effect of the operating parameters (W, U, and C) on the amplitude and location of the pressure spike is shown in Table 1. The operating parameters normally used in elastohydrodynamic lubrication line contact analysis are used here where
mass flow rate per unit length,
w
1 -
pressure, E /w/2n, Pa PS
amplitude of pressure spike, Pa
Q
dimensionless mass flow
q
The presence of a pressure spike in the outlet region of elastohydrodynamic lubricated conjunction produces large shear stresses that are localized very close to the surface. Houpert, et al. (1986) found that the pressure spike may halve the life of a non-conformation conjunction.
= -
U
l/(k+ri),
Q,U/E'R
mean surface velocity in direction of motion
W
(ua + ub)/2, m/s 1 dimensionless load parameter, w/E R
W
load per unit width, N/m
X
dimensionless coordinate, x/b
xcP
dimensionless location of center of pressure, xcp/b
XS
dimensionless locating of pressure spike, x,/b
xs,w
dimensionless width of pressure spike
xs,w/b coordinate in direction of motion, m location of center of pressure, m location of pressure spike
G
aE
=
(3)
The amplitude of the pressure spike, its width and its location are made dimensionless in the following way. Ps
absolute viscosity at gauge pressure,
1
It should be noted from the above dimensionless grouping that if the load per unit width (w) changes the dimensionless load (W) changes but not the other operating parameters. Similarly, if the mean velocity (u) is changed, the dimensionless speed parameters (U) changes.but not the other operating parameters. However, when we get to the dimensionless materials parameter (C) we find that it is not possible to change the physical properties of the solid material or the lubricant without influencing the other dimensionless parameters. Equations (1) to (3) show that as either the materials of the solids (as expressed in E') or the lubricant (as expressed in no and a ) are varied not only does the dimensionless material parameter change, but so do the dimensionless speed (U) and load (W).
width of pressure spike pressure-viscosity coefficient, m2/N
(2)
E'R
m
a b dimensionless speed parameter,
U
u = -nou
effective radius in xdirection,
(1)
E~R
kg/ (sm)
R
W
=
pS -
('H'max where (pHImax
=
-
E /W/2n,
20 1
maximvm H e r t z i a n p r e s s u r e S 'w, d i m e n s i o n l e s s s p i k e width xs,w = b
(51
where b = 2 R m , semiwidth o f H e r t z i a n c o n t a c t
x
X S = S
, dimensionless s p i k e l o c a t i o n
The w i d t h of t h e s p i k e X,,, t h e base o f t h e s p i k e .
(6)
is d e s i g n a t e d a t
I n a l l t h e data p r e s e n t e d i n t h i s p a p e r t h e nodal s t r u c t u r e i n t h e v i c i n i t y o f t h e s p i k e had a d i s t a n c e of AX = 0.001, and a t l e a s t t h i r t y nodes were u s e d t o d e f i n e t h e
o p e r a t i n g p a r a m e t e r s normally e n c o u n t e r e d i n physical s i t u a t i o n s existing i n e l a s t o h y d r o d y n a m i c l u b r i c a t i o n . T a b l e 1 shows how these performance p a r a m e t e r s are a f f e c t e d by t h e wide r a n g e o f o p e r a t i n g p a r a m e t e r s f o r r e s u l t s o b t a i n e d from c o u p l i n g t h e Reynolds e q u a t i o n w i t h t h e f l u i d r h e o l o g y and e l a s t i c i t y e q u a t i o n s . These r e s u l t s a r e d e s i g n a t e d by Ps and Xs i n T a b l e 1 . T a b l e 2 shows how t h e o p e r a t i n g p a r a m e t e r s i n f l u e n c e t h e p r e s s u r e s p i k e base-width (x, w ) . This data was u s e d i n d e t e r m i n i n g a n e h p i r i c a l r e l a t i o n s h i p f o r t h e performance p a r a m e t e r s a s a f u n c t i o n o f these o p e r a t i n g p a r a m e t e r s . The e q u a t i o n s developed f o r t h e d i m e n s i o n l e s s p r e s s u r e s p i k e a m p l i t u d e , w i d t h and l o c a t i o n can be expressed as
p r e s s u r e s p i k e . T h i s s o r t of a c c u r a c y was needed i n order t o g e t meaningful r e s u l t s . F i g u r e 1 shows t h e e f f e c t o f d i m e n s i o n l e s s s p e e d on t h e p r e s s u r e p r o f i l e throughout t h e c o n j u n c t i o n . The i n l e t t o t h e c o n j u n c t i o n is t o t h e l e f t and t h e o u t l e t i s t o the r i g h t , w i t h t h e center of t h e contact o c c u r r i n g a t X = 0 . From t h i s f i g u r e , i t is observed t h a t a s t h e d i m e n s i o n l e s s s p e e d decreases, t h e dimensionless pressure s p i k e decreases i n amplitude, the pressure s p i k e width d e c r e a s e s , and its' l o c a t i o n moves more towards t h e o u t l e t . F i g u r e 2 shows t h e e f f e c t of d i m e n s i o n l e s s speed on t h e f i l m s h a p e for e x a c t l y t h e same c o n d i t i o n a s those p r e s e n t e d in Figure 1 . J u s t as w e observed i n Figure 1 t h a t t h e p r e s s u r e s p i k e moved more t o w a r d s t h e o u t l e t a s t h e speed decreased, s o t o o i n Figure 2 we f i n d t h a t c o r r e s p o n d i n g t o t h e s h i f t of t h e p r e s s u r e s p i k e there is a comparable s h i f t i n t h e l o c a t i o n of t h e minimum f i l m t h i c k n e s s . I t is a l s o o b s e r v e d that the l e v e l of t h e f i l m thickness has decreased c o n s i d e r a b l y a s t h e s p e e d is decreased. F i g u r e 3 shows t h e e f f e c t of d i m e n s i o n l e s s p r e s s u r e i n a n EHL c o n j u n c t i o n f o r t h r e e values of dimeggfonless load while and C = 5007.2. keeping f i x e d U = 1 x 1 0 As t h e d i m e n s i o n l e s s l o a d i n c r e a s e s , t h e amplitude o f t h e p r e s s u r e s p i k e d e c r e a s e s , its l o c a t i o n moves more t o w a r d s t h e o u t l e t and t h e p r e s s u r e s p i k e w i d t h d e c r e a s e s . For e x a c t l y t h e same c o n d i t i o n s p r e s e n t e d i n F i g u r e 3 , Figure 4 shows t h e e f f e c t on d i m e n s i o n a l f i l m shape. I t is o b s e r v e d t h a t a s t h e l o a d increases t h e l e v e l o f t h e f i l m shape d e c r e a s e s and t h e l o c a t i o n o f t h e minimum f i l m t h i c k n e s s moves more towards t h e o u t l e t . These o b s e r v a t i o n s were made a b o u t t h e p r e s s u r e s p i k e as t h e d i m e n s i o n l e s s s p e e d and load were v a r i e d and are i n agreement w i t h experimental r e s u l t s p r e s e n t e d by Safa et a l . (1983). I n t h e i r e x p e r i m e n t s , t h i n f i l m manganin p r e s s u r e t r a n s d u c e r s were used i n o b t a i n i n g good r e s o l u t i o n o f t h e p r e s s u r e spike f o r various operating parameters.
Table 1 shows t h e e f f e c t o f d i m e n s i o n l e s s l o a d , s p e e d , and materials p a r a m e t e r s on t h e amplitude and l o c a t i o n o f t h e p r e s s u r e s p i k e . The twenty-two cases p r e s e n t e d i n t h i s t a b l e representing a f a i r l y complete range o f
- 51S
1
-
xS , W
=
=
2.46 W
0.199
-0.941
w -0.878
u0.206 c-0.848 uO. 347 G-O. 126
The b a r on these performance p a r a m e t e r s i n d i c a t e s t h a t a l e a s t s q u a r e s f i t was a p p l i e d t o t h e d a t a o b t a i n e d from c o u p l i n g t h e Reynolds e q u a t i o n w i t h t h e f l u i d r h e o l o g y and e l a s t i c i t y e q u a t i o n s . The v a l u e s of t h e performance p a r a m e t e r s as o b t a i n e d from t h e l e a s t s q u a r e f i t are shown i n T a b l e s 1 and 2 . I t is o b s e r v e d t h a t t h e b a r r e d and u n b a r r e d v a l u e s i n Tables 1 and 2 are i n q u i t e good agreement. Location o f Center o f Pressure A c a l c u l a t i o n very useful i n r o l l i n g t r a c t i o n s t u d i e s is t h e l o c a t i o n of t h e c e n t e r o f p r e s s u r e . The a p p r o p r i a t e e q u a t i o n is
W r i t i n g t h i s i n d i m e n s i o n l e s s form g i v e s (10)
The l o c a t i o n o f t h e c e n t e r o f p r e s s u r e i n d i c a t e s t h e p o s i t i o n on which t h e r e s u l t i n g force is a c t i n g . The f a c t t h a t t h e r e s u l t i n g force is n o t a c t i n g t h r o u g h t h e c e n t e r o f t h e r o l l e r contributes t o t h e rolling resistance. This has a significant effect i n the e v a l u a t i o n o f t h e r e s u l t i n g f o r c e s and power l o s s i n t r a c t i o n d e v i c e s and o t h e r machine elements. T a b l e 2 shows t h e e f f e c t o f t h e o p e r a t i n g p a r a m e t e r s on t h e l o c a t i o n o f c e n t e r o f p r e s s u r e a s o b t a i n e d from t h e n u m e r i c a l a n a l y s i s . A least squares f i t t o t h i s d a t a provided f o r t h e formulation o f t h e following equation
-x
CP
=
-0.800
w -1
.547 u0.589 G-0.421
(11)
The v a l u e s o b t a i n e d from t h i s e x p r e s s i o n a r e shown i n t h e l a s t column o f T a b l e 2 . I n a l l cases shown i n T a b l e 2 t h e c e n t e r o f p r e s s u r e is i n f r o n t of the center o f the Hertzian contact.
202
Table 3 shows t h e e f f e c t o f o p e r a t i n g p a r a m e t e r s (U, W, a n d C) on minimum f i l m thickness. From t h i s d a t a a l e a s t s q u a r e s f i t o f t h e d a t a was p e r f o r m e d w i t h t h e e n d r e s u l t b e i n g the f o r m u l a shown b e l o w h . min -0.142 u0.760 G0.702 H . = - = 0.0280 W (12) min R The v a l u e s o b t a i n e d from t h e l e a s t s q u a r e s f i t of t h e d a t e a r e shown i n T a b l e 3. The v a l u e s o b t a i n e d a g r e e well w i t h t h e n u m e r i c a l o b t a i n e d v a l u e s , s o t h a t a good f i t is o b t a i n e d by u s i n g e q u a t i o n (12). Mass Flow Rate
The mass flow r a t e p e r u n i t l e n g t h c a n b e w r i t t e n as (13) The f i r s t term on t h e r i g h t s i d e o f e q u a t i o n (13) d e s c r i b e s t h e Couette flow and t h e second term on t h e r i g h t s i d e is t h e P o i s e v i l l e flow term. If p
-
-
pop,
n
-
=
non, h
= 12 x while the other operating p a r a m e t e r s were h e l d f i x e d a t U = 1 x a n d G = 5007. I n a l l t h r e e f i g u r e s , t h e sum of t h e C o u e t t e a n d P o i s e u i l l e c o m p o n e n t s of t h e flow i s c o n s t a n t t h r o u g h o u t t h e I t is a l s o o b s e r v e d t h a t a s t h e conjunction. l o a d is i n c r e a s e d , t h e f l o w t h r o u g h t h e contact region remains constant except a t t h e pressure spike.
W
Minimum F i l m T h i c k n e s s
=
2 b H/R,
and x
=
bX a n d
Conclusion The i n f l u e n c e of o p e r a t i n g p a r a m e t e r s o n p e r f o r m a n c e p a r a m e t e r s was s t u d i e d f o r a elastohydrodynamic l u b r i c a t e d conjunction while assuming t h e f o l l o w i n g c o n d i t i o n s (1) n o s i d e l e a k a g e (2) f u l l y f l o o d e d c o n d i t i o n s (3) isothermal behavior ( 4 ) smooth s u r f a c e (5) N e w t o n i a n f l u i d b e h a v i o r Twenty-two cases were i n v e s t i g a t e d c o v e r i n g a c o m p l e t e r a n g e of o p e r a t i o n p a r a m e t e r s normally experienced i n elastohydrodynamic l u b r i c a t i o n c o n d i t i o n . The f o l l o w i n g f o r m u l a s were d e v e l o p e d f r o m t h e 22 cases s t u d i e d . P r e s s u r e Spike Amplitude - = 0.267 W -0.375 u0.174 G0.219 Ps P r e s s u r e S p i k e Width -0.878 "0.347 c-O. 1 26 x = 0.199 w
e q u a t i o n (13) becomes
S
(14) 2
where K
=
3a u -
(15)
4w2 E q u a t i o n (14) f a n b e r e w r i t t e n a s (16) t h e i n t e g r a t e d form of t h e Reynolds e q u a t i o n when made d i m e n s i o n l e s s i n t h e same manner a s above g i v e s (17) O b s e r v i n g e q u a t i o n s (16) and (17) we c a n write t h e following
-
Q = peHe
(18)
-
where ;,He
is pH when
dP = dX
0.
T a b l e 3 shows t h e e f f e c t of o p e r a t i n g p a r a m e t e r s (W, i, and C ) on d i m e n s i o n l e s s mass f l o w r a t e Q or peHe. Applying a least s q u a r e s f i t o f t h i s d a t a we f o u n d t h e f o l l o w i n g t o b e true
Q
=
peHe
=
1.084 w-1.148 u0.737 G0.646
The a g r e e m e n t w i t h t h e n u m e r i c a l d a t a is q u i t e good. F i g u r e s 5, 6, and 7 show t h e e f f e c t of C o u e t t e , P o i s e u i l l e and t o t a l d i m e n s i o n l e s s mass f l o w t h r o u g h t h e c o n j u n c t i o n f o r t h r e e d i f f e r e n t v a l u e s of d i m e n s i o n l e s s l o a d w = 0.1 x w = 1.3 X 10-4, and
tw
Pressure Spike Location -0.941 1 - ys = 2.460 W
u0.206 G-0.848
L o c a t i o n of C e n t e r of P r e s s u r e -1.547 ,,0.589 6-0.421 X = -0.800 W CP Minimum F i l m T h i c k n e s s -0.142 u0.760 G0.702 H . = 0.0280 W min
Mass Flow Rate Q
=
peHe
=
1.084 W
-1.148 "0.737 G0.646
References Hamrock, B. J . a n d T r i p p , J. H . , (1984). t t N u m e r i c a l Methods a n d C o m p u t e r s Used i n E l a s t o h y d r od ynami c L u b r i c a t i o n . I t P r o c e e d i n g s of t h e 1 0 t h Leeds-Lyon Symposium on T r i b o l o g y , B u t t e r w o r t h s , G u i l f o r d , E n g l a n d , pp. 11-19. H o u p e r t , L . G. a n d Hamrock, B . J . , (1986). " F a s t Approach f o r C a l c u l a t i n g F i l m T h i c k n e s s e s and P r e s s u r e s i n Elastohydrodynamically Lubricated Contacts a t High L o a d s . " T r a n s . of ASME, J o u r n a l o f T r i b o l o g y , Vol. 108, No. 3, pp. 41 1-420. H o u p e r t , L. G., I o a n n i d e s , E., K u y p e r s , J. C . , a n d T r i p p , J. (1986). "The E f f e c t of t h e EHD P r e s s u r e S p i k e o n R o l l i n g B e a r i n g Fatigue." To b e p r e s e n t e t l a t T r i b o l o g y C o n f e r e n c e (ASMEIASLE) i n O c t o b e r 1986 i n P i t t s b u r g h a n d l a t e r p u b l i s h e d i n t h e J. o f T r i b o l o g y , T r a n s . of ASME.
203
Okamura, H. (1982). "A Contribution to the Numerical Analysis of Isothermal Elastohydrodynamic Lubrication." Proceedings of the 9th Leeds-Lyon Symposium on Tribology, Butterworths, Guilford, England, pp. 313-320.
DIMENSIONLESS PRESSURE PROFILE a
Safa, M. M. A., Anderson, J. C., and Leather, J. A., (1983). "Transducers for Pressure, Temperature, and Oil Film Thickness Measurements in Bearings." Sensors and Ac? tuators, i o l . 3 (1982183). pp. 119-128.
5 X-COORDINATE, X I 8
Figure 3.
Variation of dimensionless pressure in a elastohydrodynamically lubricated conjunction for three values of dimensionless load parameters while holding the dimensionless speed and materia parameters fixed at U 1 x 10and G 5007.2.
-
-
X-COORDINATE, X I B
Figure
1.
is
DIMENSIONAL FILM THICKNESS
50 o-'
variation of dlmenslonless pressure In a elastohydrodynamically lubricated conjunction for three values of dimensionless speed parameters while holding the dimensionless load and materials parameters fixed at W 1.3 x and C 5007.2.
-
-
DIMENSIONAL FILM THICKNESS L
.
1
I
1.01 -1.5
Figure 4.
12E-03 -1.0
-0.5
Variation of dimensional film shape In a elastohydrodynamically lubrlcated conjunct ions for three values of dimensionless speed parameters while holding the dimensionless load and materials parameters fixed at W 1.3 x and C 5007.2.
-
-
5
1.0
Variation.of dimensional film shape in a elastohydrodynamically lubricated conjunction for three values of dimensionless load parameters while holding the dimensionless speed and materia parameters fixed at U 1 x 10and C 5007.2.
-
Figure 2.
0.0 0.5 X-COORDINATE, X I 8
-
+a
204
DIMENSIONLESS MASS FLOW RATE PER UNIT LENG'
10.
I
xm-2 DIMENSIONLESS 9.01.
MASS FLOW RATE PER UNIT LENGTH
1
I
E 5.0
=
3.0
Couette
2.0
p -1.0
s -2.0
I
3 -3.0
,--\
9 -4.0
Poiseuille
I
2 - 7.0
- 8~.~ .0
-a4 I -1.5
-1.0
1.0
-9-4L5
V a r i a t i o n of C o u e t t e , P o i s e u i l l e and t o t a l d i m e n s i o n l e s s mass flow r a t e through t h e c o n j u n c t i o n when t h e operating parameters are W 0.1 x U 1 x and G = 5007.2.
Figilre 5.
-
1
-0.5 0.0 0.5 X-COORDINATE, X I B
-
figure 6 .
' -1.0
-0.5 0.0 a5 X-COORDINATE, X I B
1.0
J
1.5
V a r i a t i o n of C o u e t t e , P o i s e u i l l e and t o t a l d i m e n s i o n l e s s mass flow r a t e through t h e c o n j u n c t i o n when the operating parameters ar W = 1.3 x U 1 x lo-", and G = 5007.2.
-
-0.5
0.0 0.5 X-COORDINATE, X I B
1.0
I
1.5
V a r i a t i o n of C o u e t t e , P o i s e u i l e and t o t a l d i m e n s i o n l e s s mass flow r a t e through t h e c o n j u n c t i o n when t h e o p e r a t i c parameters are W 12 x 10 U = 1 x and C 5007.2.
--
DIMENSIONLESS MASS FLOW RATE PER UNIT LENGTH
'
' -l:o
Figure 7 .
d
-1.01 -1.5
I I
I,
Y
, 10'
u
,
10'1
'.
0
1
0.1
2
0.20153
0.6160
0.6816
1 .2386
I ,2019
2
O.ZOl53
3
0.1
0.8210
0.8322
1.0690
0.9393
3
0.1
1
0.6
0.8850
0.8851
0.9681
0.8068
"
0.6
5
8.3
I
I
0.9510
0.9116
0.5955
0.6038 ~
1.0
5W7.2
I I
I
..
I
I
~~~~
I
..
.. ..
0.08306
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-0. I1 9R
-0. I 381
- 0 . I388
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~~
~~
~
-0.1011
5
1.3
0.029m
0.02919
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-0.007913
~~
6
3.0
0.9190
0.9718
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6
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0.01 331
0.011w
-0.00232R
-0.002179
7
5.0
0.9816
0.9811
0.3315
0.3611
1
5.0
0.008308
0.008939
-0.wo9393
-0.0009886
8
8.0
0.9890
0.9899
0.3039
0.3055
8
8.0
0.006318
0.005916
-0. Ow1816
-0.0001'179
t
0.001211
0.001113
-0.Do02513
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9
I
'3
t
12.0
0.25
I
0.75
I
I
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0.9932
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0.2621
9
0.9610
0.9668
0.1138
0.1202
11
0.25
0.OlIW
0.01161
-0.WZ908
-0.003101
12
0.5
0.02019
0.01851
-0.001891
-0.001669
0.02597
0.02138
-0.MMMI
-0.00592U
0.02901
0.02362
-0.007721
-0.001023
0.9520
0.9581
0.5163 ~
0.75
0.5088
~~
I
I I
0.5319
"
0.6758
0.6111
15
3.0
0.03529
0.03157
-0.01102
-0.01 P I
0.9381
0.6931
0.7079
16
5.0
0.03921
0.01126
-0.OlPl
-0.01 81 2
0.9321
0.1016
0.7683
11
8.0
0.01117
0.01856
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-0.02389
0.1?31
0.1988
10.0
0.0172 1
0.05211
-0.oai6
0.9290
-a.azm
-0.wsO78
-0.005195
1 .o
0.010
0.9558
15
3.0
0.9190
0.9116
I6
5.0
0.9110
I1
8.0
0.9350
I8
10.0
0.9310
12.0
0.5955
I0
i
I
~
19
2.6
1.0
2.0
2503.6
0.02Wl
.o
0.02046 ~
~
~
~
~~~~
5007.2
LO2901
0.02111
-0.007121
-0.007535
0.6661
7510.8
0.03133
0.03196
-0.0091 35
-0.009366
5.1178
3892.6
0.02627
0.02186
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-0.008001
20
1.3
1
21
0.8661
22
2.6
~
2,
0.8661
0.6661
1510.8
0.9576
0.9585
0.1195
0.1096
22
2.6
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3192.6
0.9685
0.9590
0.5165
0.5919
N
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207
Paper Vll(ii)
Elastohydrodynamic lubrication of point contacts for various lubricants G. Dalmaz and J.P. Chaomleffel
The s i m u l t a n e o u s measurements o f o p t i c a l f i l m t h i c k n e s s e s and t r a c t i o n f o r c e s are performed a t room temperature, for various l u b r i c a n t s , t o i n v e s t i g a t e the e l a s t o h y d r o d y n a m i c l u b r i c a t i o n o f p o i n t c o n t a c t s u n d e r f u l l y f l o o d e d c o n d i t i o n s , o n a b a l l a n d a p l a n e d i s c a p p a r a t u s . Small r e l a t i v e s l i d i n g , l a t e r a l s l i d i n g and s p i n can be imposed n e a r pure r o l l i n g c o n d i t i o n s . The measured minimum and c e n t r a l f i l m t h i c k n e s s e s a r e compared t o t h e t h e o r e t i c a l v a l u e s g i v e n by t h e Hamrock and Dowson formulae. The t r a c t i o n f o r c e s measured s i m u l t a n e o u s l y on t h e b a l l i n t h e r o l l i n g d i r e c t i o n a n d o n t h e p l a n e d i s c i n t h e l a t e r a l d i r e c t i o n a r e compared t o t h e i s o t h e r m a l v i s c o - e l a s t i c Maxwell model proposed by Johnson and T e v a a r w r k . The l i m i t i n g s h e a r s t r e s s s u g g e s t e d by Bair a n d Winer i s a l s o considered.
1
INTRODUCTION
During t h e p a s t 30 y e a r s , t h e e l a s t o h y d r o d y n a mic l u b r i c a t i o n h a s b e e n i n v e s t i g a t e d i n t e n s i v e l y b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y . I t i s now r e c o g n i z e d t h a t b a l l b e a r i n g o p e r a t i n g c o n d i t i o n s are g o v e r n e d e s s e n t i a l l y by e l a s t o h y d r o d y n a m i c f i l m t h i c k n e s s and raceway c o n t a c t s . t r a c t i o n a t b a l l and E l a s t o h y d r o d y n a m i c l u b r i c a t i o n depends on t h e coupled e f f e c t s of t h e e l a s t i c c h a n g e s i n t h e s h a p e o f t h e c o n t a c t and o f t h e r h e o l o g i c a l c h a n g e s i n t h e l u b r i c a n t c a u s e d by t h e h i g h p r e s s u r e s i n t h e c o n t a c t zone. The f i r s t p u r p o s e o f t h i s p a p e r i s t o e v a l u a t e t h e Hamrock and Dowson f i l m t h i c k n e s s equations (1) and t h e J o h n s o n an d Tevaarwerk r h e o l o g i c a l t r a c t i o n model ( 2 1 , f o r a paraff i n i c base o i l , in r o l l i n g point c o n t a c t s w i t h a d d i t i o n a l s l i d i n g , l a t e r a l s l i d i n g and s p i n . The second purpose i s t o t e s t l u b r i c a n t s of d i f f e r e n t n a t u r e ( n a p h t h e n i c base o i l , n a p h t h e n i c b a s e o i l p l u s polymer, s i l i c o n e o i l and g r e a s e ) i n r o l l i n g - s l i d i n g p o i n t c o n t a c t s . This i s done i n o r d e r t o e i t h e r c o r r e l a t e t h e l u b r i c a n t r h e o l o g i c a l p r o p e r t i e s ( t h e mean a p p a r e n t v i s c o s i t y , t h e mean a p p a r e n t s h e a r m o d u l u s and t h e mean l i m i t i n g s h e a r s t r e s s ) t o e x i s t i n g d a t a o b t a i n e d i n d e p e n d e n t l y by B a i r and W i n e r ( 3 1 , o r t o f i n d l i m i t a t i o n s i f a n y , i n t h e u s e of t h e f i l m t h i c k n e s s f o r m u l a e a n d o f t h e t r a c t i o n model u n d e r f u l l y f l o o d e d elastohydrodynamic c o n d i t i o n s . 1.1
Notations
h
plI Ul
spin contact angle
p2
=
= Wl = w2 = u2
Q
=
=
contact location r a d i i
R cosh
pl
o1 cosl3 speed components o f s u r f a c e s 1 and 2 a t p o i n t 0 a l o n g Ox a x i s P2 02
- pl 0
o1 s i n 8 speed components o f s u r f a c e s 1 and 2 a t p o i n t 0 a l o n g Oz axis
o, s i n h L
-
o, A
s p i n a n g u l a r v e l o c i t y along axis
Oy
W
norma
fX
t r a c t i o n f o r c e on t h e b a l l i n t h e d i r e c t i o n Ox
f Z
l a t e r a l t r a c t i o n f o r c e on t h e p l a n e i n t h e d i r e c t i o n Oz
hO
hm
a p p l i e d load
centre film thickness minimum f i l m t h i c k n e s s maximum H e r t z p r e s s u r e
a
r a d i u s of t h e c i r c u l a r H e r t z a r e a
v
viscosity
e
temperature
Oxyz
a x i s of c o n t a c t 0
a
pressure-viscosity coefficient
R
sphere radius
6
temperat ure-viscos i t y c o e f f i c i e n t
Poisson's r a t i o
f
shear r a t e
T
shear s t r e s s
"1,2 E
112
01,2
13
Young's modulus a n g u l a r r o t a t i o n o f specimens l a t e r a l contact angle
-T -P
mean s h e a r stress mean H e r t z p r e s s u r e
208
-P -G -
mean a p p a r e n t v i s c o s i t y mean a p p a r e n t e l a s t i c s h e a r modulus mean l i m i t i n g s h e a r s t r e s s
=a 2
EXPERIMENTAL CONDITIONS
The p o i n t c o n t a c t a p p a r a t u s used and t h e measurements (Fig. l ) have been described i n d e t a i l e l s e w h e r e (4). S i m u l t a n e o u s measurements of load, angular speeds of both specimens, i n l e t l u b r i c a n t t e m p e r a t u r e , t r a c t i o n f o r c e on the b a l l in the rolling direction, l a t e r a l t r a c t i o n f o r c e on t h e p l a n e d i s c and f i l m t h i c k n e s s by o p t i c a l interferometry are p e r f o r m e d under e l a s t o h y d r o d y n a m i c c o n d i t i o n s , a t ambiant t e m p e r a t u r e , i n t h e c a s e o f a c o n t a c t i n l e t c o n t i n u o u s l y f e d by two l a y e r s o f lubr icant T e s t s were r u n u n d er t h e f o l l o w i n g cond it i o n s : r o l l i n g speed U + U from 0 . 2 t o 8 m / s , 1 - sliding U U i n zhe r o l l i n g d i r e c t i o n Ox, f o r c o n s t a n k v a f u e s o f U + U2 and i n t h e 1 s l i d e / r o l l r a t i o range : 0.3 < (U, U2>/(U, + U2) < + 0.3, s p i n n i n g by v a r y i n g t h e b a l l a x i s o f r o t a t i o n 3" t o + 22', a n g l e A from l a t e r a l s l i d i n g by changing t h e c o n t a c t l o c a t i o n angle 8 i n t h e range : - 0.5' < Q < + lo', a p p l i e d l o a d s w from 5 t o 50 N which g i v e t h e Hertzian c h a r a c t e r i s t i c s of the contact d e s c r i b e d i n t a b l e 1. T e s t s w e r e run f o r s i x l u b r i c a n t s Ll, L2, L3, L4, L5 and L6, whose c h a r a c t e r i s t i c s a r e g i v e n i n t a b l e 2. The f i r s t e x p e r i m e n t s a r e performed w i t h t h e p a r a f f i n i c b a s e o i l L1 f o r maximum H e r t z p r e s s u r e s from 0.2 t o 2 GPa, two r o l l i n g s p e e d s 0.8 and 2 m / s and s l i d e / r o l l r a t i o s f r o m 0.3 t o + 0.3. A d d i t i o n a l l a t e r a l s l i d i n g and s p i n a r e a l s o considered. The second set o f e x p e r i m e n t s i s performed under s i m i l a r o p e r a t i n g c o n d i t i o n s , b u t p r a t i c a l l y w i t h o u t l a t e r a l s l i d i n g and s p i n , f i r s t , f o r t h e l u b r i c a n t s L2, L3 and L 4 t e s t e d b y B a i r and Winer on t h e i r h i g h p r e s s u r e r h e o m e t e r s , and s e c o n d , f o r two l u b r i c a n t s used i n b a l l bearing applications : the s i l i c o n e o i l L5 and t h e l i t h i m g r e a s e L6.
elastohydrodynamic point contacts , the Hamrock and Dowson e q u a t i o n s f o r b o t h minimum a n d c e n t r e f i l m t h i c k n e s s e s a r e t h e most complete. Under i s o t h e r m a l , s t e a d y s t a t e , and f u l l y f l o o d e d c o n d i t i o n s w i t h no s p i n and no l a t e r a l s l i d i n g , t h e Hamrock and Dowson formulae c a n b e w r i t t e n f o r t h e c i r c u l a r p o i n t c o n t a c t geometry (1) :
0.67
-=ho
-
-
-
-
-
-
-
3
RESULTS
The s i m u l t a n e o u s m e a s u r e m e n t s o f f i l m t h i c k n e s s e s by o p t i c a l i n t e r f e r o m e t r y and t r a c t i o n forces obtained i n the rolling-sliding-spinning point c o n t a c t a p p a r a t u s under f u l l y flooded e 1a s t oh yd r o d yn a m i c c o nd i t i o n s , a t amb i a n t temperature, for the s i x lubricants, a r e compared t o t h e t h e o r e t i c a l f i l m t h i c k n e s s v a l u e s g i v e n by t h e Hamrock and Dowson formulae and t h e t r a c t i o n f o r c e s c o m p u t e d w i t h t h e i s o t h e r m a l v i s c o - e l a s t i c Maxwell m o d e l s u g g e s t e d by Johnson and Tevaarwerk.
3.1
Film t h i c k n e s s
The c l a s s i c a l elastohydrodynamic t h e o r y assumes t h a t t h e l u b r i c a n t behaves a s a newtonian f l u i d with a v i s c o s i t y which i n c r e a s e s w i t h p r e s s u r e . Although o t h e r s o l u t i o n s a r e a v a i l a b l e f o r
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i s t h e reduced e l a s t i c modulus. The i n t e r f e r o m e t r i c p a t t e r n s o b t a i n e d f o r t h e s i x l u b r i c a n t s are v e r y s i m i l a r i n s h a p e . The r e s u l t s a r e p r e s e n t e d i n terms o f c e n t r e f i l m t h i c k n e s s h and minimum f i l m t h i c k n e s s and 6. h i n t a b l e s 3, 4:5, m The f i r s t s e t o f e x p e r i m e n t s p e r f o r m e d with t h e p a r a f f i n i c b a s e o i l L1 f o r maximum H e r t z p r e s s u r e s f r o m 0.2 t o 2 GPa, t h e two r o l l i n g s p e e d s 0 . 8 and 2 m / s , d i f f e r e n t s l i d e / r o l l r a t i o s and w i t h a d d i t i o n a l l a t e r a l s l i d i n g and s p i n ( t a b l e s 3, 4 and 5) shows that ; f o r t h e g l a s s - s t e e l m a t e r i a l combination, and p r e s s u r e s f r o m 0.2 t o 0 . 5 GPa, t h e o r e t i c a l and experimental film thicknesses a r e i n g e n e r a l agreement w i t h i n 2 10 per c e n t . f o r t h e s a p p h i r e - s t e e l material combination and p r e s s u r e s f r o m 1 t o 2 GPa, c a l c u l a t e d f i l m t h i c k n e s s e s a r e 5 t o 1 0 p e r c e n t lower t h a n t h e measured f i l m s . a d d i t i o n a l s l i d i n g , l a t e r a l s l i d i n g and s p i n p r o d u c e no s i g n i f i c a n t c h a n g e s i n f i l m t h i c k n e s s i n t h e t e s t e d range. The second s e t of e x p e r i m e n t s p e r f o r m e d under similar operating conditions but p r a t i c a l l y w i t h o u t l a t e r a l s l i d i n g and s p i n w i t h t h e v a r i o u s l u b r i c a n t s L2, L3, L4, L5 and L6 w i t h t h e s a p p h i r e - s t e e l m a t e r i a l comb i n a t i o n a n d p r e s s u r e s from 1 t o 2 GPa shows that : f o r t h e n a p h t h e n i c o i l L2, t h e c a l c u l a t e d c e n t r a l f i l m t h i c k n e s s e s a r e 5 t o 20 p e r c e n t l o w e r and t h e minimum f i l m t h i c k n e s s e s 20 t o 50 p e r c e n t l o w e r t h a n t h e m e a s u r e d f i l m thicknesses values. f o r t h e b l e n d e d o i l L4, t h e e x p e r i m e n t a l f i l m t h i c k n e s s e s a r e 30 p e r c e n t l o w e r t h a n t h e p r e d i c t e d v a l u e s when t h e v i s c o u s c h a r a c t e r i s t i c s o f t h e blended o i l L 4 a r e u s e d a n d 3 0 p e r c e n t h i g h e r when t h e v i s c o u s c h a r a c t e r i s t i c s of t h e b a s e o i l L3 a r e used. - f o r t h e s i l i c o n e o i l L5, t h e measured f i l m thicknesses a r e 5 t o 15 per c e n t lower t h a n the calculated values. - f o r t h e l i t h i u m g r e a s e L6, t h e measured f i l m t h i c k n e s s e s a r e 10 p e r c e n t h i g h e r t h a n t h e c a l c u l a t e d v a l u e s using t h e b a s e o i l v i s c o u s characteristics.
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3.2
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and t h e t h r e e l u b r i c a n t p a r a m e t e r s : t h e e l a s t i c shear modulus G I t h e v i s c o s i t y p and t h e r e f e r e n c e Eyring s t r e s s T In o r d e r t o c o n s i d e r t h e l i m i e i n g s h e a r s t r e s s T & a s a p r o p e r t y o f t h e l u b r i c a n t , B a i r and Winer ( 5 , 6 ) h a v e s u g g e s t e d t o c h a n g e t h e v i s c o u s s i n h term i n
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The s h e a r s t r e s s e s T and T a r e s o l u t i o n o f t h e system o f t h e abovg d i f f e r z n t i a l e q u a t i o n s . A s i m p l e and f a s t n u m e r i c a l method g i v e s t h e s h e a r s t r e s s e s when t h e l o c a l v a l u e s o f GI p and 7 which a r e f u n c t i o n s o f t h e p r e s s u r e a r e knownOTraction f o r c e s are o b t a i n e d by t h e integration of the stresses in the circular H e r t z a r e a . In o r d e r t o c o n s i d e r t h e l i m i t i n g shear s t r e s s rk rather than the reference s t r e s s T , t h e s u p p l e m e n t a r y r e l a t i o n T = 7h?/ a P i s g s e d . The f i l m t h i c k n e s s h i s s z p p o s e d t o b e c o n s t a n t i n t h e c o n t a c t a r e a and e q u a l t o the centre film thickness h given by t h e Hamrock a n d Dowson f o r m u l a at?d t h e p r e s s u r e d i s t r i b u t i o n i s h e r t zian. The f i r s t set o f e x p e r i m e n t s o b t a i n e d w i t h t h e p a r a f f i n i c b a s e o i l L1 a l l o w s u s t o compare t h i s two d i m e n s i o n a l i s o t h e r m a l model w i t h t h e e x p e r i m e n t a l t r a c t i o n c u r v e s f and f v e r s u s )/t Ul + for slide/roll ratio ( U U '22.0 m / s , d i $ f e r e n t t h e r o l l i n g speed U 1 H e r t z p r e s s u r e s and w i t h s l i d i n g , l a t e r a l s l i d i n g and s p i n . The r e s u l t s a r e p r e s e n t e d f i g u r e s 2 , 3a) and b ) , 4 a ) , b ) , c ) a n d i ) . The v a l u e s o f t h e mean a p p a r e n t v i s c o s i t y p and o f t h e mean a p p a r e n t e l a s t i c s h e a r modulus G ( 7 ) a r e deduced from t h e s l o p e i n t h e l i n e a r r e g i o n o f t h e t r a c t i o n c u r v e s shorn-figure 2 and t h e mean l i m i t i n g s h e a r s t r e s s T c o r r e s p o n d s t o t h e maximun mean s h e a r stress o! t h e s e t r a c t i o n curves (Fig.5, 6 and 7 ) . T h e s e two l a s t experimental v a l u e s are introduced i n t h e t h e o r e t i c a l model t o compute t r a c t i o n f o r c e s . The r e s u l t s show t h a t : t h e l a t e r a l t r a c t i o n f o r c e measurements c o n f i r m t h e v a l i d i t y of t h e Johnson and Tevaarwerk v i s c o - e l a s t i c t r a c t i o n model f o r two dimensional sheared f i l m s . t h e mean a p p a r e n t v i s c o s i t i e s deduced from t h e t r a c t i o n c u r v e s f o r t h e p r e s s u r e s up t o 0.5 GPa a r e i n agreement w i t h t h e v i s c o m e t r i c d a t a . t h e mean a p p a r e n t e l a s t i c s h e a r m o d u l i a r e
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s t i l l one o r d e r o f m a g n i t u d e lower t h a n expected v a l u e s . t h e mean l i m i t i n g s h e a r s t r e s s which g i v e s t h e f l a t p a r t of t h e t r a c t i o n c u r v e s a p p e a r s t o b e an i m p o r t a n t r h e o l o g i c a l p a r a m e t e r which governs t r a c t i o n . The s e c o n d s e t o f e x p e r i m e n t s a r e perfomed under s i m i l a r o p e r a t i n g c o n d i t i o n s , b u t p r a t i c a l l y w i t h o u t l a t e r a l s l i d i n g and s p i n , f i r s t , f o r l u b r i c a n t s t e s t e d by Winer a n d c o - w o r k e r s on t h e i r h i g h p r e s s u r e r h e o m e t e r s : L 2 , L 3 , L4 a n d , s e c o n d , f o r two s p e c i f i c l u b r i c a n t s L5 and L6 u s e d i n b a l l b e a r i n g a p p l i c a t i o n s . The t r a c t i o n f o r c e s f versus s l i d e / r o l l r a t i o o b t a i n e d with t h e s g l u b r i c a n t s a r e given figures 8a), b ) , c ) , d) and e ) . T h e s e r e s u l t s c o n f i r m m o s t o f t h e previous conclusions : t h e ,shape of the t r a c t i o n curves is s i m i l a r for a l l the lubricants tested. t h e use of the isotherm v i s c o - e l a s t i c t r a c t i o n model which i n c l u d e s t h e l i m i t i n g shear stress y i e l d s s a t i s f a c t o r y t r a c t i o n curves. T h e s e r e s u l t s show s i g n i f i c a n t d i f f e r e n c e s in-the v a l u e s o f t h e mean a p p a r e n t v i s c o s i t y p, o f t h e mean a p p a r e n t combined elastic s h e a r modulus o f b o t h l u b r i c a n t and s o l i d a t c o n t c c t G a n d o f t h e mean l i m i t i n g deduced from t h e t r a c t i o n shear stress T c u r v e s f o r eacPh-Jubricant. The c r i t i c a l p r e s s u r e v a l u e s P s u g g e s t e d b y H i r s t and Moore ( 8 ) t o c a r a c t e r i s e t h e t r a n 6 i t i o n b e t w e e n t h e v i s c o u s and t h e e l a s t i c b e h a x i o u r a r e g i v e n i n t a b l e 7 . The v a r i a t i o n s o f G and T~ w i t h which a r e supposed t o be l i n e s f o r p r e s s u r e s h i g h e r than c r i t i c a l p r e s s u r e P a r e g i v e n i n t a b l e 8. T a b l e 9 shows t h a t , f o r t h e l u b r i c a n t s L2, L 3 and L4, t h e l i m i t i n g s h e a r s t r e s s m e a s u r e d b y B a i r and W i n e r ( 3 ) a r e i n good agreement w i t h t h e mean l i m i t i n g s h e a r s t r e s s b a s e d on t h e s e t r a c t i o n m e a s u r e m e n t s a s d i f f e r e n c e s o f 10 t o 1 5 per c e n t a r e o b s e r v e d .
6
-
4
DISCUSSION
The e l a s t o h y d r o d y n a m i c p o i n t c o n t a c t l u b r i c a t i o n h a s been i n v e s t i g a t e d f o r t y p i c a l maximum H e r t z p r e s s u r e s f r o m 1 t o 2 GPa and r o l l i n g speeds from 1 t o 3 m / s n e a r a m b i e nt t e m p e r a t u r e . In t h i s r a n g e of o p e r a t i n g conditions surface deformations a t contact a r e i m p o r t a n t and t h e l u b r i c n t i_slsubmitbed Jp % s t o 10 8 h i g h s h e a r r a t e s from 10 d u r i n g s h o r t t r a n s i t t i m e s from 10- s t o S.
E x p e r i m e n t a l measurement o f e l a s t o h y d r o dynamic f i l m t h i c k n e s s e s i s d i f f i c u l t b e c a u s e t h e f i l m i s g e n e r a l l y less t h a n a micrometer t h i c k . The u s e o f t h e o p t i c a l i n t e r f e r o m e t r y d e v e l o p p e d by Cameron and Gohar ( 9 ) and Foord e t a1 (10) i s recognised t o be a powerful t e c h n i q u e f o r d e t e r m i n i n g f i l m thicknesses. The Hamrock a n d Dowson f o r m u l a e a r e s e e n t o p r e d i c t l u b r i c a n t f i l m t h i c k n e s s e s which a r e i n r e a s o n a b l e agreement w i t h m e a s u r e d v a l u e s f o r b o t h c e n t r e and minimum f i l m t h i c k n e s s e s . However, f o r l u b r i c a n t L2, a s found p r e v i o u s l y b y Koye and W i n e r ( l l ) , t h e m e a s u r e d f i l m t h i c k n e s s d a t a show t h a t t h e e x p e r i m e n t a l d a t a a r e r o u g h l y 30 p e r c e n t g r e a t e r t h a n t h e c a l c u l a t e d v a l u e s w i t h t h e Hamrock a n d Dowson minimum f i l m t h i c k n e s s f o r m u l a . T h i s r e s u l t
210 c a n b e a t t r i b u t e d t o t h e h&gh v a l u e o f t h e m a t e r i a l p a r a m e t e r aE = 10 which i s g r e a t e r than the o r i g i n a l v a l y s considefed in t h e t h e o r y i . e . from 2.3 10 t o 6 . 8 10 Note t h a t with a low m a t e r i a l p a r a m e t e r a E = 7 5 0 w h i c h corresponds t o a s t e e l - g l a s s c o n t a c t l u b r i c a t e d with water-glycol, good a g r e e m e n t b e t w e e n t h e o r y and experiment i s o b t a i n e d (12). Satisfactory film thickness predictions a r e a l s o o b t a i n e d with l u b r i c a n t s L4, L5 and L6 i f t h e v i s c o s i t y used i n t h e formula i s t h e v i s c o s i t y v a l u e a t t h e h i g h s h e a r r a t e s found i n t h e c o n t a c t and n o t t h e low s h e a r r a t e v a l u e g i v e n b y c l a s s i c a l v i s c o m e t e r . Shear t h i n n i n g e f f e c t s observed with t h e s i l i c o n e o i l L5 a r e less i m p o r t a n t t h a n t h o s e found in elastohyd r o d y n a m i c l i n e c o n t a c t s f o r much h i g h e r v i s c o u s s i l i c o n e f l u i d s b y Dyson and W i l s o n
.
(13). The s i l i c o n e o i l L5 which i s controll%c), t o be n e w t o n i a n up t o s h e a r r a t e s o f 2 10 on a C o u e t t e v i s c o m e t e r h a s an e f f e c t i v e v i s c o s i t y deduced from elastohydrodynamic measurements 23 p e r c e n t lower t h a n t h e n o mi n al v i s c s i y v a l u e , f o r s h e a r r a t e s o f t h e o r d e r o f 10 s The f i l m t h i c k n e s s e s o f t h e o i l L 4 compounded o f t h e b a s e o i l L3 t h i c k e n e d w i t h p o l y m e r , exceed t h a t of t h e corresponding b a s e o i l . S i m i l a r r e s u l t s are o b t a i n e d w i t h t h e g r e a s e L6 compounded of a b a s e o i l t h i c k e n e d w i t h l i t h i u m soap : t h e g r e a s e f i l m thicknesses a r e higher t h a n t h o s e o f t h e b a s e o i l . The e f f e c t i v e v i s c o s i t i e s o f L4 and L6 which m u s t be t a k e n i n f i l m t h i c k n e s s c a l c u l a t i o n s c o r r e s ond -1 t o v i g c o g f t y v a l u e s a t s h e a r r a t e s from 10 s to 10 s The n e w t o n i a n model which i s s a t i s f a c t o r y f o r f i l m t h i c k n e s s p r e d i c t i o n s i s known t o b e v e r y inadequate t o predict t r a c t i o n i n elastohydrodynamic r o l l i n g - s l i d i n g c o n t a c t s . Exper i m e n t a l t r a c t i o n c u r v e s are a l s o d i f f i c u l t t o obt ai n because p u r e r o l l i n g k i n e m a t i c c o n d i t i o n s m u s t b e - t o n t r o l l e d with a precision b e t t e r t h a n 10 and t h e r h e o l o g i c a l b e h a v i o u r of t h e l u b r i c a n t i s d i f f i c u l t t o a n a l y s e f r o m t h e shape of the t r a c t i o n curve near pure r o l 1i n g c o n d i t i o n s . However t h e maximum t r a c t i o n f o r c e which i s a t t r i b u t a b l e t o a non newtonian b e h a v i o u r o f t h e l u b r i c a n t c a n b e p r e c i s e l y o b t a i n e d . The maximum mean s h e a r stress can be c o r r e l a t e d with t h e l i m i t i n g s h e a r s t r e s s which i s a l u b r i c a n t p r o p e r t y s u g g e s t e d by B a i r and W i n e r ( 5 , 6 ) . F o r t h e t h r e e l u b r i c a n t s L2, L 3 a n d L 4 , t h e mean maximum s h e a r s t r e s s e s b a s e d on t r a c i o n m e s u r ments under h i g h s h e a r r a t e s ( 1 0 t o 10' s-') correspond t o t h e l i m i t i n g shear s t r e s s measured i n d e p e n d e n t l y by Bair and Winer o n t h e i r h i g h p r e s s u r e v ' s c o m e t e r u n d e r low s h e a r r a t e s ( 0 . 1 t o 1 s - ~ ) . Alsaad e t a 1 ( 1 4 ) h a v e a l s o shown f o r t h e o i l L3 t h a t i n t h e ranges of our operating conditions of press u r e s , t e m p e r a t u r e s and s h e a r r a t e s , t h e o i l b e h a v e s l i k e an amorphous s o l i d ( g l a s s y s t a t e ) and t h a t t h e g l a s s y s t a t e would be e x t e n d e d t o t h e t o t a l H e r t z c o n t a c t a r e a . The mean l i m i t i n g s h e a r s t r e s s a p p e a r s t o b e an i m p o r t a n t l u b r i c a n t r h e o l o g i c a l p a r a m e t e r which g o v e r n s t r a c t i o n i n e l a s t o h y d r o d y n a m i c c o n t a c t s . In t h i s v i e w , a f o u r r h e o l o g i c a l parameter model i n which t h i s l i m i t i n g s h e a r s t r e s s c o n c e p t i s included i n t h e previous v i s c o - e l a s t i c isothermal Maxwell model was proposed r e c e n t l y b y Evans and Johnson ( 1 5 ) . The v a l u e s o f t h e mean a p p a r e n t v i s c o s i t y and
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,
5
o f t h e mean a p p a r e n t e l a s t i c s h e a r modulus found with t h e m e a s u r e d s l o p e i n t h e l i n e a r r e g i o n o f t h e t r a c t i o n c u r v e s have shown t h a t above a t y p i c a l c r i t i c a l p r e s s u r e , t h e l u b r i c a n t b e h a v i o u r i s m o s t l y e l a s t i c . The mean a p p a r e n t e l a s t i c s h e a r modulus g i v e n r e p r e s e n t t h e c o m b i n e d v a l u e s o f t h e s h e a r m o d u l i i of t h e l u b r i c a n t s and o f t h e s o l j s a t s h e a r r a t e s of t h e o r d e r of lo4 s for short t r a n s i t t i m e . These v a l u e s a r e s e v e r a l o r d e r o f magnitude lower t h a n t h e expected values which a r e o f t h e o r d e r o f 3 P under t h e s t a t i c pr_fssure P and f o r s h e a r r a t e s less t h a n 1 s The l u b r i c a n t r h e o l o g i c a l s h e a r modulus v a l u e s cannot b e found from t r a c t i o n d a t a a t t h e p r e s e n t s t a t e o f t h e a r t . Although mean e l a s t i c shear modulus d a t a cannot be c o n s i dered a s l u b r i c a n t rheological values, the e l a s t i c e f f e c t s i n t h e l u b r i c a n t a r e important a t s m a l l s l i d e / r o l l speed r a t i o s . The l a t e r a l t r a c t i o n force i s obtained both experimentally and t h e o r e t i c a l l y w i t h t h e v i s c o - e l a s t i c model in rolling-sliding-spinning c o n t a c t s . These e x p e r i m e n t a l and n u m e r i c a l r e s u l t s c a n b e a p p l i e d t o t h e c a l c u l a t i o n of t h e s h e a r s t r e s s e s i n t h e b a l l and raceway c o n t a c t s o f an a n g u l a r c o n t a c t b a l l b e a r i n g .
P
.
5
CONCLUSION
The s i m u l t a n e o u s m e a s u r e m e n t s o f f i l m t h i c k n e s s e s and t r a c t i o n f o r c e s i n t h e e l a s t o h y d r o d y n a m i c l u b r i c a t i o n r e g i m e have been performed n e a r ambiant t e m p e r a t u r e f o r f u l l y f l o o d e d p o i n t c o n t a c t s , under moderate r o l l i n g s p e e d s and w i t h a d d i t i o n a l s l i d i n g , l a t e r a l s l i d i n g and s p i n . The o p t i c a l f i l m t h i c k n e s s measurements o b t a i n e d f o r s i x l u b r i c a n t s h a v e shown t h a t t h e Hamrock and Dowson formulae can be a p p l i e d with confidence i n the range of operating conditions tested. However a l i m i t a t i o n a p p e a r s when t h e h i g h m a t e r i a l p a r a m e t e r reduced (pressure viscosity coefficient 4 e l a s t i c modulus) reaches 10 In t h i s c a s e , the theoretical film thickness equations u n d e r e s t i m a t e by 2 0 t o 5 0 p e r c e n t t h e measured minimum f i l m s and by 5 t o 20 per c e n t t h e c e n t r e f i l m s t h i c k n e s s e s . Further, f a i r agreement between t h e o r y and e x p e r i m e n t i s a l s o o b t a i n e d with t h e non-newtonian l u b r i c a n t t e s t e d i f the high shear r a t e v i s c o s i t y v a l u e i s u s e d i n t h e e q u a t i o n s i n s t e a d o f t h e low shear r a t e v i s c o s i t y . The t r a c t i o n f o r c e m e a s u r e m e n t s h a v e shown t h a t t h e i s o t h e r m a l n o n - l i n e a r v i s c o e l a s t i c Maxwell model i n t r o d u c e d by Johnson and T e v a a r w e r k a n d e x t e n d e d t o i n c l u d e t h e l i m i t i n g s h e a r s t r e s s , can be a p p l i e d t o t h e s i x tested lubricants t o predict traction f o r c e s i n elastohydrodynamic l u b r i c a t i o n . The l i m i t i n g s h e a r s t r e s s which i s s a t i s f a c t o r i l y measured i n t h e elastohydrodynamic p o i n t c o n t a c t a p p a r a t u s , a p p e a r s t o be an i m p o r t a n t l u b r i c a n t r h e o l o g i c a l p a r a m e t e r a s i t governs t r a c t i o n . This t r a c t i o n i s o t h e r m a l model c a n b e used t o c a l c u l a t e t h e s h e a r s t r e s s e s in rolling-sliding-spinning contacts.
.
6
*
ACKNOWLEDGEMENT
T h i s s t u d y was p a r t l y f i n a n c e d b y t h e " D i r e c t i o n d e s R e c h e r c h e s , E t u d e s e t Tech-
211 n i q u e s " , P a r i s , F r a n c e , c o n t r a t n o 83-071 and by t h e c o n t r a t n o 8 2 4 5 , A.T.P. "C.N.R.S. F r a n c e N.S.F. U n i t e s S t a t e s " . The a u t h o r s g r a t e f u l l y t h a n k P. T a r a v e l f o r h i s h e l p i n t h e e x p e r i m e n t a l work and f o r h i s t e c h n i c a l a s s i s t a n c e .
( 1 2 ) DALMAZ, G . , " T r a c t i o n and f i l m t h i c k n e s s measurements o f a water g l y c o l and a water i n o i l emulsion i n r o l l i n g - s l i d i v point contacts", Proc. of the 7 Leed s- Ly o n Symposium on T r i b o l o g y , W est b u r y House, London, p. 231-242, 1980.
R e f e r en c e s
a n d WILSON, A . R . , "Film ( 1 3 ) DYSON, A . , th i ckne s ses e 1a s t o h yd r od ynam i c in lubrication by s i l i c o n e f l u i d s " , Proc. I n s t . Mech. Eng., v o l . 1 8 0 , p a r t 3 K , p. 97-112, 1965-66.
-
HAMROCK, B . J . , a n d DOWSON, D. , " I s o t h e r m a l e l a s t o h yd r o d yn am i c 1ub r i c a t i o n o f P O i n t contacts. Part 111. Fully flooded r e s u l t s " , T r a n s . A.S.M.E. , s e r i e s F , v o 1 . 9 9 , p. 264-276, 1977.
JOHNSON, K.L. , and TEVAARWERK,J.J. , "S h ear oil behaviour of elastohydrodynamic f i l m s " , P r o c . Roy. S O C . , London, s e r i e s A , ~ 0 1 . 3 5 6 , p . 2 1 5 - 2 3 6 , 1977. a n d W I N E R , W . O . , "Some o b s e r vations in high pressure rheology of l u b r i c a n t s " , T r a n s . A.S.M.E. , s e r i e s F , ~ 0 1 . 1 0 4 , p. 357-364, 1982. BAIR,
S.,
( 1 4 ) ALSAAD, M . , BAIR, S . , SANBORN, D.M., and WINER, W.O. , "Glass t r a n s i t i o n i n l u b r i c a n t s i t s r e l a t i o n t o elastohydrodynamic l u b r i c a t i o n " , T r a n s . A.S.M.E. , s e r i e s F , ~ 0 1 . 1 0 0 ,p. 404-417, 1978. ( 1 5 ) EVANS, C.R. , an d JOHNSON, K.L. , " R e g i m e s o f t r a c t i o n i n elastohydrodynamic l u b r i c a t i o n " , s u b m i t t e d t o I n s t . Mech. Eng.
angular b a l l DALMAZ, G . , " S i m u l a t i n g b e a r i n g l u b r i c a t e d e 1 I i p t i c a1 c o n t a c t s F i l m t h i c k n e s s and t r a c t i o n m e a s u r e m e n t s", T r i b o l o g y i n t e r n a t i o n a l , v o l . 11, p . 273-279, 1978.
.
B A I R , S . , and 'WINER, W.O. , "A r h e o l o g i c a l model f o r e l a s t o h y d r o d y n a m i c c o n t a c t s b a s e d on p r i m a r y l a b o r a t o r y d a t a " , T r a n s . A.S.M.E. s e r i e s F , v o l . 101, p . 2 5 8 - 2 6 5 , 1979. B A I R , S . , a n d W I N E R , W.O. , " S h e a r s t r e n g t h m eas u r em ent s o f l u b r i c a n t s a t h i g h p r e s series F, v o l . s u r e " , T r a n s . A.S.M.E., 1 0 1 , p.251-257, 1979.
JOHNSON, K . L . , and ROBERTS, A.D. , "Observations of v i s c o e l a s t i c behaviour of a n e I a s t o h yd r odynamic l u b r i c a n t f i Im" , P r o c Roy. S O C . , L o n d o n , s e r i e s A , v o l . 337, p.217-242, 1974.
.
a n d MOORE, A . J . , "ElastoHIRST, W . , at high hydrodynamic lubrication p r e s s u r e s : I1 n o n n e w t o n i a n b e h a v i o u r " , Proc. Roy. SOC., London, s e r i e s A , ~ 0 1 . 3 6 5 , p. 537-565, 1979. and GOHAR, R . , " T h e o r e t i c a l and experimental s t u d i e s of the o i l film l u b r i c a t e d p o i n t c o n t a c t " , P r o c . Roy. vo1.291, SOC., London, s e r i e s A, p. 520-536, 1966.
CAMERON, A . ,
(10) FOORD, C . A . ,
HAMMA", W . C . , and CAMF,RON,A. "Evaluation of l u b r i c a n t s u s i n g o p t i c a l e l a s t o h y d r o d y n a m i c s " , T r a n s . A.S.L.E. , v o l . 1 1 , p. 31-43, 1968.
(11) K O Y E ,
K.A., and W I N E R , W.O., "An e x p e r i m e n t a l e v a l u a t i o n o f t h e Hamrock a n d Dowson minimum f i l m t h i c k n e s s e q u a t i o n f o r f u l l y f l o o d e d EHD p o i n t c o n t a c t s " , T r a n s . s e r i e s F , v o l . 1 0 3 , p. 284-294, A.S.M.E. 1981.
Table 1 Maximum Hertz pressure
Load
N
1
P GP:
Me an Hertz pressure
P
G Pa
Hertz area radius
l a
P
212 Table 2
w i t h 0,
=
25'C,
p,
= lo5
Pa and
+
=
10 t o 100 a-'
The p r e s s u r e and t e m p e r a t u r e c o e f f i c i e n t s a and 6 are d e f i n e d by t h e e x p r e s s i o n s :
Table 3
Table 4
G'PX
I
M I -U
flc
ho th
P
ho ex
P
I
hm th
P
I
hm ex
P
................................................
I
Glass d i s c steel bali I3 = + 0.25', A = 18'
0.244
28.5 29.4 29.8 30.4
0
0.10 0.20 0.40
0.44 0.42 0.42 0.41
0.44 0.44 0.43 0.42
0.38 0.36 0.36 0.35
0.38 0.36 0.34 0.34
0.23
0.37 0.36 0.36 0.35
0.35 0.36 0.36 0.35
0.22 0.22
23.7 25.2 25.3 25.5
0 0.10 0.20
0.21 0.21 0.21 0.21
25.0 26.0 26.1 26.4
0
0.18 0.18 0.18 0.18
25.4 26.3 26.5 26.9
0
21.8 22.3 22.5 22.8
0
0.249
------ ------ ------------- _----------30.4 31.3 31.8 32.1
0
0.10 0.20 0.40
0.22 0.22
0.21
-__---------------- -----___---____-29.6 30.0 30.3 30.9
0
0.10 0.20 0.40
0.22 0.21
0.95 0.92 0.92 0.90
0.86 0.86 0.86 0.86
0.59 0.56 0.56 0.55
0.63 0.61 0.60 0.59
0.86 0.83 0.83 0.82
0.80 0.80 0.80 0.80
0.53 0.51 0.50 0.50
0. 50 0.46 0.46
0.79 0.78 0.77 0.76
0.83 0.80 0.80 0.80
0.48 0.47 0.47 0.47
0.50 0.44 0.44 0.42
0.55 0.54 0.54 0.53
0.59 0.55 0.48 0.46
0.32 0.31 0.31 0.31
0.36 0.33 0.32 0.30
0.50 0.48 0.48 0.48
0.46 0.46 0.44 0.39
0.29 0.28 0.28 0.27
0.30 0.27 0.24 0.22
0.40 ------------- ------ ------------- -----0.10 0.20 0.40 0.10 0.20 0.40
0.45 ------ ------------- ------
-
Sapphire d i s c steel ball IJ = + 0.25', A = 7.8'
0.955
21.4 21.5 21.7 22.0
0
18.8 19.2 19.4 19.8
0
20.4 21.3 21.7 22.1
0
0.10 0.20 0.40
0.33 0.33 0.33 0.32
0.33 0.33 0.32 0.32
0.19 0.19 0.19 0.19
0.21 0.21
0.34 0.33 0.33 0.33
0.38 0.36 0.36 0.33
0.19 0.19 0.19 0.19
0.23 0.22 0.21 0.18
0.30 0.29 0.29 0.28
0.32 0.31 0.28 0.25
0.17 0.17 0.16 0.16
0.16 0.15 0.13 0.10
0.21 0.21
................................................ 0.10 0.20 0.40
__----__-----_----------- _--__-_----_0.10 0.20 0.40
------ ------
0.10 0.20 0.40
------------- ----_-------------22.8 24.0 24.1 24.4
0
0.10 0.20
0.40
------
------
213
__-_-1.451
21.8 21.7 21.6 21.5 21.5 21.5 21.5
-0.3 -0.2 -0.1 0 0.1
0.2 0.3
0.53 0.53 0.55 0.59 0.57 0.53 0.46
0.54 0.54 0.54 0.54 0.54 0.54 0.54
0.31 0.31 0.31 0.31 0.31 0.31 0.31
1.00 1.00
0.32 0.32 0.34 0.39 0.35 0.32 0.30
0.95 0.90
__--_-
1.441
22.3 22.3 22.3 22.5 22.3 22.4 22.4
-0.3 -0.2 -0.1 0 0.1
0.2 0.3
0.53 0.53 0.53 0.53 0.53 0.53 0.53
0.46 0.48 0.50
0.56 0.56 0.48 0.46
-----------
0.30 0.30 0.30 0.30 0.30 0.30 0.30
23.4 24.1 24.5 24.8
0 0.10 0.20 0.40
0.97 0.94 0.93 0.92
1.10 1.10 1.00
0.95
0.55 0.53 0.53 0.52
20.9 21.1 21.3 21.7
0 0.10
0.20 0.40
0.26 0.26 0.26 0.25
0.32 0.30 0.30 0.26
0.15 0.15 0.15 0.15
0.22 0.20 0.18 0.17
22.7 22.7 23.1 23.4
0 0.10 0.20 0.40
0.23 0.23 0.23 0.23
0.24 0.23 0.23 0.20
0.13 0.13 0.13 0.13
0.14 0.10 0.08 0.08
22.4 22.6 23.0 23.5
0 0.10
0.22 0.22 0.22 0.21
0.23 0.21 0.19 0.18
0.12 0.12 0.12 0.12
0.08 0.07 0.06 0.05
19.8 20.4 20.8 21.3
0 0.10
0.55 0.54 0.53 0.52
0.39 0.37 0.36 0.33
0.32 0.31 0.31 0.30
0.26 0.24 0.23 0.20
21.5 23.0 23.2 23.5
0 0.10
0.48 0.47 0.46 0.46
0.33 0.30 0.26 0.24
0.28 0.27 0.26 0.26
0.18 0.15 0.12 0.11
22.8 24.2 24.5 24.8
0 0.10
0.44 0.43 0.42 0.42
0.26 0.23 0.22 0.21
0.61 0.60
1.458
-0.3 -0.2 -0.1 0 0.1
0.2 0.3
0.53 0.53 0.53 0.53 0.53 0.53 0.53
0.46 0.50 0.49 0.53 0.52 0.46 0.46
0.30 0.30 0.30 0.30 0.30 0.30 0.30
0.28 0.33 0.32 0.36 0.36 0.30 0.30
...........................
........................... 21.6 21.6 21.6 21.7 21.6 21.6 21.6
21.9 21.8 21.7 21.4 21.7 21.6 21.4
-0.3 -0.2 -0.1 0 0.1
0.2 0.3
-0.3 -0.2 -0.1 0 0.1
0.2 0.3
0.54 0.54 0.54 0.54 0.54 0.54 0.54
0.54 0.54 0.54 0.54 0.54 0.54 0.54
0.50
0.53 0.56 0.57 0.59 0.53 0.46
0.46 0.48 0.56 0.56 0.57 0.50 0.46
0.97 0.88 0.72
-----0.90 0.80 0.70 0.54
-----------
D
22.3 22.3 22.3 22.3 22.3 22.3 22.3
1.00
0.10 0.20 0.40
1.20 1.20 1.20 1.20
0
------------______------0.28 0.31 0.31 0.36 0.36 0.32 0.30
0.62 0.62
1.10 1.10 1.10 1.10
21.9 22.0 22.2 22.5
0.31 0.31 0.31 0.31 0.31 0.31 0.31
0.31 0.31 0.31 0.31 0.31 0.31 0.31
0.33 0.32 0.36 0.37 0.36 0.33 0.31
0.30 0.32 0.36 0.36 0.36 0.32 0.31
0.957
0.20 0.40
0.20 0.40
.................................. 1.390
1.810
0.20 0.40
0.20 0.40
-------------
_------------
------ ----------- -----0.25 0.24 0.24 0.24
0.12 0.10
0.08 0.06
------
214 Table 6
Mean c o n t a c t a p p a r e n t e l a s t i c s h e a r modulus
G k
I
h
-U
!c
M
O
th
h
ex
lop
P
I
hm t h
P
I
h,
ex
P
1
versus
-G =
F
for
i >
-*
P
A F - B
S i l i c o n e o i l L5 u + u2 = 3 . 0 m / s I3 - O . l l o l A = 8.2"
.----------------------------------------1.040
23.1 23.0 23.0 23.0
0 0.10 0.20 0.30
0.37 0.37 0.37 0.37
0.31 0.31 0.31 0.30
0.22 0.22 0.22 0.22
0.21 0.21 0.21 0.21
23.4 23.9 24.0 24.2
0 0.10 0.20 0.30
0.34 0.34 0.34 0.34
0.30 0.28 0.25 0.24
0.20 0.20 0.20 0.20
0.19 0.19 0.17 0.16
24.6 24.5 24.5 24.5
0
0.32 0.32 0.32 0.32
0.28 0.26 0.22 0.20
0.18 0.18 0.18 0.18
0.16 0.15 0.12 0.10
26.4 26.6 26.6 26.6
0 0.10 0.20
0.50 0.49 0.49 0.49
0.59 0.59 0.59 0.59
0.29 0.28 0.28 0.28
0.40 0.39 0.39 0.39
27.0 27.1 27.1 27.1
0 0.10 0.20 0.30
0.45 0.45 0.45 0.45
0.56 0.53 0.50 0.50
0.26 0.26 0.26 0.26
0.33 0.32 0.31 0.31
27.3 27.4 27.4 27.4
0 0.10 0.20 0.30
0.42 0.42 0.42 0.42
0.48 0.46 0.45 0.43
0.24 0.24 0.24 0.24
0.26 0.25 0.23 0.22
-____------ .-----------_-_-__---_-0.10 0.20 0.30
0.30 _-_---___-___ ..------__---___-__ _____-
-__--------- ------ ------------- ------
Table 7
L1 L2 L3 L4 L5 L6
0.40 0.25 0.35 0.35 0.65 0.30
1
0.35 0.22 0.35 0.35 0.17 0.25
0.30 0.13 0.17 0.20 0.30 0.20
L1 L2 L3 L4 L5 L6
0.083 0.152 0.069 0.073 0.121 0.098
0.027 0.029 0.021 0.021 0.020 0.019
0.105 0.092 0.106 0.109 0.135 0.097
0.029 0.011 0.016 0.021 0.037 0.018
:c.
215
fx
t
optical system
glass of sapphire d l x
0
lubrcant teed
U,
+ U2 =
2.0 m l s
= 1.L4 GPO io = 8.7'
PO
Experiments .
o p =-o.o7 o p = 4v p = 8.2' Theory :
*
e
22.i-c
eE2i.6*c
e
=~I.L*c
tractan ~UCQ nwasurement
F i g . 1 : Schematic v i e w o f t h e a p p a r a t u s
I
Fig.3 a ) : Traction force f X
U,+U?
= 2.0
.
m/s
p = + 0.25' Experiments 0
po =
1.81 GPa
, 8 = 24.L.C
0 Po = I.LOGPa , e = 22.8-C
v
p0
-
z o . 9 e w ~e.21.1-c ,
Theory
F i g . 3 b) : L a t e r a l t r a c t i o n f o r c e f
versus slide-roll Fig.2 : T r a c t i o n f o r c e f r a t i o , p r a t i c a l l y w i t h o u r s p i n and w i t h o u t l a t e r a l s l i d i n g , w i t h p a r a f f i n i c o i l L1.
.
F i g . 3 : T r a c t i o n f o r c e f and l a t e r a l t r a c t i o n force f versus slide-ro?I ratio, pratically without ;pin, f o r d i f f e r e n t lateral s l i d i n g , with paraffinic o i l Ll.
216
fx
(N U,
+ U, =
2.Ornls
Po = 1.45 GPa
p =-0.05' Experirnenrs
o &=0.7-,
>
1
.
0
e.22.4.c
o x = 21.5- 8.22.3 'c v A .-2.4*;e.21.5.~ Theory
1 -0.3
1
-0.2
-
0.5
I
-0.1
0.1
0.2
03
UI -u2 UI
+
-'
- 0.5 F i g . 4 c ) : Lateral for A = 2 1 . 5 ' .
F i g . 4 a ) : Traction f o r c e f
(N)
',
traction
force
fZ
.
!
t
0.5
-0.5
-1
F i g . 4 d) : Lateral for A = 2.4'.
-
F i g . 4 b) : Lateral for A = 8 . 7 " .
t
"2
traction
force
fZ
i
traction
force
fZ
F i g . 4 : T r a c t i o n f o r c e f and l a t e r a l t r a c t i o n force f versus slide-ro51 r a t i o , pratically without'lateral s l i d i n g , for different s p i n c o n d i t i o n s , with p a r a f f i n i c o i l L 1 .
217
1of
-
v
(Pas)
10
P
101
/
/
/
/
Eapcriments.
& = 1.81 tpa ;e = 24.8-c o Po = 1.40 GPa ; 0 = 22.5.C V Po ~ 0 . 9 8GPa; 6 = 23.4.C
o
0
Theory
0
-
High pressure viscometcr data at 25.C
0
-0.2 -0.1
n
n
nrry
0
0
0 0 0 0
0.1
-l
EHO eaperinmts
U1+U2=0.8 m l s
U, + U 2
= 2.0 m l s
1
0
0
+
P
(GPa)
F i g . 5 : Mean a p p a r e n t v i s c o s i t y v e r s u s mean pressure f o r p a r a f f i n ’ i c o i l L 1 .
u, + u 2 u1 t u ,
= 0.8 m l s I
2.0 mls
F i g . 8 a ) : Naphthenic o i l L 2 .
0
+ U, +U, = 2.0 m l s
9
Experiments : 0 Po = 1.81 GPa ,@. 23.5.C o pa = 1.39 GPO ;e = 2 3 . 4 ~
0.05
,
0 Po = 0.97 GPa 8 I 21.7.C
Q
Theory :
-0.3
-
-0.2
-0.1
F i g . 6 : Mean a p p a r e n t e l a s t i c s h e a r m o d u l u s versus mean pressure f o r p a r a f f i n i c o i l L 1 .
O I 0
0
I
16’
OJ0I
:
-0.3
p”
1
U, + U , = 2.Omls
/
0,9
I
0.5
I
I
1
1 -
P (GPa)
Fig.7 : Mean l i m i t i n g s h e a r s t r e s s v e r s u s mean pressure f o r p a r a f f i n i c o i l L1.
F i g . 8 b) : Naphthenic o i l L3.
0.2
0.3 U, 4
Y4
2
218
U,+U,
(N)
= 2.0 m/s
U,
Eaperiments 0 Po = 1.80 GPa
v
,e
I
Theory
GPa
-
2
-
F i g . 8 e ) : Lithium g r e a s e L6.
kv-
4 = 1.91 GPa ;8 = 21.5.C : :( .i.u GPO e = 2 1 . 2 ~
o po
r1.04GPa,0=23.0*C
-
2
v -0.2
GPO; 0 = 27.1.C'
F i g . 8 : Traction f o r c e f v e r s u s s l i d e - r o l l r a t i o , p r a t i c a l l y w i t h o u t s p f n and w i t h o u t l a t e r a l s l i d i n g , for the l u b r i c a n t s L2, L3, L 4 , L5 and L6.
E xperlments
Theory
GPo I 0 = 26.6.C
Theory
u,+U2 = 3.0 rn/s
vp,
GI%; 0: 27. 4.C
5 = 1.43 V 4 = 1.00 0
F i g . 8 c ) : Naphthenic o i l p l u s polymer L4.
o
0.6 rnls
8 :1.65
o
21.7.C
e = 23.5.C p0 =0.96 m,e = 2 1 . 3 ~ 2
o Po =1.39
+ U2 =
Experiments
-0.1
F i g . 8 d ) : S i l i c o n e oil L 5 .
v
0
219
Paper Vll(iii)
A numerical solution of the elastohydrodynamic lubrication of elliptical contacts with thermal effects A.G. Blahey and G.E. Schneider
This paper describes a method for solving the problem of elastohydrodynamic lubrication in elliptical contacts which includes thermal effects. The technique is based on discrete formulations and computer solutions of the fluid flow. the solid elastic deflections and the heat transfer within both the fluid and the bounding solids. Detailed fluid pressure, film thickness and temperature solutions are reported for several conditions. The results clearly indicate the departure from isothermal conditions which exists as the entrainment velocity, and particularly the sliding velocity. are increased. The inclusion of thermal effects decreases the film thickness and the magnitude of the pressure spike by significant amounts
I.
INTRODUCTION Elastohydrodynamic lubrication (EHL) exists in the successfully lubricated contacts that are formed between heavily loaded machine components such as gear teeth and rolling element bearings. This form of lubrication is characterized by an ability to generate pressures that are of sufficient magnitude to increase the lubricant viscosity considerably. and to cause local elastic deformation of the contacting surfaces. Much insight and understanding of the performance of EHL contacts can be gained through theoretical studies. However, the theory which describes this phenomena is sufficiently complex that complete analysis necessitates the use of numerical methods. In 1949, Grubin [ll] first advanced the principles of EHL for line contacts: a limiting condition which is inherently easier to examine because the fluid flow is predominantly onedimensional. By the late 1950's, the isothermal theory for line contacts had become wellestablished with the most notable contributions being made by Dowson and Higginson [5]. By the mid 1960's researchers such as Cheng and Sternlicht [4] and Dowson and Whitaker [7], and more recently, Kaludjercic [la]. had directed their efforts to the incorporation of thermal effects into the line contact theory. Owing to the complications which result from side leakage, it was not until 1975 that a full numerical solution was first presented by Ranger, Ettles and Cameron [21] for the isothermal EHL of point contacts. Hamrock and Dowson [12]. [13], [I41 presented solutions for the more general elliptical contacts, and their results are supported by the experimental comparisons of Koye and Winer 1191. More recently, Evans and Snidle [9], [lo] have developed refined solutions through an inverse solution technique. In 1982, Bruggemann and Kollmann 121 included thermal effects into the EHL of point contacts in an approximate solution which assumed a lubricant pressure distribution. However, it was not until 1984 that a full thermal solution was first presented by Zhu Dong and Wen Shi-zhu [25]. This work presents a solution for the EHL of elliptical contacts including thermal eff-
ects. While including constitutive equations for the pressure and temperature effects on lubricant properties, this analysis simultaneously solves the Navier-Stokes and continuity equations for fluid flow, the elasticity equation for surface deflection. and the energy equations which describe the heat transfer within the lubricant and both solids. Detailed solutions for pressure, film thickness and temperature are presented, and complement the results presented in [251. 1.1 Notation
BI,ke C I J k e D,]k-Coefficients for discrete equations of fluid motion or for discrete fluid energy equation. c - constant in Power Law relationship (9.3101E-10 Pa-l) C - constant in pressure-density relationship (5.83E-10 Pa-1) Cu, - speific heat of solid (460 w/kg"C) D - constant in pressure-density relationship (1.68E-09 Pab1) E - elastic modulus of solid surface (200 GPa) E f - combined elastic modulus for both solids(Pa), where 1 -1 E ' = ' ' lL- ~+ ?- ] l-vjj e
1
ET El, E,, , FIJ, G,, , HI,- coefficients for discrete
pressure equation film thickness (m) constant in Power Law relationship (22.7431) k - contact aspect ratio P - fluid pressure (Pa) P I T . w ( )-, ~atmospheric pressure (Pa) P* - equivalent fluid pressure (Pa) Q - heat flux (W/m2) R,, - radius of Hertz contact (m) U , U , w - fluid velocities in x , y, z-directions (m/s) Z - constant in Roelands equation (0 Greek cy - equivalent pressure viscosity coefficient 0, - thermal diffusivity for solid ( 1. 27E-05m2/s)
K h
220 J,
- temperature-viscosity coefficient (0.0364 "C-l)
Y
- constant in Roelands equation
6
- elastic deflection (m) - fluid viscosity (Pa.s)
(1.9609S+08 Pa)
'I
qo - fluid viscosity at atmospheric pressure
(4.11E-02 Pa.s) constant in Roelands equation (6.3lE-05 Pa.s) 0 - temperature ("c) i - temperature excess ("c) 0 v,j - ambient temperature (30°C) A, - thermal conductivity for fluid (0.145 w/moc) A, - thermal conductivity for solid (46 W/m"c) u - Poisson's ratio (0.3) p - fluid density (kg/m3) po - fluid density at atmospheric pressure (880 kg/m3) Superscripts/Subscripts z y - for the x, y-directions c - for Couette velocities f - for fluid s - for solid T - top solid B - bottom solid 'loo -
,
BASIC EQUATIONS Traditionally, the flow of fluid in thin lubricant films has been studied through solutions of the standard Reynold's equation. While this technique is appropriate for many lubrication problems, it is inadequate for the present problem of thermal elastohydrodynamic lubrication, where fluid properties are likely to vary through the film thickness as well as within the fluid film plane. Although a viable approach to overcome this shortcoming would be to achieve the fluid flaw solution through solving the Generalized Reynolds equation [6]. this work uses the Navier-Stokes and continuity equations directly. Due to the nature of the EHL film shape, and in consideration of the conditions under which it is developed, several simplifying assumptions [15] can be made which reduce the Navier-Stokes equations to 2.
where the integration takes place over the entire region. A , of significant hydrodynamic pressure. The lubricant which is trapped within the contact is subjected to a shearing action imparted by the bounding surfaces. If the shearing is adequate, and the lubricant viscosity is high, then significant heat will be generated within the trapped fluid. An order of magnitude analyses [ l ] suggests that the dominant mode of heat transfer within EHL contacts is by conduction in the direction through the film thickness. Neglecting convection and the remaining conduction terms, the heat transfer within the lubricant can be approximated by the fluid energy equation in the form of
Once the heat has left the lubricant and entered the solid. it is likely to be condut ted away from the contact zone equally in all principal directions. Additionally, since the solid surface moves relative to the contact itself, then a mechanism exists whereby heat can be convected in the direction of surface motion. Since the region of significant temperature increase occurs within a comparably small area relative to the size of the surface itself, then the surface can be modeled as a half-space. From a point attached to the moving body, the surface temperatures can be determined through the solution of
a2e +-a2e a2e 2 ay2+s-
1
ae
LyI
at
A key mechanism that enables the satisfactory operation of EHL contacts is the marked increase in lubricant viscosity -due to high A commonly-used relation is the pressure. Roelands [22] equation
I rlral
'I = rloi-=i
which can be suitably approximated with Power Law expression 193
the
'I = 'Io(l+CP)K
ap
a
Additionally, this work examines lubrication under steady-state conditions, and for this situation. the continuity equation can be expressed as
The pressures which are generated within the lubricant film have sufficient magnitude to deform the bounding surfaces elastically. since the area over which significant elastic deformation occurs is small compared to the physical dimensions of the surfaces that form the contact, it is reasonable to conduct the deflection analysis by modeling the surfaces as homogenous elastic half-spaces At any point within the contact, the total normal elastic deflection of both surfaces [24] is given by
Additionally, viscosity is assumed to vary with temperature according to [25] 'I = floe
-BTP - b l t n )
Lastly, density is modeled as being influenced by pressure according to [12] r
P = P./l
1
+
5 1
clearly, an anthytical solution to the above system of equations is not possible at present. In result, discrete methods are used to examine the present thermal EHL problem. SOLUTION DOPU\IN DISCRETION The fluid solution domain is chosen to have a rectangular shape in the x-y plane. and is bound by the solid surfaces in the z-direction. The inlet and lateral dimensions of the solution domain are sized to reflect fully flooded conditions. In keeping with the findings of Hamrock and -on 1141, these dimensions are taken as " L = 4.0 Rff Yl'..tT = 1.6 3.
w,
221
where Rfl and Rfl are the dimensions in the x and y-directions for a similar Hertz contact. For conditions of high contact speed, the region of significant pressure generation extends in the inlet region, and the inlet dimension is appropriately increased to The solution domain outlet dimension needs to be sufficiently large to contain the fluid
flow cavitation boundary. Since the shape or position of the boundary is not known apriori. it has been found convenient to make z,,
= 1.5
R;,
Having defined the extent of the solution domain. it is now necessary to establish the fashion and level with which it is discretized. Ideally, the solution domain should be discretized to provide the maximum resolution at the positions of greatest change in pressure gradient. For this reason, it is desirable to provide fine resolution in the vicinity of the Hertz circle. particularly in the regions of the pressure spike and cavitation boundary. Conversely, only coarse resolution is required for areas such as the contact inlet. where the pressure variation is gradual. To reflect the above considerations, this work uses a pressure grid discretization scheme which is uniform in the y-direction. but non-uniform in the xdirection. This is Shown in Fig. 5, where the uniform spacing in the y-direction is Ryl /9, and the minimum spacing in the x-direction is Rfl 119. It is further noted that since the lubricant pressures are symmetric about the xaxis, the solution will be computed over just the negative y-plane. At any position within the fluid film plane, the film thickness is divided into a number of equally spaced divisions. Although a non-uniform spacing may be desirable to capture the resolution of the temperature profile in this direction, no effort was made to explore this possibility. Lastly, within any principal direction, the total number of grid points was chosen so that the solution would not be affected significantly should more grid points be used in that direction. For this work, the total numbers of pressure grid points are 50 from inlet to Outlet, 15 for the contact negative y domain, and 15 through the film thickness. As always with problems of this size, the previous statement has been tempered by the computer storage requirements and solution cost considerations. 4. NUMERICAL SOLUTION TECHNIQUES 4.1 Fluid Flow This work has developed a novel numerical procedure to solve the reduced forms of the Navier-Stokes and continuity equations simultaneously. The technique is based upon a control volume formulation with a staggered grid. It is capable of considering a fully three-dimensional variation of fluid properties, while maintaining mass conservation within the plane of the fluid film for any level of solution discretization. The concept of the staggered grid is that the fluid velocities are calculated at locations which differ from the locations where the fluid pressures are solved. For this work, the velocity grid points are located midway between the pressure grid points. Although the posit-
ion of the pressure and velocity grid points remains fixed for any given fluid flow analysis, the control volume boundaries need not remain definitive during all components of the solution. As will be seen, different control volume arrangements are chosen judiciously for the analysis of each of the Navier-Stokes and continuity equations. Further information on the philosophy of the staggered grid can be found in Harlow and Welch [16], and Patankar 1201. Discrete Equation of Fluid Motion; X-Direction Consider the equation of fluid motion for the x-direction. and the appropriate fluid control volume which is shown in Fig. 1. Note that, by having a pressure grid point existing on either side of the velocity grid point, the pressure difference p,,, -P,, provides the driving force for the fluid velocity ulJk . The fluid viscosity and density grid points are chosen to be coincident with the pressure grid points. The control volume formulation integrates the reduced form of the Navier- Stokes equation over a typical control volume.
where the subscripts n,s,e,w,t,b respectively imply the north, south, east, west, top and bottom faces of the control volume. By performing the integration it is possible to arrive at Aflk U i l k - 1 -k B:jk u , J k + c:jk U i j k - I = D:jk [ p h + l ~ - p , ~ ] (1) where the coefficients .4:Ik, BfIk , c:lk , D:,k are functions of the discretization geometry and the local viscosities as detailed in [l]. The above discrete form of the equation of fluid motion cannot be solved to yield fluid velocities since the fluid pressure distribution is not yet known. However, insight to the fluid solution can be gained by examining the discrete equation of fluid motion for two specific cases.
Couette Velocities Consider the fluid domain when there exists no pressure gradient in the x-direction. Under this condition, there is no pressure driven flow, and the fluid velocities are simply Couette velocities (denoted by c superscript) The above equation may be applied to each of the kmax control volumes which comprise a column of fluid having a height equal to the local film thickness. By including the no slip velocity conditions which occur at each fluidsolid interface. it is possible to express the kmax unknown Couette velocities in terms of kmax equations. This represents a system of equations which can be solved easily and efficiently by the tri-diagonal matrix algorithm (TDHA). Pressure Sensitivity Coefficients A second specific case which provides insight into the fluid solution exists when the derivative of the discrete equation of motion is taken with respect to the local pressure difference P,+l,-P,J
.
222
where
+ Fl-ljpZ-l]
FI]pI+lJ
-(FIJ
and f l l k will be identified as a pressure sensitivity coefficient for the x-direction, since it represents the total velocity's sensitivity to change resulting from the local pressure difference in the x-direction. The x-direction velocity is linearly dependent on the indicated pressure difference. As with the Couette velocities, the above equation can be applied to each control volume within a column of fluid. since the fluid velocity is independent of the pressure gradient at each fluid/solid interface, the pressure sensitivity coefficient is set equal to zero at each surface. With this knowledge. a system of kmax equations can be written to describe the kmax unknown pressure sensitivity coefficients within any fluid column. This again is a system which can be readily solved. By examining each fluid column within the fluid solution domain, it is possible to determine the entire field of Couette velocities and pressure sensitivity coefficients. The fluid total velocity can be represented as the sum of its parts by %jk
=
U:jk
+
-p ~ ~ )
f t ~ k ( ~ ~ + l j
The above equation enables the fluid total velocity. a variable which varies in three directions, to be expressed in terms of calculable quantities and the local pressure difference which varies only in two dimensions. In this fashion, the fluid velocity is represented in a form where its dimensionality is reduced from three to two. Discrete Equation of Fluid Motion: Y-Direction The approach which is used for investigating the equation of motion in the y-direction is similar to that which was adopted for examining the x-direction. By integrating the equation of motion over a control volume as shown in Fig. 2, it is possible to arrive at
+
Fl-l,
H,J
+
J
Couette velocities in the y-direction can be determined by solving and pressure sensitivity coefficients can be resolved by applying A!jkgijk-l
+ B t k g i j k + C!jkgijk-l
= DIjk
where
which leads to
Discrete Continuity Equation In examining the continuity equation. it is convenient to consider the control volume shown in Fig. 3. With the geometry chosen in this manner, the mass flaws into and out of a control volume are well defined. By integrating the continuity equation over the control volume, and substituting equations 1 and 2, the discrete continuity equation can be written as a discrete pressure equation with calculable coefficients.
Ht]-lp~j-l
HIJ-l)ptJ
=-Ell
+ El-[]
-
G,,
+
Gg1-1
where the coefficients E l l , F , I , G I , . H , , are defined in [l] . Determination of the Pressure Distribution The fluid pressure distribution can now be described by applying the preceding discrete pressure equation to each fluid column within the solution domain. Further information is required in the form of pressure boundary conditions, and these can be defined as P = P=P{TMOS for the inlet and lateral boundaries of the solution domain. Also, since the pressure distribution is symmetric about the x-axis. aP -
aY
=O
at y=O
Lastly, the Reynolds boundary condition is assumed to describe the cavitation boundary so that a p -P=P\TMO5', a n -0 where n = normal to exit boundary. With the discrete pressure equation written for each fluid column, and by including the pressure boundary conditions. the (imax*jmax) unknown discrete pressures can be expressed in terms of (imax*jmax) equations. This work solved the system of equations by using a modified strongly implicit solution procedure (HSIP) [23], and established a technique whereby the cavitation boundary could be located at any position between adjacent pressure grid points. 4.2 Elastic Deflections The discrete elastic deflections are evaluated through a surface element technique which is described by D w s o n and Hamrock [8] and by Hartnett 1163. Briefly. the discrete elastic deflection is given as ininz
jmnz
m=l
rr=l
1] P,zn 0;"
6,) = L
f
H,jPlj+I
where 0:;" is the influence coeEficient describing the contribution to total elastic deflection at node ij. resulting from the application of a uniform effective pressure, P * , acting on the mn surface element. It should be noted that by using a nonuniform grid spacing in the x-direction, considerably more deflection influence coefficients must be calculated than would be required for a uniform grid. Hawever, it is felt that this penalty is more than offset by the corresponding improvement in resolution of the pressure solution. Additionally, this work makes the distinction that the effective pressure, Pa , differs from the grid point pressure. P . More explicitly. the effective pressure is the pressure which is required to apply an identical load to surface element mn. as would be applied by the grid point pressure field with a bilinear interpolation. The use of the effective rather than grid point pressures was found helpful in the vicinity of the pressure spike, and was felt to provide more consistent input to the conservative fluid flow solution procedure.
223 4.3 Fluid Heat Transfer
The evaluation of fluid temperatures is achieved through a control volume solution of the fluid energy equation. Based upon integrating over the control volume configuration which was used for the continuity equation,
Assuming constant thermal properties, letion of the integration yields
time. Since this work, examines EHL contacts under steady state conditions. the time integration is conducted over the limits from zero to infinity. This is a minor inconvenience which is overcome by integrating up to times which are sufficiently large such that an increase in the limit of integration does not significantly affect the magnitude of the integral. The temperature influence coefficient is expressed in terms of the error function according to
compwhere
where the coefficients A , , , B,, , C,, , DIIk are defined in [l]. A distinct advantage is noted in that the discrete fluid temperature equation is one-dimentional in z . By applying this equation to each of the kmax control volumes through the film thickness, and by incorporating the temperatures which are found at each fluid/solid interface ( = 8, at each interface), the kmax unknown fluid temperatures can be expressed in a system of kmax equations. This system is solved easily and in the same manner that was used to determine the Couette velocities and the pressure sensitivity coefficients. It is noted that the ease with which fluid temperatures are calculated would vanish if those temperatures were connected within the x-y plane : i.e. if other modes of conduction and convection were important. 4.4 Solid Heat Transfer The heat which is generated within the lubricant is conducted into the bounding solid surfaces. The task within the solid heat transfer analysis is to determine the temperature rise at the surface of a moving solid which is subjected to an arbitrary heat flux field within the solution domain. This goal is achieved by starting with Carslaw and Jaeger's [3] solution for the incremental temperature rise experienced by a moving half-space that is heated by an instantaneous point heat source. By integrating the point source over a rectangular area, and over the time during which the source exists, then a surface element method prevails where the solid temperatures are evaluated by a technique which is similar to that used for calculating elastic deflections. Thus, the temperature rise at element ij is given as i m a i jrnnr B,,= Qr,,tTYr'
ry"
r n =1
11
d@,
1,-
t3Z
=A,---
au
-
b=-
2%
bu 2a,
where a and b are the semi-dimensions of the surface element at node mn. for the x and y-directions respectively. and
where T is the time of temperature measurement which tends to infinity. Indeed, computation of all the temperature influence coefficients requires considerable effort. since a non-uniform grid spacing is used in the xdirection, and the two surfaces which form the EHL contact are likely to have different surface velocities. To this end, the coefficients are determined at the beginning of a solution and retained for all subsequent calculation. 5. NUMERICAL SOLUTION PROCEDURE At this point. the basic components of the numerical solution have been established. However, before they can be assembled into an integrated procedure, some further details must be discussed. 5.1
Definition of Film Thickness The lubricant film thickness is defined by the equation
h ( z , y ) = ho + S ( Z , Y+) ~ ( z , Y ) where S(z,y) is the separation which exists solely by virtue of the curvature of the undeformed solid surfaces. and h, is parameter defined by tlo = h, - 6, where h, and 6,. are values at the coordinate origin.
=1
where is the temperature influence coefficient representing the contribution to temperature increase at element ij due to a constant flux acting over element inn, and Q is the heat flux at the fluid/solid interface found from
Q=
-
a=-
a*,
5.2
Reduced Pressure Transformation The solutions presented in this work are obtained by solving the fluid flow equations which have been modified by the reduced pressure transformation of the fluid pressure which exists in the form P
a2
The temperature influence coefEicient is found by integrating analytically with respect to area, but is rendered in a form which must be integrated numerically with respect to
where w is a dummy variable, and 'lo is the fluid viscosity at the lubricant temperature and at atmospheric pressure.
For this work, the Roelands viscosity equation is chosen to model the temperature influence on viscosity. When substituted into the reduced pressure transformation, however. the form of this equation does not integrate to a simple algebraic relationship. To remedy this situation the Power Law viscosity model can be fitted to approximate the Roelands equation very well. FOf the constants provided in the notation, the two viscosity relations differ by less than 4% over the important pressure range of 0.05 GPa to 0 . 1 GPa. The resulting expression for g is I'
I-(l+CP)'-h h
-
C(K-1)
Substitution of the reduced pressure transformation into the fluid flow solution leads to the reduced Navier-Stokes equations
which indicates that the subsequent discrete equation coefficients are simply expressed in terms of q 0 instead of r) . The application of the reduced pressure boundary conditions is essentially unchanged. An important consequence of the reduced pressure transformation is 'I
-
4
asP-oo
u
where CU - C(K-1) for K i ? in the Power Law viscosity relationship. Thus, while trying to converge the reduced pressure field, any stray reduced pressures which exceed l / Z may be reset to just less than this limiting value. This is preferable to computing with regular pressures of corresponding magnitudes. The benefit of this cannot be overstated, since typical converged solutions have a maximum reduced pressure in the range of 0.99
5 6qmax<
1.0
.
5.3 Pressure Relaxation Computation of EHL phenomena is difficult because of the strong coupling which exists between the lubricant viscosity. pressure and local film thickness. To solve the highly non-linear system of equations which mathematically describes this problem, this work uses an iterative procedure which incorporates a relaxation of fluid pressures between consecutive iterations. This solution procedure can be conceptualized as one that starts with an assumed pressure distribution. From elasticity theory, the fluid film shape is then determined. Next, hydrodynamic theory is applied to the film shape to establish a second fluid pressure field. Through the relaxation process, the second pressure field updates the first so that any change in pressure occurs gradually. Using the superscripts n and o for new and old, this relaxation process can be viewed as qb
where
e
0<€<1.0
= g"
+ t ( Q L- 'I")
is relaxation parameter in the range
.
This work incorporated the pressure relaxation process directly into the current fluid pressure computation, rather than by applying it between consecutive iterations.
This technique is reported in [ l ] . and provides a means for the fluid pressure field to evolve gradually, with better feedback being established between the pressure and deflection calculations. 5 . 4 Solution Procedure F l o w Chart
A flow chart of the solution procedure is The process starts with detailed in Fig. 4 . any converged pressure solution which may have been obtained for a contact with similar conditions. Alternatively. if no appropriate solutions are available, then the procedure may be started with a modified Hertzian profile similar to the one specified by Evans and Snidle
191* An examination of the solution procedure flow chart reveals that the thermal BHL analysis is broken into two parts: a core analysis which computes isothermal EHL analyses, and a secondary analysis which determines the thermal influences. The advantage of this partitioned analysis is that the pressure-deflection component of EHL solutions are very slow to converge. Once satisfactory convergence has been obtained. however, the inclusion of thermal effects and the subsequent convergence of the solution are relatively rapid. For this reason, all thermal EHL results reported in this work were firstly solved as isothermal problems. Then, the thermal effects were incorporated to achieve the full solution. The isothermal component of the solution procedure is done using a "straightforward" approach (This is in contrast to the "inverse" solution procedure proposed by Evans and Snidle [ 101). This method alternates between calculations of the fluid pressures for a given set of elastic deflections and vice-versa. Resultingly. the solution procedure has two main iteration loops. Experience has shown that the calculation of elastic deflections is more computer intensive than is the determination of the fluid flow. To this end, it is the outer loop which conducts the elasticity analysis, while the inner. more frequently accessed loop completes the fluid flow. Within the inner loop. the reduced fluid pressures are calculated for a set of fixed elastic deflections. Additionally, adjustment is made by positioning one surface nearer to or farther from its neighbour to achieve the desired contact load. At no time are the surfaces located sufficiently close to each other to permit the determination of a reduced pressure greater than 1/ ( Y . The pressure distribution which emerges from the inner loop may not reflect the desired contact load if it is limited by a local reduced pressure which approximates 1 / t % . To compensate, the first step of the outer loop is to normalize the fluid pressures to produce the desired contact load ( The concept behind this approach is that it is more important to retain the shape of the EHL pressure distribution, even at the expense of having pressure magnitudes which are somewhat incorrect). With the normalized fluid pressures, the outer loop updates the elastic deflections and the pressure influence on fluid density. The secondary, or thermal analysis consists of a single iterative loop. For a given fluid flow solution and set of solid surface temperatures, the fluid energy equation is solved to establish the heat flux field which exists at
225
each fluid/solid interface. With this information, the energy equation is solved for each solid so that a new estimate is determined for each surface temperature distribution. Using a relaxation process, the surface temperature solutions are updated. thus completing a single iteration of the thermal influence loop. 6. ISOTHERIWL EHL SOLUTIONS - A VALIDATION OF THE SOLUTION PROCEDURE. The solution procedure identified in this work has been thoroughly tested by reproducing twenty isothermal EHL solutions from the wellestablished works of Ranger, Ettles and Cameron [21]. Hamrock and W s o n [12], [13], [14] and Evans and Snidle [lo]. As a sample of this validation, two isothermal EHL solutions are presented here as reproductions of the work by Hamrock and W s o n [14]. The conditions for these tests are identified as Cases 1 and 2 of Table 1, and represent conditions of fast and slow entrainment velocity. As an indication of the solution quality, Fig. 5 shows a three-dimensional plot of the pressure solution for Case 2. This plot clearly shows the smoothness of the pressure distribution, and the detail of the pressure spike which has been afforded by the use of the nonuniform pressure grid (Note that this and all subsequent graphs are plotted with normalized dimensions so that the Hertz ellipse appears circular). This same information may be viewed differently in the dimensionless pressure ( P / E ' ) contour plot of Fig. 6, where the "tic marks" indicate. the pressure grid points, and the cavitation boundary is drawn at the contact The corresponding dimenexit (right side) sionless film thickness ( h / R Z ) contour plot is shown in Fig. 7, where it is apparent that the contact central region develops a nearly constant thickness, and the minimum thickness occurs at two well-defined side lobes. Lastly, the centre-line fluid pressure and film thickness profiles are detailed together as the solid lines of Fig. 8. Shown as dotted lines are the corresponding solutions from Hamrock and Dowson [14]. The film thickness solutions are seen to be virtually identical, with the pressure solutions showing some differences, particularly in the position and height of the pressure spike. These differences are also noted in Fig. 9 which identifies the centre-line solutions for case 1. It is believed that these results provide unquestionable validation of the isothermal EHL solution.
.
THERMAL EHL RESULTS Thermal EHL solutions have been determined for three basic contact conditions identified as Cases 3 through 5 of Table 1. (It is noted that Cases 3 and 5 are thermal solutions of isothermal Cases 1 and 2 respectively). Each of the above cases was further examined for four sliding ratios, ranging from conditions of near-rolling (S=0.5) to pure sliding (S=2.0 1, where S is the sliding ratio defined by 7.
s = -UT-u/l U
anduT,uBare the velocities of the top and bottom surfaces in the x-direction. Samples of the resulting EHL dimensionless temperature ( @ X I / 'lou2 1 solutions are shown in Figs. 10 and 11. Figure 10 shows dimensionless temperature contours along the contact centre-line for Case
3. It should be noted that these plots are drawn with much greater magnification in the vertical direction than for the horizontal direction. Taking this into account, the greatest temperature gradients exist through the film thickness, suggesting significant heat conduction in this direction. Figure 10 shows that, for conditions of near-rolling, only a moderate temperature rise is experienced, with the maximum temperature occurring midway between both surfaces and coinciding with the position of pressure maximum which is located just 'before the fluid film exit constriction. The region of significant temperature rise extends further upstream than downstream of the temperature maximum. The temperature contours are nearly symmetric for the top and bottom halves of the film thickness, and this is expected since both surfaces are moving at roughly the same velocities and hence have similar heat removal characteristics. For pure sliding conditions, considerably more heat is generated within the contact and the fluid temperature rises considerably over a larger region of the solution domain. The temperature maximum still occurs near the fluid mid-plane, but the temperature contours are no longer symmetric for the top and bottom halves of the fluid film. This lack of symmetry reflects that only the bottom surface is moving. A close examination of this figure suggests a mechanism where heat is transferred from the fluid to the solid near the contact centre. The heat is convected downstream by the bottom surface, and is then conducted back into the fluid in the exit region. Figure 11 shows the temperature solutions for the top and bottom surfaces, and fluid midplane of the sliding condition for Case 4. The fluid mid-plane temperatures clearly reflect the corresponding pressure solution. suggesting that the pressure effect on fluid viscosity is a key factor in the generation of heat within EHL contacts. Significant temperatures are generated only within the contact central area, with slight increases occurring in the contact inlet. The surface temperature contours are more rounded in shape, and this results from the ability of the heat to diffuse better within the solids than within the lubricant. The influence of thermal effects on EHL pressure and film thickness distributions is shown in Figs. 12 and 13. where the corresponding isothermal solutions are also plotted for comparison. The thermal solutions are seen to approach the isothermal solutions for near rolling conditions. However. as more sliding is introduced to the contact, the thermal effects become appreciable and this is noted as a departure from the isothermal solutions. For contacts with large entrainment velocities (Case 3). the influence of thermal effects is particularly evident. As the transition from rolling to sliding occurs, the magnitude of the pressure spike decreases considerably, and the location of pressure maximum moves upstream. The corresponding film shapes develop a less pronounced exit constriction, and the central film thickness decreases to about 75% of the isothermal value. These thermal influences are also present but to a lesser extent for contacts with small entrainment velocities, ( Case 5 ) as shown in Fig. 13. The influence of thermal effects on film thickness is summarized in Fig. 14 as the ratio
226
of thermal/isothermal film thickness plotted as a function of sliding ratio for all twelve thermally influenced EHL solutions examined in this work. An exact comparison of the present results with those from Zhu Dong and Wen Shi-zhu [25] cannot be made. since some differences exist between the mathematical descriptions of the problem and in the methods by which they were solved (The work by [25] was probably not formulated with a conservative fluid flow solution, used a higher order deflection analysis, neglected conduction of heat in the x-direction within the solids. but included heat convection within the fluid). Despite the above-mentioned differences, a qualitative comparison can be made between the present solution for Case 4 with S=0.5 and a similar solution from [25] but with S=0.25. The two solutions generally show good consistency between the pressure distributions, particularly in the response of the pressure spike to thermal effects. The fluid film shapes are i n approximate agreement, except for the slight difference that [25] predicted the minimum film thickness to occur over a single crescentshaped area, while the present work found the minimum at two side lobes. The solution from [25] estimates a 1.3% reduction in isothermal central film thickness. which compares well with 2.1% for the present work. The temperature solutions also show reasonable agreement over most of the contact, except at the localized position of temperature maximum ([25] predicted 40°C temperature increase versus 19OC for present work). At the present time. it is not known whether this discrepancy reflects the differences in modeling the heat transfer. or whether it can be simply attributed to the difference in the resolution of the pressure spike and its subsequent impact on local heat generation. When the complexity of this thermal EHL problem is taken into account, it is believed that very good agreement is established between the two solutions.
9 ACKNOWLEDGEMENT The author (AGB) gratefully acknowledges financial support for this work i n the form of a Postgraduate Scholarship from the Natural Sciences and Engineering Research Council of Canada, The American Society of Lubrication Engineers Special Education Grant, and the University of Waterloo Carl Pollock Fellowship.
8. CONCLUSIONS Several conclusions may be drawn from this work. They are: 1) A novel solution procedure has been developed for examining two-dimensional Newtonian fluid flow in thin lubricant films. This technique is based on a control volume formulation which is capable of describing fluid property variations in three dimensions. Due to the nature of this formulation. conservation of mass is ensured for each fluid column. while this is not the case for c m o n non-conservative finite difference methods. 2) Based on some earlier work [3]. a surface element method has been developed for determining the temperature at the surface of a translating flat half-space which is subjected to an arbitrary heat flux field. The solution considers conduction of heat in all three principal directions. and has no restrictions on half-space velocity. 3) This work has evaluate elastic deflections by a well-established surface element solution [8], [17]. However, it has been found that effective pressures provide more consistent input for the technique than do simple grid point pressures. This supports the need for higher order deflection analysis for the accurate modeling of EHL contacts, particularly
APPENDIX References 1. B1ahey.A.G. The Elastohydrodynamic Lubrication of Elliptical Contacts with Thermal Effects, PhD.Thesis, Univ. of Waterloo. 1985. 2. Bruggemann,H. and Kollmann.F.G.,"A Numerical Solution of the Thermal Elastohydrodynamic Lubrication in an Elliptical Contact." J.Lubr.Techno1.. Vo1.104. no.3. pp.392-400. 1982. 3. Cars1aw.H.J. and Jaeger,J.C. Conduction of Heat in Solids Clarendon Press. Oxford, 1959. 4. Cheng.H.S. and Sternlicht.B.."A Numerical Solution for the Pressure, Temperature, and Film Thickness Between Two Infinitely Long. Lubricated Rolling and Sliding Cylinders, Under Heavy Loads."J.Basic Engrg., Vo1.87. n0.3, pp.695-707, 1984. 5. Dows0n.D. and Higginson,G.R., " A Numerical Solution to the Elastohydrodynamic Problem." J.Mech.Engrg.Sci., Vol.1. no.1. p.6. 1959. 6. Dowson,D., "A Generalized Reynolds Equation for Fluid-Film Lubrication," 1nt.J.Mech. Sci., V01.4, pp.159-170, 1962. 7. Dowson,D. and Whitaker.A.V.."A Numerical Procedure for the Solution of the Elastohydrodynamic Lubrication Problems of Rolling and Sliding Contacts Lubricated by a
in the vicinity of the pressure spike. 4) The present work has verified several isothermal EHL solutions presented in the literature [9]. [13]. 1141, [211. 5) A full solution for the EHL of elliptical contacts with thermal effects has been presented. Detailed temperature solutions have been calculated, and show slight temperatxe increases in the contact inlet area. This results from the enhanced fluid strain rates in the fluid recirculation zone, and manifests the mechanism of inlet shear heating. Significant fluid temperature increases were calculated within the central region of the contact, with temperature maximums occurring near the fluid mid-plane just ahead of the contact exit. The fluid temperature contours were found to follow the pressure solution closely, suggesting that the effect of pressure on fluid viscosity contributes considerably to the generation of heat within EHL contacts. For nearrolling conditions. the thermally influenced pressure and film thickness solutions were found to approach the corresponding isothermal predictions. However. as more sliding was introduced into the contact, the film thickness experienced subtle changes in profile, and reductions in magnitude by up to 25%. The pressure distributions change little over most of the contact, except for a significant decrease in the magnitude of the pressure spike. and an upstream shift in spike location. This is a detail of the present pressure solutions which brings the theoretical pressure spikes in closer agreement with the experimental observat ions.
227
Newtonian Fluid. "Proc.Inst.Mech.Eng., Lond. Vo1.180. Part3b. p.57, 1965-66. 8. Dowson,D. and Hamr0ck.B. J., "Numerical Evaluation of the Surface Deformation of Elastic Solids Subjected to a Hertzian Contact Stress," A.S.L.E. Trans., Vo1.19, no. 4 , pp.279-286, 1976. 9. Evans.H.P. and Snid1e.R.W.. "The Isothermal Elastohydrodynamic Lubrication of Spheres," J.Lubr.Techno1.. V01.103, no.4. pp.547-557, 1981. lO.Evans,H.P. and Snid1e.R.U.. "Inverse Solution of Reynolds' Equation Under Point Contact Elastohydrodynamic Conditions," J. Lubr.Techno1.. Vo1.103, no.4. pp.539-546, 1981. ll.Grubin.A.N.."Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces." in Investigation of the Contact of Machine Components, Kh. F. Ketova, ed. Translation of Russian Book No. 30, Central Scientific Institute for Technology and Mechanical Engineering. Moscow. Ch. 2, 1949. 12.Hamrock.B.J. and Dows0n.D.. "Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part 1 - Theoretical Formulation." J. Lubr.Technol., Vo1.98, no.2, pp.223-229. 1976. 13.Hamrock,B.J. and Dows0n.D.. "Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part 2 - Ellipticity Parameter Results, " J.Lubr.Techno1.. Vo1.98. no.3. pp.375-378. 1976. 14.Hamrock.B.J. and Dowson,D., " Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part 3 - Fully Flooded Results, J.Lubr.Techno1.. Vo1.99, no.2. pp.264-276. 1977. 15.Hamrock.B.J. and Dows0n.D.. Ball Bearing Lubrication, John Uiley and Sons, New York.
1981. 16.Harlow,F.H. and Ue1ch.J.E. "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface." Phys. Fluids, Vo1.8.p.2182. 1965. 17.Hartnett.M. "A General Solution for Elastic Bodies Contact Problem," in Solid Contact and Lubrication,.edited by Cheng and Kerr. A.S.M.E. publicaation number AMD-39. pp.5166. 1980. 18.Kaludjercic,A., "Thermohydrodynamic Effects in Line Contacts," PhD thesis from Imperial College, London, 1979. 19.Koye,K.A. and Uiner,W.O.. "An Experimental Evaluation of the Hamrock and Dowson Minimum Film Thickness Equation for Fully Flooded EHD Point Contacts." J.Lubr.Techno1.. Vol. 103, no.2. pp.284-294,1980. 20.Patankar.S. V. Numerical Heat Transfer and Fluid Flow, Series in Computational Methods in Mechanics and Thermal Science, McGrawHill, New York, 1980. 21.Ranger.A.P.. Ett1es.C.M.M. and Cameron,A., "The Solution of Point Contact Elastohydrodynamic Problem." Proc.R.Soc.Lond., Series A, V01.346, pp.227-244. 1975. 22.Roelands.C. J. A., "Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils," Druk.V.R. B., Groningen. Netherlands, 1966. 23.Schneider.G.E. and 2edan.M ."A Modified Strongly Implicit Precedure for the Numerical Solution of Field Problems." Num. Heat Transfer, Vo1.4, pp.1-19, 1981. 24.Timoshenko.S.P. and Go0dier.J.N. Theory of Elasticity, McGrarHill, 1970. 25.Zhu Dong and Wen Shi-zhu. "A Full Numerical Solution For the Thermoelastohydrodynamic Problem of Elliptical Contacts," J.Lubr. Technol., Vo1.106, pp.246-254, 1984.
D I M E N S I O N L E S S
CASE
P A R A M E T E R S
b
1:
1.3 4
2.5
=
'lo" ER'
5.049E-11 1.683E-11 8.414E-12 Table 1.
Fiqure 1 - Fluid Control Volume For Equation Of Fluid Motion: X-Direction
<;
=
E' ~
01
4.522E+03 4.522E+03 4.522E+03
u'=E'R=~ 7.369E-07 7.369E-07 7.369E-07
k = - R YI Rk 6.0 6.0 6.0
9, =
w%,'3
2 I!
1.7381+04 1.564E+05 6.258E+05
GW3
qT, = -
liz
7.100E+05 6.390E+05 2.556E+07
Definition of EHL Contact Conditions
Fiqure 2 - Fluid Control Volume For Equation Of Fluid Motion: Y-Direction
Figure 3 - Fluid Control Volume For Equation Of Mass Conservation
228
1
Thermal
Fiqure 4 - Solution Flow Charts
Isothermal
PRESSU(IE C O N I W R I 0 1 . 5 9 E-3 .l.*7 E-3 0 1 . 3 6 E-3
A1.13 E - 3 v 9 . 1 m E-*
.1.7# H 1.79 01.es A1.97 V2.88 *2.15 02.2m 02.33 02.47
6 s . a ~E - * 0 ' 1 . 5 5 E-*
0 2 . 2 7 E-9 oq.55 E-5
Fiqure 6
F I L n THK C O N I O U I S
-
Dimensionless Pressure Contour Plot For Case 2
E-5
E-5 E-5 E-5 E-5 E-5 E-5 E-5 E-5
Fiqure 7 - Dimensionless Film Thickness Contour Plot For Case 2
229
111 W h J N. Z h 0 . H
111 W z
=:g=3
Fiqure 8 - Contact Centre-Line Dimensionless Pressure And Film Thickness Distribution For Case 2. (Present Solution Shown As Solid Line. Hamrock and Dowson [14] Solution Shown As Dotted Line)
lor
1EW. 2.99 2.59 0 2.15 A 1.76 V 1.66
SURFRCE
4
B O T T M I SURFRCL
Sliding Ratio. S
Figure 10
F
-
-
Fiqure 9 - Contact Centre-Line Dimensionless Pressure And Film Thickness Distribution For Case 1. (Present Solution Shown As Solid Line. Hamrock and Dowson [14] Solution Shown As Dotted Line)
mnr. 0 3.92
CONlWIS
El El
3.13 2.35 A 1.96 V 1.70 +1.58 01.97 01.37
El El El
I I7 F I
CONTOURS
El El El
El E E E E
l l l l
Sliding Ratio. S = 2.0
0.5
Contact Centre-Line Dimensionless Temperature Contour Plots For Case 3 With Sliding Ratios Of S = 0.5. 2.0 (Dimensionless Ambient Temperature = 11.76) I IEU.
COIlM 12
€2 0 1 . w €2 A 1 . N €2 T 1 . 2 1 €2
* I . l S €2 0 1 . 1 2 €2 01.ms € 2
COIITOURS
2 I 2 €2
2 . u 12 0 2.11 11
A
0 2.11
CDllDUI €2
1.w €2
0 1.11 € 2
Bottom Surface Temperature Dirtr i b u t ion
Top Surface Temperature Distribution mnr.
I€*.
I.?# t 2
::::: :: 01.27 t2 0 1 . 1 s 12
Fiqure 11 - Dimensionless Temperature Contour Plots Of The Top Solid Surface, Fluid MidPlane, And Bottom Solid Surface For Case 4 With sliding Ratio of S = 2.0 (Dimensionless Ambient Temperature = 105.8)
1 Fluid Mid-Plane Temperature Distribution
230
.
0 ISOTHERflRL 5-8.5 05-1.8
A 5-1.5 V5-2.0
Fiqure 12
-
Dimensionless Centre-Line Fluid Pressure And Film Thickness Plots For Case 3 With Sliding Ratios Of S = 0.5. 1.0. 1.5. 2.0. (The Corresponding Isothermal Solution Is Also Presented For Comparison)
.
0 ISOlHERflRL .5
0 s AS V5
-2.50
-
0 ISOTHERIIRL
s
0.5 1.0 1.5 2.0
-1.e0
0 5
A S V 5
-1.10
OIflCNSIONLESS
- 40
X-POSITION
38
1.08
IX/HER:Z
RRD.1
1.70
-4
80
-
-
0.5 1.0 1.5 2.0
-1'90 - 80 - 28 qu 1 00 OIIIENSIONLESS X-POSITION I X / H E R l Z RAO 1
1 6
Fiqure 13 - Dimensionless Centre-Line Fluid Pressure And Film Thickness Plots For Case 5 With sliding Ratios Of S = 0.5, 1.0, 1.5. 2.0. (The Corresponding Isothermal solution Is Also Presented For Comparison)
\ b
case 3 h I
eb
1.h
SLIDING RATIO.
1.60
2.00
5
Figure 14 - Ratio of Thermally Influenced Central Film Thickness To Isothermal Central Film Thickness Plotted As A Function Of Sliding Ratio For Cases 3, 4. 5.
231
Paper Vll(iv)
A full E.H.L. solution for line contacts under slidding-rolling condition with a non-Newtonian rheological model S.H. Wang, D.Y. Hua and H.H. Zhang
Theoretical analysis on non-Newtonian character of lubricant and its effect on EHL is given and lubrication equation for fluid obeying power law rheological model under sliding-rolling condition is deduced. With the help of Newton-Raphson method, the lubrication equation coupled with film shape equation is solved within whole contact zone, and oil film shape and pressure distribution are obtained. Then the authors inspect the effects of rheological parameters nb,n on oil film and pressure profiles. With Dowson's minimum film thickness formula thickness as foundation, the authors fitted a new minimum film thickness formula for the power law model.
1 INTRODUCTION
Now, non-Newtonian behavior is almost invariably observed in various lubrication processes, such fluids violate the Newtonian postulate which assumes a linear relationship between shear stress and rate of shear. Various theories 'have been postulated in recent years to describe the flow behavious of non-Newtonian fluids. One of the models is that of the "power law" fluid model, in which the shear stress varies as some power of shear rate. The so-called "power law" constitutive equation is widely used because not only some fluids yeild this constitutive relation in certain condition, but also its simplisity. The power law exponent n is the rheological index. For n=l, the fluid is Newtonian, for nl, it is a dilatant fluid. Marnell and Elrod (1973) used a power law fluid approximation for the synovial fluid in their study of the tribological behavior of the human hip. Rajaligham (1979) obtained performance curves for journal bearings. In 1982,1983, Sinha and Singh published papers on power law lubricants in pure rolling conditions <2>,<3>. Then Dien and Flrod developed a lubrication equation with a equivalent power law model for pure sliding condition and used it to analyze journal bearing performances. Whereas to thoroughly research the problem, wider working conditions including slideroll of two discs; more factors including pressureviscos effects of fluids and surface deformation of discs are ought to be considered. Of course, the problem will be much complicated. In this paper, an endeavour is made to solve the controvesies by applying the power law model to the problem of non-Newtonian fluid effects in sliding-rolling line contact EHL
.
1.1 Notation
b
semiwidth of Hertzian contact
E'
effective elastic modulus of discs
G
dimensionless material parameter ( a E '
h
oil film thickness
H m,
dimensionless film thickness (hR/b2 consistency coefficient o f lubricant at atmospheric pressure
n
power law exponent of lubricant
P P
pressure dimensionless pressure (bR/bE')p
R
effective radiu of discs RlR2/(Rl+R2)
U
sum velocity of discs, (Ui+U2)/2
U1,UZ velocity of discs dimensionless equivalent velocity,
V
load
wY W
dimensionless load parameter, (w/RE')
X
coordinate in direction of motion
Y
coordinate in direction of film thickness
a
pressure-viscosity lubricant
5
slip ratio, ~ ( u Z - U ~ )/(uI+u~)
T
shear stress
coefficient
2 GENERAL ANALYSIS The one dimensional constitutive for power law model of fluid is:
of
equation
232
where m is consistency index, n is power law exponent of lubricant. Comparing equation (1) with classical Newtonian model, the equivalent viscosity can be written as:
where I=( a u/ a yl2, means the second invariant of strain tensor. So n = 11 (I) is the function of I. To derive a lubrication equation under slider-roll condition for the non-Newtonian model, we adopt the same perturbation method as Dien and Elrod's . By expanding the velocity of fluid using small amplitude parameter , we get: (3) u = &+EU1+' * * The second invariant then have the expansion:
Thus: a/ a d n , a%/ay)
=
o
(13)
a/ a y(nl a%/ aY+ n,aul/aY) = ao /ax
(14)
Because and I, are constants in y direction at every column of x direction, we can get u, easily with boundary conditions y=O,u,=U1; y=h, &=U2: U,
(15)
= Ui+(U2-U1 )y/h
From can get:
equation
(6),(9),(~2),(~5), we
(16)
a2uljay2= ao/ax/(n,n)
By integrating above equation with boundary conditions y=O, and y=h, q = O , the expression of ul is obtained:
or
I, = a%/ay = 2a%/ay
(5)
- au,/ay
(6)
Therefore, velocity expressed as follows: u =Ul+(U2 +
Again, by expanding the equivalent viscosity in the small area near I, with Taylor's expansion, we get: rl
,+
=TI
EIl(a YiaI),
equation
can
be
- Ul)y/h
1/2n no
-
(18)
ap/ax(y2-hy)
The coordinate system and physical meaning of u can be seen from Fig.1, where U2>U1 is always the case.
yl
(7)
saying
Then we examing the momentum equation: (10) aT/ a y = ap/ax In one dimensional analysis, pressure p is a constant in y direction at .every column of x direction (see Fig.1). Using Elrod's expression, the pressure gradient can be in the form :
0 // / / / / / / / / / / Y / / /
u1
-
t/ /Z X
///
Fig.1 Geometry and velosity of one dimensional sliding-rolling
E.H.L.
In fact, equation (11)leads an exact solution for Newtonian fluids, in which the velosities and their derivatives vary linearly with the pressure gradient. Considering equations (11,(2),( 3 ) , ( 7 ) and (11) the momentum equation can be in the form:
Since in one dimensional analysis side leakage is ignored and the fluid flux per unit width is a constant, we can obtain lubrication equation from equation (18)easily:
(12)
(19)
233
where p o ,h, are dencity of lubricant and film thickness at the columum of dp/dx=O INPUT PARAMETERS W,Ul,U2,n,m, etc. AND INITIAL PRESSURE DISTRIBUTION, FILM THICKNESS SOLVETHEFILMSHAPEEQUATION
In the analysis of EHL problem, the piezo-visco effects ought to be considered:
m = m,Exp(ap)
TO GET FILM SHAPE FOR GIVEN
,
DISTRIBUTION OF PRESSURE.
(20)
where m, is the consistency coefficient lubricant at atmospheric pressure. Thus, equation ( 19 becomes :
of
I
,
SOLVE LUBRICATION EQUATION
.r(
t-' 3
2
2.5
12
(21)
PRINT Hmin, Hc, V etc. DRAW PROFILES OF PRESSURE, OIL Film shape equation is a normal one: h =h, + x2/2R-2/rE'I
2 p(s)ln(x-s)ds
Fig.2
Block diagram of computational procedures
(22)
Where Xb expresses the exit of contact zone at which p=dp/dx=O. Besides, load condition must be taken into account:
4
RESULTS
4.1 The influence of lubricants' compressibility on pressure and film thickness
(23)
3 NUMERICAL PROCEDURE By introducing nondimensional parameters:
W
= Wy/E'R;
P
= p(kR/E'b); X=x/b; H=hR/b2
G = a E'
Fig.3 presented the results of pressure and film shapes obtained by considering or neglecting the lubricants' compressibility. We can see from the figure that the influence of compressibility are ignorable in engineering use except the slight diminishes of pressure spike and minimum film thickness.
HI
equations (21), (22) and (23) can be written in dimensionless forms. Then rewrite the equations by finite difference approximations which rely on the fact that a function can be represented with sufficient accuracy over a small range, and in this paper Euler's difference methoed is used. The numerical solutin is based on Newton-Raphson iterative method illustrated in detail by Okmura <4>. Schematically, the entire iterative prcedure can be shown by the block diagram of Fig.2. It takes about 6 to 8 iterations to reach& bellow 1.E-3.
Fig.3
Pressure and film shapes from different analysis
234 E f f e c t s of d i f f e r e n t parameters on minimum f i l m t h i c k n e s s
4.2
The influence of m, on minimum f i l m t h i c k n e s s i s shown i n Fig.4. With t h e i n c r e a s e of m,, t h e f i l m w i l l r i s e f o r same condition. The tendency i s l i k e that of Newtonian f l u i d s ' reference v i s c o s i t y . I n f a c t , when power l a w exponent n equals one, t h e problem becomes Newtonian and m, equals t h e r e f e r e n c e v i s c o s i t y of Newtonian f l u i d . Hmin
The e f f e c t film thickness shown i n Fig.7. l a w exponent n thickness w i l l slide-roll ratio
I Hmin
of
slip ratio on minimum d i f f e r e n t n and m, are For t h e l u b r i c a n t i t s power i s l a r g e r t h a n one, t h e f i l m r i s e with t h e increase of and v i c e versa.
at
W=2.17E-5
W=l.63E-5 G=3450
U=4.5m/s
0.e
n = l .3
nT9, ~ = 0 . 0 5 n%6, m,=3
5=0.22 0.6 0.5
0.4
Minimum f i l m t h i c k n e s s versus s l i d e r - r o l l r a t i o s a t d i f f e r e n t n and nb.
Fig.7 0.2
I Fig.4
4.3
8. E-4
2. E-4
A Formula Thickness
for
Predicting
minimum f i l m
Influence of m, on minimum f i l m t h i c k n e s s
Fig.5 and 6 show t h e r e l a t i o n s h i p s of G and H , W and hmin/R. The l a r g e r t h e material parameter G, t h e t h i c k e r t h e o i l f i l m and t h e l a r g e r t h e load parameter W, t h e t h i n n e r t h e film. These tendencies a r e a l s o t h e same a s those of Newtonian f l u i d s . The only d i f f e r e n c e i s t h e i n c r e a s e or d i c r e a s e r a t i o v a r i e s with power l a w exponent n.
I Hmin
I t seems complex from above curves t h a t t h e minimum f i l m t h i c k n e s s depends not only on W,G but a l s o on t h e r h e o l o g i c a l parameters n and Q . Besides, t h e numerical s o l u t i o n i s s o luxurious t h a t it would t a k e many t i m e t o o b t a i n a r e s u l t . So how t o p r e d i c t t h e minimum f i l m t h i c k n e s s by a simple formula f o r t h e non-Newtonian r h e o l o g i c a l model i s a very i n t e r e s t i n g problem. To develop a s o l u t i o n , t h e authors introduce a dimensionless v e l o c i t y parameter V:
'
0.4
V = U[ (U,-U, 0.3
'
0.2
*
G
3000
1000 Fig.5
I
2.5
5000
)/h,]"-' m,/E'R
(24)
where ho means t h e c e n t r a l film thickness and obtained i n numerical s o l u t i o n . Then, a l a r g e amount r e s u l t s of f i l m thickness are p l o t t e d a g a i n s t t h e dimensionless v e l o c i t y parameter V on a coordinate system paper (see F i g . 8 ) . n=2.0
Hmin
Influence of dimensionless m a t e r i a l parameter G on minimum f i l m thickness f o r d i f f e r e n t values of n and m,. hmin/Rx105
. 0.6
W=1.63E-5 G=3450
G=5000 U=4.25 0.4
5=0.1
1.5
n=2,
m,=3.9E-8
n=1,
m,=0.02
0.2
~ 0 . 6 ,~ 4 . 2 5
3 Fig.6
1.0
5
10
Minimum f i l m thickness versus dimensionless load parameter W a t d i f f e r e n t n and m,.
0.5 Fig.8
1.0
1.5
Minimum f i l m t h i c k n e s s versus V a t d i f f e r e n t n.
235
From Fig.8, we can see when n equats one, there is a Newtonian Hmin curve and Dowson's minimun film thickness formula is eligible enough for calculating the minimum film thickness in EHL condition. For the same column of V, the biger the power law exponent n, the thicker the film, but the effect is medium. This result is agree with the result of Dien and Elroad's in their research of lubrication problem on journal bearings . This suggests that we can fit a new formula based on famous Dowson's minimam film thickness formula for the power law rheological model by weighting the influence of n. For a pticular V, we can get different values of Hmin for various n. Considering Newtonian minimum thichness formula <5>, the minimum film thickness can be predicted in follow manes Hmin=0.63G0.6V 0.7. na
G
I 3450 I
3450
u
4.5
4.5
5
0.23
m, n
I I
1 3450 I 3450 I 3450 1
3450
4.5
4.5
5.5
0.23
0.23
0.23
0.18
9.E-2
l.E-5
2.E-4
3.E-2
1.5E-1 4.E-1
0.9
1.5
1.3
1.0
I
I HmiJ
0.216
I
' .'
*'
n
0.7
0.336
1
I
I 0.324 I 0.349 I
0.210
0.354
I
Hmin
0.492
(26)
0.345
The last problem is to determine the relationship between Ho and Hmin. Acoording to Dowson, Higginson and Cheng's opining, Ho is about Hmin times 1.2 to 1.3 (5). From the authors' large numbers of numerical results, the ratio of Ho to Hmin is between 1.15 to 1.32 acoording to different parameters W,G,V, etc. Of course, if have time, one can fit the relationship among the film thichness ratio and W,G,V, the minimum film thickness formula will be more accurate. But for engineering use, 1.2 is feasible. By taking the ratio into equation (27) and moving Hmin of RHS to LHS, we obtain the formula needed:
. 2. 3431/(0.3+0.7n)
0.495
0.564 IO.111
I 0.221 I I
0.561
0.109
0.223
0.478
I
0.772
0.43
0.69
I 0.502 1 0.791 1 0.44 I I
I
There is a linear function about the coefficient a. With minimum squar method, we can fit the coefficient a at different column of V. The average value of a is 0.345. Thus, equation (25) becomes: '/W
0.8
W1e13
Rewrite equation (25) as: In Hmin=ln B + a .In n
'
0.55
W Lr26E-5 11.6 3E-5 Il.63E-511.6 3E-5) 5.43 E-5) 3.26E-5)
I
Hmin=O.63c
11
I
I
I
Table 1 presents the values of Hmin which are predicted from full numerical solution and HmEn which are calculated by formula (28). we can see the difference between them is tolerant in engineering use.
0.6881
0.44
Table 1 Nondimensional minimum film thickness from different predictions COCLUSION The numerical results show that the influence of lubricants' compressibility on pressure and film profiles are slight. The effects of load parameter W, material parameter G and consistency coefficient m, on E.H.L. are much like those in Newtonian fluid analysis. In fact, when the power 16w exponent n equals 1, the problem is entirely a Newtonian one. From the results, we can see that the influence of power law exponent n on mimimum film thickness is medium. With the increace of n, the oil film becomes thicker and vise versa. For the dilatant fluids whose power law exponent n is lager than unit, the film will rise with the increase of slideroll ratio 5 For the pseudoplastic fluids, the tendency is opposite.
.
It is clearly seen from equation (21) that when slide-roll ratio equatls zero, the pressure gradiant will always be zero through out the whole contact zone. This means that the lubrication equation is not suitable for pure rolling condition. Perhaps the case is caused by neglecting the higher order terms of perturbation E *u etc,* But the introduction of the higher order terms is much more complicated especially in E.H.L. analysis. To conquer the difficulty, the authors derived a lubrication equation for pure rolling condition and make E.H.L. solution. Further research about this problem is on the way. + * - - $
where Vo=U &/E'R[ (U2-Ul)/RIn-l
0.43 I
236
References <1> Dien, I.K. and Elrod H.G. 'A Generalized
Steady-state Reynolds Equation For NonNewtonian Fluids, With Application to Journal Bearings' ASME Jour. of Lubr. Tech., Vo1.105, 1983, pp385-390. <2>
Sinha, p.and Singh,C. 'Lubrication of Cylinder on a Plane With a Non-Newtonian Fluid Considering Cavitation', ASME Jour. of Lubr. Tech., Vo1.104 1982,pp168-172.
<3> Sinha, P. and Raj,A.'Exponential Viscosity Variation in the Non-Newtonian Lubrication of Rollers Considering Cavitation'. WE&, 87, 1983,pp29-38.
<4> Okamura,H. 'A Contribution T o The Numerical Analysis of Isothermal E.H.L.', Proc. of 9 th Leeds- Lyon Symp. on Trib.
<5> Dowson, D. and Higginson, G.R.'Elastohydro-
dynamic Lubrication', Pergamon Press, 1977.
SESSION Vlll ELASTOHYDRODYNA M IC LUBR ICAT10N (3) Chairman: Professor R. Bosma
PAPER Vlll(i)
Elastohydrodynamic lubrication of grooved rollers
PAPER Vlll(ii)
The lubrication of elliptical conjunctions in the isoviscous-elastic regime with entrainment directed along either principal axis
PAPER Vlll(iii) Effect of surface roughness and its orientation on E. H.L. PAPER Vlll(iv) The elastohydrodynamic behaviour of simple liquids at low temperatures
This Page Intentionally Left Blank
239
Paper Vlll(i)
Elastohydrodynamic lubrication of grooved rollers G. Karami, H.P. Evans and R.W. Snidle
The paper is concerned with the problem which arises in the elastohydrodynamic lubrication (ehl) analysis of real surfaces under conditions where conventional ehl theory predicts a lubricant film thickness which is of the same order or less than the mean height of surface roughness asperities. A simple micro ehl model for the lubrication of rough surfaces is described and theoretical results are given which demonstrate the effect of load. The results show the transition from "isolated" asperity contact behaviour to a situation where significant pressures are generated in the valleys between asperities. 1
INTRODUCTION
Conventional elastohydrodynamic lubrication (ehl) theory, which is based upon the assumption of perfectly smooth surfaces, is applicable to engineering situations where the thickness of the lubricant film is significantly greater than the height of surface roughness asperities. But in many cases of practical importance the application of the well-known Dowson and Higginson (1) minimum film thickness formula results in a predicted film thickness which is of the same order or even less than the roughness average value. This situation can occur in toothed gearing operating at high temperatures and it appears that many gearing systems operate quite successfully in this state. Under such conditions the assumptions of conventional ehl theory are no longer valid and it becomes necessary to consider the effect of surface roughness in the ehl analysis. In a previous paper ( 2 ) the authors have described a full analysis of ehl of rough surfaces in which the roughness consists of sinusoidal grooves aligned in the direction of lubricant entrainment. For a constant value of the roughness wavelength and load, the effect of varying roughness amplitude was investigated. It was shown that two different regimes of lubrication may exist. At relatively large roughness amplitude each asperity contact behaves as an "isolated" point contact with the pressure between asperities falling to zero. At lower amplitudes, however, the pressures in the deepest parts of the valleys between the asperities increase and, as roughness is reduced further, the minimum film thickness increases towards the value predicted by the Dowson and Higginson line contact formula. The purpose of the work described in the present paper is to investigate the effect of load on pressure and minimum film thickness, all other factors (i.e. viscosity of lubricant, entraining velocity, roughness amplitude, roughness wavelength) being constant.
1.1
Notation
b
semi-width of Hertzian contact for smooth b = (4wRx/nE')f cy1inders amplitude of sinusoidal roughness
D
elastic deformation
DO
elastic deformation due to one cycle of pressure
E
Young's modulus
E'
reduced elastic modulus
h
film thickness
II
semi-wavelength of sinusoidal roughness
P
pressure
RX
radius of relative curvature in entraining direction
R Y
radius of relative curvature at tip of unloaded asperity in axial direction
U
mean velocity of surfaces in x-direction relative to conjunction
W
load per unit length in y-direction
1 E
7 =
1-y12
1-v22
El
E2
-+ -
XtY coordinate axes in entraining and axial directions, respectively
--
XfY values of x and y at which elastic deformation is required a
pressure coefficient of viscosity
Y
constant in pressure/density relationship
TI
viscosity
n0
A
viscosity at zero pressure constant in pressure/density relationship
V
Poisson's ratio
P
density
Po
density at zero pressure
Subscripts 1 and 2 refer to the two surfaces.
240
2
DESCRIPTION OF THE MODEL
The model for rough surface ehl which has been adopted in our initial studies is described in detail elsewhere ( 2 ) . Only a brief description is given here. The model consists of a circumferentially-grooved roller running in lubricated contact with a smooth .roller. The grooves are assumed to be sinusoidal in profile but the numerical analysis is not restricted to this simple shape. The situation is illustrated schematically in Figure 1.
The numerical ehl solution used here is not dependent upon any assumption regarding the shape of the dry contact areas, neither is it limited by the magnitude of the deformation in relation to the roughness amplitude. Under lubricated conditions it is assumed that the pressure distribution over each of the asperity contacts is identical and the longitudinal planes corresponding to the peaks and valleys of the rough surface are planes of symmetry in the hydrodynamic pressure distribution. These assumptions, together with the usual upstream zero pressure and downstream cavitation condition, provide the boundary conditions for the hydrodynamic analysis. The normal elastic deformation of the surface at a given point is obtained by superposition of the result for the deformation produced by the pressure acting over one wavelength of the surface. If we denote the elastic deformation obtained from this single cycle of pressure as D where 0
then the total elastic deformation at any point, due to the pressure acting upon the asperity on which the point lies and the decreasing effect of pressure acting on successively more remote pairs of asperities on either side, will be given by
_ _
-
D(x,y) = Do(x,y) + Do(x, 211-5) + Do(;,
+
D
0
(x,41-y)
+ Do(;,
Z!&+y)
4!&+y)
+ etc.
Figure 1. Schematic diagram of lubrication of grooved r o l l e r s . Top: axial f i l m thickness when undefoned. Centre: axial f i l m thickness when loaded and lubricated. Lower: corresponding areas of contact under dry conditions.
Sufficient terms of this infinite series must be considered to give convergence of the ehl analysis. Advantage can be taken of the fact that the deformation produced by the single cycle of pressure approaches that due to an equivalent point load at large distances from the loaded area. The ehl analysis involves the simultaneous solution of the elastic deformation and lubrication equations. The following form of the Reynolds equation is assumed
where Elastic deformation of the surfaces is analysed on the assumption that because the roughness amplitude is usually very much smaller than the wavelength of the roughness, the surfaces may be treated as semi-infinite elastic solids. It is assumed that the lubricated rollers are infinitely long and that there is no net flow of lubricant in the direction parallel to the axes of the rollers. Under dry conditions the contact between the rollers will consist of elongated areas of contact as shown in Figure 1. At very light loads these areas will be almost elliptical in shape and the contact areas and stresses will be close to those calculated from Hertzian theory based on the asperity tip curvatures. At loads sufficient to cause significant elastic deformation, however, the effects of a sinusoidal surface will become apparent and the contact areas are expected to depart from an elliptical shape.
n
=
no exp(ap) PO
and 3
=
1 + Ye [l + Ap)
NUMERICAL MICRO EHL SOLUTION
The maximum hydrodynamic pressures occurring in the lubrication of rough surfaces will tend to be much higher than those in the corresponding smooth surface problem operating at nominally the same load. Indeed, it is likely that in situations where "running-in", brought about by plastic deformation of asperity tips, takes place the maximum hydrodynamic pressures will be of the order of the hardness of the surfaces. For these conditions a heavy-load ehl analysis technique is required such as that devised by Dowson and Higginson (1) for line contacts. Such 'inverse' methods also have advantages in speeds of convergence over the conventional forward method under the lightly loaded conditions for which the latter method will converge.
241 The authors' micro ehl solution technique derives from their earlier solution method for heavily-loaded spherical contacts (3) and the elongated elliptical contacts found in high conformity gears of the Wildhaber-Novikov type (4), (5). The solution method, which utilises the 'inverse' solution of the Reynolds equation, is as described in the above publications except that the elastic deformation calculation must take account of the deformation produced by pressures acting on adjacent asperities, as mentioned above, and in the solution of the lubrication equations the side boundary condition is one of zero transverse pressure yradient rather than zero pressure. RESULTS
4
The purpose of the present paper is to investigate the effects of load, and in particular the way in which increasing load can produce a transition from "isolated" asperity contact behaviour, to a regime in which significant pressure generation occurs in the valleys between asperities. The conditions assumed in obtaining the numerical results are chosen to correspond to those of a typical two disc machine operating with disc bulk temperatures of about 150 deg.C with a medium viscosity mineral oil as the lubricant. The wavelength and amplitude of the sinusoidal roughness are chosen to be representative of a finely ground surface. The operating conditions are summarised in Table 1. TABLE 1 Operating Conditions R =
19mm
a
0.1 mm
=
d = 0
=
u =
-
no u =
0.125 bm 104~1 to~ 8 . 8
1.1
104~1~
5 m/s 0.0034 Ns/m2 1.5 x lo-' m2/N
y
=
2.266 x
m2/N
h
=
1.683 x
m2/N
E l = E2 = 200 x lo9 N/m2 v1 = w p =
0.3
Figure 2 shows the axial pressure and film thickness distributions for four different values of the load. At the lighter load the hydrodynamic pressure in the deepest part of the valley is relatively small compared with the value occurring on the centre line of the asperity, but at the heaviest load the valley pressure (on the transverse centre line) is over half the maximum value. There is also significant deformation of the asperity at the heaviest load considered as shown in the comparison of axial film thickness for the four different loads. A striking feature of this comparison is the small variation of asperity centre-line film thickness with load. Figure 3 gives a similar comparison of longitudinal pressure and film thickness for the four different loads. This comparison shows the transition from a typically lightly-loaded ehl pressure distribution and film thickness shape at the lightest load to a heavily-loaded condition in which the film is very close to parallel over the contact area. The pressure
distribution has the typical downstream spike with steeply falling pressure at the outlet. The longitudinal (x) axis has been nondimensionalised with respect to b, the semi width of the corresponding Hertzian line contact for dry contact of cylinders with a radius of relative curvature equal to Rx.It will be noted that as the load increases the nominal length of the contact in the x direction tends towards 2b. Figure 4 shows the pressure surface and film thickness contours for the most lightly-loaded case. The pressure plot shows the low valley pressures in relation to the asperity centre line distribution and the film thickness contours are typical of a lightlyloaded point contact. Figure 5 shows the pressure surface and film thickness contours for the most heavilyloaded case. The valley pressures in this case are much closer to the centre-line values and the film thickness contours show the increased degree of flattening of the asperity in the loaded area. 5
DISCUSSION AND CONCLUSIONS
The results presented here show that an increase in load can produce an effect which is similar to that obtained by reducing the roughness amplitude (as demonstrated in our It has been theoretically earlier paper ( 2 ) ) . demonstrated that if conditions are chosen such that the valley pressures are low in comparison with the pressures occurring on the centre-line of the asperity, then by increasing the load the valley pressures can be increased in relation to the centre line pressures, thus producing a transition from what may be described as "isolated" asperity ehl to a regime in which the valley pressures are significant. A pre-requisite. for this transition is that "isolated" behaviour occurs at the lightest load, and this would seem to require a minimum film thickness which is of the same order or less than the roughness amplitude at the lightest load. Under these conditions we might expect a transition from point contact behaviour to line shows the contact behaviour. Figure 6 variation of minimum film thickness with load. Also shown for comparison are the corresponding variations obtained from the Dowson and Higginson formula (1) for line contacts and the point contact formula of Archard and Cowking (6) for elliptical contacts. In the Dowson and Higginson formula the radius of relative curvature of the contacting cylinders is equal to Rx. In the case of the Archard and Cowking formula the radius of curvature in the entraining direction is also Rx and the radius of curvature in the transverse direction is assumed to equal the tip radius of curvature of the sinusoidal profile, i.e. Y
=
[i-] [;I2
Under conditions where the hydrodynamic pressure is concentrated about the tip of the asperities we would expect the film thickness to correlate with the Archard and Cowking formula, but as the contacts become less isolated we would expect a tendency to line contact behaviour. The results given in Figure 6 show evidence of this expected transition.
242
x10-1
XI04 6
Ib
d
5
Lu
$
3
v) v) Lu
01 a
2
0 0
I
1
-5
0
I
5
10 x10-l
Y/[ Figure 2 ( a ) Pressure distributions on x = 0; loads (a) 1.1 x lo4 N/m; (c) 4.4 x lo4 N/m;
(b) 2.2 x lo4 N/m;
(d) 8.8 x lo4 N/m;
x10-2
x10-2 35
30
. . E
Y
20
v) v) Lu
z
-+ u Y
15
10
5
0
I
0
-5
I
I
0
5
Figure 2(b) Film thickness on x = 0; l o a d s (a) 1.1 x lo4 N/m; (c) 4.4 x lo4 N/m;
( d ) 8.8 x lo4 N/m;
(b) 2.2 x lo4 N/m;
10
243
x10-1 ti
6
1
5
4
3
2
1
0
Figure 3(a) Pressure distributions on y = 0; loads (a) 1.1 x lo4 N/m;
(b) 2 . 2 x lo4 N/m;
(d) 8.8 x lo4 N/m;
(c) 4 . 4 x lo4 N/m;
E C
5
4 h
E
3
v v)
s
z
3
z
2
Y
J J=
I
0
I
1
I
I
-6
-4
-2
0
x/b
Figure 3(b) Film thickness on y = 0; loads (a) 1.1 x lo4 N/m; (d) 8.8 x lo4 N/m;
(b) 2.2 x lo4 N/m;
(c) 4.4 x lo4 N/m;
244
\ X
Figure 4(a)
Pressure surface; load = 1.1 x lo4 N/m; width of plot in y-direction = .9
maximum pressure = 0.22 GPa;
1.0
0.5
y/l0.0
-0.5
-1.0
x/b Figure 4(b)
contour thickness (elm)
1
2
.125 .15
Film thickness contours;
3
.175
load = 1.1 x lo4 N/m
5
6
7
8
.20 .25
.30
.35
.40
4
9 1 0 1 1 1 2 1 3 .45
.50
-55 -60
.65
245
Figure 5 ( a )
Pressure surface; load = 8.8 x lo4 N/m; width of plot in y-direction = k?
maximum pressure = 0.54 GPa;
1.0
0.5
y/10.0
-0.5
-1.0 - 1 15
- 1 :0
-0.5
0:0
1 :0
x/b
contour thickness
(run)
1
2
3
4
5
6
.15
.20
-25
-30
.35
.40
7 .SO
8 .60
9 1 0 1 1 .70
.80
.90
246
n
E
a
W
c a C
-
10
50
100
The orientation of surface roughness considered here corresponds to that occurring in Wildhaber-Novikov gears in which rolling takes place along the gear teeth in the axial direction. Since the gear teeth are usually produced by profile grinding the entrainment of lubricant is therefore in the same direction as the surface finish. In conventional involute profile gear teeth the entrainment of lubricant occurs across the direction of surface finish. In principle the techniques of micro ehl analysis described here can be applied to this configuration, but account must be taken of the non-steady nature of the situation. The micro ehl analysis described here is also being developed with the aim of providing a deterministic equivalent of Dyson's theory of ehl failure. In recent work on Dyson's theory ( 7 ) it has been shown that failure of the elastohydrodynamic system in the lubrication of circumferentially finished rollers (as evidenced by scuffing) correlates reasonably well with the theoretical failure of the lubricated system to generate an adequate average pressure between the rough surfaces in simulated dry contact. It remains to be seen if the equivalent effect in a deterministic model, of failure to generate adequate valley pressures, can also be shown to correlate with the effective failure of lubrication between grooved rollers. 6
ACKNOWLEDGEMENT
The work reported in this paper is supported by a Research Grant from the Science and Engineering Research Council. This support is gratefully acknowledged.
Figure 6. Variation of film thickness w i t h load. ( a ) m i n i m film thickness; ( b ) m i n i m film thickness from Dowson and Higginson's f o m l a ; ( c ) f i t m thickness from Archard and Cowking's f o m l a . One particularly interesting feature of these results is the insensitivity of the minimum film thickness to load. For the conditions considered it appears that the transition from point to line contact behaviour, brought about by an increase in the load, occurs at almost constant minimum film thickness. The analysis and results obtained in the paper are based upon a simple deterministic model of a rough surface. In principle, however, any repetitive asperity profile or even a representative length of a real surface profile can be used in the numerical analysis. It should therefore be possible to model the effects of surface profile modification due to plastic deformation and "running in" which tend to produce a non-symmetrical distribution of profile heights. The practical significance of the micro ehl approach is that it enables a detailed study of the mechanism of ehl under conditions when simple ehl theory would suggest significant interaction of surface asperities. The simple picture of a film of lubricant with asperities projecting through it and intermeshing as metal to metal contact is clearly inadequate.
References DOWSON, D. and HIGGINSON, G.R. 'Elastohydrodynamic Lubrication', Pergamon, Oxford 1966. KARAMI, G., EVANS, H.P. and SNIDLE, R.W. 'Elastohydrodynamic lubrication of circumferentially-finished rollers having sinusoidal roughness', Proc. Instn. Mech. Engrs. (in press). EVANS, H.P. and SNIDLE, R.W. 'The elastohydrodynamic lubrication of point contacts at heavy loads", Proc.R.Soc. Lond. 1982, E ,183-199. EVANS, H.P. and SNIDLE, R.W. 'Analysis of elastohydrodynamic lubrication of elliptical contacts with rolling along the major axis', Proc. Instn. Mech. Engrs. 1983, 197 C, 209-211. DYSON, A., EVANS, H.P. and SNIDLE, R.W. 'Wildhaber-Novikov circular arc gears: geometry and kinematics', Proc.R.Soc. Lond. 1986, A 403, 313-340. ARCHARD, J.F. and COWKING, E.W. 'Elastohydrodynamic lubrication at point contacts' , Symposium on Elastohydrodynamic Lubrication, Proc. Instn. Mech. Engrs. 180, 1965-66, 47. SNIDLE, R.W., ROSSIDES, S.D. and DYSON, A. 'The failure of elastohydrodynamic lubrication', Proc.R.Soc. Lond. 1984, 2 291-311.
247
Paper Vlll(ii)
The lubrication of elliptical conjunctions in the isoviscouselastic regime with entrainment directed along either principal axis R.J. Chittenden, D. Dowson and C.M. Taylor
Synopsis The study of elastohydrodynamic lubrication, in which the deformation of the contacting bodies plays an important part in the lubrication process, may be divided into two categories. Firstly, there is the regime in which the pressures generated within the conjunction cause the lubricant viscosity to increase by several orders of magnitude - the piezoviscous-elastic regime. The second regime encompasses conditions where the fluid is essentially isoviscous throughout the contact, and is therefore known as the isoviscous-elastic regime.
In the 1970's the work of Hamrock and Dowson ( 1 , 2 , 3 , 4 ) advanced the study of piezoviscouselastic point contacts considerably, although the range of solutions was restricted to conditions in which lubricant entrainment coincided with the minor axis of the contact ellipse. Subsequently, the analysis of most of the other lubrication regimes required for the construction of lubrication regime charts enjoyed comparab,le improvement. The exception to this was the analysis of isoviscous-elastic conjunctions which remained limited to conditions in which lubricant entrainment was aligned with the minor axis of the contact ellipse. These restriction was removed for piezoviscous-elastic contacts by Chittenden, Dowson, Dunn, and Taylor (5,6). The work presented in this report therefore seeks to remove the geometrical limitations existing in the analysis of the isoviscous-elastic regime, so as to allow lubrication regime charts to be constructed for a general case. A solution procedure is detailed, and from thirty new results computed for lubricant entrainment directed along the major axis of the contact ellipse, dimensionless minimum and central film thickness expressions are derived. With the inclusion of the existing results €or entrainment coincident with the minor axis of the contact ellipse (and six new solutions for these situations) equations are developed for entrainment along either principal axis. Some of the similarities and differences between solutions in the piezoviscous-elastic and the isoviscous-elastic regime are outlined, and a comparison is made with the limited number of existing isoviscous-elastic results. 1 INTRODUCTION
In recent years the analysis of isothermal point contacts has made considerable advances. Procedures have been developed to allow the simultaneous solution of the elasticity and Reynolds' equations, and have provided many numerical results from which theoretical film thickness expressions have been derived. These solutions to the elastohydrodynamic problem may be divided into two types. Firstly, where the lubricant viscosity is significantly affected by the generation of pressure within the conjunction area the conditions are known as piezoviscous or 'hard' elastohydrodynamic lubrication. Typical situations for this type of lubrication are steel bodies lubricated by a mineral oil, e.g. ball bearings. The second type of elastohydrodynamic lubrication is that where the fluid experiences very little change in its viscosity and is therefore termed isoviscous or 'soft' elastohydrodynamic lubrication. This type of lubrication would be expected where the contacting materials are of low elastic modulus (e.g. nitrile rubber) lubricated by a mineral oil or a fluid of very low pressure-viscosity coefficient. (These two regimes of lubrication may also be described as
piezoviscous-elastic and isoviscous-elastic respectively.) Numerical solutions to the line contact piezoviscous elastohydrodynamic problem became available in the 1960's with the work of Dowson and Higginson (7,8), a similar state being reached for point contacts in the late 1970's through the work of Hamrock and Dowson (1,2,3,4). In comparison the study of isoviscous elastohydrodynamic contacts has received very little consideration. Theoretical solutions to the problem of the 'soft' elastohydrodynamic lubrication of a reciprocating seal have been presented by Dowson and Swales (9) and Ruskell ( l o ) , whilst other theoretical work has been undertaken by Baglin and Archard ( I I ) , and Herrebrugh (12) who used an integral equation technique. A slightly simplified analysis of the line contact problem existing in human ankle joint lubrication was given by Medley and Dowson ( 1 3 1 , who assumed both a cylindrical and an equivalent planeinclined surface to allow squeeze effects to be included. Experimental investigations were unusual, but include those of Higginson (14) and Dowson and Swales ( 1 5 ) .
248
The amount of information available on the isothermal, isoviscous-elastic situation is even more restricted although some reports may be found in the literature. These include the theoretical work of Biswas and Snidle (16) together with the experimental studies of The most wide ranging Jamison et a1 (17). examination of this problem is probably that of They adapted their Hamrock and Dowson (18,lg). solution procedure to the piezoviscous elastohydrodynamic problem (1,2) to yield theoretical solutions for film thickness and pressure in isoviscous conjunctions. In both cases they restricted their analysis to situations where the direction of lubricant entrainment was aligned with the minor axis of the contact ellipse. Their formulae for dimensionless minimum and central film thickness, obtained by curve fitting seventeen isoviscous-elastic results, may be written as:Hmin
=
0.65 -0.21 7.43 Ue We (1-0.85EXP(-0.31k)) 0.64 w-0.22
Hcen = 7.32 Ue
e
h
film thickness
hO
notational separation of rigid solid and plane solid on the z axis ellipticity ratio (k = a/b) boundary location parameters in directions of major and minor axes (y,x) in numerical analysis
m*
critical inlet distance in Hertzian half widths
P
pressure
rAx’‘A y r r Bx’ By
radii of curvature of ellipsoids A and R in principal planes
S(X,Y)
geometric separation of contacting solid
U
mean entraining velocity (u =
e
(1-0.72EXP(-0.28k))
(UA
f
+
elastic deformation The restrictions imposed by the limited geometrical range for which equations (1) and ( 2 ) are applicable is,not particularly serious if the expressions are used only for the calculation of film thickness in the isoviscouselastic regime of lubrication, since common engineering applications tend towards line contacts, eg. elastomeric seals and some human joints. Recently, however, the geometrical restrictions associated with the other regimes of lubrication, as defined by Hamrock and Dowson (20) have been relaxed. The restrictions in the piezoviscous-rigid regime were removed by Dowson, Dunn and Taylor (22). whilst Chittenden et a1 ( 5 , 6 ) removed those in the piezoviscouselastic regimes. The work described in this paper therefore seeks to remove the last major geometrical restrictions encountered in the production of lubrication regime charts. (A detailed consideration of such charts may be found in Thirty Chittenden, Dowson and Taylor (23)). solutions to the problem of the isoviscous elastohydrodynamic lubrication of point contacts were computed for conditions in which lubricant entrainment was directed along the major axis of 1.0). Six new the contact ellipse, (R /R solutions were also comfut8d for lubricant entrainment along the minor axis, which when combined with the thirty solutions already mentioned and the seventeen presented by Hamrock and Dowson (18) allow a full range of lubrication regime charts to be constructed. Minimum and central film thickness expressions are presented for lubricant entrainment along the major axis of the contact ellipse and along either principal axis, and the similarities and differences between isoviscous and piezoviscous elastohydrodynamic results are noted.
E
Young‘s modulus of elasticity
E’
equivalent elastic constant
2 -
=-
1
E’
-
VA
n
L
+-
1 -VB
EA
L
EB
F
normal load
H cen
dimensionless central fClm thickness
-
'ten
dimensionless central film thickness parameter =
(Heen = He H min
0.65 -0.21 Heen lue we
dimensionless film thickness (H e=h/Re ) dimensionless minimum film thickness
dimensionless minimum film thickness parameter
(‘,in HO
I Hmin
/u0.65 = Hmin e
w-0.21 1 e
dimensionless film thickness (Ho = ho/Re) Dimensionless minimum film thicknessspeed parameter # u-o. 5) (Hmin = Hmin e Dimensionless load-speed parameter -0.75 1 (Mp = We Ue
Notation a,b
Cartesian coordinates
semi-major and semi-minor axes of contact ellipse
249 N1, N2
number of nodes ( T a b l e 1)
2
P
d i m e n s i o n l e s s p r e s s u r e (P = P I E ' )
Rx' Ry
e q u i v a l e n t r a d i i of c u r v a t u r e i n (x,y) directions
Re
e f f e c t i v e r a d i u s of c u r v a t u r e i n t h e d i r e c t i o n of l u b r i c a n t e n t r a i n m e n t
The i s o v i s c o u s - e l a s t i c problem w a s f o r m u l a t e d i n a similar way t o t h e piezoviscouse l a s t i c problem d e s c r i b e d by Hamrock and Dowson (1). Only t h e major f e a t u r e s of t h e a n a l y s i s are d e s c r i b e d , t h e r e f o r e , t o g e t h e r w i t h t h e m o d i f i c a t i o n s made t o cater f o r n u m e r i c a l s o l u t i o n s i n t h i s regime.
"e
D i m e n s i o n l e s s speed p a r a m e t e r (Ue =Me/E'Re)
a ) Geometry
D i m e n s i o n l e s s l o a d parameter
e'
FORMULATION OF THE PROBLEM
(We = FIE' Re2) x,y
Dimensionless C a r t e s i a n c o o r d i n a t e s ,
z1, 22
Number o f nodes ( T a b l e 1 )
n
dynamic v i s c o s i t y
The two c o n t a c t i n g e l a s t i c b o d i e s , (A) and ( B ) , shown i n F i g u r e l ( a ) were r e p r e s e n t e d by a n e q u i v a l e n t e l l i p s o i d n e a r a p l a n e , a s shown If t h e p r i n c i p a l r a d i i o f i n Figure l(b). c u r v a t u r e of t h e undeformed s o l i d s are (r r ) and ( r r i t may be shown t h a t t o same seB&a!Yon c l o s e t o t h e p o i n t of gi&'t& c o n t a c t t h e e q u i v a l e n t e l l i p s o i d has p r i n c i p a l r a d i i of c u r v a t u r e (Rx,R ) a t t h e c o n t a c t p o i n t Y g i v e n by:-
-1= - +1-
1
Rx
'Bx
'Ax
l u b r i c a n t v i s c o s i t y a t atmospheric pressure
rl
v i s c o s i t y r a t i o ({ =
V
Poisson's r a t i o
P
lubricant density
PO
l u b r i c a n t density a t atmospheric pressure
Q-)
qO
-
density r a t i o
P
P ( b = p) 0
@
I f t h e e l l i p s o i d a l b o d i e s are p r e s s e d t o g e t h e r by a normal l o a d ( F ) , t h e n a n e l l i p t i c a l c o n t a c t zone i s developed i n t h e v i c i n i t y o f t h e p o i n t of c o n t a c t w i t h s e m i major and semi-minor a x e s d e n o t e d by ( a , b ) r e s p e c t i v e l y . The p r i n c i p a l . a x e s o f t h i s c o n t a c t e l l i p s e may be d e t e r m i n e d , a c c o r d i n g t o t h e t h e o r y of H e r t z ( 2 4 ) , and t h e r e l e v a n t e x p r e s s i o n s may b e found i n Hamrock and Dowson (25). The c o n d i t i o n s c o n s i d e r e d by Hamrock and Dowson ( 3 , 4 ) were r e s t r i c t e d t o t h o s e i n which l u b r i c a n t entrainment was d i r e c t e d along the
Vogelpohl s u b s t i t u t i o n
t'
Y HER'TZIAN CONTACT ELLIPSE (a1
i
X
LUBRICANTENTRAINING VECTOR X
w
HERTZIAN CONTACT ELLIPSE
Y
J
Figure t .
(a) (b)
(bl
(bl
Geometry of P o i n t C o n t a c t s . C o n t a c t between two e l l i p s o i d a l s o l i d s E q u i v a l e n t Geometry
Figure 2.
C o o r d i n a t e Systems f o r L u b r i c a n t E n t r a i n m e n t A l i g n e d w i t h t h e Minor Axis ( a ) and Major Axis (b) of t h e Contact E l l i p s e Respectively.
250 minor a x i s of t h e c o n t a c t e l l i p s e . T h i s c o n d i t i o n i s i l l u s t r a t e d i n F i g u r e 2 ( a ) , and may be c o n t r a s t e d w i t h F i g u r e 2 ( b ) which shows the s i t u a t i o n f o r l u b r i c a n t entrainment along t h e major a x i s o f t h e c o n t a c t e l l i p s e . I t s h o u l d be n o t e d t h a t t h e major a x i s of t h e c o n t a c t e l l i p s e h a s a g a i n been a s s o c i a t e d w i t h the 'y' coordinate d i r e c t i o n allowing t h e e l l i p t i c i t y r a t i o ( k = a / b ) t o be c a l c u l a t e d c o n s i s t e n t l y f o r e i t h e r case. b) Film Shape
c
The f i l m shape w i t h i n a n e l a s t o h y d r o d y n a m i c c o n j u n c t i o n may be r e p r e s e n t e d by t h e expression:-
N I UNIFORM NOOIS
Figure 4.
The S o l u t i o n Domain w i t h Respect t o t h e Dry C o n t a c t E l l i p s e (Only half t h e region was considered b e c a u s e of t h e p r e v a i l i n g symmetry)
With r e f e r e n c e t o F i g u r e 3, ho r e p r e s e n t s t h e s e p a r a t i o n along t h e z-axis of t h e undeformed e l l i p s o i d and p l a n e , s ( x , y ) i s t h e s e p a r a t i o n o f t h e c o n t a c t i n g b o d i e s due t o t h e i r geometry and w(x,y) i s t h e e l a s t i c deformation. It s h o u l d be n o t e d t h a t h usually takes a negative value i n the computations s i n c e t h e l o c a l deformation c o n s i d e r a b l y e x c e e d s t h e f i l m t h i c k n e s s , as i s a l s o t h e case w i t h p i e z o v i s c o u s - e l a s t i c conditions.
i
+
Figure 3.
+
-----
Components of F i l m T h i c k n e s s f o r a n E l l i p s o i d a l S o l i d Near a P l a n e .
-
(X
- mb) 2
+
E l a s t i c Deformation
The e l a s t i c d e f o r m a t i o n of t h e c o n t a c t i n g s o l i d s w(x,y) was e v a l u a t e d a c c o r d i n g t o t h e method d e s c r i b e d by Hamrock and Dowson ( 1 ) . Thus, t h e p r e s s u r e d i s t r i b u t i o n qver t h e c o m p u t a t i o n a l r e g i o n was approximated by a series of r e c t a n g u l a r r e g i o n s of c o n s t a n t p r e s s u r e which allowed t h e t o t a l e l a s t i c deformation a t t h e c e n t r e of t h e r e c t a n g l e t o be c a l c u l a t e d by t h e p r i n c i p a l of s u p e r p o s i t i o n .
Remembering t h a t t h e ' y ' c o o r d i n a t e d i r e c t i o n i s a s s o c i a t e d w i t h t h e major a x i s of t h e d r y c o n t a c t e l l i p s e , t h e geometric s e p a r a t i o n c a n be approximated by:-
S(X,Y)
(y2-Rla) 2
d)
Lubricant P r o p e r t i e s
The e f f e c t o f p r e s u r e s g e n e r a t e d w i t h i n t h e c o n j u n c t i o n area was assumed t o have a n e g l i g i b l e e f f e c t upon he l u b r i c a n t v i s c o s i t y . The f l u i d was t h e r e f o r e t a k e n t o be i s o v i s c o u s . The e f f e c t of p r e s s u r e upon t h e d e n s i t y , however, was i n c l u d e d i n t h e c a l c u l a t i o n s t o g i v e an a n a l y s i s c o n s i s t e n t with those of o t he r investigators. The e x p r e s s i o n governing t h e change i n d e n s i t y w i t h p r e s s u r e w a s t h a t d e v e l o p e d by Dowson and Higginson (261, namely
-
P =
Rx
W(X,Y) Rx
c)
_ _ _ _ _ho_ _ - - -
-I
Y
where t h e l e n g t h o f t h e c o m p u t a t i o n a l zone i n t h e e n t r a i n i n g and s i d e l e a k a g e d i r e c t i o n s i s g i v e n by ( m ) and (1) r e s p e c t i v e l y , t h e c e n t r e o f t h e c o n t a c t e l l i p s e b e i n g a t ( m b , l a ) a s shown i n F i g u r e 4. I f t h e f i l m t h i c k n e s s i s then normalised w i t h r e s p e c t t o t h e r a d i u s o f c u r v a t u r e , %' t h e n e q u a t i o n ( 9 ) may be w r i t t e n as:-
0
=
+
0.58 PE' 1+1.68 PE'
(It s h o u l d be n o t e d t h a t t h e h i g-~ h pressures r e q u i r e d t o produce a n o t i c a b l e change o f d e n s i t y in an i s o v i s c o u s - e l a s t i c c o n t a c t would a l s o r e s u l t i n a v i s c o s i t y i n c r e a s e of a similar proportion.) e ) Reynolds' E q u a t i o n The Reynolds' e q u a t i o n g o v e r n i n g t h e development of p r e s s u r e w i t h i n t h e c o n j u n c t i o n
25 1
INPUT DATA
I
CALCULATE HERTZIAN CONTACT CONDITIONS DETERMINE THE ELASTIC INFLUENCE COEFFICIENTS
EVALUATE DENSITY ELASTIC DEFORMATION ANALYSIS
OVER RELAXATION OF FILM THICKNESS
DETERMINE FINITE DlFFERENCE COEFFICIENTS
I
MANUAL RESTART GAUSS-SEIDEL ITERATION FOR PHI
I
UNDER RELAXATION OF PRESSURE
(8)
I
COMPARE INPUT LOAD AND CALCULATED LOAD ADJUST FILM THICKNESS CONSTANT (Ho) EVERY TENTH CYCLE I F DIFFERENT
STOP AFTER 1800 CPU SEC.
-I Figure 5
-
I
I
EXAMINE OUTPUT
Flow C h a r t o f t h e C o m p u t a t i o n a l P r o c e s s
whose f i l m s h a p e is g i v e n by e q u a t i o n (7) may b e w r i t t e n in non d i m e n s i o n a l terms as;-
or I
I
= 12
[<]
R ue
2
a
;(
He)
.
( f o r l u b r i c a n t entrainment a l o n g t h e major a x i s of t h e c o n t a c t e l l i p s e ) .
= 12 Ue
a
(p He)
(9)
( f o r l u b r i c a n t e n t r a i n m e n t a l o n g t h e minor a x i s of t h e c o n t a c t e l l i p s e ) .
The e x p r e s s i o n i s s l i g h t l y d i f f e r e n t f r o m t h a t employed by Hamrock and Dowson ( 2 ) s i n c e Ue h a s been normalised using t h e r a d i u s of c u r v a t u r e Re, and t h e R e y n o l d s ' e q u a t i o n includes the radius r a t i o rather than the e l l i p t i c i t y r a t i o (k).
252
The Reynolds’ equation was then modified by the use of the PHI(0) substitution described by Vogelpohl ( 2 7 ) , where
f)
Dimensionless Groups
The dimensionless groups involved in this isoviscous elastohydrodynamic analysis are those of dimensionless film thickness, speed and load. They may be written as:-
Dimensionless film thickness He
=
h Re
“ 0 ue Dimensionless speed parameter U = e E’ Re
Dimensionless load parameter
We
=
F
E’ :R
It will be noticed that the dimensionless materials parameter is not involved. 3
COMPUTATIONAL PROCEDURE
a) Solution Procedure The computational process followed was based on that adopted by Chittenden et a1 (5,6), the main features of which were detailed by Hamrock (28). Zero pressure was imposed along the boundary of the computational zone, with the Reynolds’ film rupture condition being additionally applied along the trailing edge of the lubricant film. Once the computational process had been initialised by the use of a dry contact pressure distribution the calculations proceeded through a series of nested loops whose main features are set out in the flow diagram, Figure 5 . The purpose of the two main loops, the pressure loop and the load loop, was as follows:-
The convergence process was observed to differ slightly from that €ound during the computation of results in the piezoviscouselastic regime (Chittenden et a1 ( 5 ) ) . In the piezoviscous work central film thickness was found to be the best indicator to the convergence of a solution, whereas minimum film thickness was the most important variable for the isoviscous solutions. It was also necessary to increase the pressure under-relaxation parameter in order to achieve a stable iteration. The computation of the elastohydrodynamic results presented in this paper again indicated that the value assumed for the initial separation of the solids was of considerable importance. Unlike the piezoviscous results, however, the computational time required for the hydrodynamic analysis could be close to that required for the elasticity analysis. A solution was deemed to have convergea if two successive runs of the computer program produced results which differed by less than 1%. In general about sixteen runs of the computer program, each of 1800 C.P.U. secs., on the Amdahl V7 computer at Leeds University were required before this criterion was reached. b)
Mesh Structure
The selection of a suitable mesh or nodal structure was known to be important from previous theoretical elastohydrodynamic studies. An incorrect choice of inlet film location could lead to a starved condition being computed if it was too restricted, or an unnecessary use of computer time for too large a value. The number of nodes along each axis was of comparable importance since too few nodes would lead to an unstable solution procedure and too many would cause an extended computational run time. To try and minimise these problems an inlet film location was determined in a similar way to This that described by Chittenden et a1 ( 5 ) . method utilised the expression presented by Hamrock and Dowson (19) for the critical inlet distance in isoviscous-elastic contacts, ie:-
THE PRESSURE LOOP In this loop a Gauss-Seidel solution of the Reynolds’ equation.was carried out until a tight convergence criterion was met for the determination of PHI(@). Within this loop film shape and density were held constant at each nodal point. THE LOAD LOOP Once the previous loop had been completed a new pressure distribution was determined with the use of considerable under-relaxation. The elastic deformation was then re-evaluated and a new film shape was computed by the use of an over-relaxation procedure, before a new set of finite difference coefficients were determined. On every tenth cycle through the loop the input load and integrated load were compared, allowing any discrepancy to be corrected by an adjustment of the central film thickness constant (Ho).
*
where m is the critical inlet distance from the centre of contact in Hertzian half widths, and Hmin is the minimum film thickness calculated from equation (1). For cases where entrainment was directed along the minor axis of the contact ellipse, therefore, the minimum film thickness was determined from equation (1) and hence the critical inlet distance could be determined from equation (13). In the majority of the results computed, entrainment was aligned with the major axis of the contact ellipse and therefore equation ( 1 ) was not directly applicable. The inlet distance required by a circular contact (%/Re = 1.0) was therefore determined and modified with the aid of experience gained during the computation of
253
-I
(0.25 t o 0.078) [2.5 t o 51
(0.352 t o 0.854) [2 t o 1.111
(1.0 t o 1.154) [1.0 t o 1.11
Number o f Nodes i n C o n t a c t E l l i p s e Normal t o Flow (22 1
26
22
18
Number of Nodes i n C o n t a c t E l l i p s e a l o n g Flow Direction (21)
14
14
12
1.4
1.5
2.0
1.5
1.5
1.5
Number o f Nodes in Flow Direction (Nl)
38
38
38
Number of Nodes Normal t o Flow (N2)
39
33
27
I n l e t Boundary L o c a t i o n i n Hertzian half widths (1) L a t e r a l Boundary L o c a t i o n i n H e r t z i a n Half Widths (m)
Table 1
T y p i c a l Mesh S t r u c t u r e s used i n t h e Computational Procedure.
piezoviscous-elastic s o l u t i o n s (5,6). Finally, the number of nodes i n a H e r t z i a n h a l f w i d t h was chosen based o n t h e s e v e r i t y o f t h e c o n d i t i o n s , the o u t l e t boundary f i x e d a t a small number of nodes beyond t h e H e r t z i a n r e g i o n , and a small adjustment made i n t h e i n l e t d i s t a n c e t o e n s u r e f u l l y f l o o d e d c o n d i t i o n s and t o a l l o w f o r t h e possibility t h a t c e n t r a l film thickness was a f f e c t e d by s t a r v a t i o n . Examples of t h e r e s u l t i n g meshes a r e g i v e n i n Table 1.
4
RESULTS
T h i r t y - s i x new s o l u t i o n s t o t h e problem of t h e l u b r i c a t i o n of c o u n t e r f o r m a l e l l i p t i c a l c o n t a c t s i n t h e i s o v i s c o u s - e l a s t i c regime were computed. Of t h e s e , t h i r t y e x t e n d e d t h e r a n g e of g e o m e t r i c a l c o n d i t i o n s examined by Hamrock and Dowson ( 1 8 ) . These r e s u l t s i n which l u b r i c a n t e n t r a i n m e n t was d i r e c t e d a l o n g t h e major a x i s o f t h e c o n t a c t e l l i p s e a r e i l l u s t r a t e d in F i g u r e 6 (Hmtn) and F i g u r e 7 The r e m a i n i n g s i x e s u l t s were (Heen). computed f o r c o n d i t i o n s i n which l u b r i c a n t entrainment w a s a l i g n e d w i t h t h e minor a x i s o f the c o n t a c t e l l i p s e and are i n c l u d e d on Figure 8 (H,in) and F i g u r e 9 (H,,,), together with t h o s e computed by Hamrock and Dowson ( 1 6 ) and t h e new s o l u t i o n s i l l u s t r a t e d i n F i g u r e s 6 and 7.
In t h e f i l m t h i c k n e s s e x p r e s s i o n s developed by C h i t t e n d e n a t a1 ( 5 , 6 ) , f o r t h e p i e z o v i s c o u s e l a s t i c regime, o n l y one s e t of powers w a s employed on t h e d i m e n s i o n l e s s groups. T h i s a l l o w e d b o t h minimum and c e n t r a l f i l m t h i c k n e s s e q u a t i o n s t o be f u l l y c o n s i s t e n t from t h e p o i n t o f view of d i m e n s i o n l e s s a n a l y s i s and a l s o reduced t h e number o f powers w i t h o u t a n y s i g n i f i c a n t l o s s of a c c u r a c y . T h i s approach h a s a g a i n been adopted i n t h e p r e s e n t a t i o n o f t h e r e s u l t s f o r t h e i s o v i s c o u s - e l a s t i c regime, t h e s e l e c t e d powers b e i n g t h o s e g i v e n by Hamrock and Dowson ( 1 8 ) f o r minimum f i l m t h i c k n e s s . A f i l m t h i c k n e s s e x p r e s s i o n may t h e n be w r i t t e n as:He =
( c o n s t a n t ) x ( f u n c t i o n o f geometry)
where lie =
He 0.65 -0.21 "e we
A l e a s t s q u a r e s p r o c e d u r e w a s used t o d e t e r m i n e e x p r e s s i o n s f o r b o t h minimum and c e n t r a l f i l m thickness, with the geometrical f u n c t i o n i n i t i a l l y t a k e n t o be of t h e form:( ~ - E x P ( - A ( R ~ / R ~ ) ) f) o r RJR,
51
( ~ - E x P ( - A ( R , / R ~ ) ) )f o r g e n e r a l R,/R,
254 where A is a c o n s t a n t d e t e r m i n e d from t h e l e a s t squares analysis. I t was found, however, t h a t t h i s form of s i d e leakage f a c t o r d i d not provide a s a t i s f a c t o r y r e p r e s e n t a t i o n of minimum f i l m t h i c k n e s s . It is e v i d e n t from F i g u r e 6 t h a t t h e computed r e s u l t s e x h i b i t a n ‘S’ shaped c u r v e and hence a n a l t e r n a t i v e f u n c t i o n was s o u g h t ; t h e form of e x p r e s s i o n which provided t h e g r e a t e s t improvement b e i n g o f a d o u b l e e x p o n e n t i a l n a t u r e :-
I f t h e s e v e n t e e n r e s u l t s p r e s e n t e d by Hamrock and Dowson ( 1 8 ) a r e combined w i t h t h e t h i r t y r e s u l t s i n which l u b r i c a n t e n t r a i n m e n t w a s d i r e c t e d a l o n g t h e major a x i s , and t h e s i x new r e s u l t s f o r e n t r a i n m e n t a l o n g t h e minor a x i s , t h e n i t is p o s s i b l e t o g e n e r a t e g e n e r a l e x p r e s s i o n s f o r t h e p r e d i c t i o n o f minimum and c e n t r a l f i l m thickness. The a p p l i c a t i o n o f a least s q u a r e s procedure t o t h e s e f i f t y - t h r e e r e s u l t s allowed non-dimensional minimum and c e n t r a l f i l m t h i c k n e s s e x p r e s s i o n s t o be e v a l u a t e d as:-
-
( ~ - E x P ( - A ( R ~ / R) )~ )(1 - E X ~( - B ( R ~ / R)~) )
= 6.71(1.0-EXP(-0.52(R
Hmin where A and B are c o n s t a n t s t o be d e t e r m i n e d . The r e s u l t i n g e x p r e s s i o n s f o r d i m e n s i o n l e s s film thickness with l u b r i c a n t entrainment d i r e c t e d a l o n g t h e major a x i s of t h e c o n t a c t e l l i p s e were:-
Hmin
17.4(l.0-EXP(-6.40(Rs/Re)))
=
( 1 .O-EXP( -0.14 ( Rs/Re)
-Hcen
(11
4.69(1.0-EXP(-3.35(Rs/Re)))
=
These f u n c t i o n s are shown in F i g u r e s 6 and 7 , from which i t may be s e e n t h a t agreement between t h e computed v a l u e s and t h e i r a s s o c i a t e d formula is q u i t e r e a s o n a b l e . The mean and mean a b s o l u t e e r r o r s r e s u l t i n g from t h i s comparison are r e c o r d e d i n T a b l e 2.
6 00
5.50
f
5.00
----
Curve Fit
11.0-EXP I -0.14 l R ~ / ~ l I l
Ue
t
Ue + Ue 1
V
a30
a20
040
.._-_ Curve Flt
5.00
350-
4
3.00 -
5
i
1,385€-08, We
i
9261E-04 b.013E-04
i
2.bOhE-04
F i g u r e 8.
Minimum F i l m T h i c k n e s s w i t h E n t r a i n ment Along E i t h e r P r i n c i p a l Axis.
Figure 9.
C e n t r a l F i l m Thickness w i t h E n t r a i n ment Along E i t h e r P r i n c i p a l Axis.
.
V
‘
V
“ ,’V 1,
/e
3‘
-
/
Y
1.00
= L.69 11.0-€XP I -3.35 IRs/RpllI
H:en
,f
1.50 1.00
0.90
, v
200
0
0.80
r
2.50
5 0-50
0.70
lRs/Rel
: - _ -+ . --,’---v -v / _.. -_ -.- v . , .,
- 4.00 4.50
a
0.60
0.50
Minimum F i l m T h i c k n e s s R e s u l t s and Curve F i t w i t h E n t r a i n m e n t Along t h e Major Axis.
F i g u r e 6.
-y E
1.385E-08. We
1.00
Ommrionlrsi parameter
s
i i
Ue I 1.385€-08. We 4 Hamrocks rirvllr
0.00 0.10
I‘
(14)
I
2 1.50
\
86 (Rs/Re )2’3) )
These f u n c t i o n s are shown i n F i g u r e s 8 (H ), and 9 (H 1, and t h e mean and mean ab!!&ute e r r o r s @? a g a i n r e c o r d e d i n T a b l e 2. T h i s t a b l e shows t h a t t h e a c c u r a c y o € t h e s e g e n e r a l e x p r e s s i o n s is l i t t l e d i f f e r e n t from t h a t p r o v i d e d by e q u a t i o n s ( 1 4 ) and (151, a l t h o u g h t h i s is not immediately a p p a r e n t i f F i g u r e s 6 and 8, o r F i g u r e s 7 and 9 are compared. It s h o u l d be remembered, however, t h a t a l t h o u g h t h e o v e r a l l a c c u r a c i e s are s i m i l a r , e q u a t i o n s ( 1 1 ) and (12) a l l o w f i l m t h i c k n e s s t o be p r e d i c t e d w i t h g r e a t e r a c c u r a c y f o r cases where l u b r i c a n t e n t r a i n m e n t is c o i n c i d e n t w i t h t h e major a x i s o f t h e c o n t a c t ellipse.
2.00
r
= 8.28(1.O-EXP(-O.
(l3)
1.385E-08. We i 9.261E-OL + Up = 1385f-08, We = 4.813E-Ob V Up I l 3 8 5 E - 0 8 . We i 2.bOhE-04
4.00
E
(l.0-EXP(-1.70(R:/R~)2’3))
= 17.35 II.O-€XP I -6.40 IRs/ReIII
H*,,
x
-
Ecen
/ R )2/3))
+
Up Up
i i
l.30SE-08. We l.385E-08. We
i
9.2KlE-04
:
48llE-04
V Ue = 1.385E-08. We = 2 b06E-04
,
-/’
...
000 010
020
030
040
050
Dmenrmlrrs paramrler
F i g u r e 7.
060
070
a80
090
100
IR,/A,I
C e n t r a l F i l m Thickness R e s u l t s and Curve F i t f o r Entrainment Along t h e Major Axis.
255
Non-Dimensional Film T h i c k n e s s
Error ( % )
E n t r a i n m e n t A l i g n e d w i t h t h e Major A x i s
ion 14 )
-4.53
H ( f @ a t i o n 15)
-1.59
:{Gat
12.67
a. 14
Entrainment A l i g n e d w i t h e i t h e r P r i n c i p a l Axis
:$$:ation
17)
Table 2
-1.60 Summary of t h e D i s c r e p a n c i e s Between t h e Computed F i l m Thickn e s s and P r e d i c t i o n s o f t h e Formulae Developed
5
DISCUSSION
The amount o f p r e v i o u s l y p u b l i s h e d i n f o r m a t i o n on t h e o r e t i c a l and e x p e r i m e n t a l a s p e c t s of isoviscous-elastohydrodynamic lubrication is very limited, especially f o r the unusual g e o m e t r i e s c o n s i d e r e d h e r e . Several a u t h o r s have produced t h e o r e t i c a l r e s u l t s f o r the l i n e c o n t a c t problem, b u t v e r y l i t t l e e x p e r i m e n t a l work h a s been p u b l i s h e d . Higginson (141, and Dowson and S w a l e s (9,15) c o n s i d e r e d t h e problem o f t h e isoviscous-elastohydrodynamic l u b r i c a t i o n o f r e c i p r o c a t i n g seals, b o t h e x p e r i m e n t a l l y and t h e o r e t i c a l l y . A simplified t h e o r e t i c a l a n a l y s i s o f t h e l i n e c o n t a c t problem e x i s t i n g in human a n k l e j o i n t s w a s d e s c r i b e d by Medley and Dowson (13), who m o d e l l e d t h e j o i n t by both a c y l i n d e r n e a r a p l a n e and by a n equivalent plane i n c l i n e d s u r f a c e t o allow squeeze e f f e c t s t o be i n c l u d e d more r e a d i l y . The a v a i l a b i l i t y o f i n f o r m a t i o n on t h e l u b r i c a t i o n of p o i n t c o n t a c t s i s e v e n more r e s t r i c t e d , b u t t h e t h e o r e t i c a l work of B i s w a s and S n i d l e (16) and Hamrock and Dowson ( l R ) , t o g e t h e r w i t h t h e e x p e r i m e n t a l s t u d i e s of Jamison e t a1 (17) a l l o w s a c o m p a r i s o n t o be made w i t h t h e p r e s e n t work f o r c i r c u l a r c o n t a c t condi t ions. The d i m e n s i o n l e s s f i l m - t h i c k n e s s f o r m u l a e f o r isoviscous-elastohydrodynamic c o n d i t i o n s may be e x p r e s s e d i n t e r m s o f t h e d i m e n s i o n l e s s load-speed p a r a m e t e r (Mp) and t h e f i l m thicknesss-speed parameter ( by Moes and Bosma (29).
of film thickness expressions f o r a c i r c u l a r c o n t a c t t o be compared. These are r e c o r d e d i n T a b l e 3, t o g e t h e r w i t h t h e i s o v i s c o u s - r i g i d e x p r e s s i o n d e t e r m i n e d by K a p i t z a (30), and a l l o f them a r e i l l u s t r a t e d i n F i g u r e 10 ( t h e a l t e r n a t i v e r e p r e s e n t a t i o n s of t h e J a m i s o n e t a 1 r e s u l t s b e i n g shown a s d o t t e d l i n e s . ) . I t may be s e e n in F i g u r e 10 t h a t a l l o f t h e p r e d i c t i o n s i l l u s t r a t e d show good agreement w i t h e a c h o t h e r . The p r e s e n t a n a l y s i s p r e d i c t s l o w e r v a l u e s o f t h e f i l m thickness-speed parameter (H ) than a l l t h e o t h e r expressions f o r va'Pi!i&?s o f t h e d i m e n s i o n l e s s load-speed p a r a m e t e r (Mp) g r e a t e r t h a n 500. A t v a l u e s of (Mp) between 500 and a b o u t 14 t h e new r e s u l t s p r e d i c t a f i l m t h i c k n e s s - s p e e d p a r a m e t e r between t h e l o w e r bound o f t h e J a m i s o n e t a 1 r e s u l t s and t h e o t h e r expressions. For l o w e r v a l u e s o f (Mp) t h e l i n e r e p r e s e n t i n g t h e B i s w a s and S n i d l e r e s u l t s is c r o s s e d t o g e t h e r w i t h t h e u p p e r bound o f t h e J a m i s o n e t a1 r e s u l t s a t a s l i g h t l y lower v a l u e o f (Mp). The r e s u l t s o f t h e s t u d y a r e t h e r e f o r e i n good a g r e e m e n t w i t h t h e o t h e r r e s u l t s c o n s i d e r e d a t low v a l u e s o f d i m e n s i o n l e s s l o a d Its divergence a t higher s p e e d p a r a m e t e r (Mp). v a l u e s o f (Mp) i s due t o t h e magnitude o f t h e power on t h e d i m e n s i o n l e s s load-speed p a r a m e t e r . Thus a change i n t h e r e l a t i o n s h i p between dimensionless film thickness, dimensionless s p e e d p a r a m e t e r (u 1, and d i m e n s i o n l e s s l o a d p a r a m e t e r (W ) d e r i c e d by Hamrock and Dowson (18) would bg r e q u i r e d t o g i v e a g r a d i e n t c l o s e r t o t h o s e of t h e o t h e r r e s u l t s . I n o r d e r t o examine t h e a c c u r a c y o f t h e powers a s c r i b e d t o t h e d i m e n s i o n l e s s s p e e d and l o a d p a r a m e t e r s f u r t h e r s o l u t i o n s were computed f o r c i r c u l a r c o n t a c t s (R /R = I ) . T h r e e new s o l u t i o n s were o b t a i n e d ? o r e d i f f e r e n t s p e e d s a t a g i v e n l o a d and two more s o 1 u t i o n s . w e r e obtained f o r d i f f e r e n t l o a d s a t a set speed. The r e s u l t s f o r d i f f e r e n t s p e e d s are shown i n F i g u r e 11, w h i l s t t h o s e f o r d i f f e r e n t l o a d s are i l l u s t r a t e d i n F i g u r e 12. A l e a s t s q u a r e s f i t t o t h e s e t o set o f r e s u l t s g a v e t h e f o l l o w i n g r e l a t i o n s h i p s between f i l m t h i c k n e s s s p e e d , and load.
0.71 Hmin OL
0.28 We
0.59 wo.12
Hcen " e
e
LOG 10
'R 1 : IH*m,nl from K a p i t z a 119511 2 : IH*m,nl l r o m lamiron et a t 119781 Id assumed fdm-thxknrir ratio. lHmin/Hwnl : 18% Ibl assumed Iilm-thickness ratio, IHmm/Hcenl = 47% 3 : IH*mml from B w a r and Snidle I1916l L . IH*mmt f r o m Hamrock and Douron 119781 5 : Wmlnl from present malyrir . -
-1
oo LOG10
'
Load-speed p a r a m r t r r , MP = W~IJ;'"
F i g u r e 10. The use of t h e s e two p a r a m e t e r s r e s u l t s in a g e n e r a l f l m t h i c k n e s s e x p r e s s i o n o f t h e form Hmin = A.M$, and t h i s i n t u r n a l l o w s a number
Ue
The Comparison o f F i l m T h i c k n e s s E x p r e s s i o n s O b t a i n e d by D i f f e r e n t Workers.
256
LOG 10
A
Re fer ence
Expression
Biswas and Snidle (16) Jamison et a1 (17)
-0.11 Hmin=1.96 Mp
V
Notes
Hmin ken
-- Lead
squares line lor Ill .....t e a s t squarer ll"L lor 121
-
I
Hmin=1.87Mp -0.075
I
Hmin=l.llMp -0.075
Hamrock and Dowson (18)
I
Hmin=3. 21Mp-0.21
-0.21
Equation (13) Kapitza (30)
Hmin=1 1 3.6Mp
The Effect of Speed Upon Film Thickness (We = 4.81 x
Figure 1
LOG 10
Noteu Jamison et a1 only gave central film th$,lyjyijs expression, ( H ' = 2-40 MP ) and if the rat%"of ratio of minimum to central film thickness according to Hamrock and Dowson (18) is assumed (H /H = 78%) this expression may be ob'?8Yneatn As [ I ] but assuming (H /H = 47%) from equations (13) and (147.in cen Analysis of the results used in the derlvation of equations (13) and (14) shows that:-
51 -5
I
I -L
LOG10
-3
Dlmmrlonlerr Load P a r a m e t e r IWc I
Figure 12.
The Effect of Load Upon Film Thickness (Ue = 1.38 x
;;
/R )2/3)) Hmin = 7.70(1-EXP(-1.69(R ( 1 -EXP (-0.52 (Rs/Rz) Hcen = 9.50( 1-EXP (-0.86 (R:/Re)
Expression for an Isoviscous-Rigid Contact Table 3
Film Thickness Expressions for a Circular Contact
It is evident from equations ( 1 5 ) and ( 1 6 ) that the powers deduced €or a circular contact are somewhat different from those determined by Hamrock and Dowson, but their adoption was not found to produce a significant change in the accuracy of the predictions of the film thickness expressions. For this reason, and to try and minimise the proliferation of powers on these two variables, these alternative values were not incorporated into the full analysis. The shortage of experimental results €or situations in which lubricant entrainment i f i directed along the major axis precludes any comparison between experiment and theory f o r these unusual geometries. At the opposite end of the range, however, It i s possible to use the experimental line contact results from Swales (31) to investigate the accuracy of the asymptotes given by the minimum film thickness formula, equation (13). Figure 13 shows each of these experimental results and the resulting line contact asymptote in terms of the = dimensionless film thickness parameter (H ) * min
I
0
5
10
15
20
25
30
35
E x p m m m t a l Number
Figure 13.
Theory Compared with che Experimental Results of Swales (31).
All of the results fall some way below this asymptote, but if the true geometry of the contacting bodies used in this investigation is considered (R /Re = 13) then the average error is redueed to about 22%. This theoretical overestimation of the isoviscous-elastohydrodynamic film thickness may be contrasted with the findings of Chittenden, Dowson, and Taylor (32) who observed that experimental measurements of film thickness in
257
Pressure p r o f i l e :
F i l m p r o f i l e - Deformed a r e a
1
4.80 7 4.00 3.20 2.40
4.80 4.00
3.20 2.40 1.60 .80
i.60 .80
.oo
.oo 2 A X I S 010 -2
Pressure p r o f i l e : 4.80
2 Dimensionless parameters
7
4.80 4.00 3.20 2.40 1.60 .80
prof1 l e
.963E -3
=
R s /R e
(a)
_ - - Hertzlan
. 138E
We =
K
.oo
P O. 060
Ue =
=
-7
1.667 .448
Isometric Projections
l a x press a t
x
I
0.048 0.036 0.024 0.012 Max press=
0.474E-01
Mln f i l m at
I
0.000
H
Oo30
?z(
0.0024 0.001 8 0.0012 0.0006 Hcen= Hm I n=
0. 133E0.345E-04
0.0000 (b)
Figure 14.
Cross Sections and Contour Maps
Representations of Pressure and Film Thickness in an Elliptical, Isoviscous-Elastic Conjunction.
:
258 p i e z o v i s c o u s - e l a s t i c c o n t a c t s t e n d t o be l a r g e r than experimental predictions. The d i s c r e p a n c y between t h e o r e t i c a l p r e d i c t i o n s , f o r an i s o v i s c o u s l u b r i c a n t , and e x p e r i m e n t a l r e s u l t s , however, is n o t u n u s u a l s i n c e s e v e r a l w o r k e r s have found t h a t l i n e c o n t a c t t h e o r y w i l l a l s o overpredict the dimensionless film thickness i n l i g h t l y l o a d e d c o n t a c t s such a s t h e s e . I t is a l s o i n t e r e s t i n g t o compare d e t a i l s of t h e i s o v i s c o u s - e l a s t i c r e s u l t s d es cr i b ed h e r e w i t h t h o s e of t h e p i e z o v i s c o u s - e l a s t i c r e s u l t s p r e s e n t e d by C h i t t e n d e n e t a1 ( 5 , 6 ) . F i g u r e 14 shows i s o m e t r i c p r o j e c t i o n s t o g e t h e r w i t h a c o n t o u r map and c r o s s s e c t i o n o f d i m e n s i o n l e s s p r e s s u r e and f i l m t h i c k n e s s f o r a r a d i u s r a t i o This f i g u r e i s (R / R ) of 0.46, (k=1.67). t y f i i c s l of t h e r e s u l t s o b t a i n e d i n t h e p r e s e n t s t u d y , s i n c e i t shows t h a t t h e p r e s s u r e s g e n e r a t e d w i t h i n t h e c o n t a c t zone a r e v e r y s i m i l a r t o t h o s e p r e d i c t e d by a H e r t z i a n a n a l y s i s . ( I t s h o u l d be n o t e d t h a t t h e c o r r u g a t i o n s i n t h e f i l m p r o f i l e towards t h e e x i t are p r o b a b l y a f e a t u r e o f t h e n u m e r i c a l methods employed f o r t h e o r e t i c a l s o l u t i o n s , similar f e a t u r e s b e i n g e v i d e n t i n t h e work of a number of r e s e a r c h e r s . ) I n t h e case of piezoviscous elastohydrodynamic l u b r i c a t i o n i t was observed t h a t one of t h e d i s t i n g u i s h i n g f e a t u r e s of t h e r e s u l t s w i t h l u b r i c a n t e n t r a i n m e n t d i r e c t e d a l o n g t h e major a x i s o f t h e contact e l l i p s e was the rise i n c e n t r a l p r e s s u r e s above t h o s e o b t a i n e d from a H e r t z i a n a n a l y s i s . I n i s o v i s c o u s c o n t a c t s , however, t h i s d i d n o t o c c u r above a r a d i u s r a t i o (R / R ) of 0.6, and even t h e n t h e p r e s s u r e s wefe 80 more t h a n ( 1 % ) above t h e c o r r e s p o n d i n g H e r t z i a n v a l u e s , compared w i t h t h e (10%) t y p i c a l of p i e z o v i s c o u s c o n j u n c t i o n s .
pronounced e x i t c o n s t r i c t i o n , which may be seen i n t h e c r o s s s e c t i o n o f f i l m shape, F i g u r e 14. 6 ) CONCLUSIONS T h i r t y - s i x new s o l u t i o n s t o t h e e l a s t o hydrodynamic l u b r i c a t i o n problem f o r m a t e r i a l s o f low e l a s t i c modulus have been p r e s e n t e d . T h i r t y of t h e s e r e s u l t s were f o r s i t u a t i o n s where t h e d i r e c t i o n o f l u b r i c a n t e n t r a i n m e n t w a s a l i g n e d w i t h t h e major a x i s o f t h e c o n t a c t e l l i p s e . The s o l u t i o n s f o r t h e s e c a s e s were used t o y i e l d t h e f o l l o w i n g e x p r e s s i o n s f o r minimum and c e n t r a l f i l m t h i c k n e s s : -
%in
/R ) ) ( 1-EXP (-0.14 (R: / R ~) )
= 17.6(1-EXP(-6.40(R
It was then possible t o include the s o l u t i o n s computed by Hamrock and Dowson ( 1 8 ) and t o d e v e l o p e m p i r i c a l e x p r e s s i o n s t o a l l o w t h e p r e d i c t i o n of f i l m t h i c k n e s s when entrainment w a s aligned with e i t h e r principal a x i s o f t h e c o n t a c t e l l i p s e . These r e l a t i o n s h i p s are:2 // 3 3 )) = 6.71 (l-EXP(-O. 52(Rs /& ) 2 Hmin (I-EXP(-I .70(Rs /Re )) = 8.28( l-EXP(-O. 86 (Rs Heen
I n t h e p i e z o v i s c o u s case t h e p o i n t of minimum f i l m t h i c k n e s s was found t o move from a p o s i t i o n on t h e c o n t a c t c e n t r e l i n e a t t h e e x i t c o n s t r i c t i o n by d i v i d i n g and f o l l o w i n g t h e development o f s i d e l o b e s p a r a l l e l t o t h e d i r e c t i o n of entrainment. I n t h e i s o v i s co u s case t h e development o f s i d e l o b e s s t i l l t a k e s p l a c e a s t h e r a d i u s r a t i o (R / R ) is reduced b u t i n a more l o c a l i s e d form, p r 8 b a h y due t o t h e r e l a t i v e l y l a r g e c o n t a c t zone r e s u l t i n g from t h e These low e l a s t i c modulus o f t h e materials. s i d e l o b e s no l o n g e r c o n t a i n t h e p o i n t ( s ) o f minimum f i l m t h i c k n e s s which remains on t h e semi-axis o f t h e c o n t a c t e l l i p s e a t t h e e x i t . The c e n t r a l r e g i o n of a p p r o x i m a t e l y c o n s t a n t f i l m t h i c k n e s s remains, b u t t h e r e l a t i v e change i n f i l m t h i c k n e s s a l o n g i t s l e n g t h is much l a r g e r . T h i s f e a t u r e may be observed i n i s o m e t r i c p r o j e c t i o n s of f i l m s h a p e , b u t i t is n o t p a r t i c u l a r l y e v i d e n t on t h e c o n t o u r map due t o t h e c o n t o u r p o s i t i o n s . The f i l m t h i c k n e s s r a t i o (Hmi /Heen) is Shorn, as a f u n c t i o n of geometry, i n F i g u r e 15 f o r b o t h i s o v i s c o u s - e l a s t i c c o n t a c t s ( d e r i v e d from e q u a t i o n s (13) and ( 1 4 ) ) and p i e z o v i s c o u s e l a s t i c c o n t a c t s ( d e r i v e d from t h e e x p r e s s i o n s I t may be s e e n g i v e n by C h i t t e n d e n e t a1 ( 6 ) ) . from t h i s f i g u r e t h a t t h e v a l u e s o f f i l m t h i c k n e s s r a t i o are higher f o r t h e piezoviscouse l a s t i c cases, a l t h o u g h t h e a s y m p t o t i c v a l u e i s o n l y ( 6 % ) above t h a t of t h e i s o v i s c o u s - e l a s t i c conditions. A t low v a l u e s of t h e r a d i u s r a t i o (Rs/RJ t h e d i f f e r e n c e is c o n s i d e r a b l y l a r g e r , a f e a t u r e a t t r i b u t e d t o t h e development o f t h e
/
7
lo'
10' 10' 10' 10' 10 o m n r ~ o n i r s reiarhrity parameter
F i g u r e 16.
lo
10
10
w'
qE
An Example of t h e New L u b r i c a t i o n Regime C h a r t s from C h i t t e n d e n , Dowson and T a y l o r ( 2 3 ) .
259
The agreement between the predictions of these two pairs of expressions and corresponding computed solutions has been shown to be reasonable, although the point of inflection evident in the computed values of minimum film thickness results in a significant overestimate of film thickness for values of radius ratio close to 0.43.
Chittenden R.J., Dowson D., Dunn J.F., and Taylor C.M., 'A Theoretical Analysis of the Isothermal Lubrication of Point Contacts, Part I1 - General Case, with Luhricant Entrainment at Along either Principal Axis of the Hertzian Contact Ellipse o r at Some Intermediate Angle. ' , Roc. R. SOC. Lond., A=, 1985, 271-294.
The agreement between the predictions of the expressions derived and the sparse amount of other published work has been found to be good, and several similarities and differences have been identified between the isoviscous solutions presented here and those for piezoviscous-elastic conditions described by Chittenden et a1 (5.6).
Dowson D. and Higginson G.R., 'A Numerical Solution to the Elastohydrodynamic Problem.', J. Mech. Eng. Sci., 1959, 1(1), 7-15.
The film thickness ratio (Hmi /H ) for isoviscous-elastic conditions has 6eeSe?ound to be always less than the corresponding ratio for piezoviscous-elastic conditions, with the difference at low radius ratios being most marked.
Dowson D. and Swales P.D., 'An Elastohydrodynamic Approach to the Problem of the Reciprocating Seal.', The Third International Meeting on Fluid Sealing, British Hydrodynamics Research Association, 1967, 33-44.
The results of this study extend the range of conditions for which predictions of elastohydrodynamic film thickness may be made when dealing with materials of low elastic modulus. Although the application of the unusual geometries considered in this study to lubricated elastomeric machine elements may well be limited, a restriction upon the satisfactory construction of lubrication regime charts has now been removed. As an example of this, Figure 16 shows a lubrication regime chart, constructed with the aid of the film thickness expressions noted earlier, for a radius ratio (R /R = 0 . 5 ) taken from the work of Chittenden, Dgwssn and Taylor (23).
Ruskell L.E.C., 'A Rapidly Converging Theoretical Solution of the Elasto hydrodynamic problem for Rectangular Rubber Seals.', J. Mech. Eng. Sci., 1980,
Re€ erence s Hamrock B.J. and Dowson D., 'Numerical Evaluation of the Surface Deformation of Elastic Solids Subject to Hertzian Contact Stress.', N.A.S.A. TN D-7774, 1974 Hamrock B.J. and Dowson D., 'Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part I - Theoretical Formulation.', Trans A.S.M.E., J. Lubr. Technol., 98(2), 1976, 223-229. Hamrock B.J. and Dowson D., 'Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part 111 - Fully Flooded Results', Trans. A.S.M.E., J. Lubr. Technol., 99(2), 1977, 264-276. Hamrock B.J. and Dowson D., 'Isothermal Elastohydrodynamic Lubrication of Point Contacts. Part IV - Starvation Results.'. Trans. A:S.M.E., J. Lubr. Technol., z ( l j , 1977, 15-23. Chittenden R.J., Dowson D., Dunn J.F., and Taylor C.M., 'A Theoretical Analysis of the Isothermal Lubrication of Point Contacts, Part I - Direction of Lubricant Entrainment Coincident with the Major Axis of the Hertzian Contact Ellipse.', Proc. R. SOC. Lond., A K , 1985, 245-269.
Dowson D. and Higginson G.R., 'New Roller Bearing Lubrication Formula.' Engineering, Lond., 1961, 192, 158.
( 1 1 ) Baglin K.P. and Archard J., 'An Analytical
Solution of the Elastohydrodynamic Lubrication of Materials of Low Elastic Modulus.', Proc. 2nd Symposium on Elastohydrodynamic Lubrication, Institute of Mechanical Engineers, Lond., i972, 13-21. (12) Herrebrugh K., 'Solving the Incompressible and Isothermal Problem in Elasto hydrodynamic Lubrication Through an Integral Equation.', Trans. A.S.M.E., J. Lubr. Technol., =(1), 1968, 262-270. (13) Medley J.B. and Dowson D., 'Lubrication of Isoviscous Line Contacts Subject to Cyclic Time-Varying Loads and Entrainment Velocities. ', A.S.L.E. Transactions, 27(3), 1984. 243-251. (14) Higginson G.R., 'A Model Experiment I n Elastohydrodynamic Lubrlcation.', Inst. Journal Mech. Eng. Sci., 1962, 205-210. (15) Dowson D. and Swales P.D., 'The Development of Elastohydrodynamic Conditions in a Reciprocating Seal.', Fourth International Conference on Fluid Sealing, Philadelphia Pa., 1969. (16) Biswas S. and Snidle R.W., 'Elastohydrodynamic Lubrication of Spherical Surfaces Of Low Elastic Modulus.', Trans A.S.M.E., J. Lubr. Technol., 98(4), 1976, 524-529. (17) Jamison W.E., Lee C.C. and Kauzlarich J.J., 'Elasticity Effects on the Lubrication of Point Contacts.', A.S.L.E. Transactions, 21(4), 1978, 299-306.
-
260 (18) Hamrock B.J.
and Dowson D., 'Elastohydrodynamic Lubrication of Elliptical Contacts for Materials of Low Elastic Modulus, Part I - Fully Flooded Conjunction.', Trans. A.S.M.E., J. Lubr. Technol., =(2), 1978, 236-245.
(19) Hamrock B.J.
and Dowson D., 'Elastohydrodynamic Lubrication of Elliptical Contacts for Materials of Low Elastic Modulus, Part I1 - Starved Conjunction.', Trans A.S.M.E., J. Lubr. 1) ,92-98. Technol.,
m(
(20) Hamrock B.J.
and Dowson D., 'Minimum Film Thickness in Elliptical Contacts for Different Regimes of Fluid Film Lubrication.', Proc. 5th Leeds-Lyon Symposium on Tribology, 1979, 22-27, Eds. Dowson D., Taylor C.M., Godet M., and Berthe D.
(21) Brewe D.E.,
Hamrock B.J. and Taylor C.M., 'Effects of Geometry on Hydrodynamic Film Thickness.', Trans A.S.M.E., J. Lubr. 1979, 231-239. Technol., %(2),
(22) Dowson D., Dunn J.F.,
and Taylor C.M., 'The Piezoviscous Fluid, Rigid Solid Regime of Lubrication.', Proc. Inst. Mech. Engrs., Lond., E(C), 1983, 43-51.
(23) Chittenden R.J.,
Dowson D., and Taylor C.M., 'The Estimation of Minimum Film Thickness in the Design of Concentrated Contacts.', Submitted for publication to the Inst. Mech. Engrs.
(24) Hertz H., 'The Contact of Elastic Solids.', J. Reine Angew, Math., 92, 1882, 156-171. (25) Hamrock B.J.
and Dowson D., 'Ball Bearing Lubrication, The Elastohydrodynamics of Elliptical Contacts.' J. Wiley 6 Sons, New York, 1981.
(26) Dowson D. and Higginson G.R.,
'Elastohyd rodynamic Lubrication. The Fundamentals of Roller and Gear Lubrication', Pergamon, Oxford. 1966. (27) Vogelpohl G., Forsch Hft. Ver Dt. Ing., 386, 1937, 1-28. (28) Hamrock B.J.,
PhD Thesis, Department of Mechanical Engineering, Leeds University, 1976.
( 2 9 ) Moes H., and Bosma R., 'Film Thickness and
Traction in E.H.L. at Point Contacts.', Proceedings of the Second Symposium on Elastohydrodynamic Lubrication, Inst. Mech. Engrs., Lond., 1972. (30) Kapitza P., 'Hydrodynamic Theory of
Lubrication During Rolling.', Zh. Tekh. Fiz., 1 5 ( 4 ) , 1955, 747-762. ( 3 1 ) Swales P.D.,
PhD Thesis, Department of Mechanical Engineering, Leeds University, 1969.
( 3 2 ) Chittenden R.J.,
Dowson D., and Taylor C.M., 'Experimental Investigation of Elastohydrodynamic Film Thickness In Concentrated Contacts, Pert I1 - The Correlation of Experimental Results with Elastohydrodynamic Theory.', Proc. Inst. Mech. Engrs., Z(C3), 1986, 219-226.
26 1
Paper Vlll(iii)
Effect of surface roughness and its orientation on E.H.L. D.Y. Hua, S.H. Wang and H.H. Zhang
In this paper, a generalized stochastic Reynold's equation considering surface possessing Reynold's roughness of arbitrary orientation has been deduced. The full numerical solution of stochastic Reynold's equation for one dimensional, steady flow has' been obtained in the whole contact zone. Numerical results show that the film thickness and the load capacity are affected by surface roughness orientation even in full lubrication region. The experiment results by Timken test apparatus also show the effects of surface roughness orientation on EHL and the trends of scoring load are concord with the numerical solution.
1 INTRODUCTION How to consider the effect of surface roughness and its orientation on EHL is a problem which some researchers made efforts to solve. Christensen [l] used stochastic theory to hydrodynamic lubrication equation and derived a Reynold's equation with two different kinds of roughness orientation (transverse and longitudinal) on the basis of some hypotheses. This Reynold's equation was used to study bearing lubrication [ 2 ] [ 3 ] . On the basis of study by Christensen and other researchers, Elrod [4] derived a lubrication equation fit for surfaces possessing Reynold roughness of general orientation by multi-scale method. But unfortunately, solution of this equation hasn't been found. Chow and Cheng [ 5 ] gave a stochastic Reynold's equation considering that both surfaces are rough. The instantaneous effects caused by surface roughness also have been dealt with. The authors have used the equation they derived to solve the EHL problem of rollers by Grubin's analysis. Cheng and Dyson [6] gave a solution in the whole inlet zone of EHL of circumferentially ground discs. They considered the effects of surface roughness both on hydrodynamics and contact methanics. They obtained surface temperature due to interferes of asperities in the inlet zone according to Blok's surface flash temperature formula. Dyson[ 71 [ 81 compared the solution of EHL of longitudinal surface roughness with scoring tests of circumferentially ground discs. Snidle and Rossides[9] improved the theory of Dyson on the load-gap relationship of surface asperities contacts. Scoring temperature preformed is concord well with experiment results. Experiments on surface texture on disc tester have also been taken by Ku, Carper and Staph[lO] and Li[ll], etc.. On the basis of theories mentioned above
a general form of Stochastic Reynold's equation considering arbitrary surface roughness orientation has been derived. The full numerical solution shows the effects of surface roughness orietation on film thickness and pressure distribution. The numerical results
has also been compared with experiment results by Timken ring-block apparatus and discussion has been given. 1.1 Notation
hT
actual oil film thickness
hC
central film thickness, where
aP/aX =O
hmin minimum film thickness oil film pressure P U
surface velocity
V
elastic deformation of surface
G
non-dimensional material parameter
V W
non-dimensional velocity parameter non-dimensional load parameter
6
surface roughness
U
mean square deviation of 6
A'
film thickness to roughness ratio
rl
viscosity of lubricant
c(
pressure-viscosity coefficient
2 THEORY Actually, all machined surfaces are not smooth. Generally, they can be devided into two major groups: one is isotropic roughness surface and the other is striated rough surface. Up till now, surfaces of most machine elements are characterized striated roughness. Therefore, the research of the effects of surface roughness orientation on EHL is necessary. As shown in Fig.1, one surface possesses striated roughness with long narrow ridges and furrows. E, is the direction along which the amplitude of roughness is varied smoothly and slowly. 5 is perpendicular to E, That is, the amplitude of the roughness is principally depended on 3 The deviation (from mean line) of the roughness can be indicated as statistical parameter of random
.
.
262
Therefore, the stochastic charicteristic of gap function hT will surely lead to random variation of pressure P. Generally, the two stochastic processes of hT and p are related. Although all the statistical characters of hT are known, it is impossible to evaluate the expectations E(h~’ap/ax)and E(h~’ap/ax) unless introducing some assumptions.
”I
According to G.G.Hirs’1121 assumptions, for incompressible fluid and rough surfaces possessing narrow-grooves, the following statements are reasonable:
Fig.1
Striated rough surface
.
Whereas the character roughness along 5 ( 6 is of striation can be indicated as 6 and the angle of striated orientation 5 motion direction x).
(i) in the direction parallel to the roughness orientation, pressure gradient ap/ag is a stochastic variable with zero variance; (ii) in the direction perpendicular to striation, the component of unit flow 5 is a stochastic variable with zero variance. so, ap/aS and h ~ ’are independent stochastic variables; 5 and l / h ~are ~ also independent stochastic variables. Then, evaluating the expectation of terms in two sides of formula (2-2) and considering the above assumptions we get: E ( h ~ ~ e -ap/as) ~p =e-aP aP/ac
E(~T’)
X
+
u2
Fig.2.
Geometry of surfaces
As shown in Fig.2, the velocities of two contacting surfaces are u1 and u2 recpectively. The surfaces are rough. The actual oil film thickness between two surfaces is expressed as
hT = h + 61 + 6 2
( 2-11
where, h denotes the nominal smooth part of the film geometry, 6 1 ( i=1,2) denotes the part due to surface roughness as measured from the nominal level and regard as ”Reynolds roughness”. For incompressible fluid, the differential Reynolds equation is as follows,
where, u= 1/2 ( u l + u2). There is a direct relation between forming of oil film pressure and shape of the gap.
( 2-4
This is the general form of stochastic Reynolds equation considering surfaces possessing Reynolds roughness of arbitrary orientation.
263
In order to obtain the expectations in (2-4), it is necessary to give distribution function of stochastic variable 6 . Here, Christensen's approximate polynome has been chosen as the distribution function of 6 , that is -c<6
(c2-62)3
32c
( 2-5
elsewhere where, c=3o, o i s mean square deviation of 6. By integral, the expectations required in equation (2-4) can be obtained as follows, E (hT) = h
( 2-6
E ( h ~= ~ h3 ) + 1/3 hc2
( 2-7
E (hT-2) = 35/32C7[6h(c2-h2)21n(h+c/h-c) -4/5c(15h4-25c2h2+8c4) 1 (2-8) E ( h ~ ) -=~ 35/32c7[ 3( 5h2-c2) ( C2-h2) ln(h+c/h-c) +2ch(15h2-13c2) ( 2-9
Although there are some additional terms and factors due to random surface roughness and its orientation in equation (2-4), the terms occured are local mean pressure 5 and expectations of function of actual film thickness only. Therefore, on the meaning of statistical average the hydrodynamic lubrication according to this equation is the same as the smooth form. A l l the analytical and numerical methods used to evaluate the full film Reynold's equation are still applicable. Now consider two special cases: ( i) For longitudinal surface roughness, equation. (2-4) is reduced to:
a / ax[ e-aP ap/ ~ X (EhT ) I +a / ay[ e-@a p/ ay /E(~T-') 1 = 12rl0Ua/ax[E(h~)]+12n~ a/at[E(hT) 1
rl
=
no exp ( aP)
ap/ax/~(h~-3) ~+a/ay[e-~p W ~ Y
'E(h~3)l = 12rloua/aX[E(h~-~)/E(h~-~)]
3 NUMERICAL SOLUTION Considering steady flow, one cylinder with smooth surface moves with velosity u1 and the other with rough surface is static, then the item of a/at in equation (2-4) can be omited. Considering one dimensional flow, neglecting side leakage, then all the item about a/ay can be omited. So, a stochastic Reynolds' equation considering visco-pressure relation and suitable for one dimensional steady flow can be given as follows,
By integrating and introducing non-dimensional parameters
,
V=Qou/E'R G=
aE'
non-dimensional velocity parameter ; non-dimensional mterial parameter ;
,
non-dimensional film thickness;
,
H=h(R/b2)
P=p(hR/E'b), X=x/b ,
In these two cases, the stochastic Reynolods' equations are results[l].
the
same
as
Christensen's
For elastohydrodynamic lubrication,following equations are necessary. Function of the gap, h = h,
+ x2/2R + v
( 2-10 )
where, h, is central film thickness;
R
is equivalent radius ofcylinderpair
v
is elastic deformation of surfaces,
non-dimensional load parameter;
,
a/at [E(hT) 1
+%o
( 2-12
where, no is viscosity of lubricant under normal atmosphere; a is pressure-viscosity coefficient of lubricant. The stochastic Rdynolds' equation (241, gap function (2-10) and viscosity-pressure relation (2-12) are the basic equation of elastohydrodynamic lubr'ication considering surface roughness and its orientation.
W= w/E'R
( ii) For transverse surface roughness, equation.(2-4) is reduced to:
a/ax[e+
is The dynamic viscosity of lubricant 0 taken to vary with pressure by the relation
non-dimensional pressure; non-dimensional coordinate.
Then Considering the Hertzian contact, b = R m ,
half-width
of
We can obtain
a ~ / a X [ E ( H ~ ~ ) c o s ~ 8 + s i n ~ B / E1 ( H ~ - ~ ) =AeG*p{ [E(H T - ~ )/E( HT-~)-E(HT~-O )/E(HTc-~ s in28+ [ E ( HT -E (
1cos
B]
( 3-2
1
264
A
where,
= 48V(a/8W ) *
G* = G
m
non-dimensional film thickness can be expressed as follows,
( 3-3 )
where, Hw is the sum of central film thickness H, and the constant of elastic deformation. The full numerical solution of striated roughness elastohydro dynamic lubrication is more difficult, more complex and more tedious than that of smooth problem. NewtonRaphson method has been used and some of the numerical results has been shown in table 1 and pressure distribution has been shown in Fig.3. From the numerical results we may know that, (1) for one kind of roughness orientation, non-dimensional parameters W,V,G have the same effects on film thickness and pressure distribution as the smooth results. (2) on the same operation conditions, film thickness is affected by surface roughness orientation. As roughness orientation transforms from longitudinal to transverse (that is, as increases) the central film thickness hc and the minimum film thickness hmin are increased.
(3) generally, the value of pressure spike increases as decreases. The location of pressure spike moves slightly towards exit point when % increases.
(4) the effects of roughness orientation on film thickness are affected by the film thickness to roughness ratio ;\'(A'= hmin / c 1. As A'decreases, the effect of B on film thickness is larger. When A' is large (such as A ' =2), the effect of B on hminis smaller than that on hc. Whereas when is small (such as A' =1-1.5), the A' effect of % on hmin is larger than that on hC. t
H,
)
B(O
H -
90
1.472
1.249
60
1.405
1.172
45
1.469
1.092
30
1.321
1.041
0
1.245
0.993
TABLE 1 A group of numerical results W=l.lE-5 G=3500
,
, V=2.3E-ll Az1.5
(1) %=90° (2)
%=45O
(3)
B=OO
W = l . 1E-5
V=2.3E-ll G=3500
A=1.5
Fig.3.
Pressure distribution
4 EXPERIMENTS Experiments on the effects of surface roughness orientation upon lubricant film load capacity have been done by Timken test apparatus. A ring and a block are the test pieces. The hardness of test pieces is HRC58-62. The diameter of ring is 49.24mm and its width is 12.7mm. The sizes of block are 12.35 x 12.35 x 19mm. The ring is circumferentially ground and blocks have been ground with different striation direction. Surface roughness has been meassured by Talysulf 5 surface meassurement system. Viscosity of the lubricant is qe =0.217Pa.s. Criterion of the test is that scoring may be occure under a certain load. Comparative tests have been made with different roughness orientation in the velocity of both llOOrpm and 1500rpm. The results are shown in Fig.4 and Fig.5. The scoring load capacity of transverse surfaces is about 1.3 times as that of longitudinal surfaces. The scoring load capacity increases gradualy when increases. At the speed of 1500rpm the scoring load capacity of the surfaces on which is equal to 45O is the lowest of all. It seems that this is contrary to the theory, but if we notice that the surface roughness of which is larger than others' the experiment result is reasonable. Under these experimental conditions the numerical results have also been shown in Fig.4 and Fig.5. The trend of the scoring load capacity varying with the roughness orientation is concord with the numerical results of theory.
265
characteristic of EHL, such as film thickness and pressure spike, will be different due to roughness orientation. So, research on this field is useful to practical elements machining. Of course, it is necessary to do further work bo$h on theory and experiment.
x experimental theoritical
References
0 30' 60° 90' Fig.4. Comparason of experiments and theory results (1500rpm)
w
1
8
(~10-5)
0 Fig.5.
x experimental 0 theoritical
30' 60" goo B Comparason of experiments and theory results (1100rpm)
5 DISCUSSION The generallized stochastic Reynold's equation derived here may be used for contact surfaces possessing arbitrary roughness orientation. F'ull numerical solution has been given to discuss the effects of roughness orientation on EHL. Numerical results show that both central film thickness hc and minimum film thickness hmin are affected by roughness orientation and the effects of roughness orientation on film thickness is determined by film thickness to roughness ratio A ' . Cer tainly, when A ' is large enough (like A ' > 3) these effects will be reduced to zero. Roughness orientation also has some effects on pressure spike, mainly on its magnitude but slightly on its possition. Experimental results by Timken test apparatus have also shown the effects of surface roughness orientation. Lubricant film load capacity varies with different striation direction. The trends of the experimental results are concord with numerical solution. We can conclude not only from theory but also from experiment that even in the reason to range of 3>A'>1, there is no disregard the effects of surface roughness orientation on EHL. Although it is the full film lubrication region when >1 and generally regards that asperity contact will not occure,
Christensen,H. "Stochastic models for hydrodynamic Lubrication of rough surfaces", proc.IME, v.184 (1969-70), pt.1, No. 55, p.1013-1026. Christensen,H. and To nder,K."The hydrodynamic Lubrication of rough bearing surfaces of finit width", J.Lubcn.Techn., Trans.ASME, v.95(1973), n.2, p.166-172. Christensen,H. "A Theory of mixed Lubrication", proc.IME, v.186(1972), 41/72, p.421-
430. Elrod, H. G."Thin film Lubrication theory for Newtonian fluids with surface possessing striated roughness or grooving", J.Lu-b bcn. Techn. ,Trans. ASME,v.95(1973) , p.484-
489.
.
Chow,L.S.H.and Cheng,H.S "The effect of surface roughness on the average film thickness between lubricated rollers",J. Lubcn.Techn., Trans.ASME,v.98(1976), p.117124. Cheng,H.S. and Dyson,A. "EHL of circumferentially ground discs", ASLE paper, no.76-LC-U-2. Dyson,A. "The failure of EHL of circumferentially ground discs", proc.IME, v.196(1976), 52/76, p.699-711. Dyson,A. "EHL of rough surfaces with lay in the direction of motion'', proc 4th Leeds-Lyon Symp on Tribology, 1977, p.201-209. Rossides,S.D. and Snidle, R.W."surface topography and the scuffing failure of circumferentially ground discs" ASME Winter annual Meeting, Nov.16-21, 1980, p.93-
144.
[ 103 StaDh.H.E. .Ku,P.M.and Carper,H.J. "Effect of surface- roughness and surface texture A
.
on scuffing",Mech.and Mach. Theory, ~-8(1973 P.197-208* Ku,P.M. and Li,K.Y. "Effect of surface topography on sliding rolling disc scuffing", proc .bth.Leeds-Lyon Symp.on Tribology, 1977, p.210-217. Hirs,G.G."Externally Pressurized Bearing with Inherent Friction Compensation", J.Applied Mechanics, Trans.ASME, paper N0.65-APM-B.
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267
Paper Vlll(iv)
The elastohydrodynamic behaviour of simple liquids at low temperatures C. Waterhouse, G.J. Johnston, P.D. Ewing and H.A. Spikes
The r e l a t i o n s h i p between m o l e c u l a r s t r u c t u r e and e l a s t o h y d r o d y n a m i c r h e o l o g i c a l p r o p e r t i e s of One r e a s o n f o r t h i s i s t h a t normal l i q u i d l u b r i c a n t s , l u b r i c a n t s is o n l y p o o r l y u n d e r s t o o d . such a s m i n e r a l o i l s and e s t e r s , g e n e r a l l y have complex m u l t i - a t m m o l e c u l a r s t r u c t u r e s and a r e a l s o o f t e n m i x t u r e s . I t i s t h u s d i f f i c u l t t o correlate EHD performance w i t h s t r u c t u r a l d e t a i l s . This paper d e s c r i b e s a f e a s i b i l i t y s t u d y i n t o t h e t e c h n i q u e o f u s i n g v e r y low temperatures t o convert s i m p l e , low v i s c o s i t y l i q u i d s i n t o l u b r i c a n t s and t h u s s y s t e m a t i c a l l y i n v e s t i g a t e t h e i n f l u e n c e o f s t r u c t u r e o n e l a s t o h y d r o d y n a m i c (EHD) p r o p e r t i e s . An o p t i c a l E H D b a l l on p l a t e r i g has been m o d i f i e d t o s t u d y t h e l u b r i c a t i n g p r o p e r t i e s of s i m p l e A t these t e m p e r a t u r e s many s i m p l e f l u i d s are found t o a t t a i n t h e u s e f u l f l u i d s down t o -120‘C. Two classes of f l u i d have been s t u d i e d , t h e viscosity range for l u b r i c a t i o n without f r e e z i n g . p h t h a l a t e esters and , t h e a l k y l b e n z e n e s . Both gave m e a s u r a b l e E H D f i l m t h i c k n e s s e s though t h e alkylbenzenes were e f f e c t i v e o v e r a v e r y l i m i t e d a n d low t e m p e r a t u r e r a n g e .
1
INTRODUCTION
Elastohydrodynamic (EHD) l u b r i c a t i o n o c c u r s between l o a d e d , r o l l i n g o r s l i d i n g , l u b r i c a t e d c o n t a c t s where t h e geometry and hardness of t h e two s u r f a c e s produce v e r y h i g h p r e s s u r e s i n t h e contact. I n s u c h s y s t e m s , two effects of t h i s p r e s s u r e , t o e l a s t i c a l l y f l a t t e n t h e surfaces and t o g r e a t l y enhance t h e v i s c o s i t y o f t h e l u b r i c a n t , combine t o g e n e r a t e hydrodynamic o i l f i l m s of t y p i c a l l y 50-5000 nm t h i c k n e s s . Two c h a r a c t e r i s t i c s of t h e s e f i l m s have emerged a s being of p a r t i c u l a r importance i n performance terms, f i l m t h i c k n e s s and t r a c t i o n . Large f i l m t h i c k n e s s e s have been shown t o r e d u c e High wear (11, s c u f f i n g (2) a n d p i t t i n g ( 3 ) . t r a c t i o n is important i n t r a c t i o n d r i v e systems, but i n many l u b r i c a t e d c o n t a c t s low t r a c t i o n and t h u s low power l o s s and h e a t g e n e r a t i o n are sought a f t e r
.
Both f i l m t h i c k n e s s and t r a c t i o n have been shown t o depend s t r o n g l y o n t h e n a t u r e and c h a r a c t e r i s t i c s of the l u b r i c a n t u s e d . For f i l m t h i c k n e s s i t h a s been d e m o n s t r a t e d t h a t two p r o p e r t i e s of the l u b r i c a n t are i m p o r t a n t , v i s c o s i t y a t ambient p r e s s u r e , ”0, a n d p r e s s u r e Equation 1 i n d i c a t e s v i s c o s i t y c o e f f i c i e n t a. the approximate dependence of EHD f i l i n t h i c k n e s s on t h e s e p r o p e r t i e s ( 4 ) .
I t s h o u l d be n o t e d t h a t i s the v i s c o s i t y of t h e l u b r i c a n t a t t h e v e r y 6 h i g h shear r a t e o f the EHD i n l e t , t y p i c a l l y 10 -10’ s-’. a i s a composite p r e s s u r e - v i s c o s i t y . c o e f f i c i e n t over t h e p r e s s u r e i n t h e EHD i n l e t zone.
T r a c t i o n depends h e a v i l y o n t h e l u b r i c a n t u s e d , which has been shown t o v a r i o u s l y d i s p l a y e l a s t i c , nowNewtonian or Newtonian shear s t r e s s / s t r a i n r a t e beha-viour depending upon t h e d e g r e e of s l i d i n g . ( 5 ) . T r a c t i o n c o e f f i c i e n t is found t o r i s e r a p i d l y w i t h i n c r e a s i n g s l i d e - r o l l r a t i o and t h e n t o f a l l a t h i g h e r s l i d i n g d u e t o combined non-Newtonian and thermal effects. An i m p o r t a n t p r a c t i c a l c h a r a c t e r i s t i c f o r each l u b r i c a n t a t a g i v e n l o a d a n d temperature is t h e maximum v a l u e of t h e t r a c t i o n c o e f f i c i e n t This i s t y p i c a l l y between 0.02 and reached. 0.08.
C l e a r l y i t would be b e n e f i c i a l i n EHD a p p l i c a t i o n s i f t h e l u b r i c a n t c o u l d be d e s i g n e d t o g i v e t h i c k EHD films and p a r t i c u l a r l y low o r h i g h t r a c t i o n coef f i c i e n t s , d e p e n d i n g upon t h e a p p l i c a t i o n , a n d t h e r e is c u r r e n t l y a good d e a l of e f f o r t b e i n g expended i n s e a r c h i n g for h i g h A major t r a c t i o n c o e f f i c i e n t f l u i d s (6). problem is t h a t t h e r e are, as y e t , no clear g u i d e l i n e s r e l a t i n g molecular s t r u c t u r e t o properties such as pressure viscosity c o e f f i c i e n t and t r a c t i o n f o r t h e l u b r i c a n t Some o u t l i n e r u l e s have designer t o use. emerged from e m p i r i c a l s t u d i e s , as w i l l be d i s c u s s e d i n t h e n e x t s e c t i o n , b u t these have n o t y e t been f o r m a l i z e d . A p r a c t i c a l d i f f i c u l t y i s t h a t EHD f i l m t h i c k n e s s and t r a c t i o n s t u d i e s can n o r m a l l y o n l y be c a r r i e d o u t on m o d e r a t e l y v i s c o u s l u b r i c a n t s - i n t h e lOcP t o 10 Pas r a n g e . Less v i s c o u s f l u i d s c a h n o t g e n e r a t e f u l l EHD films under normal speed c o n d i t i o n s , even w i t h v e r y smooth surfaces. U n f o r t u n a t e l y v i s c o u s l u b r i c a n t s t e n d t o have q u i t e complex m o l e c u l a r s t r u c t u r e s , a n d / o r t o be m i x t u r e s . Simpler s t r u c t u r e s would
268
have lower v i s c o s i t i e s and p u r e s y s t e m s would t e n d t o c r y s t a l i z e w i t h time. T h i s means t h a t r u l e s r e l a t i n g molecular s t r u c t u r e t o EHD properties have to be derived using i n t r i ns i c a l l y corn p l ex s y s t e m s . T h i s s t u d y describes t h e f i r s t s t e p s i n an a l t e r n a t i v e a p p r o a c h , which i s t o r e n d e r s i m p l e m o l e c u l a r l i q u i d s v i s c o u s , by c o o l i n g them t o c r y o g e n i c t e m p e r a t u r e s , and t h e n t o measure t h e i r EHD p r o p e r t i e s d i r e c t l y . I t is hoped eventually thereby t o r e l a t e p r o p e r t i e s such as t r a c t i o n t o f e a t u r e s of t h e m o l e c u l a r s t r u c t u r e o f t h e f l u i d s used. F o r t u n a t e l y i t has been shown t h a t even q u i t e s i m p l e l i q u i d s , s u c h a s t o l u e n e and b u t y l i o d i d e , become v i s c o u s l i q u i d s and eventually glasses, rather than c r y s t a l l i z i n g , when c o o l e d r a p i d l y . Apart from t h i s s c i e n t i f i c r e a s o n f o r l o o k i n g a t t h e EHD of s i m p l e l i q u i d s a t low temperatures t h e r e is a l s o a p o t e n t i a l p r a c t i c a l application i n space. Dry l u b r i c a t i o n is c u r r e n t l y employed i n most low t e m p e r a t u r e s p a c e bearings. Such b e a r i n g s have l i m i t e d l o a d capacity, however, and have f i n i t e lives determined by l o s s of material through wear. R o l l i n g element b e a r i n g s c o u l d be used i f lubricants able to operate at v e r y low t e m p e r a t u r e s were a v a i l a b l e . T h i s paper d e s c r i b e s i n i t i a l s t u d i e s on t h e EHD p r o p e r t i e s of s i m p l e f l u i d s a t v e r y low t e m p e r a t u r e . Two c l a s s e s of f l u i d a r e examined thicknesses and viscosities and EHD f i l m measured down t o - 1 O O O C .
2
PREVIOUS WORK
2.1 L u b r i c a n t s a t Low Temperatures Most interest in low temperature lubrication has centred on maintaining p u m p a b i l i t y and f l o w of e n g i n e l u b r i c a n t s i n c o l d e n v i r o n m e n t s , by r e d u c i n g pour p o i n t . Mineral o i l s g e n e r a l l y have pour p o i n t s i n t h e -30 t o - 1 O O C r a n g e , b u t t h e development of s y n t h e t i c s s u c h a s s i l i c o n e s and esters has reduced t h i s t o -6OOC i n some i n s t a n c e s . There h a s been l i t t l e s y s t e m a t i c s t u d y o f liquid lubricants for use below this temperature, unsuprisingly i n view of the limited number of potential applications. Ultra-low t e m p e r a t u r e o p e r a t i o n s s u c h a s t h o s e i n s p a c e and l i q u i d f u e l h a n d l i n g have t e n d e d t o employ c o n v e n t i o n a l d r y l u b r i c a n t s . Zaretsky e t a1 ( 7 ) d e s c r i b e t h e t e s t i n g o f o l i g o m e r i c p e r f l u o r o e t h e r s i n a r o l l i n g 4 - b a l l machine From t h e n a t u r e o f between -164°C and - 50°C. the rubbed t r a c k , t h e a u t h o r s i n f e r r e d t h a t t h e l u b r i c a n t s were p r o v i d i n g EHD p r o t e c t i o n under these conditions. A r e c e n t s t u d y by Merriman and Kannel ( 8 ) i n v e s t i g a t e d t h e l u b r i c a t i n g c h a r a c t e r i s t i c s of l i q u i d n i t r o g e n i n a d i s c machine. The a u t h o r s found no e v i d e n c e of E H D f i l m f o r m a t i o n and suggested t h a t the v i s c o s i t y of l i q u i d nitrogen I t s h o u l d be does n o t i n c r e a s e w i t h p r e s s u r e . n o t e d , however, t h a t t h e t e m p e r a t u r e employed i n t h i s s t u d y was o n l y j u s t below t h e b o i l i n g p o i n t of l i q u i d n i t r o g e n .
(9) have Clynes, Evans and Johnson subjected conventional l u b r i c a n t s t o ultralow t e m p e r a t u r e s i n o r d e r t o mimic t h e e f f e c t s of h i g h p r e s s u r e and t h u s s t u d y non-Newtonian behaviour under s u c h c o n d i t i o n s
.
In c o n t r a s t to t h e paucity of l u b r i c a t i o n s t u d i e s t h e r e h a s been a good d e a l of work o n t h e v i s c o s i t y of f l u i d s a t low temperatures. Barlow, Lamb and Matheson (10) measured t h e v i s c o s i t i e s and d e n s i t i e s of a r a n g e o f s i m p l e l i u i d s o v e r a w i d e t e m p e r a t u r e r a n g e down t o -12OOC. Two classes of f l u i d were examined, t h e a l k y l b e n z e n e s and t h e p h t h a l a t e e s t e r s . Denney c a r r i e d o u t s i m i l a r measurements on a series of alkyl halides (11). Both t h e s e s t u d i e s showed t h a t these simple f l u i d s , i f c h i l l e d reasonably r a p i d l y , s u p e r c o o l e d below t h e i r normal f r e e z i n g p o i n t s w i t h o u t c r y s t a l l i z i n g , t o become more and more v i s c o u s l i q u i d s . Barlow, Lamb and Matheson r e a c h e d v i s c o s i t i e s a s h i g h a s 500 Pas w i t h t h e a l k y l benzenes. 2.2 R e l a t i o n s h i p between S t r u c t u r e and EHD F l u i d Properties The p r e s s u r e v i s c o s i t y c o e f f i e n t s Of most l u b r i c a n t s l i e between 5 O C and 40GPa-’ a t 20°C. There i s a broad c o r r e l a t i o n between h i g h and high pr essur e- v i s c o s i t y temperature-viscosity coefficient. Flexible m o l e c u l a r s t r u c t u r e s , s u c h a s s i l i c o n e s and p o l y g l y c o l s , t e n d t o have low a - v a l u e s , a s do v e r y s i m p l e and s y m m e t r i c a l molecules such a s benzene. S a t u r a t e d r i n g s and b u l k y , i n f l e x i b l e Kuss m o l e c u l e s t e n d t o have h i g h e r a-val’ies. ( 1 2 ) h a s shown t h a t i t i s i m p o s s i b l e t o o b t a i n p r e s s u r e v i s c o s i t y c o e f f i c i e n t s of up t o 8 0 ~ P a - l by u s i n g m o l e c u l e s w i t h t h r e e , 1i n k e d , s a t u r a t e d rings. The r e l a t i o n s h i p between. s t r u c t u r e and is n o t well maximum traction coefficient u n d e r s t o o d d e s p i t e a good d e a l of work ( 6 ) ( 1 3 ) (14). There i s a broad c o r r e l a t i o n between h i g h t r a c t i o n and h i g h a - v a l u e , b u t w i t h s e v e r a l exceptions. Again, r i g i d , i n f l e x i b l e and bulky m o l e c u l e s a p p e a r t o a f f o r d h i g h t r a c t i o n and g e n e r a l ideas t o e x p l a i n t h i s based o n m o l e c u l a r i n t e r l o c k i n g have been s u g g e s t e d ( 6 ) ( 1 4 ) .
3
TEST APPARATUS
Viscometric and EHD film thickness measurements were carried o u t i n a c a b i n e t c a p a b l e of m a i n t a i n i n g t e m p e r a t u r e s down t o A schematic -120OC f 1 °C for e x t e n d e d p e r i o d s . diagram of t h e c a b i n e t and c o o l i n g system i s shown i n f i g u r e 1 . The c a b i n e t c o n t a i n s a h e a t i n g e l e m e n t t o m a i n t a i n t e m p e r a t u r e s above ambient. Low t e m p e r a t u r e s are a c h i e v e d by f e e d i n g l i q u i d n i t r o g e n vapour a t a c o n t r o l l e d r a t e i n t o t h e chamber and c i r c u l a t i n g t h e vapour using a fan. V i s c o s i t i e s were measured u s i n g a r a n g e o f s t a n d a r d c a p i l l a r y v i s c o m e t e r s suspended i n t h e c o l d chamber, which had a d o u b l e g l a z e d g l a s s front to facilitate observation. Elastohydrodynamic f i l m t h i c k n e s s was measured using chromatic o p t i c a l interferometry using t h e The arrangement shown i n f i g u r e s 2 and 3. p r i n c i p l e s of t h e method a r e d e s c r i b e d e l s e w h e r e (15). A chromium-plated g l a s s d i s c i s r o t a t e d by means of a s h a f t p a s s i n g t h r o u g h t h e l i d of A 25.4mm diameter s t e e l ball t h e c o l d chamber.
269
FIGURE 1 : COLD CHAMBER FOR VISCOSITY MEASUREMENTS
Controller
-Heater I
!
'
I
I
I I
I
I
I
-Glaar
Wlndow
I I
I I
- Vlrcometer
I
I
I
I
T e m p e r a t u r e was m o n i t o r e d u s i n g two n i c k e l thermocouples chrome/ni c k e l aluminium (K- t y p e p o s i t i o n e d a t t h e rear of t h e c o n t a c t as shown i n f i g u r e 4 a n d o n e t h e r m o c o u p l e was employed t o c o n t r o l t h e chamber t e m p e r a t u r e by f e e d b a c k . To maintain temperature a c c u r a t e l y and evenly over t h e l u b r i c a t e d zone i t was f o u n d n e c e s s a r y t o p o s i t i o n a f i n n e d t r a y d i r e c t l y below t h e b a l l support system t o l o c a l i z e t h e n i t r o g e n vapour, a s shown i n f i g u r e 2 a n d t o add a p r o p e l l o r t o improve n i t r o g e n vapour c i r c u l a t i o n a r o u n d t h e test rig.
Temperature Controller
I ) II
fl
Flexible Coupling l . 4
u
' A I Y/I Solenoid
I(
I
FIGURE 2 : COLD CHAMBER
FOR FILM
THICKNESS MEASUREMENTS Chromium Coating'
bL1,2-l
(
Ball
)
~
(EN31, US52100) is loaded a g a i n s t t h e u n d e r s i d e of t h i s d i s c u s i n g a lever arm and s p r i n g balance a r r a n g e m e n t , s o t h a t t h e d i s c d r i v e s t h e The r e s u l t a n t ball i n nominally pure r o l l i n g . EHD c o n t a c t between d i s c and b a l l is o b s e r v e d through a d o u b l e - g l a z e d window i n t h e l i d of t h e cold chamber.
FIGURE 3 : OPTICAL SYSTEM
0 U Load
270
Lb b r e v i a t i o n .n t e x t
No
1
ouptes
2 3
i
I
4 5
L u b r i c a n t was s u p p l i e d t h r o u g h a h e l i c a l copper t u b e , f e d from a p i p e t t e o u t s i d e t h e chamber. The flow r a t e o f l u b r i c a n t was small so t h a t t h e l u b r i c a n t c o o l e d t o t h e t e s t temperature d u r i n g its p a s s a g e t h r o u g h t h e h e l i c a l t u b e . This s u p p l y method r e s t r i c t e d t h e maximum v i s c o s i t y of t h e l u b r i c a n t t h a t c o u l d be s t u d i e d t o a p p r o x i m a t e l y 1500O cP. R e f r a c t i v e i n d i c e s of a l l f l u i d s u s e d were measured u s i n g a n Abbe r e f r a c t o m e t e r . This g i v e s r e f r a c t i v e i n d e x measurements t o w i t h i n 2 x lo-*.
benzene
Baker
Ethyl benzene
Koch
I s o p r opy benzene Butyl benzene
I
Decyl benzene
Light
EtB
0.77
BDH
iPrB
0.90
Light
Bu B
1.16
Koch Light
DecB
4.49
6
Di-methyl p h t h a l a t e BDH
DMeP
7
Di-nBDH butyl phthalate
DBu P
22.4
8
D i - (2KO& ethyl Light hexyl) p h t h a l ate
DEtHeP
77.0
14.7
FLUIDS TESTED
The f l u i d s tested a r e l i s t e d i n t a b l e 1. These l i q u i d s were chosen because their v i s c o s i t i e s and d e n s i t i e s have a l r e a d y been Barlow, Lamb a n d e x t e n s i v e l y measured by Matheson ( 1 0 ) a n d i t i s known t h a t t h e y form glassy solids, without crystallizing, on cooling. I n i n i t i a l tests, i c e c r y s t a l s formed from s m a l l amounts of water on cooling, dissolved i n these f l u i d s . It was therefore found n e c e s s a r y t o d r y a l l t h e f l u i d s p r i o r t o u s e w i t h a m o l e c u l a r s i e v e (BDH Type 4 A ) .
5
I 0.66 Me B May a n d Methyl
-
*
4
I iscosi t y I t 2ooccp
Table 1.
L i s t of F l u i d s T e s t e d
c r u d e a p p r o x i m a t i o n , b u t a c c e p t a b l e i n view of the small change of r e f r a c t i v e i n d e x w i t h temperat-e n o t e d , o f 10.5% p e r 2OOC.
Name
NO
-
RI a t
-3. ooc
R I at 20.5oc
1
Methyl benzene
1.5095
1 .4962
2
E t h y l benzene
1 .5075
1.4952
3
I s o p r o p y l benzene
1.5025
1 .4908
4
Butyl benzene
1.5010
1 .4895
5
Decyl benzene
1.4921
1.4820
RI a t -2.5oc
R I at 21 . o o c
RESULTS
5.1 V i s c o s i t i e s F i g u r e s 5 a n d 6 show t h e v i s c o s i t i e s of t h e two claqses of f l u i d s t u d i e d as a f u n c t i o n o f temperature. The l i n e s on t h e f i g u r e s are Barlow, Lamb a n d Matheson's v i s c o s i t y r e s u l t s and t h e p o i n t s were t a k e n i n t h i s s t u d y . I t can be s e e n that t h e r e i s good agreement between t h e two. The l i n e for d e c y l benzene i s t a k e n from The f i g u r e s have o r d i n a t e r e f e r e n c e 16. l o g l o g ( v + 0.71, where v is t h e v i s c o s i t y i n c e n t i s t o k e s , and a b c i s s a log(T) (17).
6
Dimet h y l p h t ha 1a t e
1.5240
1.51 49
7
Di-n-but y l ph t ha1a t e
1.5012
1.4920
1.4952
1.4858
5.2 R e f r a c t i v e i n d i c e s . R e f r a c t i v e i n d i c e s v a l u e s were t a k e n a t o n l y two temperatures, 20.5OC and -3°C and a r e shown i n t a b l e 2. R e f r a c t i v e i n d e x i s needed t o c o n v e r t o p t i c a l t o real f i l m t h i c k n e s s i n interferometry. Values a t lower temperatures were derived by extrapolation, assuming l i n e a r i t y w i t h temperature, C l e a r l y t h i s i s a
8
-
DC(2-ethyll h e x y l p h t ha1 ate 1
T a b l e 2.
R e f r a c t i v e I n d i c e s of F l u i d s T e s t e d
271
0.6 000
0.4 00
0.2
2
0 0.0
4-
3
-
In
,-0.2
r
0
r
A 0 0
A -0.4
'
-0.6
'
-0.8
'
.->m I
-5OOC
ooc
50°C
Log T (K) FIGURE 5 : VISCOSITIES OF DIALKYL PHTHALATES
10000
1000
100-
10
--
In
r
I
m
0
m
5
1
-0.8-IOO0C
ooc
-50%
FIGURE 6 : VISCOSITIES OF ALKYL BENZENES
5.3 EHD F i l m T h i c k n es s
The o i l shown i n f i g u r e 7 i s a .~ r e f e r e n c e mineral oil w ith a pressure-viscosity c o e f f i c i e n t of 21.8 GPa-', v i s c o s i t y of 550cP T his and r e f r a c t i v e i n d e x of 1.489 a t 2OOC. datum was used to de te rm ine the p r e s s u r e - v i s c o s i t y c o e f f i c i e n t s of t h e e s t e r s i n f i g u r e 7 on t h e b a s i s t h a t , w i t h i d e n t i c a l ~
The p h t h a l a t e ester f l u i d s gave normal EHD behaviour when c o o l ed . F i g u r e 7 shows p l o t s of Log,,U where h i s t h e o p t i c a l f i l m t h i c k n e s s and U the mean r o l l i n g s p eed . I t can be s e e n t h a t th e g r a d i e n t is a p p r o x i mat el y 0.7 i n most cases, in accordance w i t h EHD t h e o r y .
272
-8.
'/ 0
-leoc
-6.
-6.
/
+looc
-8.
-6,
-1.4
-1.6
-1.2
Log U
FIGURE 7 : EM0 FILM THICKNESS
Table 3. shows the r e s u l t a n t a - v a l u e s .
Dimethyl phthalate
Temp
-9
OC
)i (2-ethylhexyl ) I h t h a l ate
(m/d
Derived a-value GPa-
A good d e a l of d i f f i c u l t y was experienced i n o b t a i n i n g measurable EHD f i l m t h i c k n e s s w i t h As w i l l be d i s c u s s e d i n t h e t h e alkylbenzenes. n e x t s e c t i o n , t h e temperature r a n g e over which three f l u i d s g i v e s u i t a b l e f i l m t h i c k n e s s a p p e a r s t o be v e r y narrow, and i s below t h e i r There 'was t h u s a normal f r e e z i n g p o i n t s . tendency for f r e e z i n g t o take p l a c e i f t h e f l u i d s were h e l d s t a t i c a t t h e t e m p e r a t u r e r a n g e of interest. EHD f i l m t h i c k n e s s e s were eventually obtained by modifying the e x p e r i m e n t a l t e c h n i q u e and m o n i t o r i n g EHD f i l m t h i c k n e s s w h i l s t s l o w l y c o o l i n g t h e whole system down. F i g u r e 8 shows EHD f i l m t h i c k n e s s e s o b t a i n e d f o r i s o p r o p y l b e n z e n e and t a b l e 4 t h e d e r i v e d a - v a l u e s . The v a l u e s i n t a b l e s 3 and 4 are k.209.
14 ~~
Dibutyl phthalate
-
FOR DIALKYL PHTHALATES
r o l l i n g speed and l o a d , f o r two f l u i d s 1 and 2 a c c o r d i n g t o EHD t h e o r y
Test f l u i d
-0.4
-0.8
-0.8
-1.0
~~
Test f l u i d -15
20
-1 8
21
-21
26
-25
30
-35
33.5
+19
20
T a b l e 3 Derived P r e s s u r e V i s c o s i t y C o e f f i c i e n t s o f P h t h a l a t e Esters
iso-Propyl Benzene
Temp
OC
Derived av a l u e GPa-'
-94
3
-99.5
46
Table 4 Derived Pressure Viscosity C o e f f i c i e n t s of i s o p r o p y l Benzene.
273
-6
iPrB -9SOC
Log (h)
(m)
Ref. Oil -6.
/ /
/
/ /
-6.
-94Oc
/
/
/ / -6.8
/ / /
/
d
.
/ -6.1
-1.6
- 1.4
-1.0
-1.2
-0.6
-0.8
Log
u (m/d
-0.4
FIGURE 8 : EHD FiLM THICKNESS FOR ALKYL BENZENES
6.
DISCUSSION
6.1 V i s c o s i t i e s Good agreement was found between the r e s u l t s of t h i s s t u d y and o t h e r l i t e r a t u r e values of v i s c o s i t y . A l l t h e f l u i d s t e s t e d were found t o g i v e g l a s s y s o l i d s on c o o l i n g . There i s c o n s i d e r a b l e l i t e r a t u r e on t h e v i s c o s i t y of some o f t h e f l u i d s examined. Thus b u t y l benzene and d i b u t y l p h t h a l a t e have been i n v e s t i g a t e d a s o r g a n i c g l a s s e s f o r matrix t r a p p i n g work by Ling and W i l l a r d ( 1 8 ) who q u o t e v i s c o s i t i e s up t o 10' Pas, a s shown f o r d i b u t y l p h t h a l a t e i n f i g u r e 9. The k i n k s i n t h e g r a d i e n t s of v i s c o s i t y versus t e m p e r a t u re f o r i s o p r o p y l benzene and methyl p h t h a l a t e were n o t ed by Barlow, Lamb and Matheson, who a c c e n t u a t e d them by p l o t t i n g (l/T)3 or logarithms of viscosity against (l/T)*. I t is i n t e r e s t i n g t o note t h a t these g r a d i e n t changes a r e s t i l l v i s i b l e on ASTM p l o t s , a t t h e same t e m p e r a t u r e s as n o t ed by Barlow and c o l l e a g u e s . Barlow a s c r i b e d t h e s e k i n k s t o a m o l e c u l ar c l u s t e r i n g which o c c u r r e d near t h e m e l t i n g p o i n t s of t h e f l u i d s . T h i s was subsequently d i s p r o v e d by Beuch and Davison I t has been s u g g e s t e d by Davies and (19). Matheson ( 2 0 ) t h a t t h e change i n g r a d i e n t r e p r e s e n t s t h e complete l o s s of mo l ec ula r r o t a t i o n above t h e g l a s s t r a n s i t i o n t e m p e r a t u r e The a u t h o r s showed a s te m p e r a t u r e is lowered. t h a t t h e s h a p e s of t h e mo l ecu l es of t h e s e f l u i d s were such t h a t t h e r e was no l o n g e r room f o r them t o r o t a t e as i n t e r m o l e c u l a r s p a c i n g d e c r e a s e d temper a t ur e v i s c o s i t y- k i n ktl be1 ow the ir Davies and Matheson s u g g e s t t h a t f l u i d s can have t h r e e v i s c o s i t y r eg i mes . A t high temperatures t h e r e i s f u l l r o t a t i o n around a l l a x i s and t h u s Arrhenius v i s c o s i t y - t e m p e r a t u r e behaviour As the t e m p e r a t u r e f a l l s and t h e mo l ecu l es come closer t o g e t h e r , non- symmet r i cal molecules
.
.
s u f f e r loss of freedom of r o t a t i o n around two Some a xe s and e n t e r a non-Arrhenius r e g i o n . l i q u i d s re a c h t h e g l a s s t r a n s i t i o n t e m p e r a t u r e i n t h i s c o n d i t i o n but o t h e r s , suc h a s i s o p r o p y l benzene, l o s e t h e f i n a l a xe s of r o t a t i o n above t h i s temperature.
1
0
0
WIIIard & Line
Currant
w h
-t
I
.
too
zoo
aoo
400
600
600
TOO
EOO
t i o 3 / ~ ) 4 (10
FIQVRL 9 : VISCOSITY OF OI6 VT IL M T W A L A T L
900
214
6.2 P r e s s u r e V i s c o s i t y C o e f f i c i e n t s The p r e s s u r e v i s c o s i t y c o e f f i c i e n t s f o r t h e p h t h a l a t e esters were i n t h e 15-35GPa-1 r a n g e , a n d , a s found w i t h most f l u i d s , i n c r e a s e d w i t h decr eas i n g temper a t ur e . There are few p r e s s u r e - v i s c o s i t y v a l u e s f o r p h t h a l a t e esters in the literature, but d i (2-ethylhexyl) p h t h a l a t e , which h a s a c o n v e n t i o n a l l u b r i c a n t v i s c o s i t y - t e m p e r a t u r e s p r e a d , was s t u d i e d i n t h e ASME P r e s s u r e - V i s c o s i t y program (21 1. Typical v a l u e s from t h a t w o r k , u s i n g h i g h p r e s s u r e v i s c o m e t r y , were 24GPa-' a t O°C and 20GPa-' a t 25OC, i n good agreement w i t h t h e c u r r e n t o p t i c a l EHD r e s u l t s . Dimethyl p h t h a l a t e a t -9OC had a considerably lower a-value than dibutyl p h t h a l a t e a t -15OC, a l t h o u g h t h e l a t t e r was, under t h e s e c o n d i t i o n s , t h e l e s s v i s c o u s f l u i d . (Figure 5 implies that t h e dimethyl p h t h a l a t e i s l e s s v i s c o u s a t -9OC t h a n t h e d i b u t y l p h t t a l a t e a t -15OC but t h i s graph u s e s k i n e m a t i c v i s c o s i t y u n i t s . S i n c e t h e methyl ester i s 20% more d e n s e t h a n t h e b u t y l , t h e o r d e r i s r e v e r s e d i n dynamic viscosity units
.
The a - v a l u e s f o r t h e i s o p r o p y l benzene a r e s u p r i s i n g l y h i g h , comparable w i t h t h e best lubricants. It i s clear from t h e r e s u l t s t h a t b o t h t h e v i s c o s i t y and t h e p r e s s u r e v i s c o s i t y is rising very rapidly with coefficient temperature i n the region s t u d i e d , which accounts f o r the extreme d i f f i c u l t y i n o b t a i n i n g A t s l i g h t l y lower m e a s u r a b l e EHD films. t e m p e r a t u r e s no f i l m i s o b t a i n e d due t o low a a n d a t h i g h e r ones the f l u i d i s s o v i s c o u s t h a t starvation occurs. No p r e s s u r e - v i s c o s i t y d a t a f o r t h e h i g h e r a l k y l benzenes c o u l d be f o u n d i n t h e l i t e r a t u r e , though Wilbur and J o n a s (22) g i v e r e s u l t s f o r methylbenzene which i n d i c a t e p r e s s u r e - v i s c o s i t y c o e f f i c i e n t s of 5.0 CPa-' a t 5OoC, 6.1 GPa-' a t O°C and 7.5 CPa-' a t -35OC. These are much lower than those for Either t h e a-value i s o p r o p y l b e n z e n e a t -94OC. of t h e i s o p r o p y l b e n z e n e i s much g r e a t e r t h a n t h a t of t o l u e n e or i t i s showing a v e r y r a p i d I n view of the i n c r e a s e a t low t e m p e r a t u r e s . v a r i a t i o n o b s e r v e d between measured v a l u e s a t -94 and -99.5OC and t h e g e n e r a l l y low a - v a l u e s of s i m p l e m o l e c u l e s under normal c o n d i t i o n s the l a t t e r seems more l i k e l y . It i s i n t e r e s t i n g t o s p e c u l a t e whether t h i s i n c r e a s e might be r e l a t e d the to the change in gradient in viscosity-temperature curve observed f o r the i s o p r o p y l benzene a t a p p r o x i m a t e l y t h e same temperature. There i s a w i d e l y n o t e d broad correlation between pressure-viscosity and temperature-viscosity i n d i c e s (21) ( 2 3 ) , and t h u s the kink s e e n i n t h e isopropylbenzene curve i n f i g u r e 6 may well c o r r e l a t e w i t h a n a-value rise. With t h e a l k y l b e n z e n e s , i f c o n d i t i o n s were h e l d s t a t i c a t v e r y low temperatures for p e r i o d s of s e v e r a l m i n u t e s , f r e e z i n g o c c u r e d . There was no e v i d e n c e o f f r e e z i n g i n t h e moving or b r i e f l y halted contacts, however, as r e p o r t e d in d r o p p i n g b a l l e n t r a p m e n t s w i t h a l k a n e s by Hirano and co workers ( 2 4 ) .
C l e a r l y the f l u i d s s t u d i e d s o f a r i n t h i s work a r e q u i t e i m p r a c t i c a l l u b r i c a n t s w i t h t h e i r
very limited useful temperature ranges. M i x t u r e s of s i m p l e f l u i d s would be needed t o extend t h i s range. However t h e a - v a l u e s found a r e q u i t e h i g h , a n d comparable t o better lubricants at conventional temperatures. a-value appears t o increase r a p i d l y with d e c r e a s i n g temperature, s o t h a t a l t h o u g h s i m p l e f l u i d s may have u n a c c e p t a b l y low a - v a l u e s a t room t e m p e r a t u r e t h i s may n o t be t h e case i n much c o l d e r c o n d i t i o n s . 7
CONCLUSIONS
I t has been shown t h a t EHD f i l m s can be g e n e r a t e d a n d measured u s i n g f l u i d s of s i m p l e Two r a n g e s of s t r u c t u r e a t low t e m p e r a t u r e s . f l u i d s have been examined. The p h t h a l a t e esters gave p r e s s u r e - v i s c o s i t y c o e f f i c i e n t s b r o a d l y i n accord w i t h t h o s e of o t h e r esters i n t h e l i t e r a t u r e . Alkylbenzenes a p p e a r t o form f i l m s o n l y i n a r e s t r i c t e d a n d v e r y low temperature r a n g e and t h e i s o p r o p y l b e n z e n e t e s t e d gave s u r p r i s i n g l y h i g h a - v a l u e s a t -94 t o -99.5OC of 33 t o 46GPa-'. Refer enc es 1.
Sastry,
Sethuramiah, A and S i n g h , s t u d y of Wear Under P a r t i a l EHD C o n d i t i o n s " . Proc 1 l t h Leeds-Lyon Symposium ItM i xed on Tribology, Leeds 1984. Lubrications and L u b r i c a t e d Wearti ed. D . Dowson e t a l . h b l . W l t t e r w o r t h s 1985. B.V.
V.R.K.,
"A
2.
"A C r i t i c a l F i n k i n , E.F., Cu, A . , Yung, L. Examination o f the EHD C r i t e r i o n f o r t h e S c o r i n g o f Gears". Trans. ASME 1974 pp 418-425
3.
" L i f e Adjustment F a c t o r s f o r B-all and R o l l e r An E n g i n e e r i n g Guide Sponsored Bearings". the Rolling-Elements Committee, by L u b r i c a t i o n D i v i s i o n of t h e ASME ( 1 9 7 1 ) .
4.
"Ball B e a r i n g Dowson, D. a n d Hamrock, B.J. Lubrication. The Elastohydrodynamics of E l l i p t i c a l Contactsii. J. Wiley a n d Sons 1981.
5.
ttShear Johnson, K L. a n d Tevaarwer k , J. L. Behaviour of Elastohydrodynamic O i l Films". Proc. R. SOC. Lond. 1977 A356 pp215-236.
6.
'IThe I n f l u e n c e o f Molecular H e n t s c h e l , K.H. S t r u c t u r e o n t h e F r i c t i o n a l Behaviour of L u b r i c a t i n g F l u i d s i t . J. Synth. Lub. 1985 1. pp 143-165.
7.
D i e t r i c h , M.W., Townsend, D.P. a n d Z a r e t s k y , E. V. tlRolling-Element L u b r i c a t i o n w i t h Fluorinated Polyether at Cryogenic Temperatures (160° t o 41O0R)It. N A S A Tech. Note TN D-5566.'
8.
Merriman, T. a n d Kannel, J. W . rtEvaiuation o f EHD Film T h i c k n e s s f o r Cryogenic F l u i d s " . ASLE Trans. 1986 2 pp 129-184.
9.
C l y e n s , S., Evans, C.R. a n d J o h n s o n , K.L. tiMeasurement of t h e V i s c o s i t y of S u p e r c o o l e d L i q u i d s a t High S h e a r Rates w i t h a Hopkinson Proc. R. SOC. Lond. 1982 A T o r s i o n Bar". 381 - p p 195-214
.
215 10. Ehrlow, A . J . Lamb, J. and Matheson, A. J. ttViscous Behaviour of S u p e r c o o l e d L i q u i d s " . Proc R. SOC. Lond. 1966 A 292 pp 322-342 1 1 . Denney, D. J. " V i s c o s i t y o f Some Undercooled L i q u i d A l k y l Halides". J. Chem. p h y s . 1959 30 pp 159-162
12. Kuss, E. "Extreme Values of t h e Pressure C o e f f i c i e n t of V i s c o s i t y t f . Angew. Chem. I n t . Ed. 1965 pp 944-950
5
19. Beuch,
F. and Davi s o n , "V i s c o s i t y- Tem per a t u r e Re1a t i ons Diethylphthalate". J. Chem. Phy 1966 4361 -4362
M. for pp
5
"Influence 20. Davies , D. B. and Matheson, A. J. o f Molecular R o t a t i o n on t h e V i s c o s i t y of L i qu i ds" J. Chem. phy 1966 pp 1000-1006.
.
21. ASME 1953.
Pressure
5
Viscosity
Reports
Vol
I1
13. G e n t l e ,
C . R. "Tract ion in Elastohydrodynamic C o n t a c t s " . FhD Thesis. Univ. of London ( 1971 1
14.
T.,
Yoshitake, H., I t o , T. and "Correlation between Flow Propel-ties and Traction of Lubricating Oilstv. ASLE Trans 1986 2 p p 102-106 Kyotani,
Tamai,
Y.
Hamman, W.C. and Cameron, A. " E v a l u a t i o n of L u b r i c a n t s Using O p t i c a l Elastohydrodynamicslf ASLE Trans 1968 p
15. Foord, C . A .
11
31
W . "On t h e Dependence o f V i s c o s i t y on Temperature f o r L i q u i d s " . Rec. d e s T r a v . Chim des Pays-Bas .1970 pp 625-635
16. Feiggen
2
17.
D.J. and J o n a s , J. "Fourier 22. W i l b u r , Transform NMR i n L i q u i d s a t High Pressure I11 S p i n - L a t t i c e R e l a x a t i o n i n T o l ~ e n e - d g ~ ~ . J. Chem. Phys., 1975, 62 pp 2800-2807.
"Viscosity-Temperature Petroleum P r o d u c t s 1 r . S e c t i o n 5. Volume 05.01.
Charts
for 0381.
Liquid ASTM
18. Ling, C . and W i l l a r d , J . E . " V i s c o s i t i e s of Classes Used a s Trapping Some Organic Matrices1! J . phys. Chem. g pp 1918-1923 (1968).
23. Roelands, C . J. A. " C o r r e l a t i o n a l Aspects of the V i s c o s i t y- temper a t ur e- pr essur e R e l a t i o n s h i p o f L u b r i c a t i n g O i l s . Croningen VRB K l e i n e ( 1 9 6 6 ) . 24.
Hirano, F., Kuwano, N. and Ohno, N. " O b s e r v a t i o n o f S o l i d i f i c a t i o n o f O i l s Under High Pressure1f. Proc. JSLE I n t . T r i b . Conf. pp 841-846,.
This Page Intentionally Left Blank
SESSION IX ELASTOHYDRODYNAMIC LUBRICATION (4) Chairman: Professor H. Christensen
PAPER IX(i)
A method for estimating the effect of normal approach on film thickness in elastohydrodynamic line contacts
PAPER IX(ii)
Transient oil film thickness in gear contacts under dynamic loads
PAPER IX(iii) A full numerical solution for the non-steady state elastohydrodynamic problem in nominal line contacts PAPER IX(iv) The lubrication of soft contacts
This Page Intentionally Left Blank
279
Paper IX(i)
A method for estimating the effect of normal approach on film thickness in elastohydrodynamic line contacts N. Motosh and W.Y. Saman
Although normal approach t a k e s p l a c e i n most of t h e machine e l e m t s o p e r a t i n g i n the elastohydrodynamic resiriic, t h e f i l m t h i c k n e s s formulas a v a i l a b l e f o r designers d o n o t t a k e i n t o account l o a d v a r i a t i o n The paper o u t l i n e s a n a n a l y i c a l method which c o n s i d e r s b o t h r o t a t i o n and normal approach. I n i t i a l r e s u l t s suegest t h a t t h e e f f e c t of normal approach i s q u i t e small i n r e l a t i o n t o t h e r o t a t i o n component
.
.
1
INTRGDUCTION
The understanding of t h e elastohydrodynamic l u b r i c a t i o n regime which t a k e s place i n h i g h l y loaded c o n t a c t s such as gear t e e t h and r o l l i n g c o n t a c t bear i n g s i s now e s t a b l i s h e d However, nos’t a n a l y t i c a l and experimental studi e s i n t h i s f i e l d have considered e i t h e r s t e a d i l y loaded r o l l i n g or s l i ding c y l i n d e r s and s p h e r e s ( I - 4 ) , o r d e a l t with s t a t i o n a r y elements i n no+ ma1 approach (5-6). I n p r a c t i c e , a l a r g e number o f machine elements o p e r a t i n g i n t h e elastohydrodynamic regime a r e influenced by t h e combined a c t i o n of r o t a t i o n and nomial approach, s i n c e t h e applied load o f t e n v a r i e s i n magnitude and sometimes in d i r e c t i o n i t p r e s e n t , t h e u s u a l method f o r c a l c u l a t i n g t h e f i l m t h i c k n e s s in such s i t u a t i o n s i s t o u s e the r e s u l t s f o r s t e a d i l y loaded c o n j u n c t i o n s . T h i s hovrever i g n o r e s t h e squeeze a c t i o n , although no a n a l y t i c a l o r experimental evidence has been p u t forward t o j u s t i f y t h i s . Very few a t t e m p t s t o conside r t h e combimed problem of r o t a t i o n anti normal approach have been made P e t r u s e v i t c h (7) have made such a study, but t h e paper i s somewhat vague as t o t h e method of s o l u t i o n . Another c o n t r i b u t i o n i n t h i s f i e l d i s t h a t of IIolland (8) who suggested a f i l m t h i c k n e s s formula for t h e combined e f f e c t by comparing t h e a v a i l a b l e f i l m t g i c k ness equations i n terms o f t h e u s u a l ~ ~ i i ~ i e n s i o n l parameters ess f o r the cases of r o t a t i o n and normal approach
.
.
.
.
By u s i n g a s e m i - a n a l y t i c a l method t h e p r e s e n t a u t h c - 7 (9) reported some f i l m t h i c k n e s s r e s u l t s which took i n t o account t h e combined e f f e c t s o f r o t a t i o n and normal approach f o r t h e c a s e s of r i g i d conjunction, isoviscous lubri c a n t and r i g i d c o n j u n c t i o n , p r e s s u s e de:>endent v i s c o s i t y l u b r i c a n t . It i s t h e aim O F t h i s paper t o o u t l i n e a method o f s o l u t i o n which a l s o . t a k e s i n t o account t h e e l e s t i c deformation of cylinders
.
1.I Notation
-
E
/
=
Equivalent modulus o f e l a s t i c i t y
h
=
Film t h i c k n e s s
R
=
iiadius of e q u i v a l e n t c y l i n d e r
P
=
Pressure
U
= P e r i p h e r a l speed
W
q
Load c a p a c i t y due t o r e t a t i o n p e r u n i t le-ilgth
=
q0=
Lubricant v i s c o s i t y a t t h e i n l e t
o( = P r e s s u r e 2
-
v i s c o s i t y exponent
ku’kLYSIS
Considering t h e geometry of t h e problem a s t h a t o f an e l a s t i c a l l y deformed c y l i n d e r on a p l a n e ( F i g u r e 1 ) and
280
u s i n g t h e IIertzian appl'oximt i o n , the t o t a l load car-:-c a p c i t y of the conjunction i s due t o t h e following mechanisms :
1. Rotation: i.nw?e,by usf~rigt h e E r t e l Grubin(l0) T i l i i -i;hic?tness formula , it may be s i o i ; ~ -that : 8
= 6 , 7 8 . ( T o . U .o()
WR
x R ~ E'. .
h-"
(1)
2. Normal approach: W e r e t h e load c a r r y i n g c a p a c i t y nay be obtained by i n t e g r a t i n g the l i e p o l d s equation Using t h e usual essumptions, t h i s reduces t o :
.
In o r d e r t o caTry out t h e i n t e g r a t h e conjunction may be devtion i d e d i n t o two load c a r r y i n g regions: 2-1.The r e g i o n o u t s i d e t h e Hertzian c o n t a c t (r'igure 1): Assuming a para b o l i c geometqy, f o r t h e purpose of s i m p l i f y i n g tlie a n a l y s i s , i.e :
2-2.Ihe Hertzion c o n t a c t region : Where t h e f i l m t h i c k n e s s v a r i e s with time only, v i z
h H = ho
t
4.w lr.€'
(5)
along t h e s e c t i o n of l e n g t h 2b ( Figure 1 ), where b i s given by
b=
[
a T f.. w €' . R ?
(61
It may be seen, on i n t e g r a t i o n , t h a t t h e p r e s s u r e in t h i s region t a k e s t h e form :
,
x2 h = b +2.R
and n o t i n t h a t the value of "C" i n equation 7 2 ) reduces t o zero a s
it may be r e a d i l y seen t h a t t h e p r e s s u r e d i s t r i b u t i o n i n t h i s region takes the f o m
The t o t a l load c a r r y i n g c a p a c i t y of t h e c o n j u c t i o n may be found ( 4 ) and (71, from equations (I), thus :
w = w,
-b + 2 - j PI.dx
-b T h i s equation can be w r i t t e n in t h e form :
Where :
I,=
j
-b (Tl
\ f, +
281
.
Figure (2) shows the was obtained pressure distribution in the oil film resulting from normal approach, at different time intervals Figure ( 3 ) shows the variation with time for the case of high appThe figure roch velocity of 20m/s also includes the load values obtained when the effect of normal approach is neglected
-b
.
-0c
.
L
.
4.
and :
f, = K . V
CONCLUSIONS
A method has been put forward for calculating the film thickness in the case of unsteady elastohydrodynamic lubrication By adding up the components due to both rotation and normal approach, the effect of load variation on film thickness under the combined action has been estabished In-spite of the assumptions employed to calculate the normal approach component, namely the parabolk approximation outside the Hertzian region and constant film thickness within it , it is believed that the analysis brings the mathematical representation of unsteady elastohydrodynamic a step nearer to the situation encountered in such important applications as gear teeth, cams and rolling contact bearings The preliminary results suggest that the effect of normal approach on film thickness is quite small and should only be taken into account when the approcach velocity is very high. This tends to justify the present practice of neglecting the effect of normal approach when estimating the film thickness in most elastohydrodynamic lubricatin applications It is hoped that more results of this analysis and some experimental validation will clarify the role played by squeeze action
.
.
.
The t2irce integrals contained in equation (9) nay be integrated analytically in t h e nanner described Fn. appendix(l), thus a r%lationship between the load By and cqproch velocity is obtained putt inG' zT= A h / A t 8
.
aquation (9) may be applied to any load variation encountered in specific opera t i n s conditions By selecting sufficiently small time increments ,dependinC; on the rate,p$ change of load, convergent film t w k n e s s results may be obtained after 'itteraion
.
.
.
.
References
3. PA'~LIIII~LLALY REXJLTS In o r d e r to demonstrate the use of this method, two cases of constant velocity of approach were considered irliere t h e iIertzian zone film thickne s s m s allowed to drop from 2.2 to 1.8 rlicrons during 2x10-8 seconds U s k g t h e values given in table 1 i o r the required data and assuming steadg conditions prior to the reduction in film thickness , the correaponclizq load variation with time
.
(1) Cheng, H.S. and Sternlicht ,: "A numerical solution for the pressure, temperature and film thickness between infinitely long, Lubricated rolling and sliding under heavy - cylinders . loads. It J.Mech. Engr. Sci. Vol 87.1965
.
282
( 2 ) Harzoclr ,B.and Dowson, D. " I s o t h e r m a l
elastohydrodynamic L u b r i c a t i o n o f p o i n t c o n t a c t s . €%.-starvation r e s u l t s It T r a n s ASME.SER.I?. Vo1.98,1976.
The r o o t s o f t h i s e q u a t i o n a r e
.
( 3 ) Dyson,A.l?ylor,H.and
Wilson A.R. : "The measurement of o i l f i l m t h i c kness in elastohydrodynamic contac t s . " Iiaoc. I n s t n . Mech.Fhgrs Vol. 180 p a r t 33, 1965 - 66
.
( 4 ) Westla?:e, P.J.and
Cameron,A. : " I n t e l - J c r o m e t r i c s t u d y of p o i n t contact Lubrication .It ElastoQdrodynamic L u b r i c a t i o n : 1972 sgnposium(1eeds). I n s t n Mech. Xngrs. paper C 39/72
The s o l u t i o n depends on wether t h e v a l u e o f f-, and hence t h a t o f w i s positive o r negative Let u s c o n s i d e r f i r s t t h e c a s e when f, > 0.
.
..
(5) Clliiisteiisen,H.:"The
o i l f i l m in a c l o s i n g gap". Proc.3oy SOC. S e r i e s A , Vo.226, 1962
.
I n t h i s case
( 6 ) Dowson.D. and Jones ,D. :"An o p t i c a l i n t e r T e r e n c e method of measurement of t h e - d e p e n d e n t elastohydrodynamic f i l u p r o f i l e . " Pr0c.b-s-tn.Mech.Engre.Vol.182, P a r t SG, I976 - 68
.
( 7 ) Petrusevi.tch,A.,Kodnir,D.,salukvadzc ,;I. ,Baliashvill .D. and schwarzmarm, V.:"!i?he i n v e s t i g a t i o n o f o i l f i l m t h i c k n e s s i n Lubricated b a l l r a c e r o l l i n g contact."Wear Vol. 19,1972
.
( 8 ) I I o l l a i d , J. :" l n s t a t i o n a e r e E l a s t o h;dr.ocl;.lzariic ' I . Kons t r u k t i o n z e itsc;ajA.'t,~O. 1978 H.9, ( 9 ) Motos:i,i;, and 3aman,W.Y. : " E f f e c t o i 1 0 c . J v u i a t i o n on f i l m t h i c k n e s s i n :G.shly loaded r i g i d conjunct.
-
ions. 'I
Proc.2l.W: leeds-lyon symposium on Tribology 1978 ( 10)Grubin ,.*.LT. and Vinogradova ,I. E. : "Central s c i e n t i f i c research i n s t i t u t e ~ O Ytechnology and mechanical enLiccc:,:h; . I t Book No. 30 Moscow.
.
Appendix 1 Method c..? F n t e g r a t i o n B o t h rt a d 1 , ( s e e e q u a t i o n (10)) can be i n t e g r a t e d a n a l y i c a l l y However IIneeds som manipulation b e f o r e analyticnl integration is possible. Consider t h e e q u a t i o n :
.
T h e r e f o r e , i n t h e c a s e when
Fl
>
0.
283
--o
Where :
T h u s t h e i n t e g r a l s I,, f , acd reduced t o t h e b a s i c f o r m :
I=
Irl
( a t b.d
1,
+C.A')dY
T h i s i s a standai-d i n t e g r a l which has value:
a i ~
284
A-
Param e t e r
Value
Units
/m2
0.16
N.S
u
4.4
m/s
a(
2.67~10-~ m2/N
R
8.4
mm
E'
2.1~10"
N/&
Table 1 : Values used in numerical example
I ,o
4
a 0.2
1
-v
x
t
285
Paper IX(ii)
Transient oil film thickness in gear contacts under dynamic loads A.K. Tieu and J. Worden
I n t h i s p a p e r , t h e f i l m t h i c k n e s s i n g e a r c o n t a c t s under dynamic l o a d s w a s c a l c u l a t e d by t h e rrethod o u t l i n e d by V i c h a r d (1) f o r i s o t h e r m a l c o n d i t i o n and neglecting s i d e leakage. The e f f e c t s o f f r e q u e n c y , r a n g e o f l o a d and s p e e d on t h e f i l m thi.ckness f o r d i f f e r e n t g e a r r a t i o s a r e c o n s i d e r e d .
1 INTRODUCTION
The s t a r t i n g p o i n t f o r many EHD s t u d i e s can be t r a c e d back t o t h e t h e o r e t i c a l t r e a t m e n t s by Ertel-Grubin ( 2 ) and Dowson and Higginson ( 3 ) . They have been used widely f o r l i n e c o n t a c t problems b u t i n most c a s e s , t h e s q u e e z e f i l m e f f e c t i s not included i n t h e Reynolds' e q u a t i o n . The a p p l i c a t i o n of EHD f i l m t h i c k n e s s t o gears i s an a r e a t h a t became i n c r e a s i n g l y popular from e a r l y 1 9 7 0 ' s up t o a r o u n d 1 9 8 0 . Akin ( 4 ) ( 5 ) a t t e m p t e d t o p r o v i d e a broad b a s e d s e t of r e l a t i o n s h i p s from which t h e p r o b a b i l i t y o f s c u f f i n g o f s p u r , h e l i c a l and b e v e l g e a r s c o u l d be p r e d i c t e d . Gu ( 6 ) p r o p o s e d a c r i t e r i o n applying s t e a d y - s t a t e e l a s t o h y d r o d y n a m i c theory t o g e a r s . He i n d i c a t e d t h a t i t i s not a p p l i c a b l e t o h e a v i l y l o a d e d g e a r s due t o t r a n s i e n t e f f e c t s . F i n k i n e t a 1 ( 7 ) examined t h e a v a i l a b l e r e s u l t s t o e s t a b l i s h t h e v a l i d i t y of minimum E H D f i l m thickness as a basis for the l u b r i c a t i o n f a i l u r e of g e a r s . They s t a t e d t h a t t h e i s o t h e r m a l EHD c r i t e r i o n a t t h e p i t c h p o i n t was i n a d e q u a t e a s a g e n e r a l guide t o p r e v e n t t h e s c o r i n g of g e a r s . Wellauer and Holloway ( 8 ) s t u d i e d a number of t h e a v a i l a b l e methods t o c a l c u l a t e EHD f i l m t h i c k n e s s i n g e a r s , e s p e c i a l l y t h e combined e f f e c t s of lambda line velocity and factor, pitch p r o b a b i l i t y of g e a r d i s t r e s s . Jackson and Rowe ( 9 ) produced an EHD c r i t e r i o n t o g e a r i n g system, making u s e of s p e c i f i c f i l m t h i c k n e s s and l u b r i c a n t p a r a m e t e r s .
I n t h e above s t u d i e s , t h e s q u e e z e f i l m term has been n e g l e c t e d mainly due t o t h e small i n f l u e n c e of l o a d on f i l m t h i c k n e s s i n t h e c a s e of s t e a d y s t a t e l o a d s . The e f f e c t of suddenly a p p l i e d o r s t e p l o a d on g e a r f i l m t h i c k n e s s is an a r e a i n which l i t t l e h a s been p u b l i s h e d . Vichard (1) i n c l u d e d t h e s q u e e z e f i l m t e r m s i n t h e Reynolds' e q u a t i o n , and made u s e of is a Grubin's approximation. This comprehensive paper which included
t h e o r e t i c a l and e x p e r i m e n t a l t r e a t m e n t of t r a n s i e n t c o n t a c t s u n d e r dynamic l o a d , w i t h and w i t h o u t e l a s t i c d e f o r m a t i o n . H e e s t a b l i s h e d t h a t t h e squeeze f i l m added t o t h e wedge e f f e c t p o s s e s s e s t h e c h a r a c t e r i s t i c s o f v i s c o u s damping i n as well as hydrodynamic e l a s t o - h y d r o d y n a m i c l u b r i c a t i o n . Motosh and Saman (10) c o n s i d e r e d t h e e f f e c t of l o a d v a r i a t i o n on f i l m t h i c k n e s s a t g e a r tooth contact without any e l a s t i c d e f o r m a t i o n . The e f f e c t s of changing l o a d i n g e a r s w a s c o n s i d e r e d by Wang and Cheng (11) a n d t h o s e o f v i s c o e l a s t i c i t y and f l u c t u a t i n g l o a d s by Rohde e t a 1 ( 1 2 ) . The i n f l u e n c e o f dynamia l o a d s i n e l a s t o h y d r o d y n a m i c j o u r n a l b e a r i n g have been t r e a t e d i n d e t a i l s by many a u t h o r s , s u c h a s t h o s e b y Rohde and Oh (13) , Oh and Goenka (14), LaBouff and Booker ( 1 5 ) . Dowson e t a 1 ( 1 6 ) p r e s e n t e d t h e o r e t i c a l study of isothermal hydrodynamic l u b r i c a t i o n o f r i g i d c y l i n d r i c a l and p o i n t c o n t a c t s , w i t h combined r o l l i n g and normal m o t i o n . I k e u c h i and Mori ( 1 7 ) s t u d i e d t h e s q u e e z e f i l m e f f e c t due t o s t e p l o a d on elastomer, without t h e wedge e f f e c t . The s t u d y i n t h i s p a p e r was c a r r i e d o u t t o d e t e r m i n e t h e v a r i a t i o n of f i l m thickness i n a p a i r of i n v o l u t e gear u n d e r e x c e s s i v e dynamic l o a d s . The dynamic l o a d s , i n t h e o r d e r of 6 t o 9 t i m e s t h e s t e a d y - s t a t e load, w e r e r e s u l t s of t o r s i o n a l v i b r a t i o n with backlash i n a r o l l i n g m i l l d r i v e . The s t u d y h e r e considered t h e isothermal Reynolds equation without s i d e l e a k a g e . The method o u t l i n e d i n (1) was used h e r e . 1.1
Notation
A
=
R/Ri
C
=
centre distance
El
=
reduced E l a s t i c Modulus
286
G
=
CLE'
h
=
film thickness
H
=
h/Ri
The Reynolds equation is then reduced to:
HlrH3.5rH5 steady-state film thickness with load factor 1,3.5 and 5 By introducing dimensionless variables, the above equation yields: PB
=
length of contact
p
=
pressure in lubricant film
P
=
Q
=
R
=
load per unit width 1 - e-aP G local equivalent R1R2/ (R1+R2)
The load is obtained by: log (1-GQ) dX
RblrRb2 base radii of pinion and gear R1,R2
local radii cylinders
Ri
equivalent radius of curvature at pitch point
=
of
=-J
equivalent
(3)
G
radii
For heavily loaded lubricated contacts, the surface deformation of the solid can be assumed to conform to the Hertzian dry contact. Accordingly, the film thickness is equal to the sum of unknown level Ho, which is a function of time, and the Hertzian deformation He, which is a function of X and T. Thus
S
distance along length of contact
S
squeeze number
T
dimensionless time =at
U
mean rolling speed U
H S
=
=
Ho(T) + He(X,T)
The function He is
V
=
for
He = 0 (U1+U2)/2
He =2W[ 2
pitch line velocity
W
dimensionless load W
X
dimensionless distance
=
P/EIRi
a
pressure-viscosity coefficient
PO
viscosity
XE 2
W3'*/GN
! &
I XI <
2
given by:
jTK
%kc G
4WA
-log (-
x
for
s/Ri
XP
(4)
OR/U
(5)
-2JWA
Equation (2) is now integrated twice with boundary conditions from the Grubin's approximation of Xm = 0 and Q (X,) = 1/G. From Vichard (1), the resulting equation can be derived:
U
THEORY
"@ GN
The following assumptions are made to simplify the analysis: the fluid is incompressible side leakage is neglected the oil film is isothermal the viscosity can be expressed as p=poeaP
- G
G,+2W'G3-H; =
fi =
$)l
(6)
Ho
where A'
(G,+4
dA/dT=
CA AX
287
W’
=
dW/dT
d e t e r m i n e t h e damping c h a r a c t e r i s t i c s of t h e o i l f i l m i s considered h e r e .
AW
= E H’,
S
=
3
dH,/dT
=
OR/U
=
V cos@ Ri
The r e s u l t s o b t a i n e d w e r e v e r i f i e d a g a i n s t t h o s e o b t a i n e d by G r u b i n ’ s method f o r steady s t a t e cases.
R
The l e f t hand t e r m s i n e q u a t i o n ( 6 ) governs t h e s t e a d y s t a t e f i l m t h i c k n e s s , whereas t h e r i g h t hand terms measure t h e squeeze f i l m . The f o u r G i f u n c t i o n s can be approximated b y . e x p o n e n t i a 1 f u n c t i o n s a s shown i n ( 1 ) . A l l t h e d i f f e r e n t i a l s i n equation ( 6 ) a r e with r e s p e c t t o T . I t was more c o n v e n i e n t t o r e p l a c e t h e dimensionless t i m e t e r m T with a d i m e n s i o n l e s s d i s p l a c e m e n t t e r m X
along t h e p a t h of c o n t a c t . T h e r e f o r e t h e d/dT t e r m s of e q u a t i o n ( 6 ) become d/dX terms. The unknown i n
RESULTS AND DISCUSSION
T h e r e a r e many r e a s o n s t h a t t h e l o a d b e t w e e n two g e a r t e e t h c a n n o t b e c o n s i d e r e d c o n s t a n t a l o n g t h e l e n g t h of c o n t a c t . F i r s t t h e r e i s t h e v a r i a t i o n of t o o t h s t i f f n e s s along t h e l e n g t h of c o n t a c t . The l o a d r i s e s g r a d u a l l y when t h e l o a d i s s h a r e d between 2 p a i r s of t e e t h i n c o n t a c t . A s soon a s t h e l e a d i n g p a i r disengages , t h e load r i s e s s h a r p l y u n t i l all t h e l o a d i s c a r r i e d by one p a i r of t e e t h . A t a d i s t a n c e a f t e r t h e p i t c h p o i n t , t h e s i t u a t i o n r e v e r s e s . Moreover t h e r e a r e e x c e s s i v e dynamic l o a d s due t o i n e r t i a s and s t i f f n e s s e s o f t h e d r i v e t r a i n s , b a c k l a s h e s and p r o f i l e e r r o r s of the gears. However i n t h i s p a p e r ,
t h e dynamic l o a d input e i t h e r a s a s t e p i n p u t o r a s i n e wave t o cater f o r any of t h e above dynamic loads.
w i l l be s i m u l a t e d by an a r b i t r a r y
F i g . 1 shows t h e r e s u l t of an i m p u l s e change of a l o a d t w i c e t h e s t a t i c load of 65.4 k N / m between t h e 200th and 400th s t e p . The f i l m t h i c k n e s s show d i s t i n c t damping c h a r a c t e r i s t i c s , as it t r i e s t o follow t h e l o a d change. The film t h i c k n e s s reaches t h e steady s t a t e value f o r l o a d change of t w i c e t h e i n i t i a l l o a d , b u t not h i g h e r .
e q u a t i o n ( 6 ) i s HIo which
e q u a l s AHo/Ax, t h e change i n f i l m t h i c k n e s s f o r a c o r r e s p o n d i n g change i n d i s p l a c e m e n t a l o n g t h e l i n e of a c t i o n . With known G I N,W, A, G i f u n c t i o n s , Po,
The dynamic l o a d i s r e p r e s e n t e d by a s i n e wave a s shown i n F i g . 2 . The s e v e r i t y of t h e dynamic l o a d s c a n b e imposed by changing t h e frequency a s w e l l as t h e maximum and minimum l o a d s .
t h e dynamic f i l m t h i c k n e s s can now be c a l c u l a t e d from e q u a t i o n ( 6 ) by Runge-Kutta method.
3.1
AP/AX,
The s t e p s i z e x w a s made s u f f i c i e n t l y small, w i t h 1 0 0 0 s t e p s a l o n g t h e p a t h of c o n t a c t . Cases of 1 0 0 and 1 0 0 0 0 s t e p s were a l s o u s e d . I t was found t h a t t h e y made no s i g n i f i c a n t d i f f e r e n c e t o t h e r e s u l t s . The f i l m t h i c k n e s s of g e a r c o n t a c t s depends on t h e i n i t i a l v a l u e chosen a t t h e s t a r t o f t h e n u m e r i c a l scheme. I t can a f f e c t t h e r e s u l t s over a distance a f t e r the i n i t i a l contact, e s p e c i a l l y f o r h i g h s p e e d and l i g h t l o a d s . For a l l t h e r e s u l t s p r e s e n t e d h e r e , t h e i n i t i a l f i l m t h i c k n e s s were chosen c l o s e t o t h a t of s t e a d y s t a t e load c a s e . A p a i r of g e a r s of r a t i o 1 and 5 w i l l b e c o n s i d e r e d h e r e . Data of t h e c a l c u l a t i o n a r e shown i n t h e Appendix. The f i l m t h i c k n e s s v a r i a t i o n a l o n g t h e l e n g t h of c o n t a c t a r e t o be compared t o t h o s e o b t a i n e d by s t e a d y - s t a t e l o a d s . The p a r a m e t e r s X p and XE shown i n (1) t o
E f f e c t of frequency
range 1 t o 3 . 5 i s now a p p l i e d t o a p a i r of t e s t g e a r s w i t h r a t i o R = l , and t h e f r e q u e n c y i s v a r i e d t o s t u d y i t s e f f e c t . The t r a n s i e n t f i l m t h i c k n e s s i s compared w i t h t h a t due t o s t a t i c l o a d s of 65.4 k N / m and 2 2 9 k N / m . A dynamic l o a d of
F i g . 3 shows t h e e f f e c t of a frequency on t h e s y s t e m . On t h e s e f i g u r e s , t h e maximum and minimum f i l m t h i c k n e s s H1 and H 3 . 5 c o r r e s p o n d t o t h e above s t a t i c l o a d s . A t a f r e q u e n c y of 4 , t h e s q u e e z e f i l m h a s had l i t t l e e f f e c t on t h e magnitude of t h e o i l f i l m t h i c k n e s s . The s i n u s o d a l f i l m t h i c k n e s s o s c i l l a t e s between t h e H1 and H3. c u r v e . When t h e f r e q u e n c y i s i n c r e a s e d t o 2 0 and 4 0 , t h e t r a n s i e n t r e s p o n s e t o t h e dynamic l o a d f a i l e d t o t o u c h t h e H1 c u r v e , and i s s h i f t e d below t h e H3.5 c u r v e .
288
3.2
E f f e c t o f load m a g n i t u d e
at thicker f i l m t h i c k n e s s i n - t h e o r d e r o f 3 p.m, a n d t h e loads v a r i e d between 65.4 t o 196.2 kN/m, the transient film thickness e x c e e d s t h e H1 c u r v e . When t h e l o a d i n c r e a s e s t o 400-1200 kN/m, t h e f i l m t h i c k n e s s varies w i t h i n t h e bound o f t h e H1 a n d H3 c u r v e ( F i g . 4 b ) . I n t h e s e t w o A s c a n be o b s e r v e d i n F i g . 4a,
the x factor v a r i e s between P 0.006-0.32 ( f o r minimum a n d maximum loads i n F i g . 4a) t o 0 . 0 9 2 - 0 . 4 7 8 (Fig. 4 b ) . Moreover, t h e f i l m t h i c k n e s s c u r v e H tends t o d w e l l f o r longer duration t o w a r d s t h e H1 c u r v e t h a n a t t h e H3 curve.
cases
When t h e r a n g e of load i s i n c r e a s e d from 6 5 . 4 kN/m t o 1 . 7 MN/m, a s i n F i g . 4c, t h e s h a p e o f t h e H c u r v e becomes l e s s s i n u s o i d a l a n d t u r n s more t o t h e dome shape t y p e , w i t h t h e f l a t p l a t e a u close t o t h e H1 c u r v e . T h i s r a n g e a n d m a g n i t u d e o f load i s i n c l u d e d h e r e t o a l l o w f o r t h e e f f e c t o f h i g h r a t e of c h a n g e of load t o be s t u d i e d . A s i g n i f i c a n t load c h a n g e c a n be b r o u g h t a b o u t b y i n c r e a s i n g t h e magnitude o f t h e load which i s a p p l i e d o v e r a g i v e n period of t i m e . A s mentioned above, t h e reduced magnitude of the film thickness H curve i s
a s s o c i a t e d w i t h t h e parameter
$.
Fig. 5 d e m o n s t r a t e s t h e combined e f f e c t o f h i g h v a l u e of xp = 0 . 9 2 - 4 . 7 8 , XE = 0 . 0 2 3 - 0 . 0 3 9 a n d t h e f r e q u e n c y from 4 t o 4 0 . I n t h e s e h e a v y d u t y cases, t h e load v a r i e s b e t w e e n 0 . 4 t o 1 . 2 MN/m ( r a n g e of l o a d 1-3) , a n d p i t c h speed 3.14 m / s . It i s noted here t h a t t h e o r d e r of m a g n i t u d e of f i l m t h i c k n e s s i n t h i s case i s s m a l l , b u t t h e t r e n d of f i l m t h i c k n e s s c h a n g e i s q u i t e e v i d e n t . The f i l m t h i c k n e s s v a r i a t i o n a r e s i g n i f i c a n t l y r e d u c e d b e t w e e n t h e H1 a n d H3 c u r v e s . A t t h e f r e q u e n c y o f 40, t h e H c u r v e f a l l s below t h e H3 c u r v e ( F i g . 5 c I 5 d ) . A t f i l m t h i c k n e s s i n t h e order o f 8 p m , t h e H c u r v e f a l l b e l o w t h e H3 curve, b u t t h e amplitude is n o t changed i n t h i s case i s s i g n i f i c a n t l y . The X P 0 . 0 2 - 0 . 1 1 . The r e d u c t i o n i n a m p l i t u d e of t h e H curve i s n o t always a s s o c i a t e d w i t h v a l u e s , as shown i n F i g . 5 e where the x P t h e loads imposed v a r y between 2 t o 6 MN/m. = 1 t o 5 . 3 and xE=0.050 to 0.087).
(xp
3.3.
E f f e c t o f Gear R a t i o
I n t h i s case t h e e f f e c t o f r a d i u s of curvature also influence the f i l m thickness. The g e a r r a t i o i s now 5 : l a n d t h e l o a d 2 7 . 3 - 8 1 . 9 kN/m ( r a n g e 1-3 i n F i g . 6 ) a n d 2 7 . 3 - 1 3 6 . 5 kN/m ( r a n g e 1-5 i n F i g . 7 ) , F i g . 8 shows t h e i n f l u e n c e o f
f r e q u e n c y on t h e f i l m t h i c k n e s s . S i m i l a r t r e n d s t o t h e case of gear r a t i o 1 : l i s o b s e r v e d . The f i l m t h i c k n e s s c u r v e H g i v e s s l i g h t l y lower v a l u e s t h a n t h e H5
xp
c u r v e i n F i g s . 7 a n d 8 . The factor i n these figures is higher than t h a t i n Fig.6. 4
CONCLUSIONS
film effect has some The squeeze i n f l u e n c e o n t h e m a g n i t u d e o f EHD f i l m thickness, e s p e c i a l l y with high frequency o f l o a d c h a n g e s . The f i l m t h i c k n e s s a t s u s t a i n e d h i g h l o a d f r e q u e n c y i s lower t h a n t h o s e o b t a i n e d a t t h e maximum s t e a d y load. The r e d u c t i o n i n a m p l i t u d e of t h e f i l m t h i c k n e s s due t o squeeze f i l m effect i s a s s o c i a t e d w i t h h i g h v a l u e s o f Xp a t a v e r a g e load. The t h e o r y i n (1) o f f e r s a s i m p l e method t o e v a l u a t e t h e t r a n s i e n t r e s p o n s e of l i n e c o n t a c t s i n gears.
References (1) V I C H A R D , J . P . 'Transient Effects i n T h e L u b r i c a t i o n of H e r t z i a n C o n t a c t s ' , J . M e c h . S c i . , V O l 13, N o 3 , 1 9 7 1 .
( 2 ) GRUBIN, A. N . 'Fundamentals of t h e H y d r o d y n a m i c T h e o r y of L u b r i c a t i o n of H e a v i l y Loaded C y l i n d r i c a l S u r f a c e s ' D.S.I.R. Booklets, London (1949), T r a n s l a t i o n N o 337. ( 3 ) DOWSON,D. and HIGGINSON,G.R., E l a s t o h y d r o d y n a m i c L u b r i c a t i o n Pergamon P r e s s , Oxford (1966)
.
( 4 ) AKIN, L. S. 'An I n t e r d i s c i p l i n a r y L u b r i c a t i o n Theory f o r Gears', T r a n s . ASME, J . of Eng. F o r I n d . S e r i e s B, N o 4 , November 1973, 1178-1195. 'EHD Lubricant Film (5) AKIN,L.S. Thickness Formulas For Power Transmission Gears', J.Lub.Tech., Trans.ASME, J u l y 1974, 426-431.
( 6 ) GU,A.'Elastohydrodynamic L u b r i c a t i o n of I n v o l u t e G e a r s ' J . of Eng. f o r I n d . , Trans.ASME, S e r i e s B, Vol 95, N o 4 , November 1973, 1164-1170. GU,A. and YUNG,L. 'A ( 7 ) FINKIN,E.F., Critical Examination of The Elastohydrodynamic C r i t e r i o n f o r t h e of Gears I , T r a n s . ASME, Scoring J.Lub.Tech., J u l y 1974, 418-425.
( 8 ) WELLAUER ,E . J . a n d HOLLOWAY, G . A . ' A p p l i c a t i o n of EHD O i l F i l m T h e o r y t o I n d u s t r i a l Gear D r i v e s ' , T r a n s . ASME, J . o f Eng. f o r I n d . , Vol 98, s e r i e s B, 1 9 7 6 , 626-634. ( 9 ) JACKSON,A and P r e p r i n t 800670, 1 9 8 0 .
ROWE,C.N.
,
SAE
289
(10) MOTOSH and SAMAN 'Effect Of Load Variation On Film Thickness In Rigid Highly Loaded Conjunctions' Elastohydrodynamics and Related Topics, Eds. by D . Dowson et all Mechanical Engineering Publication 1979. (11) WANG,K.L. and CHENG,H.S. 'Thermal Elastohydrodynamic Lubrication of Spur Gears' NASA Contract Rept.3241, Feb 1980.
(12) ROHDE, S .M., WHICKER, D . and BOOKER J.F. 'Elastohydrodynamic Squeeze Film: Effects of Viscoelasticity and Fluctuating Loads' Trans. ASME, J.Lub.Tech, Vol 101 Jan 1979, 74-80. (13) ROHDE,S.M. and OH,K.P. 'A Unified Treatment of Thick and Thin Film Elastohydrodynamic Problems by Using Higher Element Method ' Proc. Royal.SOC. Series A (London), Vol 343, 1975, 315-331. (14) OH. K . P . a n d GOENKA, P .K . 'The Elastohydrodynamic Solution of Journal Bearings Under Dynamic Loading', Trans. ASME, J. of Tribology, July 1985, 389-395.
(15) LABOUFF,G.A. and BOOKER J.F. 'Dynamically Loaded Journal Bearings: A Finite Element Treatment of Rigid and Elastic Surfaces' Trans. ASME, J.of Tribology, Vol 107, 505-515. (16) DOWSON,D, MARKH0,D.A. and JONES,D.A. 'The Lubrication of Lightly Loaded Cylinders in Combined Rolling Sliding and Normal Motion. Part 1:Theory' Trans. ASME, J.Lub.Tech.,Vol 98, No 4 Oct 1976, 509-516. (17) IKECHU,K and MOR1,H. 'Squeeze Film on Compliant Surfaces Under Step Load', Bull.JSME, Vol 27, No 231, Sep 1984. 5
APPENDIX
Data of Case Studies Spur gear module Centre Distance Gear ratio Viscosity
5.08mm 0.3m 1:l or 5:l 0.075 Pa.S unless otherwise stated
Pressure-Viscosity Coefficient Young's modulus
1.6~10-~/Pa 2 1 0 ~ 1 0 9Pa
I.0y
*48t I
I
L
-1.0
I
-.8
1
1
-.6
1
1
-.4
,
1
-.2
(a) P
1
.O
,
1
.2
,
1
.4
,
1
.6
,
1
X/PB
FIG. 1 FILM THICKNESS AGAINST IMPULSE LOAD (BASIC LOAD 65,4 KN/M, @.075 (PITCH
VELOCITY
p
load factor = 1 ,
3,14
M/S
PA.s)
-8
xE=.009
xp=.239
XE".017
*gl -vv .e
2
I
xp=.036
,
65.4-229 kN/m, V=15.7m/s =0.025 Pa.S
=
xp=.012
XE'.0O9
xp=.080
xE=.017
~
1.0
m W z
1.6
p
( b ) Same as (a) except
x
= 0.075Pa.S
u
1
:::I
(c) Same as (b)
1.6
x
=.080
1
1
xE=.017
( d ) Same as (b) 1
-1.0
-.8
1
1
1
-.6
1 -.4
1
/
-.2
,
.O
1
1
.2
1 1 .4
1
.6
1
1
.8
X/PB
FIG, 3 FILM
ThICKNESS AT DIFFERENT AND FREQUENCIES
A/'PL
FIG. 2
SINUSOIDAL
LOAD
VlscosITy
1
1
1.0
290
4.0E
x
1.a
=.006
xE=.009
xp=.032
XE'.0O9
2.8
-x
( a ) P = 65.4-196.2
kN/m,
x m
z
=.092
xp=.478
W
x
V = 31.4 m / s
x
=.023
--
-
xE=.039
2.4
-r
0
(b) P = 400-1200 k N / m ,
+
x
Y
V =31.4m/s
m
Xp=4.78
XES.O39
=.023
(b) Same as (a)
W
2.4
x
.40
m z
k
Xp=.92
x
xp=.92
1.6
~~'1.613
xE=.047
-
LL
-48
'
P
xE=.023
.
*
W
Xp'4.78
W
xE=.039
.40
(c)
Same as ( a )
Xp'.02
3.1
FIG. 4 FILM THICKNESSVARIATION W I T H LOAD RANGE (p = 0.075 PA.s)
7 e'.
xE=.023
E
-
.9
(d) P = 400-1200 k N / m ,
f
V=131.42m/s
W
3-21
~~'5.341
xE=.087
2.0
(el P - 1 .L0
-.8 I
1
- 1. 6
FIG. 5 EFFECT
L
FIG, 6 FILM THICKNESSVARIATION W I T H GEAR RATIO 5:l ( P =27,3 -81.9 KN/M, v = 3.14 M/S,jl=0.075PA.S)
1 -..I 1 1
OF
=
2000-6000 kN/m,
-.2 1
1
1. 2 ,
1.tl 1
PARAMETER xp
ON
1.
J /
v
=
1 .6,
31.424s 1 .8 ,
l/ . U
F I L M THICKNESS
' "E .e
.st
.? .a
m m W
z
.6
-
.5
-
.4
-
.3
-
L
~
-1L
a-q.0
-.a'
-14'
-'.i: 0 '
!2'
!4,
l.6'
! 8 * I!,
X/PB
FIG,8 FILM THICKNESSVARIATION W I T H LOAD FREQUENCY 40 FOR GEAR RATIO 5 :1 (SAME DATA AS Frc.7)
291
Paper IX(iii)
A full numerical solution for the non-steady state elastohydrodynamic problem in nominal line contacts Wu Yong-wei and Yan Sheng-ming
I n t h i s paper, a f u l l numerical s o l u t i o n f o r t h e non-steady-state EHD problem i n l i n e c o n t a c t s i s p r e s e n t e d and t h e method of computation a l s o described. The press u r e d i s t r i b u t i o n s and f i l m shapes are o b t a i n e d f o r a number of t h e normal-to -ent r a i n i n g - v e l o c i t y r a t i o s . The dynamic behavior of l u b r i c a n t f i l m , f o r b o t h normal approaching and s e p a r a t i o n motion i s i n v e s t i g a t e d and d i s c u s s e d . Based on t h e numerical results, a formula f o r estimating t h e dynamic f i l m t h i c k n e s s i s developed.
1. INTRODUCTION There are many machine elements such a s r o l l e r bearings, gears and cams, whose l u b r i c a t i o n between t h e c o n t a c t s u r f a c e s belongs t o the non-steady-state EHL. For r o l l e r b e a r i n g s , t h e c o n t a c t zone i s sub-
j e c t t o t h e non-steady-state load. I n the gears, not only load, but t h e r a d i i of c u r v a t u r e and e n t r a i n i n g v e l o c i t y v a r y throughout t h e zone of a c t i o n . The t i m e dependent parameters make l u b r i c a n t f i l m e i t h e r t h i c k e n i n g o r t h i n n i n g . Therefore i n o r d e r t h a t EHL t h e o r y can e x p l a i n t h e t r u e p i c t u r e of l u b r i c a t i o n regimes i n t h e c o n t a c t , i t i s necessary t o extend tho e x i s t e n t s t e a d y - s t a t e EHL t h e o r y t o t h e non-steady-state f i e l d . Recently t h e non-steady-state EHD problem has a t t r a c t e d r e s e a r c h e r ' s a t t e n $ion. I n 1 9 n , Vichard (1) made a n a t t a c k on t h i s k n o t t y problem, a d a p t i n g t h e Grubin approximations f o r f i l m shape and pressure d i s t r i b u t i o n . I n 1972, Petrousev i t c h e t al. 121 solved a l s o approximat e l y t h e non-steady-state i s o t h e r m a l EHD problem. I n 1981, a f u l l numerical solut i o n was o b t a i n e d by s o l v i n g simultaneousl y time-dependent Reynolds and e l a s t i c i t y equations from Wada e t . al. (31. They have r e v e a l e d a p a r k of c h a r a c t e r i s t i c s on non-steady-state EIID problem, b u t t h e o r d e r of magnitude i n parameters i s n o t i n t h e p r a c t i c a l range of e n g i n e e r i n g a p p l i c a t i o n . T h e i r s t u d y d i d n o t take t h e e f f e c t of normal s e p a r a t i o n i n t o account. I n f a c t , when l u b r i c a n t f i l m is thickening g r a d u a l l y , i t w i l l s u p p o r t t h e e x t e r n a l l o a d under combined e n f l u ence of t h e e n t r a i n i n g and normal separation velocities.
For t h e t i m e being, there have
e x i s t e d two well-know formulae f o r dynamic f i l m t h i c k n e s s , one of which was developed by P e t r o u s e v i t c h e t al., t h e o t h e r proposed by Holland (4) , b u t b o t h w e r e n o t based on t h e f u l l numerical s o l u t i o n . I n t h i s paper w e i n t e n d t o set up a new formula based on t h e r e s u l t s of t h e f u l l numerical s o l u t i o n and i n c l u d i n g t h e e f f e c t s of b o t h norm a l approaching and s e p a r a t i o n v e l o c i ties. 1.1 N o t a t i o a
b
h a l f - l e n g t h of H e r t z i a n l i n e c ont act reduced e l a s t i c modulus E dimensionless materials parameter, G dE h o i l f i l m thickness dimensionless f i l m thickness,h/R €I hco c e n t r a l f i l m t h i c k n e s s he f i l m t h i c k n e s s at l o c a t i o n where dp/dx=O hm minimum f i l m t h i c k n e s s f i l m pressure P dimensionless p r e s s u r e , p/E P maximum H e r t z i m p r e s s u r e Ptim PWm dimensionless maximum H e r t z i a n p r e s s u r e ,pHm/E reduced p r e s s u r e dimensionless reduced p r e s s u r e , d E reduced r a d i u s of c u r v a t u r e R e n t r a i n i n g v e l o c i t y , ( u I + u2 )/2 U U dimensionless e n t r a i n i n g v e l o c i t y , %u/ (ER) dimensionless normal v e l o c i t y V Z( dhm/dt / (ER) dimensionless l o a d parameter W Wen dimensionless l o a d c a p a c i t y f o r pure e n t r a i n i n g motion
a"
292
wsq X
X X
in xe d
t )lo
h
J
2.
dimensionless l o a d c a p a c i t y f o r pure squeeze motion c o o r d i n a t e i n t h e d i r e c t i o n of rolling dimensionless c o o r d i n a t e , x/b i n l e t coordinate c o o r d i n a t e st t h e end of o i l f i l m pressure-viscosity coefficient lubricant viscosity ambient l u b r i c a n t v i s c o s i t y velocity r a t i o a u x i l i a r y coordinate
10 '
the f i l m thickness equation
i s w r i t t e n as
GOVERNING EQUATION3
The time-dependent Reynolds e q u a t i o n
used f o r l i n e c o n t a c t s i s g i v e n as D i v i d i n g x axis i n t o uneven s t e p - l e n g t h ( F i g . l ) , and by n o r m a l i z a t i o n and d i s c r e t i z a t i o n , t h e e q u a t i o n (5) becomes I n t h e v e r y narrow c o n t a c t zone, i t is g e n e r a l l y assuied t h a t t h e s u r f a c e v e l o c i t i e s u , , ua and t h e rate of c h q e of t h e f i l m t h i c k n e s s w i t h time are independent t o x a x i s . D e f i n i n g
i""6+~,~~~+~""i Q,=0
, i=2,3;..,
M-1
(I/)
where C,i=(
.?-AL Di 1Di
c,i =~~i(D~-D~-l)-Z)(Di-i.+D~~ and d i s c u s s i n g t h e problem o n l y at some i n s t a n t , w e can r e w r i t e the e q u a t i o n (1) as o r d i n a r y d i f f e r e n t i a l e q u a t i o n
C,i
= (2+
Ai
DL-I ) Di-I
(3)
By adopting t h e power p r e s s u r e - v i s c o s i t y relation
and i n t r o d u c i n g t h e reduced p r e s s u r e function q, the equation ( 3 ) , togathe r w i t h t h e Reynolds boundary c o n d i t i o n , becomes
I t i s w e l l known t h a t t h e H e r t z i a n
S i m i l a r l y t h e e q u a t i o n ( 9 ) becomes (12) Hi = Nco + Nr where Nr=N,(Xi) + N(0) -N(Xi)
293
i s a l o o p c o r r e c t i n g t h e central. f i l m t h i c k n e s s H, i n t h e calculating procedure. I n t h e beginning c y c l e s , t h e corr e c t i o n i s achieved by
The equation (11) i s a t r i d i a q o n a l equation t o be solved e s i l y . The v a l u e s of a, b and c are g i v e n by t h e f o l l o w i n g interpolation function
where K i s t h e cycle-index. Once W j u s t l i e s between W "(k-1) and ~ ' ( k ) , t h e c o r r e c t i o n i s made by t h e f o l l o w i n g interpolation
pts,-p, ( SI = 3a9t 2 bs t c , xj-r< S<Xj+,
3.
NETHOD OF SOLUTION
Fig. 2 i s t h e flow chart f o r t h e numer i c a l s o l u t i o n of t h e coupled e q u a t i o n s (11) and ( 1 2 ) w i t h d i r e c t i t e r a t i v e m e t hod. The i n i t i a l v a l u e of t h e p r e s s u r e d i s t r i b u t i o n c o n s i s t s of three segments. The middle one i s a p a r t of H e r t a i a n pressure curve and t h e two s i d e s are concave-up p a r a b o l a s , as shown i n Mg.3. They e r e r e p r e s e n t e d as, r e s p e c t i v e l y ,
the rest
I n the l a t e r c y c l e s , t h e i n t e r p o l a t i o n r e g i o n c o n t a i n i n g t h e t r u e v a l u e of H w i l l become narrow c o n t i n u a l l y , untilCo t h e v a l u e of Hco t o be s u i t e d f o r W i s obtained. F o r o b t a i n i n g t h e converged v a l u e s of p r e s s u r e and t h e f i l m t h i c k n e s s , t h e correction t o the pressure d i s t r i b u t i o n i n each c y c l e can be c a r r i e d out by
where k) is a heavy under-relaxation f a c t o r , being of t h e f o l l o w i n g form
( 131
where t h e v a l u e s of x b and X C are p r e scribed according t o t h e parameters W and U, w h i l e Xe i s c a l c u l a t e d by
xa=xb + ( I t 0
~ _ _
_-
.(J-s
. 5 ~ ~ -1.5 (ZSXb J-x:
+arcsirXb-msinXc
)I/= (14)
s o t h a t t h e r e s u l t obtained by i n t e g r a t ing t h e i n i t i a l p r e s s u r e curve o v e r can be equal t o t h e l o a d parameter W. The s o l u t i o n of t h e e q u a t i o n (11) v a r i e s w i t h m. I n o r d e r t o f i n d t h e sol u t i o n s a t i s f y i n g t h e boundary c o n d i t i o n dQ/dX=O, i t i s n e c e s s a r y t o s h i f t t h e p o s i t i o n of t h e f i l m r u p t u r e p o i n t . If the c o n d i t i o n s , b o t h Qi>O; i= Z,..., k f o r m=k and Qw&O f o r m=k+l, are satisf i e d , we judge d ~ , / a ~ = oand i d e n t i f y xk w i t h the f i l m r u p t u r e p o i n t The i n t e g r a t e d result, W*, obtained from t h e c a l c u l a t e d p r e s s u r e P*must be equal t o t h e load parameter W,so t h e r e
With the i n c r e a s e i n t h e i t e r a t i v e t i m e , t h i s f a c t o r w i l l decrease g r a d u a l l y . If t h e s e l e c t i o n f o r t h e c o n s t a n t s A and B is s u i t a b l e , t h e i t e r a t i v e procedure
w i l l converge at f a s t e r speed w i t h i n some s p e c i f i e d range of t h e performance parameters.
4.
RESULTS ANTI DISCUSSION
As a s p e c i c a l c a s e of non-steady-state, EHD problem, w e g i v e a s e t of s o l u t i o n s f o r s t e a d y - s t a t e problem, t a b u l a t e d i n Table 1. They conform w e l l w i t h t h e d a a o b t a i n e d by Dowson's formula
&.
It confirms t h a t t h i s method of computation is feasible.
294
Some t y p i c a l computed r e s u l t s of non-steady-state EHD problem are shown i n Fig. 4 t o Fig. 7. We can f i n d followi n g p o i n t s from them. D u r i n g normal approaching f o r A < 0 , t h e f i l m shape h a s t h e characteristics of b o t h t h e pure squeeze f i l m and t h e pure entrainment f i l m . A cave i s formed i n t h e m i d d l e r e g i u n of t h e c o n t a c t zone and a necking i n t h e e x i t region. Under t h e i n f l u e n c e of t h e squeeze e f f e c t , t h e p r e s s u r e d i s t r i b u t i o n d i s p l a y s t h e char a c t e r i s t i c of pure squeeze f i l m t o a greater o r lesser e x t e n t . It looks n e a r l y l i k e a b e l l when t h e squeeze e f f e c t p l a y s a l e a d i n g r o l e . But w i t h t h e d e c r e a s e i n normal approach v e l o c i t y , t h e c h a r a c t e ristics of entrainment f i l m become oufs t a n d i n g , and the p o s i t i o n of t h e maximum p r e s s u r e s h i f t s towards t h e e x i t g r a d u a l l y too. When t h e entrainment e f f e c t i s v e r y s t r o n g , the second press u r e s p i k e may g e n e r a t e i n t h e pressure: distribution. During normal s e p a r a t i o n f o r A T 0 , t h e entrainment e f f e c t must be s t r o n g e r t h a n t h a t under s t e a d y s t a t e c o n d i t i o n , t o compensate t h e l o s s i n l o a d c a p a b i l i t y by t h e s e p a r a t i o n . I n t h i s c a s e , t h e s u r f a c e s of s o l i d s are evener i n t h e middle r e g i o n of t h e c o n t a c t zone. The p r e s s u r e curve shows t h e characteristic o f ' t h e s t e a d y s t a t e EHD l u b r i c a t i o n f o r heavier load. ' Through t h e m u l t i l i n e a r r e g r e s s i o n a n a l y s i s , we developed a formula f o r minimum f i l m t h i c k n e s s , f o r t h e range of data t a b u l a t e d i n Tables 2 and 3, as
t h e assumption t h a t
w
=
where Wen and W sq t i v e l y by
(22) are determined respec-
~ m = l . 6 @ 0 ' 6 U e 7 / W ~ pure entraince +i.h ~
U n f o r t u a n t e l y , t h e r e s u l t s obtained by t h e e q u a t i o n ( 2 2 ) are d i f f e r e n t from o u r numerical s o l u t i o n s as shown i n Table 5. The e q u a t i o n (22) i s based on t h e p r e d i c t i o n t h a t t h e p r e s s u r e d i s t r i b u t i d n is simple s u p e r p o s i t i o n of p r e s s u r e c u r v e s g e n e r a t e d by normal approach and entrainment. However i t a p p e a r s t h a t t h i s does not comply w i t h t h e o r y of d i f f e r e n t i a l e q u a t i o n v e r y w e l l , n o r leads t o a c a v i t a t i o n boundar y condition c o n s i s t e n t with current u n d e r s t a n d i n g of f i l m r e p t u r e . I n f a c t , f o r t h e l u b r i c a t i o n of t h e l i g h t l y loaded c y l i n d e r s i n combined r o l l i n g and normal motion, t h e formula of t h e l o a d c a r r y i n g c a p a b i l i t y based on t h e a c c u r a t e numerical s o l u t i o n s has been w e l l established by Dowson 1 5 1 i n 1976 It can be r e w r i t t e n as f o l l o w s
where
\
wen + w s q
f = *(6 a 1.87439-1.707 6 3 -4.62039 3"
(200)
The range of data i s not wide enough, s o t h e formula i s u n s a t i s f a c t o r y . But s t i l l , i t could be used i n wider range according t o t h e comparison w i t h t h e numerical r e s u l t s published by Wada. I n Table 4 , it i s r e a s o n a b l e t h a t t h e minimum f i l m t h i c k n e s s o b t a i n e d by formula (20) is about 80 p e r c e n t of the t h i c k n e s s at t h e f i l m r u p t u r e p o i n t by Wada. Rewriting the formula (201, t h e rate of chanlre o f t h e minimum f i l m t h i c k n e s s w i t h time can be r e p r e s e n t e d as
,
[ 'z,( E
-
Hn w 0'04T 0.f47
4.706 6"09sU
>,,, (211
where Hmt i s t h e minimum f i l m t h i c k n e s s under s t e a d y - s t a t e c o n d i t i o n f o r t h e same v a l u e s of performance parameters as t h a t considered i n determining H m . I n 1978, Holland has proposed a formula f o r dynamic f i l m t h i c k n e s s on
and W are determined by, respec99 tively,
Wen
I n t h i s formula, t h e v a l u e of t h e coeff i c i e n t f i s dependent on Hm and A , b u t not alwa s e q u a l t o 1.0 as i n t h e e q u a t i o n (22y. F o r example, f=O.5786 f o r ~ = 1 0 - 3 , H,=Io-~ and u=10-l1. It i m p l i e s t h a t t h e v a l i d i t y of t h e equat i o n ( 2 2 ) h a s y e t t o be checked.
5.
CONCLUSIONS
295
The f u l l numerical s o l u t i o n f o r t h e non-steady-state EIID problem has been obtained f o r Newtonian l i q u i d s and t h e moderate l o a d s i n l i n e contacts. The computed r e s u l t s have confirmed Wada's p r e d i c t i o n , namely t h a t t h e second p r e s s u r e s p i k e may g e n e r a t e i n t h e p r e s s u r e d i s t r i b u t i o n s under t h e non-steady-state c o n d i t i o n s . A formula f o r e s t i m a t i n g dynamic f i l m t h i c k n e s s has been proposed, whose v a l i d i t y has y e t t o be checked and modified by o u r f u r t h e r theorec t i c a l and e x p e r i m e n t d r e s u l t s .
Petrousevitch,A.I., e t al. 'The inv e s t i g a t i o n o f o i l film t h i c k n e s s i n l u b r i c a t e d ball-race r o l l i n g c o n t a c t ' , Wear, vol. 19, 1972, 36 9-389 Sanm Wada, e t a l . 'Elastohydrodynamic squeeze problem of two r o t a t i n g c y l i n d e r s ' , B u l l e t i n of t h e JSME, vol. 24, No. 190, 1981. Holland, J D i e i n s t a t iontire elastohydrodynamik' , Konstruktion, 30, NO. 9 , 1978. Dowson ,D , Markho ,P. H , Jones ,D A. 'The l u b r i c a t i o n of l i g h t l y loaded c y l i n d e r s i n combined r o l l i n g , slid i n e and normal motion'. J o u r n a l of r u b r i c a t i o n Technoloby, vol. 98, 1976 509-523 Ranger,A.P., e t a l . 'The s o l u t i o n of t h e p o i n t c o n t a c t elasto-hydrodynamic problem', Proc. R.Soc. Lond. A, 346 P. 227-244, 1975.
.
.
REFERENCES (1) Vichard,J.P.
'Transient e f f e c t s i n t h e l u b r i c a t i o n of IIertai8n cont a c t s ' , J o u r n a l mechanical engineeri n g s c i e n c e , vol. 13, No. 3, 1971.
/Specify
parameter W,G,U
r
I
IpPi+w(Pr -Pi)
I
/
and A
i.
.
1
C a l c u l a t e reduced P r e s s u r e Qi from PL
I
[ C a l c u l a t e deformation Nrfrom Pi t
I
1
I
*
Modify reduced p r e s s u r e : Qic Qi+0.3(Qi and o b t a i n new v a l u e s Calculate l o a d W
Fig. 2
*
i
*
pi
from
- Qi)
Qi
*
corresponding t o Pi
Flow c h a r t f o r t h e s o l u t i o n procedure
.
296
Fig. 3
Fig. 1 Division of X a x i s
I n i t i a l pressure distribution
PXIO’
w- I 5x 10-5
Fig. 4
Pressure d i s t r i b u t i o n s 2:ld f i l m thicknesses
Fig. 5
Pressure d i s t r i b u t i o n s a h 2 f i l m thicknesses
291
I pxro’
i x
B
I4 1.6
I
-20 - 1.0
Fig. 6
1.8
Fig. 7 P r e s s u r e d i s t r i b u t i o n s an3 film t h i c k n e s s e s
P r e s s u r e d i s t r i b u t i o n s and film t h i c k n e s s e s Table 1.
Comparison w i t h Dowson’s formula (A=O, Xin=5.0)
!J
G
(10-5)
(lo3)
2.0 2.0 2.0 2.5 2.5 2.5
1.5 1.5 2.0 2.0 2.0 2.5
U (10”’) 1.0 2.0 1.0 2.0 10.0 10.0
error
gm (10-5) this Paper
1.13 1.80 1.26 2.02 6.02 6.87
Dowson’s f orrnula 1.12 1.82 1.31 2.06 6.37 7.18
(PI 0.89 1.1
3.8 1.9
5.5 4.3 I
298
--- - - -------
---.
--
I . -
w
(10-5) this formula paper (22) 1.0 686 .O 1.0 1.15 1.0 4.55 1.0 1.34 1.5 100.0 1.5 166.0 1.5 6.10 1.5 0.19 2.0 79.6 2.0 0.85 2.0 79.3 2.0 1.96 2-5 74.5 2.5 55.2
-
(103)
-14
( 1 0 - l ~ ) (10-3)
(10
c
-
0.2
5 .O
1.0
0.5
0.2
0.1
1.0
1.0 2.0
0.8
2.0
0.5
0.4 0.4 0.8
0.4
0.2 2.0 1.0
4.0
0.8 1.0 0.1 0.8
1.5 1.2 1.0
2.0
1
Table 3.
2.0
1.5 1.2
0.5 0.1 0.8 0.2
1.5
1.0 2.0
1.2
0.2
5 -0
1.0
0.5
0.2 1.0
1.5
1.0
0.5
1.2
0.4
0.4
-
I
(20)
(P)
1.69 0.933 1.54 2.07 1.44 1.51
2.87 0.32 7.69 0.98 7.46 4.43 3.74 13.7 2.65 15.5 8.77 15.7 6.38 3.77
2.22
0.975 1.94 1.69 1.56 0.869
2.00
1.71 0.751 1.88
2.00
1-59
1.65
Comparison of numerical r e s u l t s w i t h formula (20)
V ( 10-14)
1.o
1.0
2.0
1.0
2.5
1.5 1.5 1.5 1.5
1.0 1.2
10.0 10.0
1.5 1.5
2.0 2.0
2.5 2.5 2.5 2.5
2.5 Table 4.
W
10.0 1.0
0.4
2.78
5.0
0.3
0.4
0.9
2.0
0.2 0.2 0.8
0.4
1.82 3.00
4.0 3.0 5.0
2.0
0.5
6.4 1.6 0.6 1.5
0.4 0.2 0-3 1.0
10.0
1.08
2.45 1.79 2.95 1.67 3-05 4.33 2.05 2.14 3.31 5.26
1.68 3.06 4.33 1.97 2.06 3.39 5.53
0.8
10.0
error 5.19 4.86 0.52
2 -83
2.52
2.71
2.45
2.0
0
1 I
*
1.65 1.67 0.60 0.33
o
4.06 3.88 2.36 4.88
Comparison of formula ( 2 0 ) w i t h Wada's r e s u l t s G
U
-A4
(lo3) 1.06 1.03 1.03 1.70
H,,, numerical formula results (20) 1.54 1.46 5.76 6.04 3.86 3.84 1.41 1.45
1.2 1.2
8.0
1.5
0.8
10.0
0.3
4 .O
2.0 2.0 1.0
0.4 1.0
1.o
4.0 3.0 3.0
2.0 1.o
2.0 2.0 2.0
I
2.0
1.5
t
numerical results ---1.74 0.930 1.43 2.05 1.34 1.58 2.14 1.13 1-89
1.0 0.1 1.0 0.8
1.6
- -. formula _ I
----H_rnE-?)-- e__rror
-v
-A
U
G
3.0 3.0 3.0
3.0 3.0
-v (l o -l g )
He (10-7) (Wada)
7.50 3.58 3-58 8.33 11.67
1.0
1.0
4.0 1.0 1.0
0.75 0.358 1.43 0.833 1.167 I
3.0 2.0
3-0 2.9 4.0
Hm (1~'7) formula( 20)
2.35 1.68 2 -5 7 2.41 2.86
299
Paper IX(iv)
The lubrication of soft contacts C.J. Hooke
S o f t c o n t a c t s t y p i c a l l y h a v e f a r l a r g e r s u r f a c e d e f o r m a t i o n s t h a n are e n c o u n t e r e d i n t h e familiar h a r d E.H.L. c o n t a c t . T h i s h a s two e f f e c t s . F i r s t , t h e c o n t a c t s are g e n e r a l l y n o n - H e r t z i a n and f o r c o n t a c t s w i t h a r e l a t i v e l y t h i n , s o f t , s u r f a c e l a y e r i t may b e shown t h a t a n a d d i t i o n a l regime of l u b r i c a t i o n i s p r e s e n t between t h e r i g i d a n d e l a s t i c regimes found i n H e r t z i a n c o n t a c t s . Second, f o r many of t h e c o n t a c t s t h e f i l m t h i c k n e s s e s are m i n u t e i n c o m p a r i s o n w i t h t h e m a c n i t u d e s of t h e s u r f a c e d e f o r m a t i o n s . I n t h e s e c o n t a c t s t h e f i l m t h i c k n e s s i s c o n t r o l l e d by a n a r r o w i n l e t r e g i o n and t h e s t a t i c o r dynamic a n a l y s i s of t h e i n l e t i n i s o l a t i o n e n a b l e s t h e c l e a r a n c e s t h r o u g h o u t t h e contact t o be r e a d i l y determined. 1
INTRODUCTION
L u b r i c a t i o n of s o f t e l a s t o h y d r o d y n a m i c c o n t a c t s may b e d i s t i n u i s h e d from t h a t of t h e more familiar h a r d c o n t a c t s b y two f e a t u r e s . The first is t h e a b s e n c e of a n y s i g n i f i c a n t p i e z o v i s c o u s e f f e c t i n t h e f l u i d b e c a u s e of t h e low peak p r e s s u r e s . The s e c o n d is t h e e x t r e m e l y high deformations g e n e r a l l y encountered. Because of t h e h i e h d e f o r m a t i o n most s o f t contacts are non-Hertzian and cannot be t h e conjunction of two approximated by c y l i n d e r s . The familiar r e s u l t of, f o r example, of a s o f t H e r r e b r u g h [ l ] f o r t h e E.H.L. Hertzian c o n t a c t w i l l , t h e r e f o r e , have o n l y very l i m i t e d a p p l i c a t i o n . A s e c o n d effect of t h e non-Hertzian geometry i s t h a t , i n some t y p e of c o n t a c t s , t h e t r a n s i t i o n from t h e r i g i d regime t o t h e e l a s t i c regime, as t h e d e f o r m a t i o n r a t i o i s i n c r e a s e d , is complex w i t h a n a d d i t i o n a l ' i n v e r s e ' regime b e i n g i n t e r p o s e d between t h e ' r i g i d ' a n d t h e ' e l a s t i c ' regimes. T h i s e x t r a regime i s f o u n d p r i m a r i l y i n t h o s e c o n t a c t s which h a v e a t h i n soft s u r f a c e l a y e r on a hard s u b s t r a t e . When compared w i t h h a r d E.H.L. c o n t a c t s i t i s a l s o found t h a t t h e m a j o r i t y of s o f t E.H.L. c o n t a c t s h a v e a f a r greater r a t i o of s u r f a c e d e f o r m a t i o n t o f i l m t h i c k n e s s . For example, a very h e a v i l y loaded r o l l i n g c o n t a c t b e a r i n g w i l l h a v e a r a t i o of s u r f a c e d e f o r m a t i o n t o minimum f i l m t h i c k n e s s of a r o u n d 100. V a l u e s n e a r e r 10 are, however, more common. A t y p i c a l elastomeric s h a f t seal w i l l h a v e a r a t i o of d e f o r m a t i o n t o f i l m t h i c k n e s s of between 1000 and 10000, some 100 times greater. T h i s v e r y large d e f o r m a t i o n r a t i o a n d nonH e r t z i a n n a t u r e of t h e c o n t a c t s means t h a t t h e for the analytical techniques developed c a l c u l a t i o n of f i l m t h i c k n e s s i n h a r d c o n t a c t s cannot be used d i r e c t l y . I n s t e a d , t e c h n i q u e s which t a k e t h e h i g h l y deformed n a t u r e of t h e s u r f a c e s i n t o a c c o u n t h a v e t o b e employed. 1.1 N o t a t i o n El
E q u i v a l e n t e l a s t i c modulus 2/E' I ( l - V l ) / E i + (1-V2)/E2
a
Rate of c h a n g e of e n t r a i n m e n t v e l o c i t y
al, a2
C o e f f i c i e n t s of h a l f power p r e s s u r e s e r i e s
b
semi-contact width
h
Clearance
h,
C l e a r a n c e a t maximum p r e s s u r e
hmin Minimum c l e a r a n c e k
Dry c o n t a c t p r e s s u r e c o e f f i c i e n t
P
Pressure
R*
E q u i v a l e g t r a d i u s of c u r v a t u r e 1 / R I 1/R1 + 1/R2
t
time
t
layer thickness
u
Entrainment v e l o c i t y
x
Distance
7
viscosity
2
H I G H L Y DEFORMED CONTACTS
If t h e c l e a r a n c e s and p r e s s u r e s i n a h i g h l y soft contact, with a constant deformed e n t r a i n m e n t v e l o c i t y , a r e examined, i t may be s e e n t h a t o v e r t h e m a j o r i t y of t h e c o n t a c t t h e p r e s s u r e d i s t r i b u t i o n l i e s close t o t h a t found under d r y , f r i c t i o n l e s s conditions. S i m i l a r l y , t h e c l e a r a n c e u n d e r t h e c o n t a c t , see F i g . 1 , is n e a r l y u n i f o r m and w e l l o u t s i d e t h e c o n t a c t is o n l y changed s l i g h t l y from t h e d r y p r o f i l e . T h e n o n - d i m e n s i o n a l c l e a r a n c e , F, u s e d i n t h a t i s t h a t of ref. [2]. The only figure s i g n i f i c a n t d e p a r t u r e s from t h e d r y c o n t a c t
300
values occur i n a localised region a t t h e e n t r a n c e t o t h e c o n t a c t a n d i n a r a t h e r more r e s t r i c t e d zone a t t h e e x i t . However, t h e f l u i d f i l m is generated i n t h i s r e s t r i c t e d e n t r y r e g i o n and i s t h e n c o n v e c t e d i n t o t h e c o n t a c t .
as p r e s s u r i s e d seals, t h e v a l u e s of k a t t h e two e n d s of t h e c o n t a c t may d i f f e r . The c o n d i t i o n t h a t t h e f i r s t term i n t h e p r e s s u r e a n d c l e a r a n c e d i s t r i b u t i o n i s dominant i n the i n l e t region appears t o be s a t i s f i e d for most s o f t c o n t a c t s b u t may b e checked [ 3 ] by comparing t h e p r e s s u r e g i v e n by e q u a t i o n ( 2 ) with the actual pressure distribution a t a v a l u e of x g i v e n by: x = 1.23
E:]
0.4
1
---
k0.6
(3)
point, the percentage deviation i n from t h e h a l f power form of e q u a t i o n ( 2 ) w i l l b e c l o s e l y e q u a l i n magnitude, but o p p o s i t e i n d i r e c t i o n , t o t h e p e r c e n t a g e error i n t h e calculated clearance. The f a c t t h a t t h e p r e s s u r e s and c l e a r a n c e s i n t h e end r e g i o n s of a l l h i g h l y deformed c o n t a c t s c a n be e x p r e s s e d i n t h e same f r m e n a b l e s a s i n g l e r e s u l t t o be o b t a i n e d [ 2 ] for t h e c l e a r a n c e u n d e r t h e c o n t a c t , h,: At
that
pressure
01
I
0 x/b
-1
Fig. 1
1
Clearances under a soft, highly deformed c o n t a c t . g3 = 100.
By e x a m i n i n g a l o c a l i s e d r e g i o n a r o u n d t h e i n l e t , i t is p o s s i b l e t o p r e d i c t t h e clearances t h a t a r e g e n e r a t e d t h e r e and h e n c e t o e s t a b l i s h t h e c l e a r a n c e s u n d e r t h e c e n t r e of t h e c o n t a c t . The minimum f i l m t h i c k n e s s o c c u r s i n t h e e x i t r e g i o n where t h e c l e a r a n c e s g e n e r a t e d a t t h e i n l e t are m o d i f i e d t o form a n e x i t r e s t r i c t i o n a n d , a g a i n , t h i s m o d i f i c a t i o n c a n be s t u d i e d by a n e x a m i n a t i o n of t h e e x i t r e g i o n i n i s o l a t i o n from t h e r e m a i n d e r of t h e c o n t a c t . I f t h e end of a s o f t e l a s t i c c o n t a c t is examined u n d e r d r y c o n d i t i o n s i t may be shown [2] that the pressure and clearance d i s t r i b u t i o n c a n b e e x p r e s s e d i n t h e form: P
I
E’ [ 3al xO.5 x > o
+
5a2 x l . 5
+
....
and
(4)
F o r c o n t a c t s which h a v e e q u a l v a l u e s of k a t t h e i n l e t a n d e x i t . and w h e r e there is n o sealed p r e s s u r e , t h e r a t i o of t h e minimum f i l m t h i c k n e s s i n t h e e x i t r e g i o n t o hm may b e o b t a i n e d [ 2 ] as: (5) Where t h e v a l u e s of k d i f f e r or where t h e a t the e x i t to the contact is pressure s u f f i c i e n t l y high t o prevent cavitation, t h e r e s u l t s p r e s e n t e d [ 2 ] i n F i g . 2 must b e u s e d .
I
1
(1)
h
8
[ at (-x)la5
-
.... 1
a2 ( - x ) ~ + * ~
x < o
...
depend o n where t h e c o e f f i c i e n t s a l , a2, t h e g e o m e t r y of t h e c o n t a c t i n g s u r f a c e s a n d may be found a n e l a s t i c a n a l y s i s of t h e s u r f a c e . T h i s r e s u l t i s g e n e r a l a n d c a n b e shown t o apply t o a l l elastic c o n t a c t s e x c e p t t h o s e , s u c h a s s q u a r e s e c t i o n seals, which h a v e a d i s c o n t i n u i t y i n t h e s u r f a c e geometry a t t h e e d g e of t h e c o n t a c t z o n e . P r o v i d e d t h a t t h e e x t e n t of t h e i n l e t sweep i s s u f f i c i e n t l y small, t h e f i r s t of t h e terns i n e q u a t i o n 1 w i l l dominate t h e d r y contact d i s t r i b u t i o n i n the localised region a r o u n d t h e e n t r a n c e t o t h e c o n t a c t where t h e fluid film is generated. Under these c i r c u m s t a n c e s e q u a t i o n ( 1 ) may b e t r u n c a t e d , giving: P = El (kx)OS5
x > o (2)
x < o v a l u e of k i s a c h a r a c t e r i s t i c of t h e p a r t i c u l a r d r y c o n t a c t b e i n g examined a n d must b e d e t e r m i n e d from a n e l a s t i c a n a l y s i s . I t may b e n o t e d t h a t f o r non-symmetric c o n t a c t s , s u c h
The
0
Fig. 2 3
Effect of non-symmetric end p r o f i l e s o n t h e minimum f i l m t h i c k n e s s r a t i o .
DYNAMIC BEHAVIOUR
A similar a p p r o a c h may b e a d o p t e d when d e a l i n g w i t h t h e n o n - s t e a d y m o t i o n of h i g h l y deformed s o f t c o n t a c t s . A s before, t h e f i l m t h i c k n e s s w i l l b e g e n e r a t e d i n a small e n t r a i n m e n t zone a t t h e l e a d i n g e d g e of t h e c o n t a c t . T h i s clearance i s then convected through t h e c o n t a c t at a s p e e d close t o the instantaneous e n t r a i n m e n t v e l o c i t y , c h a n g i n g s l i g h t l y as i t d o e s s o , t o emerge a t t h e e x i t . A t t h e e x i t a s h o r t e x i t minimum forms. I n a n a l y s i n g t h i s
301 t y p e of motion i t i s c o n v e n i e n t t o d i v i d e t h e c o n t a c t zone i n t o t h r e e r e g i o n s ; t h e i n l e t , t h e e x i t and t h e c e n t r a l p a r t of t h e c o n t a c t a n d t o examine them s e p a r a t e l y . I n t h e c e n t r a l s e c t i o n of t h e c o n t a c t t h e procedure d e v e l o p e d by H i r a n o [ 4 1 a n d H i r a n o and Kaneta [51 may b e u s e d and Reynolds' equation: a(uh)
ah
+
-at
may be r e p l a c e d by two e q u i v a l e n t first equations:
(6)
order
(7a)
The f i r s t e q u a t i o n , e q u a t i o n ( 7 a ) , d e f i n e s a l i n e h a v i n g a c o n s t a n t v a l u e of some new coo r d i n a t e , v, w h i l e t h e second d e f i n e s t h e rate of change o f h alone; t h a t l i n e . If the p r e s s u r e , P , c a n b e t a k e n a s b e i n g close t o t h e dry contact pressure distribution these e q u a t i o n s are r e a d i l y i n t e g r a t e d t o f i n d t h e v a l u e s of c l e s r a n c e , h., t h r o u g h o u t t h e c o n t a c t . The v a l u e s of c l e a r a n c e m u s t , of c o u r s e , b e d e f i n e d a t t I 0 a n d a l s o when t h e i n t e g r a t i o n p a t h s e n t e r t h e c o n t a c t from t h e i n l e t zone. Provided s u f f i c i e n t time i s a l l o w e d f o r t h e i n i t i a l c l e a r a n c e s t o b e c o n v e c t e d from t h e contact b e f o r e examining t h e r e s u l t s , the conditions a t t - 0 can be chosen a r b i t r a r i l y . The c l e a r a n c e s e n t e r i n g from t h e i n l e t c a n be determined 161 by a n a n a l y s i s of t h a t r e g i o n a l o n e by m o d i f y i n g t h e s t e a d y a n a l y s i s t o i n c l u d e t h e dynamic terms. S i m i l a r l y , w h e r e c l e a r a n c e s a r e c o n v e c t e d from t h e c o n t a c t i n t o t h e e x i t , t h e y form a boundary c o n d i t i o n f o r t h e c a l c u l a t i o n of t h e c l e a r a n c e s i n t h a t zone. The d e t a i l s of t h e i n l e t a n d e x i t a n a l y s i s are g i v e n i n d e t a i l i n [61 a n d follow a b r o a d l y similar p r o c e d u r e t o t h a t g i v e n below f o r t h e c o n s t a n t a c c e l e r a t i o n problem a n d n o d e t a i l s w i l l be g i v e n here.
m o t i o n i s s t o p p e d f o r a s h o r t time and t h e n restarted i n the opposite direction. The r e s u l t s are f o r a v a l u e of g3 of 100 a n d t h e l e n g t h of t h e v e r t i c a l l i n e c o r r e s p o n d s t o a v a l u e of (see ref [ 2 ] ) of 1. Before t h e motion is stopped, t h e c l e a r a n c e corresponds t o t h e standard c l e a r a n c e p r o f i l e for steady m o t i o n , w i t h e n t r a i n m e n t a t t h e l e f t hand end and a n e x i t minimum a t t h e r i g h t hand end. When t h e m o t i o n i s s t o p p e d , a n o t h e r minimum i s r a p i d l y formed i n t h e o l d e n t r y r e g i o n a n d b o t h t h i s a n d t h e e x i t r e s t r i c t i o n grow s t e a d i l y w i t h time. When t h e m o t i o n i s r e s t a r t e d i n t h e o p p o s i t e d i r e c t i o n , t h e minimum, i n what i s now t h e e x i t drops rapidly, while t h e r e s t r i c t i o n from t h e r i g h t hand e n d i s c o n v e c t e d t h r o u g h t h e c o n t a c t a t a s p e e d close t o t h e e n t r a i n m e n t v e l o c i t y d e c r e a s i n g v e r y s l i g h t l y a s i t moves. it merges w i t h the new exit Finally r e s t r i c t i o n , producing a s h o r t l i v e d decrease in clearance before leaving the contact e n t i r e l y . After t h i s t h e c l e a r a n c e a d o p t s t h e s t e a d y s t a t e p r o f i l e o n c e more.
F i g . 4.
C l e a r a n c e s u n d e r a n O - r i n g seal i n r e c i p r o c a t i n g motion. .
A s e c o n d example, F i g . 4 , shows t h e c l e a r a n c e s c a l c u l a t e d f o r a n '0' r i n g seal. The seal is o s c i l l a t i n g s i n u s o i d a l l y with an e n t r a i n m e n t a m p l i t u d e e q u a l t o 1.5 times t h e t o t a l c o n t a c t w i d t h . A n i p of 7% h a s b e e n assumed a n d a seal p r e s s u r e of 0.2 E a c t s o n The v e r t i c a l line t h e r i g h t of c o n t a c t . c o r r e s p o n d s t o a v a l u e of h ' of 1 where:
h* =
bh
Fig.
3.
Clearances under a Hertzian contact with a dwell before reversal.
A s a n example of t h e t y p e of r e s u l t t h a t may b e o b t a i n e d , F i g . 3 shows t h e v a r i a t i o n i n clearance under a H e r t z i a n c o n t a c t i n which t h e
c
--___123u0b E' 1 0 . 5
a n d uo i s t h e maximum e n t r a i n m e n t v e l o c i t y . I t s h o u l d b e n o t e d t h a t small d i s p l a c e m e n t t h e o r y h a s b e e n u s e d f o r t h e c a l c u l a t i o n of t h e p r e s s u r e d i s t r i b u t i o n rather t h a n t h e large d i s p l a c e m e n t , small s t r a i n , a n a l y s i s s t r i c t l y r e q u i r e d a n d t h e v a l u e s of k u s e d may b e s l i g h t l y i n error. T h i s error w i l l h a v e a marginal effect o n t h e m a g n i t u d e of the c l e a r a n c e s but t h e o v e r a l l behaviour w i l l be unchanged. E n t r a i n m e n t is o n t h e r i g h t f o r t h e first h a l f c y c l e and on t h e left for t h e s e c o n d . F i g . 5 shows t h e minimum c l e a r a n c e s i n t h e c o n t a c t p l o t t e d a g a i n s t time. The f u l l l i n e s show minima a r i s i n g a t t h e l e f t hand end of t h e c o n t a c t , t h e d a s h e d l i n e s minima a r i s i n g a t t h e r i g h t hand end. The t y p e of c l e a r a n c e p r o f i l e p r e d i c t e d is similar t o t h a t f o u n d e x p e r i m e n t a l l y by Blok a n d Koens [ 7 1 a n d by
302
F i e l d a n d Mau [81.
e f f e c t s of c l e a r a n c e v a r i a t i o n s on p r e s s u r e may be ignored i n t h e c e n t r a l region.
\
\
0.3 -
,1
Fig. 5
0.5
1.0 Cycles
1.5
1
\
I
'L
0
\
1
I
2.0 Cycles
Minimum f i l m t h i c k n e s s e s f o r a r e c i p r o c a t i n g O - r i n g seal Fig. 7
The a n a l y s i s o u t l i n e d above i s approximate i n t h a t i t i g n o r e s t h e effect of t h e v a r i a t i o n
i n c l e a r a n c e under t h e c o n t a c t on t h e p r e s s u r e t h i s appears to be distribution. While acceptable for s t e a d y motion where the c l e a r a n c e s a r e f a i r l y u n i f o r m a n d where i n v e r s e t h a t t h e effect of any theory suggests v a r i a t i o n i n p r e s s u r e w i l l simply b e t o s h i f t t h e l o c a t i o n of a p a r t i c u l a r c l e a r a n c e , t h i s is less obviously the case under dynamic c o n d i t i o n s . Here t h e c l e a r a n c e d i s t r i b u t i o n is non-uniform w i t h l o c a l i s e d r e s t r i c t i o n s b e i n g c o n v e c t e d across t h e c e n t r e of t h e c o n t a c t . I t is probable t h a t t h e s e r e s t r i c t i o n s will p r o d u c e a m a t c h i n g , l o c a l i s e d , p r e s s u r e anomaly t h a t w i l l tend t o d i f f u s e t h e r o s t r i c t i o n . A d e t a i l e d a n a l y s i s of t h i s r e q u i r e s t h e e f f e c t of c l e a r a n c e o n p r e s s u r e t o b e a s s e s s e d a t e a c h s t e p of t h e i n t e g r a t i o n of e q u a t i o n s ( 7 a ) a n d ( 7 b ) and t o b e i n c l u d e d i n t h e a n a l y s i s . T h i s may b e done b u t g r e a t l y l e n g t h e n s t h e a n a l y s i s ( a f ' a c t o r of 50 i n c o m p u t e r time i s t y p i c a l 1 a n d does n o t a p p e a r t o g r e a t l y a f f e c t t h e r e s u-l t s o b t a i n e d .
Minimum c l e a r a n c e s , i n c l u d i n g t h e e f f e c t s of d i f f u s i o n .
One p a r t i c u l a r f e a t u r e of t h i s t y p e of r e c i p r o c a t i n g c o n t a c t t h a t may b e n o t e d i s t h e f o r m a t i o n of a minimum i n t h e e n t r a n c e r e g i o n a s t h e entrainment v e l o c i t y f a l l s . T h i s is s e e n most d i s t i n c t l y i n t h e f i r s t of t h e c u r v e s p r e s e n t e d i n F i g . 4 . T h i s c u r v e shows t h e a t t h e e n d of t h e s t r o k e w i t h clearance e n t r a i n m e n t o n t h e l e f t of t h e c o n t a c t . I t may b e s e e n t h a t t h e c l e a r a n c e s produced i n t h e e n t r a i n m e n t r e g i o n a r e lower t h a n t h o s e i n t h e e x i t . T h i s r e s u l t seems t o b e t y p i c a l of a l l r e c i p r o c a t i o n c o n t a c t s w h e r e there i s a smooth r e v e r s a l of m o t i o n . I t arises d u e t o t h e f o r m a t i o n of lower c l e a r a n c e s i n t h e e n t r y region with reducing entrainment velocities. The r e d u c t i o n i n clearance is e v e n t u a l l y h a l t e d by s q u e e z e f i l m e f f e c t s w h i c h s u s t a i n t h e f i l m u n t i l l a r g e r c l e a r a n c e s are c o n v e c t e d back o u t of t h e c o n t a c t a s t h e m o t i o n r e v e r s e s . 3.1 C o n s t a n t A c c e l e r a t i o n A g e n e r a l e x p r e s s i o n f o r t h e minimum c l e a r a n c e formed a s t h e i n l e t c h a n g e s t o a n e x i t c a n b e d e t e r m i n e d i f t h e a s s u m p t i o n i s made t h a t t h e
r a t e of c h a n g e of e n t r a i n m e n t v e l o c i t y c a n b e t a k e n a s u n i f o r m , a t l e a s t o v e r t h e time t a k e n f o r t h e f l u i d f i l m t o p a s s t h r o u g h t h e end region. Under c o n d i t i o n s of c o n s t a n t a c c e l e r a t i o n , Reynolds' e q u a t i o n becomes:
"
"I
_ _ ___ _ _ ax a 19ax
Fig. 6
C l e a r a n c e s i n c l u d i n g t h e e f f e c t of diffusion
6 a n d 7 show t h e c l e a r a n c e s f o r t h e recalculated including t h i s diffusion e f f e c t a n d i t may b e s e e n t h a t t h e effect of diffusion on t h e c l e a r a n c e s a s t h e y are c o n v e c t e d t h r o u g h t h e c o n t a c t i s minimal i n t h i s example a n d may be s a f e l y i g n o r e d w i t h o u t s i g n i f i c a n t loss of a c c u r a c y . Similar r e s u l t s h a v e been o b t a i n e d i n a l l t h e c o n t a c t s t h a t have b e e n examined a n d i t is s u g g e s t e d t h a t t h e
Figs
'0' r i n g
= - at
_a _h + b--h ax a t
(81
where ' a r i s r a t e of c h a n g e of t h e e n t r a i n m e n t v e l o c i t y a n d t h e time, t , i s measured from t h e p o i n t a t which t h e v e l o c i t y r e v e r s e s . R e y n o l d s ' e q u a t i o n may c o n v e n i e n t l y b e rew r i t t e n i n n o n - d i m e n s i o n a l form by making t h e substitutions: f = kx G O a 5
303
and uref = ( a / k I o e 5 giving:
--[J ! ?
a
=
;3
iv,
- E __ ax
1
0 -1
+
0
aii
--
at
I
1
1 2 Distance into contact ii
I
3
I
4
(9) F i g . 8 End c l e a r a n c e s u n d e r u n i f o r m acceleration
s u b s t i t u t i o n a l s o allows t h e d r y c o n t a c t pressures and c l e a r a n c e s i n t h e end r e g i o n t o be e x p r e s s e d i n t h e form: The
I t may b e n o t e d t h a t t h e minimum c l e a r a n c e p r e d i c t e d o n t h i s basis d e p e n d s o n l y o n t h e rate of c h a n g e of e n t r a i n m e n t v e l o c i t y a t t h e end of t h e s t r o k e a n d n o t o n t h e peak v e l o c i t y reached. 4
The effect of t h e f l u i d f i l m i n t h e end region may be allowed for [91 by superpositioning solutions t o the e l a s t i c i t y e q u a t i o n s of t h e t y p e g i v e n i n e q u a t i o n ( 1 0 ) . I t is n e c e s s a r y , of c o u r s e , t o e n s u r e t h a t t h e s u p e r p o s i t i o n e d p r e s s u r e s a n d c l e a r a n c e s merge w i t h t h e d r y c o n t a c t d i s t r i b u t i o n away from t h e c o n t a c t end. T h i s l e a d s t o e x p r e s s i o n s f o r t h e p r e s s u r e and c l e a r a n c e i n t h e form:
(11)
2
where t h e weight f u n c t i o n B ( s ) must s a t i s f y t h e
conditions: 6 s ) d s
r
I
MODERATELY DEFORMED SOFT CONTACTS
Although t h e most w i d e l y u s e d soft c o n t a c t , t h e elastomer seal, has a v e r y h i g h r a t i o of d e f o r m a t i o n t o f i l m t h i c k n e s s , a number of o t h e r c o n t a c t s s u c h a s t h e elastomer c o v e r e d rollers u s e d i n t h e p r o c e s s i n d u s t r i e s t e n d t o o p e r a t e w i t h lower r a t i o s . T h e s e c o n t a c t s may, however, s t i l l h a v e large enough d e f o r m a t i o n s f o r t h e w i d t h of t h e c o n t a c t z o n e t o b e c o m p a r a b l e w i t h some d i m e n s i o n of t h e c o n t a c t s u r f a c e , s u c h as t h e t h i c k n e s s of t h e s u r f a c e layer. Where t h e soft surface of t h e c o n t a c t is of l i m i t e d t h i c k n e s s compared w i t h t h e c o n t a c t width, t h e dry contact pressure d i s t r i b u t i o n c h a n g e s i t s h a p e from t h e H e r t z i a n e l l i p s e t o a more peaked p r o f i l e as t h e r a t i o of c o n t a c t w i d t h t o s u r f a c e t h i c k n e s s i s i n c r e a s e d . The p r e s s u r e d i s t r i b u t i o n s for a c o n t a c t w i t h a s o f t s u r f a c e l a y e r were d e t e r m i n e d by Meijers [ I 0 1 a n d are shown i n Fig. 9.
1
sB(s)ds = 0
-0b
i n o r d e r t h a t c o n d i t i o n s away from t h e end r e g i o n may r e m a i n u n a l t e r e d . These e q u a t i o n s may be s o l v e d u s i n g t h e procedure o u t l i n e d i n [ 6 ] t o obtain the v a r i a t i o n of c l e a r a n c e i n t h e end r e g i o n w i t h time. F i g . 8 shows t h e c l e a r a n c e s p r e d i c t e d d u r i n g t h e t r a n s i t i o n of a n e n t r y z o n e t o a n e x i t . I t may b e s e e n t h a t t h e c l e a r a n c e minimum drops as t h e entrainment v e l o c i t y f a l l s and t h a t i t reaches i t s lowest v a l u e j u s t after t h e d i r e c t i o n of m o t i o n c h a n g e s . After t h a t , i t i n c r e a s e s o n c e more. The lowest c l e a r a n c e f o u n d corresponds t o a non-dimensional c l e a r a n c e , h , of j u s t below 1 a n d leads t o a n e x p r e s s i o n f o r minimum f i l m t h i c k n e s s of:
0.6 0.4
-
02 0
02
0.6
0.4
08
1.o
x/b
Fig.
9 Dry c o n t a c t p r e s s u r e d i s t r i b u t i o n s f o r a layer contact
F o r v a l u e s of t h e r a t i o of l a y e r t h i c k n e s s t o s e m i - c o n t a c t w i d t h a b o v e 0.5, t h e p a r a b o l i c profile is essentially retained with a
304 c o n t i n u o u s d e c r e a s e i n p r e s s u r e g r a d i e n t from t h e i n f i n i t e v a l u e a t t h e c o n t a c t e n d s t o zero a t t h e c e n t r e . Below t h i s r a t i o , a p o i n t of inflection develops i n t h e pressure curve r o u g h l y h a l f way between t h e end a n d t h e c e n t r e of t h e c o n t a c t . C o n s i d e r , f i r s t , t h e b e h a v i o u r of c o n t a c t s with v a l u e s of t / b a b o v e 0.5: at low e n t r a i n m e n t v e l o c i t i e s ( h i g h v a l u e s of g3 1 t h e i n l e t sweep w i l l b e small a n d t h e h i g h l y deformed r e s u l t w i l l apply. Then a s t h e e n t r a i n m e n t v e l o c i t y i n c r e a s e s , t h e e x t e n t of the i n l e t sweep a l s o i n c r e a s e s u n t i l it overrides t h e dry contact p r o f i l e completely and t h e l u b r i c a t i o n b e h a v i o u r i s e s s e n t i a l l y t h a t of a r i g i d c o n j u n c t i o n . T h i s b e h a v i o u r p r o d u c e s a smooth b l e n d i n g from t h e r i g i d regime t o t h e e l a s t i c regime a s shown i n F i g . 10 where t h e c e n t r a l f i l m thickness is presented.
found u s i n g t h e p a r a b o l i c i n l e t a n a l y s i s f o r a number of v a l u e s of c3. A s t h e e n t r a i n m e n t v e l o c i t y rises ar?d g3 d e c r e a s e s , t h e maximum p r e s s u r e E r a d i e n t i n t h e i n l e t sweep f a l l s eventually reaching a value equal t o t h a t a t t h e p o i n t o f maximurn p r e s s u r e g r a d i e n t on t h e f l a n k of t h e d r y c o n t a c t c u r v e . C o n t r o l o f t h e f i l m thickness then switches abruptly t o t h a t p r e s s u r e g r a d i e n t a n d i t s v a l u e must, b e found from i n v e r s e t h e o r y , [ 1 6 ] . Finally a t still h i g h e r e n t r a i n m e n t v e l o c i t i e s , t h i s p o i n t of i n f l e c t i o n i s i t s e l f o v e r r i d e n and t h e c o n t a c t shows a s e c o n d t r a n s i t i o n t o the r i g i d regime.
L
\ 0.5 Dlstanco from cent re of contact w / b
Fig.
h,,
Fig.
10
C l e a r a n c e s a t maximum p r e s s u r e i n layer contact
a
The p l o t of t h e r i g i d regime f i l m t h i c k n e s s a g a i n s t t h a t of t h e e l a s t i c regime h a s b e e n a d o p t e d t o e n a b l e t h e regime c h a n g e t o b e shown c l e a r l y . L i n e s of c o n s t a n t g3 are a l s o g i v e n t o e n a b l e a c t u a l c l e a r a n c e s t o b e determined. For t h e H e r t z i a n c o n t a c t case, t / b I OD the b l e n d i n g c u r v e i s t h a t of H e r r e b r u g h [ l j , f o r o t h e r v a l u e s t h e r e s u l t s of G u p t a [ l l ] , B e n n e t t and H i g g i n s o n 1 1 2 1 , Cudworth 1131 a n d Varnam [ 141 have been extended by additional c a l c u l a t i o n s [ 151. F o r lower v a l u e s of t / b , t h e b e h a v i o u r is more complex a n d may most e a s i l y b e u n d e r s t o o d by examining the dry contact pressure g r a d i e n t s . T h e s e a r e p r e s e n t e d i n F i g . 11 which shows t h e g r a d i e n t s f o r t h r e e v a l u e s o f t / b , 0.5, 0.2 and 0.1, and i l l u s t r a t e s the development of t h e i n f l e c t e d p r e s s u r e c u r v e a s t / b d e c r e a s e s . I t a l s o shows, c l e a r l y , how t h e end r e g i o n i n which t h e p r e s s u r e d i s t r i b u t i o n is approximately parabolic decreases with d e c r e a s i n g t /b. At low entrainment velocities, the c l e a r a n c e i s c o n t r o l l e d by t h e p a r a b o l i c r e g i o n of t h e d r y c o n t a c t c u r v e a n d F i g . 1 1 shows t h e maximum p r e s s u r e g r a d i e n t s i n t h e i n l e t r e g i o n
17
I
Pressure gradients i n a dry layer contact
The p r e s e n c e of t h e p o i n t of i n f l e c t i o n i n t h e d r y contact pressure d i s t r i b u t j o n l e a d s t o a n a d d i t i o n a l regime of l u b r i c a t i o n shown by t h e s l o p i n g p a r t of t h e c u r v e s f o r v a l u e s of t / b b e l w 0.2 i n F i g . 10 a n d h a s been l a b e l l e d 'inverse' t o d i s t i n g u i s h i t from t h e f a m i l i a r i n l e t c o n t r o l l e d e l a s t i c regir?e. S i m i l a r r e s u l t s are l i k e l y t o occur f o r a l l c o n t a c t s i n which t h e r e i s a r e s t r i c t e d d e p t h of s o f t r i a t e r i a l a n d , i n G e n e r a l , where t h e d r y c o n t a c t p r o f i l e shows a p o i n t of i n f l e c t i o n o n t h i n l e t s i d e of t h e c o n t a c t there w i l l be t h i s additional 'inverse' regime interposed between t h e r i g i d a n d elastic regimes. References HERRERRUGH, K . ' S o l v i n g t h e i n c o m p r e s s i b l e and isothermal problem in elastohydrodynamic l u b r i c a t i o n through an i n t e g r a l e q u a t i o n ' , T r a n s . ASME, 1968, 262-270. HOOKE, C. J. a n d O'DONOGHUE, J . P. Elastohydrodynamic l u b r i c a t i o n of s o f t , h i g h l y deformed c o n t a c t s ' J . Mech. FnF. S c i . , 1972, 3, 3 4 - 4 8 . HOOKE, C. J. 'A note on the elastohydrodynamic l u b r i c a t i o n of s o f t c o n t a c t s ' , P r o c . I n s t . Mech. E n g r s , 1986, 200C, 189-194. H I R A N O , F. 'Dynamic i n v e r s e problems i n hydrodynamic l u b r i c a t i o n ' , 3 r d I n t . Conf. o n F l u i d S e a l i n e , C a m b r i d p , 1967, F 1 116.
-
305
(5)
HIRANO, F. and KANETA, 14. 'Dynamic behaviour of flexible seals for reciprocating motion', 4th Int. Conf. on Fluid Sealing, Philedelphia, 1969, Session 1 11-19. ( 6 ) HOOKE, C. J. 'The elastohydrodynamic lubrication of soft, highly deformed contacts under conditions of nonuniform motion', Trans ASME F. (in press) (7) BLOK, H. and KOENS, H. J. 'The breathing film between a flexible seal and a reciprocating rod', Proc. Inst. Mech. Engrs, 1966, 180 (pt3b1, 221-223. (8) FIELD, G. J. and NAU, B. S. 'The effects of design parameters on the lubrication of reciprocating rubber seals' , 7th Int. Conf. on Fluid Sealing, Nottineham, 1975, C1 1-13. (9) HOOKE, C. J. 'The elastohydrodynamic lubrication of heavily loaded contacts', J. Mech. Eng. Sci., 1977, 19, 149-156. (10) MEIJERS, P. 'The contacrproblem of a rigid cylinder on an elastic layer', Appl. Sci. Res., 1968, Is,353-383. ( 1 1 ) GUPTA, P. K. 'On the heavily loaded elastohydrodynamic contacts of layered solids', J. Lubr. Technol., 1976, 2, 367374. (12) BENNETT, A. and HIGGINSON, G.R. 'Hydrodynamic lubrication of soft solids', J. Mech. Eng. Sci., 1977, 3,189-192. (13) CUDWORTH, C.' J. 'Finite element solution of the elasto-hydrodynamic lubrication of a compliant layer in pure sliding', Proc. 5th Leeds-Lyon Symp. on Tribology, Leeds, 1978, (Inst. of Mech. Engrs, [,ondon, 19791, 375-378. (14) VARNAM, C. J. 'Equilibriun and transients in the lubrication of a non-Hertzian elastohydrodynamic contact', Ph. D. Thesis, University of Birmingham, 1978. (15) HOOKE, C. J. 'The elastohydrodynamic lubrication of a cylinder on an elastomeric layer', Wear (in press). (16) BLOK, H. 'Inverse problems in hydrodynamic lubrication and design directives for lubricated flexible surfaces', Proc. Int. Symp. on Lubrication and Wear, Berkley, 1963, (McCutchan Publishing Corporation, Houston, TX, 1964), 1-151.
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SESSION X LUBRICANT RHEOLOGY Chairman: Professor A. Cameron PAPER X(i)
Pressure viscosity and compressibility of different mineral oils
PAPER X(ii)
Measurement of viscoelastic parameters in lubricants and calculation of traction curves
PAPER X(iii) High-shear viscosity studies of polymer-containing lubricants PAPER X(iv) Properties of polymeric liquid lubricant films adsorbed on patterned gold and silicon surfaces under high vacuum
This Page Intentionally Left Blank
309
Paper X(i)
Pressure viscosity and compressibility of different mineral oils P. Vergne and D. Berthe
H i g h p r e s s u r e m e a s u r e m e n t s on v a r i o u s l u b r i c a n t s a r e p r e s e n t e d . U l t r a s o n i c r e s u l t s are g i v e n and show t h a t some o f t h e i n v e s t i g a t e d f l u i d s c h a n g e f r o m a l i q u i d s t a t e t o a s o l i d - l i k e p h a s e . P r e s s u r e v i s c o s i t y r e s u l t s a r e a l s o r e p o r t e d and a good c o r r e l a t i o n between u l t r a s o n i c and v i s c o s i t y r e s u l t s i s n o t e d . F u r t h e r m o r e , some u n s t a t i o n a r y e f f e c t s o b t a i n e d o n p a r a f f i n i c b a s e o i l s a r e p r e s e n t e d and d i s c u s s e d . 1
INTRODUCTION
I t i s now w e l l e s t a b l i s h e d t h a t knowledge o f lubricant behaviour b r i n g s valuable information t o p r e d i c t s a t i s f a c t i n g r u n n i n g c o n d i t i o n s of most o f l u b r i c a t e d mechanisms. High p r e s s u r e l u b r i c a n t r h e o l o g y h a s known a considerable extend since several years. A f t e r t h e w e l l known w o r k o f Bridgman c1-31, Barlow e t a 1 have p u b l i s h e d c4-71 many r e s u l t s a b o u t t h e dynamic s t u d y o f f l u i d s u n d e r p r e s s u r e . The a u t h o r s h a v e p r o p o s e d t h e u s u a l B.E.L. m o d e l a n d t h e r e d u c e d v a r i a b l e s method t o describe t h e v i s c o e l a s t i c f l u i d behaviour i n shear. H u t t o n e t a 1 C8-111 h a v e s t u d i e d many types o f l u b r i c a n t s with s e v e r a l t e c h n i q u e s : u l t r a s o n i c t e c h n i q u e i n l o n g i t u d i n a l or i n s h e a r mode, s h e a r f l o w i n c o n c e n t r i c c y l i n d e r s . Viscoelastic behaviour i n shear but a l s o in compression h a v e b e e n shown and t h e d i f f e r e n c e s between %(shear modulus a t i n f i n i t e f r e G ( a p p a r e n t s h e a r modulus deduced quency), f r o m t r a c t i o n c u r v e s ) and G ( m e c h a n i c a l s h e a r modulus) h a v e b e e n c l e a r l y p o i n t e d o u t . More r e c e n t l y , W i n e r e t a 1 [12-181 h a v e produced many r e s u l t s a b o u t l u b r i c a n t r h e o l o g y . P i e z o v i s c o s i t y , g 1a s s t r a n s i t i o n , y i e l d s h e a r s t r e s s h a v e b e e n s t u d i e d , and t h e a u t h o r s h a v e of p r o p o s e d a non l i n e a r v i s c o u s model l u b r i c a n t s s h e a r under p r e s s u r e . I t ' s worth n o t i n g t h a t t h e chronology o f t e n d e n c i e s o b t a i n e d from h i g h p r e s s u r e r h e o l o g y experiments a g r e e s with t h e l u b r i c a n t behaviour models used t o c a l c u l a t e t h e t r a c t i o n c o e f f ic i e n t by n u m e r i c a l s i m u l a t i o n . T h r e e p a r a m e t e r s a r e commonly u s e d t o d e s c r i b e t h e l u b r i c a n t ' s behaviour : pressure, t e m p e r a t u r e and t i m e . These p a r a m e t e r s must n o t b e c o n s i d e r e d i n d e p e n d e n t l y : t h e l a s t one c a n be e x p r e s s e d b y t h e s h e a r r a t e ( i n v i s c o s i t y measurement) or b y t h e e x c i t a t i o n f r e q u e n c y ( i n okcillating experiment), but a l s o by t h e t r a n s i t t i m e or t h e p r e s s u r e d r o p t i m e i n a n EHD c o n t a c t . I n t h i s paper we p r e s e n t a high pressure study of v a r i o u s l u b r i c a n t s , p e r f o r m e d on a f a l l i n g b o d y v i s c o m e t e r i n which t h e knowledge of t h e plunger p o s i t i o n i s c o n t i n u o u s l y o b t a i n e d by u l t r a - s o n i c t e c h n i q u e s .
2
EXPERIMENTAL SET UP AND FLUID DESCRIPTION
2.1 ExDerimental s e t - u o T h e e x p e r i m e n t a l d e v i c e used i n t h i s work h a s The h y d r o b e e n d e s c r i b e d i n [19] a n d [20]. s t a t i c p r e s s u r e c a n r e a c h 0.7 GPa and i s known The t e m p e r a w i t h a r e l a t i v e e r r o r o f 0.5 t u r e c a n v a r y from ambient up t o 80' C and i s measured w i t h a n a b s o l u t e e r r o r o f k 0 . 2 ' C . T h e s e two p a r a m e t e r s a r e d i r e c t l y measured i n t h e main c e l l by s t r a i n gauges and thermocouples. F o r t h e f a l l i n g body d e t e c t i o n , a n u l t r a s o n i c t r a n s d u c e r (2.25 MHz, l o n g i t u d i n a l waves t r a n s m i t t e r - r e c e i v e r ) i s put o u t s i d e t h e h i g h p r e s s u r e v e s s e l . The i n c i d e n t w a v e s p a s s t h r o u g h a m e t a l l i q u i d i n t e r f a c e , are r e f l e c t e d by t h e p l u n g e r a n d g o b a c k t o w a r d s t h e t r a n s d u c e r . The a c o u s t i c impedance o f t h e steel vessel is f a i r l y c o n s t a n t i n our e x p e r i m e n t s b u t t h e f l u i d impedance v a r y i n a s i g n i f i c a n t way w i t h p r e s s u r e a n d t e m p e r a t u r e : t h e sound v e l o c i t y o f l u b r i c a n t s c a n b e m u l t i p l i e d by a f a c t o r o f 2 i n o u r p r e s s u r e r a n g e . Due t o o u r t e c h n i q u e , ( p u l s e m e t h o d with l o n g i t u d i n a l waves), t h e v a r i a t i o n o f s o u n d v e l o c i t y must b e known t o c a l c u l a t e t h e d i s t a n c e between t h e p l u n g e r and t h e u l t r a s o n i c t r a n s d u c e r . T h i s i s done i n a c a l i b r a t i o n e x p e r i m e n t which r e q u i r e s a f i x e d body i n s i d e t h e c e l l , a t a n imposed d i s t a n c e from the transducer.
t.
,
2.2 F l u i d s i n v e s t i g a t e d T h i s w o r k i s c o n c e r n e d b y t h e s t u d y o f 13 l u b r i c a n t s whose d e s c r i p t i o n a n d a m b i e n t p r o p e r t i e s are r e p o r t e d T a b l e I : 7 l u b r i c a n t s have m i n e r a l o r i g i n and 6 are s y n t h e t i c fluids. Among t h e m i n e r a l f l u i d s , 3 p a r a f f i n i c b a s e o i l s , 3 n a p h t e n i c b a s e o i l s and a b l e n d e d p a r a f f i n i c h a p h t e n i c f l u i d h a v e been t e s t e d . T h e s e f l u i d s r e p r e s e n t t h e m o s t common l u b r i c a n t s used i n s t a n d a r d a p p l i c a t i o n s . Note t h a t p a r a f f i n i c p a r t s a r e characterized by
310
p 25'C Nature
p 50'C Pa.s
0.075 0.145 0.26
3.2 3.67 4.0
0.34
4.7
I :ig I
0.04 0.405 0.802
3 5.6 5.94
.822
0.044
2.6
0.078 0.061 0.050 0.049
1.74 1.72 1.60 1.56
.872 .881 .884
P a r a f f i n i c base o i l s
50
2
2
50
napht.
I
paraff.
.895
I
I I
I
I
.915
Naphtenic b a s e o i l s
'r i - t e t r a - p e n t a m e r e
I
of decene
1.023 1.022 0.985 0.962
Iethylchlorophenylpolysiloxane :etrachlorophenylmethylpolysiloxane 'henylmethylpolysiloxane limethylpolysiloxane I-bis
(m-phenoxy phenoxy) benzene
I
I
1.2
I
2.5
15.5
T a b l e I : F l u i d d e s c r i p t i o n and p r o p e r t i e s a t ambient p r e s s u r e l i n e a r o r branched molecular chains while n a p h t e n i c p a r t s a r e c h a r a c t e r i z e d by c y c l i c c h a i n s . As m a t e r i a l s p r e s e n t e d h e r e a r e o b t a i n e d from p e t r o l e u m d i s t i l l a t i o n o r c o ld f i l t e r i n g , we assume t h a t t h e t w o p a r t s and may b e a l s o aromatic f r a c t i o n s a r e p r e s e n t i n each l u b r i c a n t . In any c a s e a dominant w e l l d e f i n e d p a r t i s p r e s e n t i n t h e concerned f l u i d . S y n t h e t i c l u b r i c a n t s c o v e r a wide r a n g e o f a p p l i c a t i o n s , u s u a l l y under s e v e r e c o n d i t i o n s . The f i r s t s y n t h e t i c l u b r i c a n t p r e s e n t e d i s a v e r y l o w w e i g h t p o l y m e r u n l i k e t h o s e used as a d d i t i v e s . I t shows good o x y d a t i v e a n d 45' C ) . t e m p e r a t u r e s t a b i l i t y , (pour point We have a l s o c h o s e n t o t e s t s i l i c o n e f l u i d s , w h i c h a r e u s e d i n l i g h t l y loaded a p p l i c a t i o n s when p r e c i s i o n and s t a b i l i t y a r e o f i m p o r t a n c e . We p r e s e n t f o u r f l u i d s w h i c h c o v e r t h e most usual molecular s t r u c t u r e s o f t h e s e l u b r i c a n t s . The l a s t s y n t h e t i c f l u i d i n v e s t i g a t e d i s t h e 5P4E m-bis (m p h e n o x y p h e n o x y ) b e n z e n e , g e n e r a l l y c a l l e d p o l y p h e n y l e t h e r . Although t h i s f l u i d i s u s e d i n a r e s t r i c t e d number o f a p p l i c a t i o n s , i t i s known f o r i t s " m a g n i f i y i n g e f f e c t s " i n EHD p r o b l e m s a n d h a s e x t e n s i v e l y been s t u d i e d [8-9-14-16] f o r t h i s r e a s o n . On T a b l e I , we have a l s o r e p o r t e d d e n s i t y and v i s c o s i t y o f f l u i d s a t ambient p r es s u re . With t h e s e f i r s t r e s u l t s , o n e c a n o b s e r v e t h e e x i s t e n c e o f f u n d a m e n t a l d i f f e r e n c e s between the two f l u i d origins: mineral l u b r i c a n t s have d e n s i t i e s l y i n g b e t w e e n 0.87 and 0.92 a n d v i s c o s i t y r a t i o s (p 25' C/p 50'C) l y i n g between 3 and 6 w h i l e s y n t h e t i c l u b r i c a n t s c o v e r wider r a n g e s , from 0.82 t o 1.2 f o r d e n s i t y and f r o m 1.6 t o 15.5 f o r t h e v i s c o s i t y r a t i o . I t ' s t h e proof t h a n l u b r i c a n t c h e m i s t r y can provide s u c c e s s f u l l y many t y p e s o f l u b r i c a n t s for particular applications.
-
3
ULTRASONIC RESULTS
F i r s t t o e v a l u a t e t h e confidence i n t e r v a l of t h e u l t r a s o n i c v e l o c i t y measurements , we h a v e p e r f o r m e d e x p e r i m e n t s w i t h p u r e water. This l i q u i d was c h o s e n b e c a u s e i t s s o u n d v e l o c i t y doesn't vary with u l t r a s o n i c frequency [23-241. Our r e s u l t s h a v e b e e n com a r e d w i t h p r e v i o u s d a t a found i n l i t e r a t u r e 121-221 and show a good a g r e e m e n t up t o 0.5 GPa a t various temperatures. D u e t o t h e number of f l u i d s i n v e s t i g a t e d , r e s u l t s are r e p o r t e d i n T a b l e 11-a a n d T a b l e 11-b and some t y p i c a l c a s e s are p l o t t e d i n f i g u r e s 1 a n d 2. A d i s c o n t i n u i t y i n t h e s o u n d v e l o c i t y v e r s u s p r e s s u r e c u r v e s c a n be o b s e r v e d i n some l u b r i c a n t s ( f i g . 2). T h i s d i s c o n t i n u i t y i s i n t e r p r e t e d [24] b y t h e a p p a r i t i o n o f an amorphous phase ( a s o l i d l i k e p h a s e ) i n t h e s a m p l e . Due t o t h e h y d r o s t a t i c p r e s s u r e , m o l e c u l e s a r e c o m p r e s s e d and t h e f r e e v o l u m e a v a i l a b l e i s a l s o reduced. U l t r a s o n i c waves are s e n s i t i v e t o t h i s e v o l u t i o n : l o n g i t u d i n a l waves introduced l o c a l p r e s s ur e f l u c t u a t i o n s and t h e s p e e d o f p r o p a g a t i o n i s d e p e n d a n t o f t h e d e n s i t y and t h e m o l e c u l a r s t a t e o f t h e t e s t e d sample. We r e c a l l t h a t p h a s e t r a n s i t i o n i s a t y p i c a l v i s c o e l a s t i c e x p r e s s i o n [14-17-23-24] o f t h e f l u i d ' s c o m p r e s s i o n a l b e h a v i o u r . It is t i m e dependent : sample h i s t o r y , s o l l i c i t a t i o n t i m e and o b s e r v a t i o n t i m e a r e parameters of importance t o d e s c r i b e t h e a p p a r i t i o n of a g l a s s y p h a s e . In o u r c a s e , t h e u l t r a s o n i c p e r i o d i s less t h a n 1 p s 1 t h e o b s e r v a t i o n t i m e l o n g e r t h a n lo3 s a n d t h e s a m p l e h i s t o r y c o r r e s p o n d s t o i s o t h e r m a l compression.
311
Lubricant
T ' C
H 8303
I
P,
25 40.3 60
Phase t r a n s i t i o n s h a v e been m a i n l y observed i n m i n e r a l f l u i d s (Table 11-b). A t c o n s t a n t temperature, we n o t e t h a t p a r a f f i n i c b a s e o i l s have h i g h e r p r e s s u r e t r a n s i t i o n s t h a n n a p h t e n i c b a s e o i l s . The mixed o i l H 8303 h a s n o t c l e a r l y shown a p r e c i s e d i s c o n t i n u i t y i n t h e c u r v e s b u t some f l u c t u a t i o n s c e r t a i n l y due t o t h e d i f f e r e n t components have been o b s e r v e d . The p o l y p h e n y l e t h e r 5 P 4 E i s t h e o n l y s y n t h e t i c l u b r i c a n t w h i c h h a s shown a n a p p a r e n t s o l i d i f i c a t i o n i n o u r experimental c o n d i t i o n s : t h e t r a n s i t i o n p r e s s u r e s observed f o r t h i s f l u i d a r e weaker t h a n f o r t h e o t h e r lubricants We h a v e r e p o r t e d , i n f i g u r e 3 , t h e v a r i a t i o n o f t r a n s i t i o n p r e s s u r e as a f u n c t i o n of t h e temperature.
GPa
.38 .50 .56 I
PA0 6
19.5 40.4 60
.56 .52 .58
s 1
20.7 24.5 29.4
.46 .50 .52
s 2
24.4
.46
.
4
s 3
24.4
.48
s4
24.4
.45
T a b l e 11-a : L u b r i c a n t s i n which no t r a n s i t i o n Pm i s t h e h a s been o b s e r v e d maxim- reached p r e s s u r e .
.
Lubricant
T 'C
Pt GPa
Pm GPa
200 NEUTRAI
22 40 60 80
.404 .54 .655
.58 .69 .74 .65
350 NEUTRAI
22
.41
.62
600 NEUTRAI
24 40
--
.5
.5 .53
60
R 620 15
750 PALE
26.2 40.3 60 80
.355 .425
---
.52 .54 .52 .62
20.6 40.3
.237 .283 .355
.46 .46 .52
.24 .336
.4 .5
60
40
1300 PALE
5P4E
60 80
18 23 40.4 60
--
.lo6 .128 .21
--
.5 .17 .21
.30 .30
T a b l e 11-b : T r a n s i t i o n p r e s s u r e Pt f o r v a r i o u s lubricants
VISCOSITY RESULTS
F i g u r e 4 shows t y p i c a l r e s u l t s obtained f o r v a r i o u s l u b r i c a n t s a t 40' C a n d 29" C. A global overview of our v i s c o s i t y pressure r e s u l t s i s g i v e n f i g u r e 5 : i n t h i s f i g u r e , we h a v e r e p o r t e d t h e v a r i a t i o n of t h e isothermal s e c a n t p i e z o v i s c o s i t y c o e f f i c i e n t ( a s ) as a f u n c t i o n of t h e t e m p e r a t u r e . I n o u r i n v e s t i g a t i o n , as h a s v a r i e d between 11 GPa-l and 35 GPa-l. I n t h i s c h a r a c t e r i z a t i o n , one c a n show t h a t each type of l u b r i c a n t p r e s e n t s t y p i c a l r e s u l t s . For example, i t i s e v i d e n t t h a t p a r a f f i n i c b a s e o i l s g i v e smaller c o e f f i c i e n t s t h a n n a p h t e n i c b a s e o i l s : it i s a l s o e v i d e n t that synthetic l u b r i c a n t s give r e s u l t s i n a l a r g e r s p e c t r u m t h a n m i n e r a l f l u i d s . The polyphenyl e t h e r 5 P 4 E and t h e p o l y m e r i c f l u i d PA06 r e p r e s e n t t h e e x t r e m e c a s e s of our s t u d y . Under o u r e x p e r i m e n t a l c o n d i t i o n s , t h e s i l i c o n e f l u i d s i n v e s t i g a t e d h a v e n o t shown noteworthy d i f f e r e n c e e x c e p t f o r t h e c h l o r o p h e n y l c l a s s w h i c h seems t o b e . a l i t t l e more performant. Although, the secant isothermal p i e z o v i s c o s i t y c o e f f i c i e n t i s u s e d i n many l u b r i c a t i o n problems, we show ( T a b l e 111) t h a t it doesn't represent exactly the pressure v i s c o s i t y v a r i a t i o n i n our experimental i n v e s t i g a t i o n . I n T a b l e 111, we h a v e a l s o r e p o r t e d at , t h e tangent p i e z o v i s c o s i t y c o e f f i c i e n t a t maximum p r e s s u r e a n d a m ,t h e b e s t mean s q u a r e a p p r o x i m a t i o n o f o u r e x p e r i mental p o i n t s a t a l l t h e p r e s s u r e s t e p s . A g e n e r a l comment c a n b e made o n m i n e r a l f l u i d v a l u e s : i n e a c h c a s e as, t h e s e c a n t c o e f f i c i e n t i s v e r y c l o s e t o a,,,, t h e b e s t mean s q u a r e a p p r o x i m a t i o n of e x p e r i m e n t a l r e s u l t s and a t , t h e t a n g e n t c o e f f i c i e n t i s s m a l l e r t h a n %. A more p r e c i s e o b s e r v a t i o n o f t h e s e v a l u e s c a n d i s t i n g u i s h between two t y p e s o f m i n e r a l f l u i d s : t h o s e which show a l a r g e d i f f e r e n c e b e t w e e n at and a,,, and t h e o t h e r s . For t h e f i r s t t y p e , we f o u n d t h e n a p h t e n i c b a s e o i l s , e s p e c i a l l y R 620 15 a t a l l temperat u r e s i n v e s t i g a t e d , 750 PALE and 1300 PALE a t l o w t e m p e r a t u r e s . A t t h e o p p o s i t e , t h e second c l a s s i s composed o f t h e p a r a f f i n i c b a s e o i l s and t h e compounded o i l H 8303. As mentionned a b o v e , s y n t h e t i c f l u i d s h a v e g i v e n a w i d e r a n g e o f r e s u l t s and i t seems t h a t each f a m i l y of s y n t h e t i c l u b r i c a n t s shows t y p i c a l r e s u l t s . The p o l y m e r PA0 f o r example g i v e s r e s u l t s comparable t o p a r a f f i n i c o i l v a l u e s . We have r e p o r t e d T a b l e I V a n o t h e r i n t e r p r e t a t i o n o f o u r e x p e r i m e n t a l d a t a : we have chosen t h e power l a w t o r e p r e s e n t
312
FLUID
T "C
-
at
a,
GPa -.
(GPa)!OO NEUTRAL
21.3 40. 60. 80.
11.62 11.09 12.45 10.86
2.104 1.956 1.454 1.464
.465 .62
20.16 15.17 11.77 10.25
21.17
16.90 13.79 11.37
16.98 13.94 11.48
;50 NEUTRAL1
22.
.205
21.01
18.89
21.07
100NEUTRAL
24. 40. 60.
6.42 13.51 9.07
4.529 1.636 2.409
24. 40. 60.
.125 .29 .37
23.00 18.05 15.53
20.45 16.96 12.36
23.09
'A0 6
19.5 40.
6.66 6.54
2.997 2.703
-
15.69
23. 40. 60.
.185 .25 .37
27.24 22.20 17.82
25.84 20.48 15.29
27.27 22.25 17.88
26.3 40. 60. 80.
.26 .31 .415 .52
27.29 24.03 19.59 15.91
27.08 23.74 19.88 16.04
27.28 24.00 19.60 15.90
21. 40. 60.
.I75 .205 .32
31.15 26.61 22.33
30.82 24.61 19.68
31.31 26.65 22.38
40. 60. 80.
.205 .29 .335
27.37 22.45 18.63
26.57 20.43 16.84
27.35 22.51 18.71
19.5 40.
.445 .415
12.83 11.87
10.39 9.00
13.03 11.99
20.6 24.4 29.4
.35 .35 .4
16.60 16.17 15.54
18.4 17.7 16.6
15.94 15.44 14.83
24.4
.45
16.86
19.7
16.23
s3
24.4
.45
15.95
20.4
15.42
s4
24.4
.35
13.05
12.6
12.21
5
5 P4E
40. 60.
.1 .16
35.55 25.56
39.72 27.89
34.46 25.52
T h e s e e f f e c t s h a v e been observed on t h e t h r e e p a r a f f i n i c base o i l s a t d i f f e r e n t temperatures and f o r h i g h e r p r e s s u r e s than those reported f o r v i s c o s i t y measurements. On f i g u r e 6 , w e h a v e r e p o r t e d t h e v a r i a t i o n of v i s c o s i t y d u r i n g a long p e r i o d a t c o n s t a n t t e m p e r a t u r e and p r e s s u r e . A f t e r p r e s s u r e a n d t e m p e r a t u r e s t a b i l i z a t i o n , we o b s e r v e a d e c r e a s e o f v i s c o s i t y w i t h number of s o l l i c i t a t i o n , b u t i f we s t o p t h e e x p e r i m e n t f o r a long t i m e , and s t a r t a g a i n , we n o t e t h a t t h e phenomenon i s r e p e a t a b l e .
200 NEUTRAI
21.3 40. 60. 80. 22.
350 NEUTRAl
600 NEUTRAl
H 8303
-
L
620 I!
'50 PAL1
1300 PALE
-
s1
s2
-
T a b l e I11 : P r e s s u r e
I
1
-
-
-
18.10
viscosity results
1
11.87
I
2.144
T a b l e I V : Power l a w p a r a m e t e r s f o r p a r a f f i n i c f l u i d s and PA0 6 p a r a f f i n i c o i l s a n d PA0 r e s u l t s . T h i s l a w needs t w o p a r a m e t e r s a and n,and i s w r i t t e n a s follow :
For t h e s p e c i f i e d l u b r i c a n t s , t h i s l a w g i v e s a b e t t e r f i t of experimental r e s u l t s t h a n t h e c l a s s i c a l e x p o n e n t i a l l a w b u t it w i l l b e more c o m p l i c a t e d t o i n t r o d u c e t h i s model i n c a l c u l a t i o n s , d u e t o t h e e x i s t e n c e o f two parameters. R e s u l t s o b t a i n e d on t h e 5P4E a r e d i f f i c u l t t o e v a l u a t e b e c a u s e o f t h e small p r e s s u r e r a n g e i n v e s t i g a t e d and t h e h i g h ambient v i s c o s i t y . I n comparison t o Winer's [17] r e s u l t s , w e have noted t h a t higher pressures m u s t be reached t o s t a b i l i z e t h e i n c r e a s e o f v i s c o sity. Silicone f l u i d s h a v e shown some f l u c t u a t i o n s a p a r t from t h e e x p o n e n t i a l law. We h a v e o b s e r v e d f i r s t a d e c r e a s e o f t h e v i s c o s i t y r i s e i n t h e medium p r e s s u r e r a n g e ( a r o u n d 0.2 GPa) and s e c o n d l y an i n c r e a s e o f t h i s r a t e f o r h i g h e r p r e s s u r e ( > 0.35 G P a ) . These f l u c t u a t i o n s c a n b e found i n t h e as i s d i f f e r e n c e s b e t w e e n a,, a t and am always g r e a t e r t h a n g, b u t smaller t h a n at. These d e v i a t i o n s f r o m e x p o n e n t i a l l a w and e s p e c i a l l y t h e o b s e r v a t i o n t h a t a t i s g e n e r a l l y d i f f e r e n t t o a,,, p o i n t o u t t h e d i f f i c u l t y t o estimate viscosity outside e x p e r imen t a 1 c o n d i t i o n s .
.
TIME EFFECTS I N PRESSURE VISCOSITY MEASUREMENTS
313 The s e c o n d e x p e r i m e n t i n which time effects have been observed c o n s i s t s i n measuring v i s c o s i t y i n f u n c t i o n of the t i m e a f t e r a p r e s s u r e s t e p . For each p o i n t r e p o r t e d i n f i g u r e 7 , c o r r e s p o n d s a w a i t i n g time between reaching t h e f i n a l p r e s s u r e and s t a r t i n g t h e e x p e r i m e n t . Between two measurements, p r e s s u r e is decreased t o a t m o s p h e r i c p r e s s u r e and t h e f l u i d i s a t r e s t d u r i n g more t h a n one h o u r . Due t o t h e l a r g e t i m e s c a l e of o b s e r v a t i o n and t h e l a r g e v i s c o s i t y v a r i a t i o n , the axes plotted i n figure 7 are logarithmic. After reaching t h e f i n a l p r e s s u r e , we o b s e r v e a n i n c r e a s e o f v i s c o s i t y , then a p l a t e a u , i n f u n c t i o n o f t h e t i m e d e l a y . The t i m e n e e d e d t o a c h i e v e a c o n s t a n t v a l u e i s a b o u t 10 m i n u t e s . Due t o t h e time s c a l e o f our e x p e r i m e n t s , t r a n s i e n t v i s c o s i t y due t o c o m p r e s s i o n n a l v i s c o e l a s t i c i t y c a n n o t b e a d v a n c e d t o e x p l a i n t h e two phenomena. We r e c a l l t h a t o n l y p a r a f f i n i c b a s e o i l s h a v e shown t h e s e e f f e c t s i n o u r e x p e r i m e n t a l c o n d i t i o n s . These f l u i d s are m a i n l y composed o f l i n e a r c h a i n s and t h e i n c r e a s e d p r e s s u r e can prevent m o l e c u l e movements. Due t o t h e s u c c e s s i v e crossing of t h e plunger i n t h e c e l l , m o l e c u l e s c a n b e o r i e n t e d and d u e t o t h e pressure,heavier molecules (from naphtenic p a r t s ) are in a f r o z e n s t a t e a n d c a n k e e p i n memory t h e o r i e n t a t i o n d u r i n g a long t i m e . W e know from pour p o i n t e x p e r i m e n t s t h a t o n l y a small f r o z e n p e r c e n t a g e i s needed t o r e a c h an The t i m e amorphous s t a t e i n a l l t h e v o l u m e e f f e c t o b s e r v e d i n f i g u r e 7 can b e a t t r i b u t e d t o t h e t i m e needed f o r a f r a c t i o n t o a t t a i n a vitrous state. The phenomena shown f i g u r e s 6 and 7 a r e r a t h e r c o n c e r n e d w i t h m a t e r i a l memory t h a n t h i x o t r o p y or c o m p r e s s i o n a l v i s c o e l a s t i c i t y . T h e s e phenomena m u s t b e r e l a t e d t o a n o n homogeneity i n t h e sample, a t a m i c r o s c o p i c s c a l e ( m o l e c u l a r s t r u c t u r e ) or a t a macroscopic s c a l e ( f r o z e n or amorphous p h a s e ) .
2. 3.
4.
.
5.
6. 7.
8.
9.
.
10.
11. 6
CONCLUSION
I n t h i s p a p e r , we have reported high pressure m e a s u r e m e n t s o n v a r i o u s l u b r i c a n t s . Some e s s e n t i a l p o i n t s have been shown :
12.
-
typical results have been obtained i n function of the molecular n a t u r e of t h e l u b r i c a n t s and i n d e p e n d e n t l y o f t h e c h a r a c t e r i za t i o n , s y n t h e t i c l u b r i c a n t s p r o p e r t i e s cover a l a r g e r range t h a n m i n e r a l l u b r i c a n t s , some f l u c t u a t i o n s n e a r t h e e x p o n e n t i a l law have been n o t e d f o r a l l l u b r i c a n t s e x c e p t f o r naphtenic o i l s , i t is not reasonable t o e x t r a p o l a t e v i s c o s i t y measurements o u t s i d e t h e e x p e r Lnenta 1 conditions, time e f f e c t s c a n b e found a t lower p r e s s u r e t h a n c l a s s i c a l EHD p r e s s u r e and a r e a t t r i b u t a t e d t o t h e l u b r i c a n t composition.
13.
-
14.
-
-
15.
References 16.
1.
BRIDGMAN P.W.;"The E f f e c t o f P r e s s u r e on the Viscosity of Forty-Three Pure L i q u i d s " , P r o c e e d i n g s American Academy o f Arts and S c i e n c e , v o l 61, n o 3,p56-99,1926.
BRIDGMAN P.W. ; " V i s c o s i t i e s up 30000kg/cm2 " , P r o c e e d i n g s A m e r i c a n 4cademy o f A r t s and S c i e n c e ,v o l 77 ,n o 4 ,p l l 5 - 1 2 8 ,1949. BRIDGMAN P.W.;"Further Rough Compressions to 40000kg/cm2,Especially Certain L i q u i d s " , P r o c e e d i n g s American Academy o f A r t s and S c i e n c e , v o l 7 7 , n ' 4,p129146 ,1949. BARLOW A. J. ,LAMB J. ,MATHESON A. J. ,PADMINT P.R.K.L. a n d RICHTER J . ; " V i s c o e l a s t i c Relaxation of Supercooled Liquids ( I ) " ,Proc Roy. SOC ,London ,A29 8 ,p467 480,1967. BARLOW A. J. ,ERGINSAV A. and LAMB J. ; " V i s coelastic R e l a x a t i o n of Supercooled Liquids (II)",Proc. Roy. S O C .,London,A298,p481-494,1967. I R V I N G J . B . and BARLOW A.J.;"An Automatic High P r e s s u r e V i s c o m e t e r " , J o u r n a l P h y s . Eng. Sc I n s t r . , n o 4,p232-236 1971. BARLOW A.J. ,HARRISON G. , I R V I N G J . B . , K I M M.G.,LAMB J. and PURSLEY W.C.;"The E f f e c t o f P r e s s u r e on t h e V i s c o e l a s t i c P r o p e r t i e s o f L i q u i d s " , P r o c . Roy. S o c . , L o n don,A327,p403-412,1972. HUTTON J . F . a n d PHILLIPS M.C. ; " S h e a r Modulus o f L i q u i d s a t E l a s t o h y d r o d y n a m i c L u b r i c a t i o n Pressures",Natural Physical S c i e n c e , v o l 238,p141-142,1972. HUTTON J . F . a n d PHILIPS M.C. "High p r e s s u r e v i s c o s i t y of a P o l y p h e n y l E t h e r M e a s u r e d w i t h a new C o u e t t e Viscometer", Natural P h y s i c a l S c i e n c e , v o l . 245, p. 15-16, 1973. HUTTON J.F.,PHILLIPS M.C.,JESSIE ELLIS,POWELL G. a n d WYN-JONES E.; " V i s c o e 1a s t i c i t y S t ud i e s o f L u b r i c a n t s and o t h e r L i q u i d s i n S h e a r and Bulk D e f o r m a t i o n a t Various F r e q u e n c i e s , P r e s s u r e s and T e m p e r a t u r e s " , P r o c e e d i n g s o f the f i f t h Leeds-Lyon Symposium o n T r i b o l o g y , p173-187,1978. HUTTON J . F . ; "Reassessment o f R h e o l o g i c a l P r o p e r t i e s o f LVI 2 6 0 O i l M e a s u r e d i n a D i s k M a c h i n e " , ASME, J o u r n a l o f Tribol o g y , v o l . 106, 1984, p. 536. NOVAK J . D . and WINER W.O. ;"Some Measurements o f High P r e s s u r e l u b r i c a n t R h e o l o g y " , T r a n s A.S.M.E. ,J o u r n a l o f L u b r i c a t i o n Technology, v o l 90,n' 3,p580-591,1968. JONES W.R. ,JOHNSON R.L. ,WINER W. 0. and SANBORN D.M. ; " P r e s s u r e V i s c o s i t y Measurem e n t s f o r s e v e r a l L u b r i c a n t s t o 5 . 5 108 Newton Per Square Meter and 149'C",A.S.L.E. T r a n s . , v o l 1 8 , n ' 4,p249262 ,1974. ALSAAD M. , B A I R S. a n d WINER W.O.;"Glass Transition i n Lubricants:its R e l a t i o n t o E 1a s t o h yd r o d ynamic Lub r i c a t i o n " ,Trans. A.S.M.E. , J o u r n a l o f L u b r i c a t i o n Technol o g y , v o l 1 0 0 , p 404-417,1978. B A I R S. a n d W I N E R W.O.;"A Rheological Model f o r E l a s t o h y d r o d y n a m i c C o n t a c t s Based o n Primary L a b o r a t o r y D a t a " , T r a n s . A.S.M.E. ,J o u r n a l o f L u b r i c a t i o n Technol o g y , v o l lOl,p258-265,1979. B A I R S. a n d WINER W.O.;"Shear Strength M e a s u r e m e n t s o f L u b r i c a n t s a t High Pressure",Trans. A.S.M.E. Journal of L u b r i c a t i o n T e c h n o l o g y , v o l 101, p251257,1979.
.
-
.
,
,
314 B A I R S. a n d WINER W.O.;"Some O b s e r v a t i o n s i n High P r e s s u r e R h e o l o g y o f L u b r i cants",Trans. A.S.M.E. , J o u r n a l of Lubrication Technology,vol 1 0 4 , p357364,1982. YASOTUMI S.,BAIR S . a n d W I N E R W.O.;"An A p p l i c a t i o n o f a Free-Volume Model t o L u b r i c a n t Rheology ( I ) Dependence o f V i s c o s i t y on Temperature and P r e s s u r e ( I 1 ) V a r i a t i o n i n V i s c o s i t y o f B i n a r y Blended Lubricants",Trans A.S.M.E. , J o u r n a l o f T r i b o l o g y , v o l 106,p291-312,1984. VERGNE P h . , BERTHE D. a n d FLAMAND L. "Glassy t r a n s i t i o n of v a r i o u s l u b r i c a n t s " , i n Mixed L u b r i c a t i o n and L u b r i c a t e d W e a r , L e e d s , September 1984. VERGNE P h . " C o n t r i b u t i o n ?I l ' e t u d e d u : Deformation d e s s u r f a c e s , c o n t a c t E.H.D. caracterisation haute pression des l u b r i f i a n t 8 " . These d e D o c t e u r - I n g e n i e u r s o u t e n u e ?I I ' I N S A d e Lyon l e 2 1 F e v r i e r 1985. STUEHR J . a n d YEAGER E.;"The P r o p a g a t i o n o f U l t r a s o n i c Waves i n E l e c t r o l y t i c S o l u t i o n s " i n P h y s i c a l Acoustics,chapter 6 , v o l I1 A , e d i t e d by MASON W.P. , A c a d e m i c Press,1965. PETITET J.P.,TUFEU R . a n d LE N E I N D R E B . ; " D e t e r m i n a t i o n o f t h e Thermodynamic P r o p e r t i e s o f W a t e r from Measurements of t h e SDeed o f Sound i n t h e T e m D e r a t u r e Range' 251.15-293.15 K and t h e P r e s s u r e Range 0.1-350 MPa" In t e r n a t i o n a 1 J o u r n a 1 o f Thermophysics,vol 4,n' l,p35-50,1983. HARRISON G.;"The Dynamic P r o p e r t i e s o f S u p e r c o o l e d Liquids",Academic P r e s s , 1 9 7 6 . MATHESON A. J. ;"Molecular A c o u s t i c " , W i l e y Interscience,l971.
17.
18.
19.
20.
3000 long. speed (mls)
2000
,
21.
,
22.
,
23. 24.
o
PA0 6
19.5 'C
A
PA0 6
60.C
+
51
20.7.C
.*
51
29.4.C
1
I
I
I
0.2
I 0.4
I
I 0.6
pressure (GPa) Fig. 1 Variation ot longitudinal speed as a tunction of pressure witnout observed transition
3000 Img.
sped (mk)
-k/ 2000
o 200 NEUTRAL 6O'C + 350 NEUTRAL A
R 620 15
o
5P4E
22'C 26'C 23'C
o.2
tI /
pR 620 15
d
x isobaric
cooling (141
0 0
40
80 temperature ('C)
Fig. 3
0
0.2
0.4
0.6 pressure ( GPa)
Fig 2
Variation of longitudinal speed a s a function of
pressure with observrd transition
Transition pressure as a function attwnpemture
b
315
40 d
x 350 NEUTRAL
( GPO-')
5P4E 1171
A R620 15 1171
10'
.
A 52
tb'P0
s3 SL
10
0
o
A
,
5 P4C 13XI Pale 40.c PA0 6 200 Neutral
200 NEUTRAL
PA06
20
40
pressure ( GPa )
Fig 4
Variation ot viscosity as a tunctlon o t pressure vuriou s lubricants
tor
vlscoslty (Ftr 5 )
80
40 20
10
60
80
temperature ( ' C )
Fig 5 . Pressure viscosity coetticient as a t u n c t m of temperature
4
..
11)
.
140 18)
200
.
t 2,5
m . ~
l P
I
5
I
viscosity
.
-
t.'24hrs M
F lg 6
I
1.48
1 hour
Variation o t vismsity as a tunction at time
hrs
a t constant pressure
This Page Intentionally Left Blank
317
Paper X(ii)
Measurement of viscoelastic parameters in lubricants and calculation of traction curves P. Bezot and C.Hesse-Bezot
In this paper, optical techniques (Fabry Perot and self beating correlation) are first described and used to measure the viscoelastic parameters of 5P4E as a function of the pressure and temperature. These results represent an extension of the data available in the literature. Then, various rheological models for fluids behaviour in an elastohydrodynamic contact (E.H.D.) are described. At least, using previous viscoelastic results, the linear part of traction curves, in a linear E.H.D. contact, is computed.
1
INTRODUCTION
It is well known that, in elastohydrodynamic (E.H.D.) experiments, the lubricants experience a large pressure jump in a short transit time. It is now recognized that under these conditions, the usual hypothesis of Newtonian behaviour is, strictly speaking, no longer valid ; particularly for traction curve simulations. However, for film thickness calculations, it has been shown that itis not necessazy to take into account this non Newtonian behaviour (1). The purpose of this paper is to focus atten tion on the linear part of the traction curve by taking into account (or not) the fact that the fluid behaviour is viscoelastic. A s our aim is also to compare numerical results obtained from various rheological models (viscoelastic or not), it has been necessary, in a first stage, to choose one particular lubricant as a test fluid and measure precisely, on as a large range of pressure and temperature as possible, its viscoelastic parameters. The first part of this paper will be devoted to an extensive study of the viscoelastic properties of 5P4E (m bis (m-phenoxy ,phenoxy) benzene) : values of shear and compressional moduli together with the structural relaxation time obtained by means of polarized and depolarized light scattering (L.S.) techniques, as a function of the pressure and the temperature, are given. These results represent a necessary extension of the data already available in the literature. In the second part, we breafly describe a sofisticated "viscoelastic with retarded compression" ncdel (V.R.) whose basis is due to Trachman and Harrison ( 2 ) and more recently to the beautiful work of Montrose and a1 ( 3 ) . It is then easy to canpare other models to the V.R. model. Lastly, computational results using the measured viscoelastic parameters of 5P4E are given for the various models under different experimental conditions
.
2 LIGHT SCATTERING STUDY OF 5P4E 2.1
General recall
When transparent liquid samp es are irradiated by polarized electromagnetic laser light, they
scatter light waves in all directions. This scattered light is due to static and dynamic dielectric constant inhomogeneities. Moreover, these fluctuations in dielectric constant are directly related to density and temperature fluctuations for its isotropic part and to anisotropic local order or rotational movement of molecules for its anisotropic part. A s a consequence, it is easy to see that the spectral analysis of the polarized spectra (I: (k,w)) will give informations on density fluctuations and particularly, on propagative longitudinal hypersonic waves and the isotropic part of local order fluctuations. In the same way, if the molecules are anisotropic, the depolarized spectra (It! (k,a)) will contain information on propagative shear waves (4). 2.2
Experimental setup and Method of analysis
The 5P4E samples where purchased from Monsanto Co and directly used in the pressure cell. The exciting laser source was a COHERENT INOVA 90-3 either in a single moie (500 mW) or multimode (1 W) output at 5145 A depending on the analysis technique. The scattered light was analyzed either by high resolution Fabry Perot Spectroscopy for the propagative mode study (longitudinal and transverse) or by photon correlation spectroscopy for the nonpropagative mode study (structural relaxation process). Details of the former technique together with the high pressure optical system (0-400 MPa) have recently been described (5). Concerning the correlation photon technique, we used a digital correlator : the scattering volume is directly imaged onto an aperture of 0.1 mm in front of the detector, a 9563 E.M.I. photomultiplier (P.M.). The resolved photodetections emerging from the back of the P.M. are then directly correlated (after amplification, discrimination and standardization ) by a Langley Ford digital correlator. More details on the latter experimental arrangement can be found in ( 6 ) . Fig. 1 shows typical spectra with both depolarized (small satellites corresponding to shear waves) and polarized (large satellites) components at T = 41°C and P = 0.1 GPa, obtained
318
by the Fabry Perot technique (frequency domain). Note that it is possible to obtain separately either polarized or depolarized components by an appropriate choice of the polarization. First we consider the satellites analysis : the apparatus function has been found to adjust to a Lorentzian line with a sufficiently good least square criterion, the finess being nearly 100. Consequently, the deconvolution procedure is greatly simplified. Both polarized and depolarized displaced lines are generally well described by the expression [7]: I(O)=
Ari/((mfy)2
+ Ti2) +
where d; values correspond to the frequency of the longitudinal or transverse propagative wave at the selected scattering wave number k = 4 W n / X s i n ( 0 1 2 ) defined by the experimental geometry ( 8 i s the scattering angle, n the refractive index, and the wavelength of the laser light in vacuum). At high pressure, however, as B becomes negligible compared to A, we used only pure displaced lorentzian lines. Now, concerning the central component of the polarized part of the spectra shown in Fig. I , at very high viscosity it is generally composed of a broad line ( 1 M Hz) related to the thermal fluctuations and a narrower one called the Mountain line, which is related to the structural relaxation (8). This latter line is too narrow, in this high viscosity domain to be resolved by the Fabry Perot interferometer. It is thus necessary to use the correlation technique In our homodyne correlation experiment, we directly obtain C(t)=a( l+b@(t)), in which b is a spatial coherence factor and a depends on the average number of photocounts in the sampling time. The 0(t) correlation function is the Fourier Transform of the central line of Fig. 1. This correlation function will be supposed latter to have a particular form depending on two parameters characterizing the relaxation process. The density values at various pressures and temperatures were obtained from the 9 and n values at normal conditions from the LorentzLorentz formula and from the refractive index values at various pressures and temperatures. The viscosity data at P = 0.1 MPa, and its variation with temperature were either obtained from the literature (8,9) or measured by Ph. Vergne (10). At various P and T, the viscosity values were deduced from the W.L.F. equation whose parameters have been least square adjusted in the literature by Winer ( 1 1 ) .
.
2.3 Moduli determination as a function of Temperature and Pressure From the depolarized spectra analysis 2.2 , the frequency C d s = d i is obtained and the velocity Vs= as/k calculated at various pressutes and temperatures. At very high viscosity, near the glass transition, the shear modulus Gis directly related to the velocity values by the simple relation Vs=(Gm / 9 )ll2. As the viscosity decreases (lower pressure and/ or higher temperatures) and if the c3, frequency is still larger than Zm-'(where 'Z,=v/G= is the Maxwell relaxation time), it is possible to greatly improve this relation ( 1 2 ) , by using
the more complexe expression : Vs = (Goo / y ) * / ' f(G)%) 123 As 7 andf are independently measured quantities, Goo is the only unknown parameter that can be calculated by a least square adjustment of equation 121 on our measured Vs data. From the polarized part of the spectra, and r p were also computed and Vp =wp/k calculated. As in the case of the depolarized spectra, at very high viscosity near the glass transiwhere , M o o i s the lontion : Vp = (MmF) ]I1 gitudinal modulus. At lower viscosity, this relation is not applicable. It must be replaced by more complex expression (12) that requires the knowledge of both the widthrp and the frequency shift up. The Goo values together with Mwand Kw, the compressional modulus (=Moo -4/3Gm) are reported in Fig. 2 at P = 0.1 GPa as a function of the temperature. The moduli values obtain d as a function of the pressure at various temperatures are reported in Fig. 3, 4 and 5 for Gm , Moo and K a , respectively. The M modulus, not reported here, has been obtained Yrom high temperature longitudinal velocity measurements. To our knowledge, the 5P4E shear and longitudinal moduli have also been measured by Litovitz (8) and Winer ( 1 3 , 14) by means of L.S. technique. However, in the first paper, experiments were performed at only 22°C as a function of the pressure. In the second one, the longitudinal moduli were measured at various temperatures and generally at higher pressures than ours. In both papers, no experimental data of GOO as a function of the temperature and pressure are reported. Our experimental values are in good agreement with Winer's data in the overlap region (for example their values at 24.4"C at 0.1 and 0.2 GPa are to be compared with our values at 26°C and the same pressures). They are generally slightly lower however than the values given by Litovitz. 2.4
Structural relaxation process study.
At very high viscosity, in the vicinity of the glass transition, the Mountain line has been studied by correlation photon spectroscopy that compute directly the correlation function
cexp
(t)
.
At each temperature and pressure study in the high viscosity region, the analytical expreshas been adjusted sion C(t) = a ( 1 + b0 '(t) to the digital experimental correlation function Cexp (t). Here, 8(t) has been taken to be equal to the empirical Williams Watt (15) relaxation function exp(-(t/z )Q ) , in which 04841 is related to the width of the distribution function and 2 is nearly the time of its maximum amplitude which decays asymmetrically on both sides. This relaxation function has generally proved to adequately represent the experimental data obtained from viscous liquids. Experimental results on 2 together with q/GOp , at + 27'C and + 50'C are reported in Fig. 6 . as a function of the temperature. Note that although we don't know the exact form of the relaxation function used by Litovitz et al, our results seem to be in qualitative agreement with theirs. It is also worth noting here that
319
i t i s 7 and n o t [ ? > t h a t w i l l be used i n one of t h e f o l l o w i n g models f o r t h e t r a c t i o n c u r v e s . 3
Maxwell model w i t h o u t r e t a r d e d compres3.2.1. s i o n (MWI)
NUMERICAL SIMULATION OF TRACTION CURVES
We i n t e n d now t o compare, i n t h e c a s e of 5P4E, p r e d i c t i o n s of v a r i o u s models, f o r t h e c a l c u l a t i o n o f t h e l i n e a r p a r t of t r a c t i o n c u r v e s i s t h e mean f r i c t i o n i n t h e U = f ( i ), where c o n t a c t and t h e s t r a i n r a t e . For reasons of s i m p l i c i t y we o n l y p r e s e n t t h e s i t u a t i o n of a linear contact. In each m?del, t h e s l o p e of t h e t r a c t i o n c u r v e a t small € ( i . e . t h e mean v i s c o s i t y 9 ) i s computed. The v i s c o e l a s t i c r e t a r d e d model w i l l be p r e s e n t e d i n p a r t 3 . 1 and compared t o o t h e r ones R e s u l t s of n u m e r i c a l s i m u l a t i o n s f o r i n 3.2 v a r i o u s models a r e g i v e n and compared t o g e t h e r i n p a r t 3.3.
-
a
.
3.1
i n t o account.
I n e x p r e s s i o n [3}, z s ( t ) i s t a k e n now t o be e q u a l t o \ j ( t ) / G a , where ( t ) = qo e x p U P ( t ) [ d Here, P ( t i s e q u a l t o t h e e x t e r n a l H e r t z p r e s s u r e a t e a c h i n s t a n t of t h e t r a n s i t t i m e . The mean v i s c o s i t y i s s t i l l computed from expression [6] Note h e r e t h a t i t seems t h e r e i s some c o n t r a d i c t i o n s i n u s i n g e x p r e s s i o n 171 t o g e t h e r w i t h r e l a t i o n 1 3 1 i n which$(t) i s taken t o be e q u a l t o a Maxwell f u n c t i o n .
.
3.2.2.
Newtonian model
The s:ress, a t e a c h c o n t a c t p o i n t , i s g i v e n by : G ( t ) = e 7 0 e x p ( o ( P ( t ) ) and, from [61 ,
-
V i s c o e l a s t i c r e t a r d e d model ( 2 , 3)
t o
=(~o/to0 ) ~ e x( op ( P ( t ) d t . The mean s t r e s s through t h e c o n t a c t h a s t o be computed ; t h a t i s t o s a y t h a t G ( t ) must be known a t each i n s t a n t i n t h e c o n t a c t . I t i s supposed t h a t , a t t = O , t h e f l u i d element i s a t t h e c o n t a c t e n t r a n c e . One g e n e r a l form o f G ( t ) may be e x p r e s s e d as :
I t can be e a s i l y s h o w n t h a t i t i s t h e l i m i t i n g c a s e of t h e V.R. model i n which a l l t h e r e l a x a t i o n times a r e suppose t o be s h o r t e r t h a n to.
E l a s t i c model (E.L.)
3.2.3.
a t each c o n t a c t p o i n t , i s w r i t t e n , we g e t != 0.5& to. T h i s e x p r e s s i o n i s o b t a i n e d from t h e V.R. model by assuming t h a t , i n e x p r e s s i o n s l 3 3 and 1 6 1 , a l l t h e r e l a x a t i o n times are l o n g e r t h a n to. The stress.,
a_s G ( t ) = ElGm t and, from [6]
This e x p r e s s i o n t a k e s i n t o a c c o u n t t h e r e l a x a t i o n a l e f f e c t s through t h e # f u n c t i o n . In t h i s model, ' Z s , s h e a r r e l a x a t i o n t i m e , i s supposed t o v a r y through t h e c o n t a c t . The main problem i s t h e n t o compute ? , ( t ) . Note t h a t i t i s n o t e q u a l t o t h e s h e a r r e l a x a t i o n time i n t h e f l u i d element a t e q u i l i b r i u m w i t h t h e external Hertz pressure. It i s supposed t h a t :
3.2.4. Maxwell model w i t h one mean r e l a x a t i o n time (MW2) L e t us d e f i n e a mean v i s c o s i t y ( t h e e q u i l i b r i u m viscosity) :
-
?s ( t 1/ 2, ( 0 1=?( t ) /?(0 ) =exp ( Vc ( 1 / v f ( t 1- 1 /V (0 ) ) L4] where V f ( t ) and V a r e r e s p e c t i v e l y t h e f r e e volume and t h e cfose-packed volume o f t h e f l u i d element i n t h e c o n t a c t , and 2 ( t ) i s t h e s t r u c t u r a l r e l a x a t i o n t i m e measured by L.S. According t o Montrose e t a l , V f ( t ) i s supposed t o follow t h e e v o l u t i o n e q u a t i o n : Vf ( t ) = V f (O)+Jdt
If(t
0
and a mean Deborah number : DEB= Ve4G@to) From e x p r e s s i o n 1 3 2 i n w h i c h $ ( t ) i s r e p l a c e d by e x p ( - ( t G w ) / y e ) and e x p r e s s i o n c 6 1 , w e obtain :
-
= I)
'to
= qoexp ( K P ) , where P=t:fP(t)dt
(3Vf ( t I ) / 3 P ) d P / d t ' 3.3.
ve
(I+DEB(exp(-1/DEB)-l))
Numerical r e s u l t s
The v i s c o e l a s t i c p a r a m e t e r s of 5P4E deduced from p a r t 2 and u s e f u l f o r t h e s i m u l a t i o n s are : where d ( t ) = Ko(t)/Km ( t ) i s supposed t o be independant of t. After mathematical t r a n s f o r m a t i o n s on 14) and 153 and u s i n g t h e p r e v i o u s l y measured v i s c o e qo and o( , w e l a s t i c parameters I f = K , / k , G,, have computed, i n v a r i o u s e x p e r i m e n t a l condit i o n s G ( t ) and t h e mean v i s c o s i t y :
2
0
I:61
where to i s t h e t r a n s i t t i m e . 3.2
Comparaison w i t h o t h e r s models
A l l t h e f o l l o w i n g models have been more o r less used i n t h e p a s t by many a u t h o r s ( 1 6 , 1 7 ) . They a l s o o f t e n c o n s t i t u t e a p a r t of more g e n e r a l models i n which non l i n e a r e f f e c t s were t a k e n
' y o = 1.3
Pas, G Q =
0.68 GPa
25, Ko/Km= 0.59 a n d o ( = 45 GPa-' The mean l o a d s , t r a n s i t times and c o n t a c t l e n g t h s a r e t a k e n from t h e l i t e r a t u r e experimental conditions ( 1 8). I n a l l t h e s i m u l a t i o n s , i t i s supposed t h a t : - The a p p l i e d p r e s s u r e h a s a n H e r t z i a n p r o f i l e . - The G o o modulus i s c o n s t a n t a l o n g t h e c o n t a c t , - The r e l a x a t i o n f u n c t i o n f ( t ) i s t h e Williams Watt f u n c t i o n . $ ( t ) = e x p - ( t / 2 ) p ) w i t h 8 = 1, or 0.5. I n a f i r s t s t e p , we s h a l l focus our a t t e n t i o n on t h e f l u i d b e h a v i o u r a l o n g t h e c o n t a c t , a f t e r w a r d s i t i s t h e mean v a l u e o f t h e v i s c o sity, , t h a t w i l l be considered.
7
320
In the V.R. model, we can define an effective pressure, corresponding to "the free volume state" of the fluid in the contact (of course different from the external applied pressure) :
The Hertz and effective pressure profiles are presented-in Fig. 7 in the experimental conditions of P = 0.302 GPa and U = 0.168 ms-l and 3.99 ms-1. This figure shows distinctly the effect of the mean fluid velocity on the effective pressure beiaviour Its mean valLe (PE ) has also been computed in the case of P = 0.302 GPa. We obtain PE = 0.244 GPa for U = 0.188 ms-l and -& = 0.213 GPa for U = 3.99 ms-l. In figs. 8 and 9, we have reported the time dependant behaviour of the viscosity, 9 (t), in the contact, for the V.R. and MW1 models. It is to be noted that, although its profile is nearly the same in both models, its absolute value is quite different, particularly at low speed. To test more precisely the rheological models, the computed mean viscosity will be presented either for a constant applied pressure as a function of the speed, or for a constant speed as a function of the applied pressure : The fig. 10 and 1 1 show the mean viscosities obtained for two rolling speeds (U = 2.2 ms-l and 0.6 ms-I), as a function of the applied pressure. The differences between the viscosity values are the most significant (up to a factor of 2 or 3) under low pressure and velocity conditions. Fig. 12 and 13 show the experimental (18)and computed mean viscosities as a function of the rolling speed for the mean applied pressure P = 0.302 CPa and P = 0.371 GPa. At high speed, all the models give nearly the same valuesin relatively good agreement with experimental ones. But at low speed, the discrepancy between experimental and computed values becomes significant. However the V.R. model values are in better agreement with the experimental ones.
.
-
-
3.4
Concluding remarks
Results of our nunerical simulations have clearly demonstrated the interest of studying the linear part of traction curves to understand the lubricant behaviour in an E.H.D. contact. For example, it has been shown that various fluid behaviours (viscoelastic, viscous or elastic) lead to significantly different computed mean viscosities. The difference among the slopes of the traction curves becomes more important at low pressure and low rolling speed. The most complete viscoelastic model (V.R.) needs only the knowledge of the viscoelastic parameters (GOD ,K /Km and? , the structural relaxation time) obeained in static conditions together with d , the piezoviscosity parameter. This work has shown that light scattering spectroscopy is quite appropriate to measure these parameters, especially in the case of anisotropic lubricants. With isotropic lubricants, a complementary ultrasonic technique is needed to measure G a
.
4 ACKNOWLEDGEMENT The authors thank Dr Berthe, G. Dalmaz and Ph. Vergne for stimulating discussions.
References For a general review, for example : CHENG, H.S., 198, CRC Handbook of lubrication (1984) 11, 139-162. HARRISON, G. and TRACHMAN, E.G.'The role of compressional viscoelasticity in the lubrication of rolling contacts', ASME J. Lub. Tech. 1972, 94, 306-312. HEYES, D.M. and MONTROSE, C.J. 'The use of line and point contacts in determining lubricant rheology under low slip elastohydrodynamic conditions',ASME J. Lub. Tech. 1983, 105, 280. BERNE, B.J. and PECORA, R. 'Dynamic light scattering',l976, Wiley. BEZOT, P.; HESSE-BEZOT, C. and PRUZAN, Ph. 'viscoelastic properties of liquid pentachlorobiphenyl under pressure using depolarized light scattering', Can. J. Phys. 1983, 9, 1291. BEZOT, P.; HESSE-BEZOT, C. and QUENTREC, B. 'Diffusion de la lumisre et ordre local dans les liquides visqueux', Mol. Phys. 1981, 43, 1407. BEZOT, P. and HESSE-BEZOT, C. 'Viscoelastic properties of tri(o.toly1)phosphate from light scattering and ultrasonic tech niques', J. of Mol. Phys. 1984, 29, 1 1 1 . DILL, J.F.; DRAKE, P.W. and LITOVITZ, T.A. 'The study of viscoelastic properties of lubricants using high pressure optical techniques', ASLE. 1975, 18 (3), 202. HUTTON, J.F. and PHILLIPS, M.C., Nature Physical Science, 1973, 245, 15. VERGNE, Ph. 'Contribution 2 l'i5tude du contact E.H.D., de formation des surfaces, caractsrisation haute pression des lubricants', Thesis, I.N.S.A. Lyon 1985. YASUTOMI, S.; BAIR, S. and WINER, W.O. 'An application of a free volume model to lubricant rheology Iand II', ASME J. Lub. Tech. 1984, 106, 291. ALLAIN-DEMOULIN, C.; LALLEMAND, P. and OSTROWSKY, N. 'Theoretical study of light scattering spectra of a pure relaxing fluid', Mol. Phys. 1976, 31, 581. ALSAAD, M.; BAIR, S.; SANBORN, D.M. and WINER, W.O. 'Glass transition in lubricants : Its relation to elastohydrodynamic lubricant', ASME J. Lub. Tech. 1978, 100, 4 0 4 . ALSAAD, M.; WINER, W.O.; MEDIA, F.D. and O'SHEAR, D.C. 'Light scattering study of the glass transition in lubricants', ASME J. Lub. Tech. 1978, 100, 418. WILLIAMS, G. and WATTS, D.C. Trans. Farad. SOC. 1970, 66, 80. JOHNSON, K.L. and TEVAARWERK, J.L. 'Shear behaviour of elastohydrodynamic oil film', Proc. R. SOC. A. 1977, 356, 215. BAIR, S. and WINER, W.O. 'A rheological model for elastohydrodynamic contacts based on primary laboratory data', ASME J. Lub. Tech. 1979, 101, 258. BIRST, W. and MOORE, A.J. 'The elastohydrodynamic behaviour of polyphenyl ether: Proc. R. SOC. London. A. 1975, 344, 403.
321
Gpa 7
6
5
4
v
Fig. 1 I, t = 41°C.
+ It
spectrum at P = O.IGPa and 1
1
0
hi-
.
49'C
Fig. 2 X :G, tion of t'C.
; *:Ma
and A :K,as
a func-
26'c 41'C 0 49-c 0 59'C x 68-C A 78'C 0
Gd (GPa 0.1
0.2
P (tipa)
1.5 Fig. 4 M o o as a function of the pressure at various temperatures.
1 a 26.C
.
0
11'C 49-c
0.5 0. I
0.2
P (GFW
I 0.1
I
1
0.2
0.3 P (GPa)
Fig. 5 KCC, as a function of the pressure at various temperatures.
Fig. 3 G o o as a function of the pressure at various temperatures.
)I
322
1Et02 T 1Et01 1Et00 1E-0 1
2
1E-02 1E-03
+
I
+ +
+
* + * *
+
+
+
*
*
*
*
*
* * * *
+ *
*
+ w + *
0.6
Fig. 6
'kk
and sc : and 0 :
+
at 27'C at 27'C
and 5OoC and 50°C
Fig. 8
*: 7 , +: q ,
0
.
0
W
+
+
W
+
*
+
+
+
*
*
*
.*
+
+. *
*
+
*
*
*
* * *
*
*
* t
*
*
w
0
Fig. 7
+
0
+
0.1
+
+ + + + + +
*
+
+ 0
0.31
I
I ) , U=3.99 m/s U=3.99 m/s
+
. *+
(p=
V.R. model MW1 model,
0
W
I
0.8
+ +
P (GPa)
0.2
+
+
+
I/
+
0.2 + : PE
*
0.4
0.6
; U = 0.168 m/s
: P, ; u = 3.99 m/s : Hertzian Pressure P
0.8
1 Fig. 9
Idem Fig. 8 with
U
=
0.168 m/s
323
: 1
I+ X
+
*-
4 .
*
+
1
6--
+
'
4-* k
1 P (GPa)
t
2 9-
Fig. 10
I
-9
I
V = 2 . 2 m/s ; :MW2 model ; + :MWI model ; Y and -:V.R. model @ = 1 and 0.5)
* + *
5
o r + O*
* 0
0
Fig. 12
f
r
*
0
10
t
r
n
1
-
-
7
for P = 0 . 3 0 2 GPa ; tMW2 model ; +:MWI model ; % :V.R. model ( 8 = I ) ; r :EL model ; 0 : experiment.
Pas)
1/
4.
*-
+
2t
+
x,
't Fig. I I
.
for
2.5
0.5 0 :3
:I
+X
*-
+
X
+
* P (GPa) I
I
0.4
0.5
Idem Fig. 10
with U
0 .'6=
0.6 m/s
0.7 Fig. 13
-'I
P
for P = 0 . 3 7 1 GPa ; m :MU2 model ; + :MW1 model ;*sV.R. model (9 = 1 ) ; 0 : experiment.
This Page Intentionally Left Blank
325
Paper X(iii)
High-shear viscosity studies of polymer-containing lubricants J.L. Duda, E.E. Klaus, S.C. Lin and F.L. Lee
T h i s paper i n t r o d u c e s an e x p e r i m e n t a l t e c h n i q u e and t h e a s s o c i a t e d d a t a a n a l y s i s procedure studying t h e shear dependent v i s c o s i t y of o i l s containing polymers a t shear r a t e s u p t o l o 6 s-l t e m p e r a t u r e s u p t o 17OoC, c o n d i t i o n s which a r e t y p i c a l l y experienced i n operating engines. experimental apparatus is a c a p i l l a r y viscometer w i t h a c o n t r o l l e d n i t r o g e n gas p r e s s u r e a s
for and The the d r i v i n g f o r c e and u s e s c o n s t a n t volume e f f l u x b u l b s t o measure t h e o i l flow r a t e . The main c o n t r i b u t i o n of t h i s work i s t h e c o u p l i n g of e x p e r i m e n t a l c a l i b r a t i o n m e a s u r e m e n t s a n d a t h e o r e t i c a l a n a l y s i s t o d e v e l o p a c a l i b r a t i o n p r o c e d u r e which can be u s e d t o e x t r a c t accurate high-shear r a t e v i s c o s i t y d a t a from c a p i l l a r y viscometer measurements when t h e f l u i d v i s c o s i t y i s being influenced by temperature, pressure, and shear r a t e . 1
INTRODUCTION
The a d d i t i o n of polymers t o l u b r i c a n t s i n order t o r e d u c e t h e i n f l u e n c e of t e m p e r a t u r e on l u b r i c a n t v i s c o s i t y h a s become an e x t e n s i v e commercial p r a c t i c e . Like most polymer s o l u t i ons , t h e s e pol ymer-contai n i ng l u b r i cants exhibit a reversible shear-thinning behavior. McMillan ( 1 ) and o t h e r s ( 2 ) have s t r e s s e d t h e i m p o r t a n c e of measuring t h e v i s c o s i t y c h a r a c t e r i s t i c of o i l s under conditions which a r e c h a r a c t e r i s t i c of e n g i n e s . General consensus indicates that the shear-rate conditions i n t h e engine r a n g e from l o 5 t o 107 s-l and t e m p e r a t u r e s from l l O ° C t o 17OOC. I t i s c l e a r t h a t l u b r i c a n t v i s c o s i t i e s m u s t be measured a t h i g h t e m p e r a t u r e s and shear r a t e s i n order t o develop e n g i n e o i l s and t o a s s e s s t h e i r performance. Most l u b r i c a n t s which c o n t a i n polymers w i l l show a c o n s t a n t v i s c o s i t y N e w t o n i a n b e h a v i o r u p t o a c r i t i c a l s h e a r r a t e . Above t h a t shear r a t e , t h e v i s c o s i t y of t h e s e polymer s o l u t i o n s d e c r e a s e s w i t h i n c r e a s i n g shear r a t e and a s t h e f i r s t a p p r o x i m a t i o n t h i s s n e a r - t h i n n i n g r e g i o n can be represented by a A t very high power-law r h e o l o g i c a l m o d e l . s h e a r r a t e s , polymer s o l u t i o n s w i l l a l s o c h a r a c t e r i s t i c a l l y approach an upper Newtonian limit.
V a r i o u s techniques have been u s e d t o s t u d y t h e v i s c o s i t y of l u b r i c a t i n g o i l s a t h i gh-shear rates. Engine t e s t s o r m o d i f i e d engine tests have been used t o g i v e i n f o r m a t i o n c o n c e r n i n g t h e e f f e c t of v i s c o s i t y of l u b r i c a n t s i n t h e high-shear r a t e regions of t h e e n g i n e . D i r e c t measurements of v i s c o s i t y a t high-shear r a t e s involve u t i l i z i n g e i t h e r r o t a t i o n a l v i s c o m e t e r s or v i s c o m e t e r s based on flow i n c a p i l l a r i e s or other conduits. S e v e r a l i n v e s t i g a t o r s have developed r o t a t i o n a l viscometers f o r t h e s t u d y of s o l u t i o n s a t high shear rates ( 3 - 6 ) . R o t a t i o n a l v i s c o m e t e r s can b e c o n s t r u c t e d t o c l o s e l y reproduce t h e geometry of t h e high-shear r a t e regions i n e n g i n e s b u t t h e y do r e q u i r e v e r y p r e c i s e and r e l a t i v e l y complex
m e c h a n i c a l s y s t e m s . Also, i t can be d i f f i c u l t t o measure and c o n t r o l t e m p e r a t u r e s i n r o t a t i o n a l viscometers. V i s c o s i t y measurements w i t h c a p i l l a r y v i s c o m e t e r s a r e c o n v e n i e n t and s i m p l e t o p e r f o r m , and numerous i n v e s t i g a t o r s have contributed t o t h e development of c a p i l l a r y - t y p e viscometers f o r utilization a t high-shear r a t e s (7-15). A 1 t h o u g h me a s u r emen t s w i t h c a p i 11a r y v i s c o m e t e r s a r e c o n v e n i e n t and s i m p l e t o perform, t h e i n t e r p r e t a t i o n of measurements a t h i g h - s h e a r r a t e s a r e complicated by s e v e r a l phenomena: a ) The excess pressure drops due t o v i s c o u s f l o w and changes i n k i n e t i c energy i n t h e entrance and e x i t r e g i o n s of t h e c a p i l l a r y can be s i g n i f i c a n t . b ) The l a r g e d i f f e r e n c e i n pressure a c r o s s t h e c a p i l l a r y needed t o produce h i g h - s h e a r r a t e s can s i g n j f i c a n t l y i n f l u e n c e t h e v i s c o s i t y of t h e f l u i d a s i t f l o w s t h r o u g h the capillary. c ) T e m p e r a t u r e changes associated with viscous heating can s i g n i f i c a n t l y i n f l u e n c e t h e v i s c o s i t y of t h e l u b r i c a n t i n b o t h t h e r a d i a l and a x i a l d i r e c t i o n s i n t h e c a p i l l a r y . Consequently, although i t is r e l a t i v e l y e a s y t o d e s i g n a c a p i l l a r y v i s c o m e t e r i n which a c c u r a t e measurements of pressure drop and flow r a t e can be d e t e r m i n e d a t e l e v a t e d t e m p e r a t u r e s and high-shear r a t e s , i t i s d i f f i c u l t t o a n a l y z e t h i s b a s i c d a t a t o determine accurate v i s c o s i t y data a t a s p e c i f i e d shear r a t e , temperature, and pressure. 1 . 1 Notation
CP
S p e c i f i c heat capacity
k
Thermal conductivity
L
C a p i l l a r y length
n
Power-law index
P
Pressure
AP
Pressure drop a c r o s s t h e capi 1l a r y
Apexcess Excess p r e s s u r e d r o p Observed p r e s s u r e d r o p Apobs
Q
Flow r a t e
R
Capillary radius
r
Radial d i s t a n c e from c a p i l l a r y a x i s
Re
R e y n o l d s n u m b e r , pQ/unR
T
Temperature
TO
Bath temperature
Vr
Radial v e l o c i t y
VZ
Axial v e l o c i t y
2
Axial d i s t a n c e from c a p i l l a r y i n l e t
P
Density
u
V i scos i t y
PO
Viscosity a t zero-shear rate
lJaPP
Apparent v i s c o s i t y
i.
Shear rate
i0
C h a r a c t e r i s t i c shear r a t e w h e r e d e v i a t i o n from Newtonian b e h a v i o r occurs
2 EXPERIMENTAL
C o n s i d e r a b l e w o r k o v e r t h e p a s t f o u r decades has been conducted i n t h e Petroleum R e f i n i n g L a b o r a t o r y a n d t h e D e p a r t m e n t of C h e m i c a l E n g i n e e r i n g a t The P e n n s y l v a n i a S t a t e U n i v e r s i t y o n t h e d e v e l o p m e n t a n d e v a l u a t i o n of c a p i l l a r y v i s c o m e t e r s f o r u s e a t b o t h low a n d h i g h r a t e s of s h e a r o v e r a w i d e r a n g e o f operating conditions. This study h a s incorporated and extended the previous c a p i 11a r y v i s comet e r d e v e l o pm e n t s of t h e s e researchers (7,16-18). The main contribution o f t h i s w o r k i s t h e c o u p l i n g of e x p e r i m e n t a l c a l i b r a t i o n measurements and a theoretical a n a l y s i s t o d e v e l o p a data a n a l y s i s procedure which c a n be u s e d t o e x t r a c t a c c u r a t e h i g h - s h e a r r a t e v i s c o s i t y d a t a from c a p i l l a r y viscometer measurements when t h e f l u i d v i s c o s i t y i s b e i n g i n f l u e n c e d by p r e s s u r e , v i s c o u s h e a t i n g a n d t h e shear f i e l d . A schematic d i a g r a m o f t h e h i g h - s h e a r c a p i l l a r y v i s c o m e t e r u t i l i z e d i n t h i s s t u d y is presented i n Figure 1. This apparatus is a m o d i f i c a t i o n of t h e v e r s i o n o r i g i n a l l y d e s c r i b e d by F e n s k e , K l a u s a n d D a n n e n b r i n k ( 7 1 , a n d d e t a i l s o f t h e e x p e r i m e n t a l a p p a r a t u s and p r o c e d u r e h a v e b e e n p r e s e n t e d b y Lee ( 1 9 ) . A l s o , d e t a i l s of t h e c a p i l l a r y a s s e m b l y a n d t h e nitrogen pressure control system have been p r e s e n t e d by Graham and c o - w o r k e r s ( 2 0 ) . The w o r k i n g p o r t i o n o f t h e v i s c o m e t e r r e s e m b l e s t h a t of a c o n v e n t i o n a l O s t w a l d - t y p e v i s c o m e t e r . The c o m p l e t e a p p a r a t u s c o n s i s t s of a l i q u i d feed r e s e r v o i r , a c a p i l l a r y , and e f f l u x bulbs i n series. The c o m p l e t e u n i t is submersed i n a constant-temperature o i l bath. The d r i v i n g h e a d which forces t h e l i q u i d t h r o u g h t h e c a p i l l a r y c o n s i s t s of a c o n t r o l l e d n i t r o g e n gas p r e s s u r e s y s t e m w h i c h is n o t s h o w n i n F i g u r e 1 . To i n i t i a t e a n e x p e r i m e n t , t h e f l u i d r e s e r v o i r w h i c h h a s a c a p a c i t y of 2 0 0 m l i s f i l l e d w i t h t h e s o l u t i o n t o be s t u d i e d , a n d t h e f l u i d r e s e r v o i r c a p w i t h a n O - r i n g s e a l is put into place. S t i r r i n g is provided t o facilitate heat t r a n s f e r t o t h e sample solution.
I FLUID RESERVIOR 2 STIRRER 3 FLUID RESERVIOR CAP
4 O-RING 5 BAFFLE DISK
7 FLUID EXIT LINE B CAPILLARY
6 HIGH PRESSURE Nt INLET LINE
9 EFFLUX BULB
Fig. 1 Schematic diagram of high-shear c a p i l l a r y viscometer.
After t h e d e s i r e d t e m p e r a t u r e is a t t a i n e d , t h e m o t o r is r e m o v e d from t h e s t i r r i n g s h a f t a n d t h e n i t r o g e n ballast system is connected t o t h e high-pressure i n l e t l i n e . To i n i t i a t e a n e x p e r i m e n t , a v a l v e b e t w e e n t h e b a l l a s t s y s t e m a n d t h e f l u i d r e s e r v o i r is opened and t h e time r e q u i r e d t o f i l l a c o n s t a n t - v o l u m e e f f l u x b u l b is d e t e r m i n e d . The recorded p r e s s u r e d r o p , APobs, i s t h e p r e s s u r e d i f f e r e n c e b e t w e e n t h e b a l l a s t system and t h e e f f l u x b u l b a t room p r e s s u r e . Several e x p e r i m e n t s c a n b e d e t e r m i n e d from o n e f i l l i n g of t h e r e s e r v o i r s i n c e a n e f f l u x b u l b w i t h a maximum v o l u m e o f o n l y 50 m l was r e q u i r e d t o o b t a i n p r e c i s i o n d a t a a t t h e maximum s h e a r rates. T h e c a p i l l a r y c o n s i s t s o f a s t a i n l e s s - s t e e l t u b i n g w i t h one end soldered into a fitting. D e t a i l s of t h i s c a p i l l a r y assembly are a v a i l a b l e (19,201. Various e f f l u x b u l b s and c a p i l l a r i e s can be u t i l i z e d i n o r d e r t o o b t a i n a c c u r a t e data o v e r a r a n g e of s h e a r r a t e a n d v i s c o s i t y . T h e e f f l u x b u l b s r a n g e d i n v o l u m e from 0.5 m l f o r l o w - s h e a r r a t e m e a s u r e m e n t s t o 50 m l f o r h i g h - s h e a r rate measurements, and t h e c a p i l l a r y d i m e n s i o n s v a r i e d f r o m 0.1 t o 0 . 3 mm i n r a d i u s a n d 10 t o 130 mm i n l e n g t h . Important steps i n the experimental p r o c e d u r e s a r e f i l t r a t i o n and d e g a s s i n g of t h e l i q u i d s . The b a f f l e d i s k a t t h e t o p o f t h e f l u i d r e s e r v o i r was i n c o r p o r a t e d i n o r d e r t o e l i m i n a t e t h e d i r e c t i m p i n g i n g of t h e n i t r o g e n g a s stream i n t o t h e l i q u i d . Under these c o n d i t i o n s , t h e i n t r od u c t i o n o f d i ss o l v e d nitrogen and n i t r o g e n bubbles i n t o t h e f l u i d was m i n i m i z e d . D i s s o l v e d or e n t r a i n e d n i t r o g e n c o u l d b e d e t e c t e d by t h e p r e s e n c e o f b u b b l e s i n Also, t h e r e g i o n of t h e e f f l u x b u l b s . c a v i t a t i o n c a n be d e t e c t e d when t h e p r e s s u r e Data drop a c r o s s t h e c a p i l l a r y is t o o h i g h .
327
a r e n o t c o l l e c t e d when t h e s e c o m p l i c a t i o n s are
.
observed T h e most c r i t i c a l a s p e c t i n t h e d e s i g n of t h e a p p a r a t u s is d e t e r m i n i n g t h e d i m e n s i o n s of t h e c a p i l l a r y s o t h a t the desirable high-shear r a t e s c a n be a t t a i n e d a t m o d e r a t e p r e s s u r e d r o p s t o m i n i m i z e t h e i n f l u e n c e of v i s c o u s dissipation while maintaining the excess p r e s s u r e d r o p s associated w i t h t h e e n t r a n c e a n d e x i t s of t h e c a p i l l a r y t o b e l o w 1 0 % o f t h e o v e r a l l p r e s s u r e d r o p . T h i s c o n s t r a i n t was d i c t a t e d by t h e d a t a a n a l y s i s p r o c e d u r e w h i c h is p r e s e n t e d i n t h e n e x t s e c t i o n .
3.
Momentum E q u a t i o n i n r a d i a l d i r e c t i o n : ap =
ar
Energy Equation:
Boundary C o n d i t i o n s : Capillary center line, r = 0
THEORY
A u n i q u e a s p e c t o f t h i s development is t h e u s e
of a computer s i m u l a t i o n t o p r e d i c t t h e p r e s s u r e d r o p a s a f u n c t i o n of f l o w r a t e for laminar flow i n a c a p i l l a r y under c o n d i t i o n s where v i s c o u s h e a t i n g and t h e i n f l u e n c e o f p r e s s u r e and shear r a t e o n v i s c o s i t y a r e important. A s discussed i n the next section, t h i s computer simulation is u s e d i n t h e a n a l y s i s o f t h e data and t o check t h e c o n s i s t e n c y of t h e e x p e r i m e n t a l r e s u l t s . The c o m p u t e r s i m u l a t i o n i n v o l v e s a f i n i t e - d i f f e r e n c e s o l u t i o n of t h e f i e l d e q u a t i o n s which describe t h i s problem w h i c h i n v o l v e s c o u p l i n g of f l u i d m e c h a n i c s w i t h pressure and temperature fields. The l u b r i c a t i o n approximation is u t i l i z e d t o e l i m i n a t e t h e momentum e q u a t i o n i n t h e r a d i a l d i r e c t i o n . T h i s s i m p l i f i c a t i o n is r e a s o n a b l e for t h e flow f i e l d s c o n s i d e r e d i n t h i s s t u d y s i n c e t h e f o l l o w i n g c r i t e r i a a r e met (21 1: R
<<
Re <<
L,
L 3 (E)
The other major a s s u m p t i o n s employed are:
F u l l y developed flow e x i s t s a t the e n t r a n c e of t h e c a p i l l a r y . The f l o w f i e l d is s y m m e t r i c a l a b o u t the c a p i l l a r y a x i s . The d e n s i t y , thermal c o n d u c t i v i t y and s p e c i f i c heat are i n d e p e n d e n t of p r e s s u r e and t e m p e r a t u r e . A s t e a d y , l a m i n a r flow f i e l d e x i s t s a t a l l positions i n the capillary. Cons ist ent with t h e l u b r i c a t i o n approximation, t h e shear f i e l d i n t h e a x i a l d i r e c t i o n is c o n s i d e r e d n e g l i g i b l e compared t o t h e shear i n t h e r a d i a l d i r e c t i o n , and heat conduction i n t h e a x i a l d i r e c t i o n is n e g l i g i b l e compared t o heat c o n d u c t i o n i n the radial d i r e c t i o n . Under these c o n s t r a i n t s , t h e problem is r e p r e s e n t e d by t h e f o l l o w i n g e q u a t i o n s a n d associated b o u n d a r y c o n d i t i o n s . C o n t i n u i t y Equation:
’
- -a [rvr] r ar
+ avZ
-
az
o
=
Mbmentum E q u a t i o n i n a x i a l d i r e c t i o n :
avZ
avZ PVr
+
o
PVZ
=
-
ap
az
aT ar 0 , ar avZ
=
=
O*
C a p i l l a r y Wall, r
vz
=
0,
vr
=
=
R
0
A d i a b a t i c Wall Case: Isothermal Wall Case:
Capillary Entrance, Z T
=
=
aT = ar
0
T = To
0
To
VZ(r)
=
( f u l l y developed v e l o c i t y p r o f i l e )
T h e s e e q u a t i o n s can be s o l v e d to predict AP, a s a f u n c t i o n of t h e v o l u m e t r i c flow r a t e , Q , i f t h e v i s c o s i t y is known a s a f u n c t i o n of p r e s s u r e , t e m p e r a t u r e a n d s h e a r r a t e . Two r h e o l o g i c a l m o d e l s a r e employed i n t h i s s t u d y f o r t h e f u n c t i o n a l r e l a t i o n s h i p b e t w e e n v i s c o s i t y a n d shear rate. I n the d a t a analysis procedure, calculations are p e r f o r m e d for Newtonian f l u i d s i n which the v i s c o s i t y i s i n d e p e n d e n t of s h e a r r a t e . I n t h e c o n s i s t enc y check p r o c e d u r e , the experimental r e s u l t s are c o r r e l a t e d w i t h a t r u n c a t e d p o w e r - l a w model i n w h i c h t h e f l u i d b e h a v e s as a N e w t o n i a n f l u i d below a c h a r a c t e r i s t i c s h e a r r a t e , Y?, a n d e x h i b i t s s h e a r - t h i n n i n g b e h a v i o r above t h i s s h e a r rate: the pressure drop,
.. l.l(Y2Yo)
= Po
(2y-l
70 T h e s e e q u a t i o n s were s o l v e d u t i l i z i n g a f i n i t e d i f f e r e n c e t e c h n i q u e similar t o t h a t d e v e l o p e d by A p p e l d o o r n a n d co-workers ( 1 5 , 2 2 , 2 3 ) . The major d i f f e r e n c e b e t w e e n t h i s a n a l y s i s a n d t h a t p r e s e n t e d by A p p e l d o o r n is t h a t i n t h i s s t u d y t h e i n e r t i a l terms a r e i n c l u d e d i n t h e a x i a l momentum e q u a t i o n s i n c e t h e R e y n o l d s n u m b e r s of t h e f l o w f i e l d s c o n s i d e r e d i n t h i s study are s i g n i f i c a n t l y h i g h e r t h a n t h o s e a n a l y z e d by A p p e l d o o r n a n d co-workers. I n c o n c l u s i o n , t h e pressure drop r e q u i r e d for a g i v e n v o l u m e t r i c flow r a t e f o r l a m i n a r flow i n a c a p i l l a r y a t high-shear rates c a n be t h e o r e t i c a l l y d e t e r m i n e d i f t h e c a p i l l a r y character i s t i c s and the appropriate p h y s i c a l p r o p e r t i e s of t h e f l u i d are k n o w n . A s i n d i c a t e d i n t h e a b o v e b o u n d a r y c o n d i t i o n s , two cases c o n c e r n i n g t h e thermal c o n d i t i o n s a t t h e
328
c a p i l l a r y w a l l were c o n s i d e r e d ( a d i a b a t i c wall and isothermal w a l l ) . I n t h e f o l l o w i n g s e c t i o n i t w i l l be shown t h a t t h e experimental c o n d i t i o n s of t h i s s t u d y v e r y c l o s e l y a p p r o x i m a t e t h e adiabatic wall boundary condition. 4 DATA ANALYSIS PROCEDURE
e x p e r i m e n t a l procedure and t h e data a n a l y s i s p r o c e d u r e are complexly c o u p l e d a s s h o w n i n F i g u r e 2. The basic c a p i l l a r y v i s c o m e t e r m e a s u r e m e n t s (APobs a n d Q) m u s t b e Corrected for t h e influence o f viscous heating and p r e s s u r e o n t h e v i s c o s i t y , i n a d d i t i o n t o c o r r e c t i o n s for t h e excess pressure drop associated w i t h t h e e n t r a n c e a n d e x i t r e g i o n s of t h e c a p i l l a r y . The i n i t i a l s t e p i n v o l v e s t h e c a l i b r a t i o n of t h e c a p i l l a r y w i t h N e w t o n i a n f l u i d s i n o r d e r t o d e t e r m i n e t h e r a d i u s of t h e c a p i l l a r y . These e x p e r i m e n t s are c o n d u c t e d a t low-shear rates so t h a t t h e influences of t h e p r e s s u r e and v i s c o u s h e a t i n g are n e g l i g i b l e , and t h e e x c e s s p r e s s u r e d r o p associated with flow n e a r t h e e n d s of t h e c a p i l l a r y a r e a l s o s m a l l a n d c a n be a p p r o x i m a t e d by t h e p r o c e d u r e p r e s e n t e d below. The
1
I1
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-I
OETERMINE FLUID CHARACTERISTICS
PIP. 1. i.01, k. p. c p
CALIBRATE CAPILL*RY
OF PRESSURE A N 0 APPARENT VISCOSITY 0 1 NEWTONIAN EXPS b I COMPUTER SIMULATION
EXIT AND ENTRANCE CORRECTIONS
t CORRECT paPp FOR EFFECT OF PRESSURE A N 0 VISCOUS HEATING
"(excess) CALCULATE APPARENT VISCOSITY. popp
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[+I2 TR
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I OETERMINE
rate data are o b t a i n e d . The o b s e r v e d t o t a l p r e s s u r e d r o p n o t o n l y r e s u l t s from t h e p r e s s u r e d r o p associated w i t h t h e f l o w t h r o u g h t h e c a p i l l a r y , but also includes excess p r e s s u r e d r o p s associated w i t h t h e e n t r a n c e a n d e x i t r e g i o n s of t h e c a p i l l a r y . A t the e n t r a n c e , s i g n i f i c a n t g r a d i e n t s i n p r e s s u r e may b e r e q u i r e d t o i n c r e a s e t h e k i n e t i c e n e r g y of t h e f l u i d from t h e e s s e n t i a l l y s t a g n a n t s t a t e i n t h e r e s e r v o i r t o t h e h i g h v e l o c i t i e s which are c h a r a c t e r i s t i c o f f l o w i n t h e c a p i l l a r y . I n p r i n c i p l e , some o f t h i s p r e s s u r e d r o p c o u l d be r e c o v e r e d when t h e f l u i d e x i t s from t h e c a p i l l a r y t o t h e r e l a t i v e l y low v e l o c i t y regime i n t h e t u b i n g and connectors between t h e c a p i l l a r y and the efflux bulb. However, b e c a u s e of t h e b l u n t c o n f i g u r a t i o n o f t h e e x i t , i t i s a s s u m e d t h a t a l l of t h e p r e s s u r e d r o p associated w i t h c h a n g e s i n t h e k i n e t i c e n e r g y i s d i s s i p a t e d . I n a d d i t i o n t o t h e i n f l u e n c e of the kinetic energy, the pressure drop per l e n g t h o f c a p i l l a r y i n t h e entrance and e x i t r e g i o n s c a n be l a r g e r t h a n t h a t a s s o c i a t e d w i t h The developed flow i n t h e c a p i l l a r y . converging and d i v e r g i n g l a m i n a r f l o w f i e l d i n t h e e n t r a n c e a n d e x i t s of c a p i l l a r i e s or t u b e s is p e r h a p s t h e most e x t e n s i v e l y s t u d i e d non-viscometric flow f i e l d . Numerous experimental and theoretical i n v e s t i g a t i o n s have focused on determining t h e excess pressure drop associated w i t h t h e converging or d i v e r g i n g stream l i n e s f o r t h e flow of N e w t o n i a n a n d non-Newtonian f l u i d s ( 2 5 - 2 9 ) . A c r i t i c a l r e v i e w of t h e s e s t u d i e s i n d i c a t e s t h a t the total excess pressure d r o p associated w i t h t h e e n t r a n c e a n d e x i t r e g i o n s is a p p r o x i m a t e l y 1 . 1 7 times t h e p r e s s u r e d r o p r e q u i r e d t o increase the kinetic energy of the fluid e n t e r i n g t h e c a p i l l a r y when t h e R e y n o l d s n u m b e r i s g r e a t e r t h a n 100:
COMPUTER SlMULATl
c ( i )
F i g . 2 E x p e r i m e n t a l and data a n a l y s i s procedures. O n c e t h e l e n g t h a n d r a d i u s of t h e capillary have been determined, t h e basic c h a r a c t e r i s t i c s o f t h e f l u i d must be d e t e r m i n e d . I n a d d i t i o n t o t h e u s e of e x p e r i m e n t s o r c o r r e l a t i o n s t o determine t h e density and thermal p r o p e r t i e s , a c o n v e n t i o n a l c a p i l l a r y viscometer s u c h a s a Cannon-Fenske viscometer is u s e d t o d e t e r m i n e t h e v i s c o s i t y o f t h e f l u i d a s a f u n c t i o n of t e m p e r a t u r e i n t h e N e w t o n i a n , low-shear, region. These v i s c o s i t y measurements n o t o n l y provide information c o n c e r n i n g t h e i n f l u e n c e of t e m p e r a t u r e o n t h e N e w t o n i a n v i s c o s i t y of t h e polymer s o l u t i o n a t low-shear rates, but t h e s e m i - e m p i r i c a l c o r r e l a t i o n of So a n d K l a u s ( 2 4 ) is u t i l i z e d t o estimate t h e i n f l u e n c e of p r e s s u r e o n t h e v i s c o s i t y from t h e s e data. Next, high-shear r a t e e x p e r i m e n t s are c o n d u c t e d a n d t o t a l p r e s s u r e d r o p v e r s u s flow
C o n s e q u e n t l y , t h i s q u a n t i t y was s u b t r a c t e d from t h e observed total p r e s s u r e d r o p t o o b t a i n a n e s t i m a t e o f t h e p r e s s u r e d r o p associated w i t h This developed flow i n t h e c a p i l l a r y . c o r r e c t i o n is o n e o f t h e m o s t c r i t i c a l i n t h e data a n a l y s i s p r o c e d u r e a n d a s a p r e c a u t i o n , t h e h i g h - s h e a r e x p e r i m e n t s were d e s i g n e d s o t h a t t h i s e x c e s s p r e s s u r e d r o p was l e s s t h a n 1 0 % of t h e t o t a l p r e s s u r e d r o p i n t h e s y s t e m . The v a l i d i t y of t h i s c o r r e c t i o n is s u p p o r t e d b y t h e c o n s i s t e n c y c h e c k p r o c e d u r e d i s c u s s e d below. The p r e s s u r e d r o p due t o t h e c h a n g e i n k i n e t i c e n e r g y is of c o u r s e related t o t h e velocity profile i n the c a p i l l a r y which is i n f 1u e n ced by t h e non-Newtonian c h a r a c t e r i s t i c s of t h e f l u i d a n d r a d i a l v a r i a t i o n s i n t e m p e r a t u r e associated w i t h v i s c o u s h e a t i n g . Calculations investigating these effects i n d i c a t e that t h e d i f f e r e n c e s i n k i n e t i c energy d u e t o t h e s e m o d i f i c a t i o n s of t h e v e l o c i t y p r o f i l e are n e g l i g i b l e f o r f l u i d s w h i c h have t h e c h a r a c t e r i s t i c s of p o l y m e r - m o d i f i e d l u b r i c a n t s a s l o n g a s t h e 10% maximum p r e s s u r e - d r o p c o r r e c t i o n c r i t e r i a is m a i n t a i n e d . A s F i g u r e 2 shows, t h e corrected pressure d r o p , AP, a n d t h e o b s e r v e d v o l u m e t r i c f l o w r a t e , Q, are used t o determine an apparent v i s c o s i t y , paapp. T h i s c a l c u l a t i o n is b a s e d o n t h e Hagen-Poiseuille equation describing steady l a m i n a r flow of a N e w t o n i a n f l u i d i n a c a p i l l ar y:
'am
=
-
4 nAPR 8QL
T h i s e x p e r i m e n t was d e v e l o p e d t o d e t e r m i n e polymer s o l u t i o n s at t h e t e m p e r a t u r e o f t h e b a t h , o n e a t m o s p h e r e of p r e s s u r e , a n d a t t h e shear r a t e o f i n t e r e s t . Several factors caused the apparent viscosity t o d e v i a t e from t h e a c t u a l v i s c o s i t y . The V i s c o s i t y i s a f u n c t i o n of p r e s s u r e , a n d t h e r e l a t i v e l y large p r e s s u r e s n e a r t h e e n t r a n c e of t h e c a p i l l a r y can cause the apparent viscosity t o be higher t h a n t h e v i s c o s i t y associated w i t h o n e a t m o s p h e r e of p r e s s u r e . S e c o n d l y , t h e w o r k associated with f o r c i n g t h e f l u i d t h r o u g h t h e c a p i l l a r y is d i s s i p a t e d a s thermal e n e r g y i n the capilldry. This viscous heating can r e s u l t i n f l u i d t e m p e r a t u r e s which are s i g n i f i c a n t l y h i g h e r t h a n t h e t e m p e r a t u r e of t h e e x p e r i m e n t a n d , consequently, w i l l reduce the apparent I t s h o u l d be n o t e d t h a t t h e viscosity. d e c r e a s e i n t h e a p p a r e n t v i s c o s i t y c a n be larger t h a n t h e decrease associated w i t h t h e adiabatic rise i n temperature. The a d i a b a t i c temperature rise represents t h e a v e r a g e i n c r e a s e i n t e m p e r a t u r e of t h e f l u i d as i t flows t h r o u g h a c a p i l l a r y w h i c h does n o t e x c h a n g e thermal e n e r g y w i t h t h e o u t s i d e media. T h e a c t u a l t e m p e r a t u r e r i s e of some e l e m e n t s of the f l u i d i n the h i g h - s h e a r r a t e r e g i o n s n e a r t h e c a p i l l a r y w a l l c a n be s i g n i f i c a n t l y higher than the adiabatic t e m p e r a t u r e rise. C o n s e q u e n t l y , t h e c h a n g e i n v i s c o s i t y of a f l u i d i n the critical r e g i o n n e a r t h e wall c a n be s i g n i f i c a n t l y h i g h e r t h a n t h a t p r e d i c t e d from t h e a d i a b a t i c t e m p e r a t u r e r i s e . A c r i t i c a l s t e p i n t h e data a n a l y s i s p r o c e d u r e i s t h e c o r r e c t i o n of t h e c a l c u l a t e d a p p a r e n t v i s c o s i t y f o r t h e i n f l u e n c e s of p r e s s u r e a n d v i s c o u s h e a t i n g . The key e l e m e n t i n t h i s p r o c e d u r e is t h e a p p r o x i m a t i o n t h a t t h e f r a c t i o n a l c h a n g e i n t h e v i s c o s i t y of t h e polymer s o l u t i o n d u e t o p r e s s u r e a n d v i s c o u s h e a t i n g effects is e q u a l t o t h e corresponding f r a c t i o n a l c h a n g e i n t h e v i s c o s i t y of a h y p o t h e t i c a l Newtonian f l u i d w h i c h e x h i b i t s t h e same r h e o l o g i c a l p r o p e r t i e s a s t h e p o l y m e r s o l u t i o n a t low-shear r a t e s where i t a p p r o a c h e s Newtonian b e h a v i o r . The h y p o t h e t i c a l N e w t o n i a n f l u i d would have a c o n s t a n t v i s c o s i t y a t a l l shear r a t e s e q u a l t o t h e v i s c o s i t y o f t h e p o l y m e r s o l u t i o n a t low-shear r a t e s , a n d t h e i n f l u e n c e of t e m p e r a t u r e a n d p r e s s u r e o n t h e v i s c o s i t y of t h i s Newtonian f l u i d would be i d e n t i c a l t o t h e i n f l u e n c e of t h e s e v a r i a b l e s o n t h e l o w - s h e a r r a t e v i s c o s i t y of t h e polymer solution. Based o n t h i s a p p r o a c h , t h e a p p a r e n t v i s c o s i t y o f t h e polymer s o l u t i o n , p a p p r c a n b e c o r r e c t e d i f t h e a p p a r e n t v i s c o s i t y of t h e corresponding hypothetical Newtonian f l u i d f l o w i n g i n t h e same c a p i l l a r y w i t h t h e same t o t a l p r e s s u r e d r o p i s k n o w n . T h e r e a r e two procedures t o determine the apparent viscosity The direct of such a Newtonian f l u i d . experimental procedure is t o measure t h e a p p a r e n t v i s c o s i t y of t h e a p p r o p r i a t e N e w t o n i a n f l u i d i n t h e high-shear c a p i l l a r y viscorneter. T h i s e x p e r i m e n t a l c a l i b r a t i o n t e c h n i q u e was e m p l o y e d b y G r a h a m a n d c o - w o r k e r s (20). A l t h o u g h t h i s e x p e r i m e n t a l t e c h n i q u e is d i r e c t , i n p r a c t i c e i t i s d i f f i c u l t t o p e r f o r m . I t is d i f f i c u l t t o f i n d a Newtonian f l u i d with the i d e n t i c a l r h e o l o g i c a l p r o p e r t i e s as e x h i b i t e d by t h e p o l y m e r s o l u t i o n a t l o w - s h e a r r a t e s . the viscosity of
This experimental procedure u s u a l l y involves conducting e x p e r i m e n t s w i t h s e v e ra1 N e w t o n i a n f l u i d s and i n t e r p o l a t i n g the r e s u l t s t o obtain the correction factor. As F i g u r e 2 i n d i c a t e s , t h e s e c o n d p r o c e d u r e h a s a t h e o r e t i c a l b a s i s a n d does n o t i n v o l v e a d d i t i o n a l e x p e r i m e n t s . As i n d i c a t e d i n t h e Theory s e c t i o n of t h i s paper, a computer p r o g r a m was d e v e l o p e d t o s i m u l a t e t h e l a m i n a r flow of a N e w t o n i a n f l u i d w i t h a p r e s s u r e a n d temperature-dependent viscosity i n a capillary u n d e r c o n d i t i o n s of h i g h - s h e a r r a t e by s o l v i n g t h e a p p r o p r i a t e f i e l d equations. Consequently, t h e i n f l u e n c e of p r e s s u r e a n d v i s c o u s h e a t i n g o n t h e a p p a r e n t v i s c o s i t y c a n be d e t e r m i n e d t h e o r e t i c a l l y by s o l v i n g t h e e q u a t i o n s f o r a N e w t o n i a n f l u i d w h i c h h a s t h e rheological p r o p e r t i e s e x h i b i t e d by t h e p o l y m e r s o l u t i o n a t low-shear r a t e s . By u s i n g o n e o f t h e s e two p r o c e d u r e s , t h e v i s c o s i t y of t h e p o l y m e r s o l u t i o n a t t h e s h e a r conditions t h a t exist i n the capillary, p ( T , P , Y ) i s d e t e r m i n e d from t h e a p p a r e n t v i s c o s i t y by c o r r e c t i n g f o r t h e combined e f f e c t s of p r e s s u r e a n d v i s c o u s h e a t i n g . As m e n t i o n e d above, the p r e s s u r e w i l l have t h e tendency t o c a u s e t h e a p p a r e n t v i s c o s i t y t o be too high a n d t h e v i s c o u s h e a t i n g w i l l have a t e n d e n c y t o r e d u c e t h e a p p a r e n t v i s c o s i t y . For most c a s e s c o n s i d e r e d i n t h i s s t u d y , t h e v i s c o u s h e a t i n g effects dominate and the a p p a r e n t v i s c o s i t y is l e s s t h a n t h e a c t u a l The v i s c o s i t y of the polymer s o l u t i o n . a c c u r a c y of t h i s c o r r e c t i o n p r o c e d u r e w i l l b e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n . The o n l y remaining element i n t h e d a t a a n a l y s i s p r o c e d u r e is t o d e t e r m i n e t h e shear r a t e i n t h e high-shear c a p i l l a r y e x p e r i m e n t . A first a p p r o x i m a t i o n i s t h a t t h e s h e a r r a t e is e q u a l t o ( 4 Q ) / ( n R 3 ) . However, t h i s p r o c e d u r e i g n o r e s t h e f a c t t h a t t h e s h e a r - t h i n n i n g b e h a v i o r of t h e p o l y m e r s o l u t i o n m o d i f i e s t h e shear r a t e r e a l i z e d a t a g i v e n v a l u e of flow r a t e a n d I f t h e p o l y m e r s o l u t i o n is capillary radius. c o n s i d e r e d t o b e a p o w e r law f l u i d , t h e shear rate i n t h e c a p i l l a r y can be c a l c u l a t e d i f t h e p o w e r - l a w i n d e x , n , is known. To d e t e r m i n e t h e power-law i n d e x , a s e r i e s of e x p e r i m e n t s were c o n d u c t e d a t d i f f e r e n t p r e s s u r e d r o p s a n d n was d e t e r m i n e d from t h e s l o p e of a p l o t of l o g p a s a f u n c t i o n o f l o g ( 4 Q ) / ( 7 r R 3 ) . The shear f i e l d i n the c a p i l l a r y t h e n can be c o r r e c t e d f o r t h e s h e a r - t h i n n i n g b e h a v i o r of t h e f l u i d . For t h e flow o f a p o w e r - l a w f l u i d i n a c a p i l l a r y , t h e shear r a t e is g i v e n by: n
.
y=-(--4Q
nR 3
4n 3n+l
If i t i s a s s u m e d t h a t t h e i n f l u e n c e of pressure and temperature on the v i s c o s i t y of t h e p o l y m e r s o l u t i o n i s i n d e p e n d e n t of shear rate, then t h i s experimental t e c h n i q u e r e s u l t s i n v a l u e s of t h e v i s c o s i t y of t h e polymer solution as a function of pressure, t e m p e r a t u r e , a n d shear rate. F i n a l l y , t h e c o n s i s t e n c y of t h i s o v e r a l l d a t a a n a l y s i s The computer t e c h n i q u e c a n be c h e c k e d . simulation presented i n t h e previous section was u s e d t o p r e d i c t t h e flow of a non-Newtonian f l u i d w h i c h o b e y s t h e t r u n c a t e d power-law m o d e l . T h e p a r a m e t e r s i n t h i s model were d e t e r m i n e d by c o u p l i n g t h e h i g h a n d low s h e a r r a t e m e a s u r e m e n t s of v i s c o s i t y . Once t h e v i s c o s i t y o f t h e polymer s o l u t i o n has b e e n d e t e r m i n e d a s a
330 f u n c t i o n o f p r e s s u r e , t e m p e r a t u r e a n d shear r a t e , t h i s r e s u l t c a n be p u t i n t o t h e c o m p u t e r s i m u l a t i o n a n d p r e s s u r e d r o p a s a f u n c t i o n of flow rate f o r a given c a p i l l a r y i n t h e a b s e n c e o f e x i t a n d e n t r a n c e c o r r e c t i o n s c a n be determined. A c o m p a r i s o n of t h e o b s e r v e d p r e s s u r e d r o p , w h i c h has been corrected f o r excess pressure drops at the ends of t h e c a p i l l a r y , a n d t h e p r e s s u r e d r o p p r e d i c t e d by t h e computer s i m u l a t i o n c a n t h e n b e c o m p a r e d . Agreement between t h e c a l c u l a t e d and observed pressure drop is a good v a l i d a t i o n o f t h e o v e r a l l e x p e r i m e n t a l and data analysis procedure. 5.
RESULTS A N D DISCUSSION
One of t h e c r i t i c a l s t e p s i n t h e d a t a a n a l y s i s is t h e a b i l i t y t o t h e o r e t i c a l l y c a l c u l a t e t h e i n f l u e n c e o f p r e s s u r e a n d v i s c o u s h e a t i n g on t h e flow o f a N e w t o n i a n f l u i d w i t h s p e c i f i e d r h e o l o g i c a l p r o p e r t i e s at high-shear rates. A comparison of e x p e r i m e n t a l r e s u l t s and t h e o r e t i c a l c a l c u l a t i o n s f o r a Newtonian f l u i d are p r e s e n t e d i n F i g u r e 3 . T h e l i n e s i n t h i s f i g u r e r e p r e s e n t t heor e t i c a l c a l c u l a t i o n s f o r t h r e e d i f f e r e n t cases. I n t h e P o i s e u i l l e f l o w case, i t is assumed t h a t t h e v i s c o s i t y of the f l u i d is n o t i n f l u e n c e d by p r e s s u r e or v i s c o u s h e a t i n g . I n t h e isothermal wall c a s e , i t is assumed t h a t t h e w a l l of t h e c a p i l l a r y i s m a i n t a i n e d c o n s t a n t a t the i n i t i a l temperature of t h e f l u i d a n d some o f t h e t h e r m a l e n e r g y p r o d u c e d by v i s c o u s d i s s i p a t i o n c a n be conducted o u t i n t o t h e s u r r o u n d i n g media. F i n a l l y , i n t h e a d i a b a t i c w a l l c a s e t h e heat
f l u x a t t h e c a p i l l a r y i s assumed t o b e z e r o a n d a l l t h e e n e r g y from v i s c o u s d i s s i p a t i o n r e m a i n s The c o m p a r i s o n of t h e i n the fluid. e x p e r i m e n t a l measurements w i t h t h e theoretical c a l c u l a t i o n s a s shown i n F i g u r e 3 i n d i c a t e t h a t t h e e x p e r i m e n t a l c o n d i t i o n s of t h i s s t u d y v e r y c l o s e l y a p p r o x i m a t e t h e a d i a b a t i c wall b o u n d a r y conditions. T h e e x p e r i m e n t a l a p p a r a t u s was s p e c i f i c a l l y designed so t h a t t h i s b o u n d a r y The c a p i l l a r y is c o n d i t i o n was a p p r o x i m a t e d . s u r r o u n d e d by a n a n n u l u s o f s t a g n a n t o i l a n d c a l c u l a t i o n s i n d i c a t e t h a t t h e heat t r a n s f e r coefficient due t o n a t u r a l convection i n t h i s c o n f i g u r a t i o n w o u l d b e v e r y small and t h e adiabatic c o n d i t i o n would be e s s e n t i a l l y realized. C a l c u l a t i o n s i n d i c a t e t h a t i t would b e more d i f f i c u l t t o e x p e r i m e n t a l l y a t t a i n c o n d i t i o n s which a p p r o x i m a t e t h e isothermal wall case. The r e s u l t s p r e s e n t e d i n F i g u r e 3 c l e a r l y d e m o n s t r a t e t h a t t h e a d i a b a t i c wall is t h e a p p r o p r i a t e boundary c o n d i t i o n a n d t h a t t h e computer s i m u l a t i o n a c c u r a t e l y d e s c r i b e d the flow of a N e w t o n i a n f l u i d u n d e r t h e s e c o m p l e x conditions. A l l t h e polymer s o l u t i o n r e s u l t s presented i n t h i s p a p e r were o b t a i n e d w i t h a s o l u t i o n of 1 5 . 5 5 w e i g h t p e r c e n t of an o l e f i n c o p o l y m e r concentrate i n a m i n e r a l o i l b a s e stock. The p h y s i c a l p r o p e r t i e s of t h i s s o l u t i o n a t t h e t e m p e r a t u r e s of t h e e x p e r i m e n t a r e s u m m a r i z e d i n T a b l e 1. Table 1 P h y s i c a l P r o p e r t i e s of P o l y m e r - S o l u t i o n Used i n t h i s S t u d y (15.55 w e i g h t p e r c e n t o l e f i n copolymer c o n c e n t r a t e )
14
121 o c
149OC -
3.56~10-3
2.31~10-3
114
114
7.52~10-3
4.66~10-3
Base O i l 12
Viscosity Viscosity Index
Polymer S o l u t i o n
10 P 0
Viscosity ( i = O )
9' P X
a
8
0 (L
Viscosity Index
156
156
Power-Law I n d e x , n
0.94
0.95
10x103
23x1 0 3
0
;o(s-l
W
U
3 6
D e n s i t y , (gm/cm3)
v)
w (L n
0.8128
S p e c i f i c Heat ( J / g ° K ) 2.34 Thermal C o n d u c t i v i t y (W/cm O K ) 1 .35x10-3
4
0.7966 2.47 1.32~10-3
( a l l v i s c o s i t i e s a r e g i v e n i n u n i t s of Pams) 2 I
0
I
I FLOW
I
I
I
I
R A T E , cm3/8
F i g . 3 Comparison of t h e o r e t i c a l c a l c u l a t i o n s w i t h c a p i l l a r y v i s c o m e t e r data f o r a N e w t o n i a n o i l a t 37.6OC. C a p i l l a r y c h a r a c t e r i s t i c s : L = 45.4 mm, D = 0.26 rnm. Fluid properties at 37.6%: 11 = 0.0693 P a - s , V i s c o s i t y i n d e x = 8 3 , p = 0.870 g/m3, Cp = 1.97 J / g O K , k = 1 . 4 4 ~ 1 0 - 3 W/cm O K .
T y p i c a l e x p e r i m e n t a l r e s u l t s a t two t e m p e r a t u r e s from t h e h i g h - s h e a r c a p i l l a r y v i s c o m e t e r t e c h n i q u e a r e s h o w n i n F i g u r e 4. T h e s e r e s u l t s show t h a t t h e a d d i t i o n o f p o l y m e r i n c r e a s e s t h e v i s c o s i t y of t h e o i l and t h a t the r e s u l t i n g v i s c o s i t y decreases w i t h i n c r e a s i n g shear rate. This p l o t i n d i c a t e s that the viscosity-shear rate b e h a v i o r is approximated by t h e power-law m o d e l . The v i s c o s i t y b e h a v i o r over a complete range o f shear r a t e c a n be d e t e r m i n e d by c o u p l i n g t h e s e r e s u l t s w i t h measurements of polymer-solution v i s c o s i t i e s a t l o w - s h e a r r a t e s i n t h e r e g i o n where Newtonian T h e e x t e n s i o n of t h e behavior is e x h i b i t e d .
331
E
I
l
1 CAPILLARY CHARACTERISTICS DIAMETER lmml
LENGTH lmml
A
45.4 127.5
0 0
I
I I I
I
I
2
0.260 0.21 e
SHEAR
RATE I
I
I I I I I
I
6
e
lo-'.
8.'
4
20
10
F i g . 4 C a p i l l a r y viscometer measurements f o r t h e polymer s o l u t i o n d e s c r i b e d i n T a b l e 1. power-law l i n e a s shown i n F i g u r e 4 u n t i l i t i n t e r s e c t s ,the low-shear v i s c o s i t y w i l l d e t e r m i n e , Y o , t h e c r i t i c a l shear r a t e for t h e o n s e t of s h e a r - t h i n n i n g b e h a v i o r . The rheological parameters f o r the truncated power-law model c o r r e l a t i o n f o r t h e s e s o l u t i o n s are p r e s e n t e d i n T a b l e 1 . If t h e r e s u l t i n g r h e o l o g i c a l b e h a v i o r f o r t h i s o l e f i n copolymer s o l u t i o n a r e c o r r e c t , t h e n a computer s i m u l a t i o n o f t h e f l o w o f a f l u i d w i t h t h e s e r h e o l o g i c a l properties should r e s u l t i n pressure drop versus flow r a t e b e h a v i o r w h i c h is i n agreement w i t h t h e observed measurements a f t e r t h e y h a v e b e e n c o r r e c t e d f o r end e f f e c t s . F i g u r e 5 shows such a c o m p a r i s o n f o r t h i s p o l y m e r s o l u t i o n where t h e l i n e s were o b t a i n e d from t h e t h e o r e t i c a l 14
I
1
I
I
I
I
04
06
08
1.0
1.2
10 0
a
;?
8
a
0
a
0
3 v)
w
a a 4
2
C
02
6
ACKNOWLEDGMENT
The a u t h o r s w i s h t o acknowledge t h e s u p p o r t of t h e P e n n s y l v a n i a Crude O i l A s s o c i a t i o n f o r t h i s project
.
References 'What s h o u l d an engine o i l v i s c o s i t y system do?', Opening Address a t SAE Open Forum, D e t r o i t , Michigan, February 26, 1976. Workshop I , High-Temperature, High-Shear Viscometry, Workshop Summary, The 1981 I n t e r n a t i o n a l Conference on t h e Viscometry of Automotive L u b r i c a n t s , October 1981. D E N H E R D E R , M.J., HARNACH, J . W . and WESTER, D.W. ' O i l v i s c o s i t y a t h i g h shear r a t e s measured by a f l o a t i n g j o u r n a l b e a r i n g ' , SAE 1978, Paper No. 780375. P I K E , W.C., BANKS, F.R. and K U L I K , S. ' A s i m p l e h i g h shear viscometer - Aspects of c o r r e l a t i o n w i t h engine performance', SAE 1978, Paper N O . 780981. DU PARQUET, J . and CODET, A. 'Assessment of l u b r i c a t i o n c o n d i t i o n s i n a big-end b e a r i n g by t e m p e r a t u r e measurements c o r r e l a t i o n w i t h high s h e a r v i s c o s i t y ' , SAE 1978, Paper No. 780980. ROUSSEL, C . and DU PARQUET, J. 'Development of a f u l l y a u t o m a t i c viscometer f o r o i l r h e o l o g y i n a broad r a n g e o f shear r a t e s ' , SAE 1982, Paper No. 821249. FENSKE, M . R . , KLAUS, E.E. and DANNENB R I N K , R.W. ' V i s c o s i t y - s h e a r behavior of two non-Newtonian polymer-blended o i l s ' , Am. SOC. T e s t i n g Materials 1950, Sp. Tech. P u b l i c a t i o n No. 1 1 1 , 3-23. TZENTIS, L.S. ' A two-way c a p i l l a r y v i s c o m e t e r ' , AIChE J . 1966, 12, No. 1 , 45-49.
McMILLAN, M.L.
12
*
model u s i n g t h e p h y s i c a l p r o p e r t i e s l i s t e d i n Table 1 as i n p u t . The a g r e e m e n t between c a l c u l a t e d and measured r e s u l t s s t r o n g l y supports the v a l i d i t y of t h i s technique. An a l t e r n a t i v e e v a l u a t i o n o f t h i s e x p e r i m e n t a l t e c h n i q u e is a l s o p r e s e n t e d i n The a g r e e m e n t between experimental F i g u r e 4. r e s u l t s o b t a i n e d from two d i f f e r e n t c a p i l l a r i e s i s additional evidence to support the a p p l i c a b i l i t y of t h i s e x p e r i m e n t a l procedure and t h e associated d a t a a n a l y s i s technique. Extensive s t u d i e s o v e r a wide r a n g e of e x p e r i m e n t a l c o n d i t i o n s show t h a t i n t h e c o n s i s t e n c y check, t h e agreement between t h e observed p r e s s u r e drop and t h e calculated p r e s s u r e drop i s t o w i t h i n f 2% as l o n g as t h e excess p r e s s u r e d r o p c o r r e c t i o n i s maintained below 1 0 % o f t h e t o t a l p r e s s u r e d r o p a n d t h e c o r r e c t i o n of t h e a p p a r e n t v i s c o s i t y f o r the i n f l u e n c e of p r e s s u r e and v i s c o u s h e a t i n g i s m a i n t a i n e d below 8 % . A modified d a t a a n a l y s i s procedure is being developed t o e x t e n d t h e r a n g e o f a p p l i c a b i l i t y of t h i s experimental t e c h n i q u e . The l i m i t a t i o n of a n 8% c o r r e c t i o n i s probably due t o a breakdown of t h e assumption t h a t t h e f r a c t i o n a l r e d u c t i o n i n t h e a p p a r e n t v i s c o s i t y due t o t h e i n f l u e n c e of p r e s s u r e and v i s c o u s h e a t i n g i s i d e n t i c a l f o r a non-Newtonian f l u i d and a Newtonian f l u i d which e x h i b i t t h e same r h e o l o g i c a l p r o p e r t i e s a t low-shear rates.
1.4
FLOW RATE, cm3/r
F i g . 5 Comparison of t h e o r e t i c a l c a l c u l a t i o n s f o r an a d i a b a t i c c a p i l l a r y w i t h viscometer d a t a . The p r o p e r t i e s of t h e polymer s o l u t i o n are presented i n Table 1.
332
(9)
JOHNSON, R . H . and W R I G H T , W . A . 'The r h e o l o g y of a m u l t i g r a d e d motor o i l ' , SAE T r a n s a c t i o n s 1968, Paper No. 680072. JOHNSON, T.W. and O'SHAUGHNESSY, M.T. 'Measurement o f temporary and permanent shear with t h e i n s t r o n c a p i l l a r y r h e o m e t e r ' , SAE 1977, Paper No. 770377. McMILLAN, M.L. and M U R P H Y , C . K . 'Temporary v i s c o s i t y l o s s and i t s r e l a t i o n s h i p t o j o u r n a l b e a r i n g perf o r m a n c e ' , SAE 1978, Paper No. 780374. PHILLIPOFF, W. ' V i s c o s i t y measurements on polymer-modified o i l s ' , ASLE T r a n s a c t i o n s 1958, 1,82-86. R E I N , S.W. and ALEXANDER, D . L . 'Development of a h i g h s h e a r r a t e c a p i l l a r y viscometer f o r engine o i l s ' , SAE 1980, Paper No. 800363. HEWSON, W.D. and C A R E Y , L.R. 'High s h e a r v i s c o s i t y o f e n g i n e o i l s ' , SAE 1980, Paper No. 801394. APPELDOORN, J . K . and DEVORE, D . I . 'Limitations i n c a p i l l a r y viscometry', SAE 1980, Paper NO. 801389. J O L I E , R.M. ' S t r u c t u r a l e f f e c t s on t h e v i s c o s i t y b e h a v i o r o f polymer s o l u t i o n s ' , Ph.D. T h e s i s , The P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1 957. A N G E L O N I , F.M. 'Development o f a c a p i l l a r y h i g h s h e a r v i s c o m e t e r ' , M.S. T h e s i s , D e p t Chem., The P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1959. BADCLEY, R.S. 'Measurement of h i g h s h e a r v i s cos i t i es o f polymer - b l ended 1ubr i c a n t s b e f o r e and a f t e r mechanical d e g r a d a t i o n ' , M.S. T h e s i s , Dept. Chem. Eng., The P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1981. LEE, F.-L. 'Measurement and a n a l y s i s o f h i g h s h e a r v i s c o s i t i e s o f polymerc o n t a i n i n g l u b r i c a n t s ' , Ph.D. T h e s i s , Dept. Chem. Eng., The P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1984. GRAHAM, E . E . , KLAUS, E.E. and BADCLEY, R.S. ' D e t e r m i n a t i o n o f t h e v i s c o s i t y s h e a r behav i o r o f polymer - c o n t a i n i ng f l u i d s u s i n g a s i n g l e p a s s , high-shear c a p i l l a r y v i s c o m e t e r ' , SAE 1984, Paper No. 841391. SCHLICHTING, H. 'Boundary Layer T h e o r y ' , Seventh E d i t i o n 1978 (McCraw-Hill). C E R R A R D , J . E . , STEIDLER, F.E. and APPELWORN, J.K. 'Viscous h e a t i n g i n c a p i l l a r i e s - t h e a d i a b a t i c c a s e ' , I&EC No. 3 , 332-339. Fund. 1965, G E R R A R D , J . E . , STEIDLER, F.E. and APPELWORN, J.K. ' V i s c o u s h e a t i n g i n c a p i l l a r i e s - t h e i s o t h e r m a l wall c a s e ' , I & E C Fund. 1966, 5 , No. 2 , 260-263. SO, B.Y.C. and KLAUS, E.E. ' V i s c o s i t y p r e s s u r e c o r r e l a t i o n o f l i q u i d s ' , ASLE T r a n s . 1980, 23, No. 4, 409-421. KESTIN, J . , SEOLOV, M . and WAKEHAM, W. 'Theory o f c a p i l l a r y v i s c o m e t e r s ' , Appl. S c i . Res. 1973, 27, 241-264. VRENTAS, J . S . , D E A , J . L . and HONG, S.A. 'Excess p r e s s u r e d r o p s i n e n t r a n c e f l o w s ' , J. o f Rheology 1982, 26, 347-357. SYLVESTER, N . D . and ROSEN, S . c 'Laminar flow i n t h e e n t r a n c e r e g i o n o f a c y l i n d r i c a l t u b e ' , AIChE J . 1970, 16, 964-972. BOGER, D . V . , CUPTA, R . and T A N N E R , R . I . 'The end c o r r e c t i o n f o r power-law f l u i d s i n t h e c a p i l l a r y r h e o m e t e r ' , J. Non-Newt. F l u i d Mech. 1978, j,, 239-248.
77,
(10)
(11)
(12) (13)
(17)
.
(1 8)
( 1 9)
(20)
(21 1 (22)
i,
(23)
(28)
(29)
and VRENTAS, J.S. ' E n t r a n c e flows o f non-Newtonian f l u i d s ' , T r a n s . SOC. Rheol. 1973, 3, 89-108.
DUDA, J.L.
333
Paper X(iv)
Properties of polymeric liquid lubricant films adsorbed on patterned gold and silicon surfaces under high vacuum Michael R. Philpott, lngo Hussla and J.W. Coburn
The distribution, configuration and orientation of polymeric lubricant molecules on surfaces of well defined morphology is an area of great scientific interest due to the very different physical and chemical interactions existing amongst polymers and surfaces compared t o small molecules and surfaces. This area is also one of immense technological importance because of its relevance t o the understanding of elastohydrodynamic and boundary lubrication between moving surfaces. The vibrational properties of model lubricants, perfluorinated polyethers, on planar and patterned Au and SiOz/Si surfaces have been studied by polarization modulated Fourier transform infrared reflection-absorption spectroscopy and the polymer distribution profile has been examined simultaneously by ion-induced volatilization. Patterned surfaces consisting of 2D square lattices of 2 pm radius holes on 8 and 16 pm centers were lithographically etched into wafers. Due to their extremely low vapor pressure at 298K, stable liquid films could be adsorbed on surfaces under HV. Measurement of the vibrational spectrum and volatilization rate as a function of temperature, lubricant thickness and surface morphology has yielded information concerning the mobility, configuration and distribution of these large molecules as well as insights into the mechanism for film spreading out of microporous structures.
1 INTRODUCTION
Perfluorinated polyethers of high molecular weight are finding increasing uses in demanding environments where resistance t o extremes of temperature, corrosion and oxidation are important (1). They are also amongst the most widely used lubricants in magnetic storage devices for the reduction of friction. and wear of rigid magnetic disk media by the sliders that carry the read-write heads (2-3). A knowledge of the configuration, orientation and distribution of these molecules in ultra thin liquid films would add to understanding of how they modify the tribological properties of sliding interfaces. These properties are also of great scientific interest since at present very little is known about the interactions between large molecules and surfaces compared t o the relative wealth of information available by modern surface science spectroscopies about even submonolayer concentrations of small polyatomics on well characterized surfaces (4). This paper briefly describes progress made in measuring the properties of thin liquid films of perfluorinated polyethers on gold and silicon surfaces using ion-induced-volatilization (IIV) to measure lubricant thickness and distribution (5-6) and Fourier transform infrared (FTIR) spectroscopy (7) t o monitor changes in chemical bonding and orientation. Particulate magnetic media have some degree of surface roughness and volume porosity that acts as a reservoir for lubricant. We have modelled this reservoir capability by lithographically etching into polished Si( 111) wafers a two-dimensional array of cylindrical holes approximately 1 pm deep and 2 pm in diameter on 8 pm and 16 pm centers. These wafers were used either with their native oxide (SiO,/Si) surface or coated with approximately 100 nm polycrystalline gold (Au surface). The model liquid lubricant used in this work was perfluoropolypropylene oxide available commercially as Fomblin-Y25 (Montefluos). The chemical formula of this material hence forth referred to as the lubricant or lube is
- [CF(CF3)CF20],
- (CF,O),
-
where m was approximately twenty times greater than n. The average molecular weight of the lubricant samples was approximately 104 Daltons. Space filling models showed the polymer chain t o be 0.7-0.8 nm thick, have 60 units and an end-to-end length of approximately 2 0 nm. A t room temperature the polymer is a colorless chemically inert liquid with extremely low vapor pressure. The glass transition occurs a t approximately -65OC. 2 ION INDUCED VOLATILIZATION
The lubricant possessed a remarkable and very useful property under argon ion radiation in the few keV range. It decomposed into small molecular weight fragments that were completely volatile. Essentially no residue remains. A t 1 keV energy the measured yields were typically 104 amu per argon ion impacting the lubricant. The depolymerization process was probably initiated by homolytic fission of the chain C-C bond. Since the gain was so high, argon ion bombardment coupled with mass spectra analysis of the residual gas in a vacuum chamber could be used to quantitatively measure the amount of lubricant volatilized from and thereby control the amount of lubricant on a surface. Penetration depth of low keV argon ions is less than 5 nm. For argon ions with 1 keV energy and current density 0.3 pa/cm2 the sputter rate was approximately 15 nm/min a t 25OC and 10 nm/min a t -65OC. Figure 1 shows a comparison of the mass 69 (CF3+ fragment) signal recorded on the mass spec during argon ion bombardment of a hand lubricated plane Si wafer and one patterned with 2 pm diameter holes on 8 pm centers. The signals were normalized to that from a very heavily lubricated sample (lube lake), in order to measure the fractional surface area covered by lubricant more than 5 nm thick. The area under the curve gave the total amount of lubricant volatilized once the beam current and etch rate were calibrated using a faraday cup and an oscillating quartz
334
crystal microbalance. The peak intensities were comparable indicating similar coverages however the time dependence was strikingly different since the patterned surface signal showed a much slower fall off in time. This was interpreted as due to lubricant migration from the cylindrical holes to the flat areas during the volatilization process. The result of volatilization at -6SoC, the glass transition of the polymer, is shown in Figure 2. A t this temperature the polymer molecules are essentially frozen. In contrast to the room temperature experiment where all the lubricant was removed in a few minutes, a very long time plateau approximately 60 minutes long was observed, corresponding to sputtering lubricant from 1 pm deep holes. The high initial peak which occurred during the first few minutes was due to the removal of surface molecules. Although the fragmentation yield was 30% lower a t -65OC it was still very high and sufficient for the IIV technique t o again give quantitative results. The inset in Figure 2 shows the result of IIV after allowing the sample t o warm to room temperature. A little residual lube was detected (note the factors ten in time scales). The quantity of lubricant spreading from the holes to bare surface was determined by sputtering the surface lube at -6SoC, warming t o room temperature, cooling t o -65OC and again removing only surface lubricant by IIV. This procedure was repeated until essentially all the lubricant was removed from the holes. One result from this series of experiments is shown in Figure 3. It plots lubricant mass found on the surface after a temperature cycle against the amount of lubricant in the pore before the cycle. The SiOz/Si and Au surfaces appeared to behave in a similar way, approximately one half of the lubricant in the pore spread to the surface during each temperature cycle. 3 FOURIER TRANSFORM INFRARED SPECTROSCOPY Carbon fluorine bonds absorb very strongly in the infrared (IR) typically in the range 7 pm to 10 pm. Consequently, Fourier tranform IR is a powerful tool with which to detect and measure the properties of fluorinated lubricants. A custom apparatus capable of ultra high vacuum was built t o use IIV and FTIR on the same samples. The ion gun directed a beam of argon ions a t normal incidence while the IR light impinged on the sample in the reflection mode at near grazing incidence. The IR technique was a modification of the method developed by Golden, et al. (8-9) in which the polarization of the incident IR beam was switched between s- and p- polarization a t 75 kHz using a ZnSe photoelastic modulator. By means of custom built data acquisition electronics the spectra were recorded as the ratio I=(Ip-Is)/(Ip+IJ where I, is the reflected light intensity polarized x=s or p. The chief attribute of this technique for the present study was that randomly oriented dipoles absorbing IR photons in the beam path were automatically cancelled leaving only the spectrum of IR active dipoles oriented normal t o the sample surface. Further details of this apparatus are available elsewhere (10). Figure 4 displays representative IIV and FT-IRRAS (inset) for a polycrystalline gold surface patterned with holes on 8 pm centers and covered with approximately 30 nm of lubricant, as determined by microellipsometry. The experiment was performed a t room temperature. The IIV spectrum shows both the reference taken from the lube lake (A) and the time dependence for complete volatilization of lubricant from the patterned surface (B). Note that the surface was incompletely lubricated otherwise the initial IIV signal would have been more comparable to that of the lube lake. However, the general similarity to the time dependence evident for the similarly patterned surface shown in
Figure 1 implies no qualitative difference in the process. This conclusion was supported by the IR spectrum which also showed no evidence interpretable as due t o partial surface coverage. The FT-IRRAS spectrum shown was obtained by ratioing the spectrum taken before the with the one taken after sputtering away all the lubricant. Ratioing spectra taken after sputtering or taken before and after a second sputtering showed no absorption bands due to lubricant or decomposition products. The features marked by arrows are all infrared active vibrational bands of bulk liquid lubricant (11) a t 25°C. These IR spectra did not differ greatly from the spectra of thickener films implying that the majority of the molecules feel an environment similar to that of the bulk liquid. There were however, some small differences associated mainly with the bands near 1300 cm-1 and 1000 cm-1 which are associated with the CFj-functionality (12). These are the subject of a continuing study, it suffices here t o mention that the effects may be due to reorientation of the polymer chains due to surface forces. 4 CONCLUSIONS
Through the use of patterned surfaces it has been demonstrated that at the glass transition temperature IIV can be used to measure the vertical distribution of a perfluorinated polyether lubricant as well as the total amount present. Topical application of lubricant t o patterned surfaces followed by rubbing into the surface results in liquid films that were essentially continuous across the surface with thickness in the range 10-20 nm on the planar regions and in filled pores. Coinplete delubrication without the formation of significant residues was verified by the novel application of FT-IRRAS. By use of temperature cycling it was also established that lubricant can migrate from pores t o planar regions. During IIV at room temperature lubricant migration occurred with sufficient facility t o essentially deplete the entire irradiated area. The use of IR spectroscopy to monitor films 10 nm thick has been established for the patterned surfaces and with sufficient signal-to-noise to show the film t o be liquid-like with regard to the positions, widths and shapes of the absorption bands.
5 ACKNOWLEDGMENTS The advice and help of H. Seki, V. Novotny, B. Hoenig and R. Sadowski are warrdy acknowledged. References SIANESI, D., ZAMBONI, V., FONTANELLI, R. and BINAGHI, M. Wear, 1971, 18.85-100. BAGA'ITA, P.U. "Fomblin perfluorinated polyethers. Lubricants for surface lubrication of magnetic media", Montefluos Technical Literature, 1985. YANAGISAWA, M. Amer. Sac. Lubrication Engineers, 1985, SP-19, 16-. See for example: BELL, A.T and HAIR, M.L. "Vibrational spectroscopies for adsorbed species," American Chemical S0cie.y Symposium Series 137 (Washington D.C., 1985). COBURN, J.W. and WINTERS, H.F. J. Appl. Physics, in press (1986). WINTERS, H.F., COBURN, J.W. and CHUANG, T.J. J. Vac. Sci. Technol., 1983, B1,469-480. AU-YEUNG, V. IEEE Trans. Magnetics, 1983, hIAG-19(5), 1662-1664. GOLDEN, W.G., DUNN, D.S. and OVEREND, J. J. Catalysis, 1981, 11,395-404. GOLDEN, W.G. and SAPERSTEIN, D. J. Electron Spectrosc. and Related Phenomena, 1983, 30,43.
335 (10) HUSSLA, I. and PHILPOTT. M.R. J. Electron Spectros. and Related Phenomena. 1986, 39,255-263. J.. ENGLAND, C.D. and (11) PACANSKY. WALTMAN, R. Appl. Spectroscopy, in press (1986). (12) HUSSLA, I., NOVOTNY, V. and PHILPOIT. M.R., unpublished observations (1985).
S i0, /S i
j
a,
3
0.8
25°C
0.6
0.4
,"
A Si A Au
01 0
1
lx2x16prn lx2x16prn
I
I
I
I
25
5 10 15 20 Lubricant Mass in Pores (arb. units)
Fig. 3 Mass of lubricant leaving pores of a patterned surface versus mass of lubricant initially residing in the pore.
0.2
0
0
2 4 6 Sputter Time t (min.)
8 t t Au
1xZx8pm
Fig. 1 Comparison of IIV of lubricant from plane and patterned surface a t room temperature.
25°C
I
I
1400
I
1300 1200
1100
1000
Wavenumbers cm-'
L
~~
500
1000
1500
2000
Sputter Time
Fig. 4 Time dependence of the IIV signals a t 25OC.; A lube lake; B lubricant on patterned gold surface. Inset: the ratio of lT-IRRAS spectra taken before and after IIV showing the IR spectrum of the volatilized lubricant.
Sputter Time t (min.) Fig. 2 IIV of lubricant from a patterned surface a t a temperature near the glass transition.
This Page Intentionally Left Blank
SESSION XI BEARING DYNAMICS (1) Chairman: Dr. O.R. Lang PAPER Xl(i)
Identification of fluid-film bearing dynamics: time domain or frequency domain?
PAPER Xl(ii)
The influence of grooves in bearings on the stability and response of rotating systems
PAPER Xl(iii) Theoretical and experimental orbits of a dynamically loaded hydrodynamicjournal bearing PAPER Xl(iv) The effect of dynamic deformation on dynamic properties and stability of cylindricaljournal bearings
This Page Intentionally Left Blank
339
Paper Xl(i)
Identification of fluid-film bearing dynamics: time-domain or f requency-domain? John Mottershead, Riaz Firoozian and Roger Stanway
In this paper the authors describe experiments which compare time domain and frequency domain approaches to the estimation of the four linearised damping coefficients associated with a squeezefilm vibration isolator. It is shown that while both algorithms satisfy a least-squares error criterion, the resulting coefficients can have quite different numerical values. Whilst the feasibility of both techniques has now been clearly established it is concluded that further work is required to ensure a more direct comparison. 1
INTRODUCTION
One of the most challenging problems in tribology is the determination of mathematical models to describe the dynamics of bearing oilsometimes referred to as films in vibration the problem of bearing identification. The central problem of bearing - identification involves the determination of the linearised coefficients associated with a journal bearing However, there are are other oil-film problems currently of interest, notably the identification of the linearised dynamics of the squeeze-film vibration isolator. It is this latter problem which will be considered here. The difficulties associated with obtaining reliable estimates of oil-film coefficients are well-known: the main problems arise from constraints on the signal used to excite the oil-film, contamination of measured responses and ill-conditioning of the equations from which the coefficient estimates are derived. To overcome these problems there has been much recent work on the development of computerbased algorithms for processing measured vibration data. Some of this work has employed a time-domain algorithm which does not impose special constraints on the form of the excitation signal, operates directly on contaminated time-series data and can accommodate various disturbances and nonlinearities (Stanway, Mottershead and Firoozian (1)). Alternatively a frequency-domain approach can be adopted (Burrows and Sahinkaya (2)): this does require a suitable multiple-frequency excitation signal and Fourier transformation of input and output signals before parameter estimation can be performed. However the frequency domain approach does appear better suited than the time-domain for estimating large numbers of parameters in linear vibrating structures (Burrows and Sahinkaya (3)). In an attempt to provide some meaningful comparisons between the two approaches, the authors have reformulated the time-domain filter described in reference ( 1 ) to accept frequencydomain data. In the same way as the time-domain filter accepts data samples at each time increment so the frequency-domain filter accepts real and imaginary components of the frequency
-
.
response spectrum at each frequency increment. In each case the coefficient estimates are updated in sequential fashion. In this paper the two approaches are applied to experimental data from a recentlyconstructed squeeze-film isolator. Some typical results are presented and their significance is discussed. 1.1
Notation
Cxx, etc.
non-dimensional squeeze-film damping coefficients
C
radial clearance (rn)
--
__
vector functions associated with parameter estimation J
non-dimensional stiffness of supporting shaft
-P
error covariance matrix non-dimensional group
Qs
T
time interval non-dimensional applied forcing
ux’‘y
vertical displacement of damper ring (m)
X
x1 x2
x3 x4
= x/c I
z
displacement ratio, x-axis
d(x/c)/dr axis
E
velocity ratio, x-
= y/c = displacement ratio, y-axis d(y/c)/dr = velocity ratio, yaxis state variables corresponding to unknown parameters
X -
state vector
340
-Z(T ,z(w) -r(T
vector of observations
-f(x,r) -
=
x2
-ESx1
weighting matrix
-
Q x X 9 2 5
-
error vector
W
radian frequency
n
frequency interval
0
T
non-dimensional time
0
E
-QSx2x7
-
0
2.1
0
Statement of Problem
For this study, the mathematical model of the squeeze-film isolator is assumed to take the form (Burrows and Stanway (4)):
-QsCxx dr
0
h(x,r) + -
Z(T) -
X X
s 4 8
+ U
y
(observation noise)
(3)
As a typical example, setting 0
-Q C
-i;
-Q
C
YX The dynamics of the squeeze-film are characterised by four non-dimensional damping and C and fluidcoefficients: C , C , C film inertia an3xstiF?nesgxterms XFe assumed to be negligible. The problem in hand is to estimate the four damping terms from measured records of the available state variables (xl,..,x4 ) associated with the damper ring's response to imposed excitation (ux,uy).
2.2
Q
The "physical" state variables (i.e. damper ring displacements and velocities) which can be measured are held in a vector of observations Z(T) where -
0
0 s
Q x X s46+'x
x4
-ksX3 -
2 BRIEF REVIEW OF THEORY
1
-
Time-Domain Identification
indicates that time-series records of the damper ring's displacement responses are available for processing. Given this formulation (and assuming that the system is observable) it has been shown (Stanway ( 5 ) ) that knowledge of the vector of observations over a suitable time span ( 0 < T < T) enables the correspond behaviour of the state vector 5 to be deduced. Since the state vector x contains the four damping terms, then the estimation of 5 automatically implies estimation of the linearised squeeze-film dynamics. An estimate of the state vector (denoted which evolves with time can be obtained through the solution of the non-linear differential equations:
-
i
One seemingly indirect technique for tackling the problem of fluid-film identification was described by Stanway ( 5 ) . This approach involves the reformulation of the problem as one of combined state and parameter estimation (Detchmendy and Sridhar (6)). Essentially the four unknown damping coefficients are defined as additiona A "paramet r" state var'ables, i.e. x C , x6 = C 2= C a n d x b C These &atexx v%iatIes a?$ subst?t&e8Y;nto equation ( 1 1. It is assumed that the damping coefficients remain constant during the course of an experiment so that the time derivatives of states x x8 are zero. Finally if it is noted tha? dx /dr 1 x2 and dx /dr I x4 then a set of eight nonlinear di?ferential equations can be written:
4
,...,
In practice such an estimate is obtained through the numerical solution of equation (4) using a technique such as Runge-Kutta-Merson with input provided by sampled records of the vector of observations z(T). Thus the input data is processed sequentially with the estima of 2 being updated at each time step. Because of the inherentlv non-linear must a1 formulation, the weighting matrix ;(TI be updated at each time step. The key component of L(T) is the error covariance matrix p(T) which evolves according to the equation dP
aff9.T)
~
~
ar-T (2.~)
where
x and
= [x 1 2' 3'
x4 x5 6' '7 ' 8 T
where H(Z,T)A=
ah(?,T)
-a% -
and
2
is a weighting
matrix. Fluid-film identification using equations (4) and (5) is not economical in computer time but has proved effective for estimating the four squeeze-film damping terms from contaminated
341
records of the damper ring's displacement responses to synchronous unbalance excitation. However time-domain approaches do seem to have difficulty in estimating additional parameters (i.e. inertia, stiffness) which might be significant in fluid-films. For this reason a frequency-domain equivalent of the time-domain filter has been developed.
2.3
Frequency Domain Identification
Frequency domain techniques have been applied previously to the problem of estimating the four squeeze-film damping terms (Burrows and Sahinkaya (2)). In reference (21, damper ring responses to multiple frequency excitation were gathered, Fourier transformed and then processed to yield coefficient estimates. Unlike the time-domain algorithm described in Section 2.2 frequency data was accepted sequentially, i.e. all the data had to be gathered before processing could begin. However there is no reason why a sequential frequency domain filter cannot be derived and indeed it follows that such a filter can be developed directly by an appropriate change of variable in equations (4) and (5). A full derivation of the frequency domain filter is given by Mottershead and Stanway (7). Essentially the unknown parameters are defined as state variables in much the same way as in the time-domain formu'lation,i.e. x = C , x = C , x = C and x = C Howeve? unlfze the tfxe d&nainYPormulagion, eh!' physical variables (i.e. displacements and velocities) are not included in the state vector, which obviously reduces its dimension, i.e.
.
m
x -
= cx a b' c'
l ' d
I f the four coefficients are assumed to be independent of frequency w then we can write dx
_ dw -
0 -
(7)
which is the frequency domain equivalent of equation (4). Measured response spectra in the frequencydomain will, in general, be complex and thus the measurement equation
-z(w)
= h(x,w) + (observation noise) -
(8)
will have both real and imaginary parts, i.e.
where E is an error vector. Tzking note of the implications of equations (7) and ( 8 1 , the frequency-domain filter can be written down directly from the time-domain version (equations (4) and (5)1 simply by substituting the frequency interval n in place of the time interval T. The frequency-domain filter is supplied sequentially with the real and imaginary parts of the observations in ~ ( w )and the filter equations are then solved at each frequency step using any suitable numerical technique. Note that in order to supply response spectra in this form the observed input and output time-series
must undergo Fourier transformation.
3 EXPERIMENTAL APPARATUS The model squeeze-film isolator used in the experiments is shown in Fig. 1. The isolator consists of a non-rotating damper ring, symmetrically supported by a flexible shaft. Damping is provided by a film of lubricant between the damper ring and a bearing housing. This housing contains two plain lands separated by a central circumferential groove. Lubricant is supplied to the annulus by a pump which feeds inlet holes at the top and bottom of the circumferential groove. No end seals are fitted and the lubricant is free to discharge into a reservoir, prior to re-circulation. The critical bearing and suspension parameters are: bearing land length 12.0 mm; damper-ring radius 60.0 mm; radial clearance 254 pm; damper-ring mass 4.5 kg per land and radial stiffness of the supporting shaft 250 kN1m. The static equilibrium position of the isolator is set by adjusting the bearing housing and then locking it in the desired position. Details of the adjustment mechanism are given by Stanway, Firoozian and Mottershead ( 8 ) . Dynamic forcing about the chosen equilibrium position is provided by two electromagnetic shakers mounted at right angles to each other, as shown in Fig. 1. Using this arrangement any desired pattern of forcing can be provided. In particular if each shaker is supplied with a sinusoidal signal with the same frequency and amplitude then synchronous forcing from an unbalanced rotor is readily simulated. Alternatively multiple frequency signals (for example, those derived from a Schroeder-phased harmonic sequence) are readily injected.
3.2 Measurements and Instrumentation The static eccentricity and attitude of the damper ringlbearing housing are monitored by mechanical clock gauges. Lubricant pressure is measured at the inlet to the housing and lubricant temperature is measured as it discharges from the annulus. Instrumentation for the generation of dynamic forces and the monitoring of responses is shown in Fig. 2. Forces applied to the damper-ring are measured by quartz load cells connected to suitable charge amplifiers. The displacement responses of the damper ring are measured by two sets of non-contacting capacitance probes, two in the vertical plane and two in the horizontal plane. Input forces and output displacements are monitored using a real-time, dual-channel FFT analyser. Suitable sequences of inputloutput data are gathered for subsequent processing by a data-acquisition system comprising a 12-bit analogue-to-digital converter controlled by a digital micro-computer. Data are stored on floppy discs before being transferred to a mainframe computer for parameter estimation. 4 EXPERIMENTAL PROCEDURE AND RESULTS
4.1
Preliminaries
Prior to each individual experiment the desired static eccentricity ratio and attitude angle was selected and the apparatus locked in position. The lubricant (tlShell"Tellus 27) was pumped
through the isolator until a steady operating temperature was reached. A typical temperature was 28°C corresponding to a viscosity of 0.06 Nslm. There was no significant temperature drop across the squeeze-film.
4.2
Time-Domain Estimation
For the time-domain estimation experiments the damper-ring was perturbed about the chosen equilibrium position by sinusoidal forces of equal amplitude and displaced in phase by 90' to simulate synchronous forcing. A forcing some 15 Hz below the frequency of 20 Hz resonant frequency of the damper ring on its was found to give a good supporting shaft signal to noise ratio. At this frequency a sampling interval of 300 ps and 1000 samples from each sensor provided approximately six cycles of steady-state vibration data. A series of tests was performed with an attitude angle of zero and static eccentricity ratios of 0.0, 0.2, 0.4, 0.6 and 0.8. Some typical results are shown in Fig. 3. These results were obtained at an eccentricity ratio of 0.4 and using peak forcing amplitudes of 25 N which produced corresponding peak displacement amplitudes of some 5-10 per cent of the radial clearance. Figure 3 shows the evolution (with increasing time) of the estimates of each of the eight state variables. Figure 4 shows a typical set of frequency responses predicted on the basis of the steady coefficient values shown on Fig. 3. Superimposed on the plots are estimates of the various frequency response functions obtained from FFT analysis of the inputloutput data.
-
-
4.3
Frequency-Domain Estimation
For the frequency-domain estimation experiments the damper-ring was perturbed using a signal derived from a Schroeder-phased harmonic sequence (SPHS). The SPHS signal contained components of equal strength from 1 to 100 Hz with a resolution of 1 Hz. The signal strength was chosen to produce peak displacement amplitudes of some 30 per cent of the radial clearance. The inputloutput responses were processed using an FFT algorithm before being processed by the frequency-domain equivalent of equations (4) and (5). For an identical set of static conditions to those described in Section 4.2 a series of frequency domain estimation experiments was performed. Typical results (again for an eccentricity ratio of 0.4) are shown in Fig. 5. These results show the evolution (with increasing frequency) of the estimates of the four parameter states. The levels of forcing for these experiments was chosen to provide peak displacement amplitudes corresponding to some 30 per cent of clearance. Predictions of the various frequency response functions using both the estimated coefficients and FFT analysis are shown in Fig. 6. Finally the steady values of the coefficient estimates obtained from the tine domain and frequency domain experiments are given in Table I. 5
DISCUSSION OF RESULTS
The results from time-domain estimation (as typified by Fig. 3 ) show that steady estimates
of the four squeeze-film damping terms can be obtained from small-amplitude synchronous responses generally less than 500 time steps are required. In most cases the orbit predicted by the estimated coefficients is indistinguishable from the orbit used as input and thus, in a least-squares sense, an adequate fit has been obtained. Moreover the identified coefficients are able to predict (with a fair degree of success) the various frequency response functions both amplitude and phase of the squeeze-film isolator. The results in Fig. 4 show how the coefficients reflect the strong direct coupling and relatively weak cross-coupling in the squeeze-film. In the frequency domain, convergence of some damping coefficient estimates to steady values is more ponderous than in the time doma n and care is required in the choice of scaling and weighting factors. With reference to Fig. are much 5, the estimates of Cx and C slower to converge thaX thoseY8f Cxx and C Also prediction of the frequency response YX amplitudes was generally poor at lower frequencies, for example see Fig. 6. However prediction of the phase angles was good over a wide frequencv range. Discrepancies between numerical values of the estimated damping coefficients is shown in Table I. The frequency domain approach invariably produced an indication of strong cross-coupling terms which was not confirmed either by the time-domain results or by direct observation of the damper ring response. The authors believe that the squeeze-film was probably over-excited during the frequency domain tests and consequently driven out of the linear regime. The presence of significant nonlinear stiffness effects in the squeeze-film could account for the discrepancies which have been observed.
-
-
-
-
6 CONCLUSIONS In this paper the authors have presented a comparison of time and frequency domain algorithms for estimating the four linearised damping coefficients associated with a squeezefilm isolator. The time-domain algorithm operates directly on displacement data and can make use of singlefrequency excitation. In the formulation presented here the coefficients are assumed to be constant during the course of an experiment but time-dependence and geometric nonlinearities can be accommodated if necessary. Perhaps the most powerful feature of the technique is that it is extremely effective at rejecting measurement noise and thus can operates on small displacement amplitudes. The frequency domain algorithm requires Fourier transformation of the displacement responses to multiple-frequency excitation before parameter estimation can begin. It is assumed that the coefficients are independent of frequency but it should be possible to examine frequency-dependence if certain modifications are made to the algorithm. When processing real data the noise rejection properties of the algorithm are not, as yet, as well established as those attributed to the time-domain formulation. Also the choice of various scaling and weighting factors seems to be important and the rates at which coefficient estimates converge to steady values can vary considerably.
343
The results presented in the paper extend previous work in that the feasibility of employing a frequency domain filter has been demonstrated. However further work (preferably using common data) is still required to provide a direct comparison between the time and frequency domain approaches. 7
Non-dimensional Coefficient
Static Eccentricity Ratio
C XY
cXX
C
C
YX
YY
0.0
1.05 1.9
0.02 2.5
0.10 1.7
0.90 3.5
t.d. f.d.
0.2
1.25 2.8
0.05 1.7
0.12 1.1
1.0 3.2
t.d. f.d.
0.4
1.4 2.3
-0.10 3.2
0.25 1.6
1.0 4.5
t.d f.d.
1.9 2.2
-0.25 2.7
0.25 0.6
1.2 3.2
t.d.
0.6
0.8
13 4.2
0.2 2.6
4.5 4.5
t.d. f.d.
ACKNOWLEDGEMENTS
The authors wish to acknowlege the assistance of the rotor-bearing group at the University of Strathclyde. Thanks are due to Professor C. R. Burrows and Dr. M. N. Sahinkaya and especially to Mr. N. Kucuk who participated in the frequency-domain estimation experiments. They also wish to thank Elaine Mooney and Frank Cumins for their help in preparing the manuscript.
2.4 2.6
References
TABLE I STANWAY, R., MOTTERSHEAD, J. E. and FIROOZIAN, R. "on-linear identification of a squeeze-film damper', presented at Fourth Workshop on Rotordynamic Instability Problems in High Performance Turbomachinery, Texas A and M University, June 2 - 4 t h , 1986. BURROWS, C. R. and SAHINKAYA, M. N. 'Frequency domain estimation of linearised oil-film coefficients', Trans. ASME, J. Lub. Tech., 1982, 194, 210-215. BURROWS, C. R. and SAHINKAYA, M. N. 'Parameter estimation of multi-mode rotorbearing systems', Proc. Roy. SOC., 1982, 3 7 9 ( A ) , 367-387.
BURROWS, C. R. and STANWAY, R. 'A coherent strategy for estimating linearised oil-film coefficients', Proc. Roy. SOC., 1980, 3 7 0 ( A ) , 89-105.
STANWAY, R. 'Identification of linearised squeeze-film dynamics using synchronous excitation', Proc. I. Mech. E., 1983, 197(C), 199-204.
DETCHMENDY, D. M. and SRIDHAR, R. 'Sequential estimation of states and parameters in noisy, non-linear dynamical systems', Trans. ASME, J. Bas. Eng., 1966, 88, 362-368.
MOTTERSHEAD, J. E. and STANWAY, R. 'Identification of structural vibration parameters using a frequency domain filter', to appear in J. Sound and Vibration. STANWAY, R., FIROOZIAN, R. and M)TTERSHEAD, J. E. 'Estimation of the linearised damping coefficients of a squeeze-film vibration isolator', submitted for publication to Proc. I. Mech. E.
f.d.
344
E E I 9 -:
U
Fig. 1
General arrangement of squeeze-film vibration isolator.
Variable phase oscillator
I
Power amplifier
b
Shaker
amplifier 4
1
Selector 4
amplifier
Squeeze -film isolator
e-
F F T analyzer and data acquistion system
Fig. 2
I
I
1
amplifier
1
Shaker
I
I
Instrumentation used in identification experiments.
m
rr rr
(D
J
rr
srr
r.
m
J
rr
0 c,
m
W
3 0
c?
m
J
23
m
1
?
?I I-.
N
Y
I
3
s 0
c-
0
& 0
-.
n Y
..
Y
I
0
e
0
s
Phase Angle(degrees1
-
0
I
0
E E'
--
-
P O0
7
i
0
1 1 1
Or 0 d
Amplitude Ratio
Estimates of "Parameter" State Variables
4'-
sqI
Estimates of "Physical" State Variables 00 ii
Jv*" O0
vl
s
I
346
l,q
r-
3[
t
x,(=Goc)
L
I I
L
1.Oy OO
50
I
I
I
J
50 L
-aJ -150
A
1501
L
lor OO
50
I-
F
OO
200
lterati ons
Fig. 5 Frequency-domain: evolution of the four elements of the state vector.
L
OO
50
Frequency(Hz1
I
E
I
I
I
I
50
-1 50
'4 I
-1501 Frequency(Hzl
Fig. 6 Frequency domain: prediction of frequency-response functions.
-from ...... from
estimated coefficients F.F.T. analysis
347
Paper Xl(ii)
The influence of grooves in bearings on the stability and response of rotating systems P.G. Morton, J.H. Johnson and M.H. Walton
Two t y p e s of dynamic problem have t o be overcome i n r o t o r b e a r i n g systems. The f i r s t r e l a t e s t o i n s t a b i l i t y a r i s i n g from t h e b e a r i n g c h a r a c t e r i s t i c s , sometimes i n c o n j u n c t i o n w i t h d e s t a b i l i z i n g i n f l u e n c e s s u c h a s r o t a t i n g damping o r aerodynamic mechanisms. The s e c o n d p r o b l e m i s t h e s e n s i t i v i t y of t h e s y s t e m t o f o r c i n g , i n p a r t i c u l a r t h a t a r i s i n g from u n b a l a n c e . In orthodox One way of m o d i f y i n g t h e dynamic b e a r i n g d e s i g n s t h e s e two r e q u i r e m e n t s a r e i n c o m p a t i b l e . c h a r a c t e r i s t i c s o f hydrodynamic b e a r i n g s is t o i n t r o d u c e e i t h e r a x i a l o r c i r c u m f e r e n t i a l g r o o v i n g . A g r o o v i n g s t r a t e g y i s p r o p o s e d which overcomes some of t h e i n h e r e n t l i m i t a t i o n s of e x i s t i n g b e a r i n g d e s i g n s and e n a b l e s improvements i n b o t h s t a b i l i t y a n d s e n s i t i v i t y t o b e made. T h e o r e t i c a l p r e d i c t i o n s are c o n f i r m e d e x p e r i m e n t a l l y on a 355 mm d i a m e t e r b e a r i n g .
1.
INTRODUCTION
The circular profile hydrodynamic oil l u b r i c a t e d bearing is simple t o manufacture, b u t i t p o s s e s s e s c e r t a i n l i m i t a t i o n s when used t o support f l e x i b l e shafts. I t is t h e m o s t u n s t a b l e of a l l b e a r i n g s , t h a t is, t h e t h r e s h o l d s p e e d a t which s h a f t s e x h i b i t 'oil w h i r l ' , is lower when s u p p o r t e d by c i r c u l a r b e a r i n g s t h a n f o r any o t h e r t y p e of b e a r i n g of comparable clearance. In contraits synchronous damping distinction, c a p a b i l i t y is e x c e l l e n t . This property l i m i t s t h e peak r e s p o n s e t o unbalance o r o t h e r synchronous e x c i t a t i o n phenomena of a r o t o r bearing system running a t a system resonance ( c r i t i c a l speed). The problem of improving t h e s t a b i l i t y behaviour w h i l s t a t t h e same time r e t a i n i n g t h e s y n c h r o n o u s damping p r o p e r t y is o n e t h a t h a s e x e r c i s e d many w o r k e r s i n t h i s f i e l d . T h i s t a s k is d i f f i c u l t , f o r a s Morton (1) p o i n t s o u t , synchronous damping and stabilizing p r o p e r t i e s m i l i t a t e a g a i n s t each other. A l l b e a r i n g p r o p e r t i e s depend on o i l f i l m p r o f i l e and t h u s on t h e p o s i t i o n of t h e j o u r n a l within t h e bearing. Strategies for i m p r o v i n g t h e c h a r a c t e r i s t i c s of any b e a r i n g at c o m p r i s i n g c i r c u l a r a r c s are aimed controlling the shaft position relative t o the bearing arcs. Derating the bearing (2) or e i t h e r by i n c r e a s i n g c l e a r a n c e c i r cumferentially grooving i t , i n c r e a s e s t h e e c c e n t r i c i t y r a t i o and t h e r e f o r e s t a b i l i z e s t h e bearing. It a l s o s t i f f e n s t h e bearing and reduces i n a l l c a s e s t h e synchronous damping c a p a c i t y (3). Grooving strategies are attractive b e c a u s e of t h e i r s i m p l i c i t y of m a n u f a c t u r e . A x i a l g r o o v i n g , f o r example, c h a n g e s a c t i v e a r c l e n g t h a n d d i s p o s i t i o n (4). "Dammed grooves'' (5,6,7,8) a r e a means o f i m p o s i n g p a r a s i t i c l o a d i n g on t h e j o u r n a l t o i n c r e a s e i t s e c c e n t r i c i t y r a t i o i n t h e main l o a d bearing arc. Multiple a r c designs ( 3 , 4 , 9 ) p r o v i d e a n a l t e r n a t i v e method of i n c r e a s i n g
t h e e c c e n t r i c i t y r a t i o i n t h e main b e a r i n g a r c s by means of p a r a s i t i c l o a d s . Even t h e t i l t i n g pad d e s i g n n e c e s s a r i l y i n c r e a s e s t h e e c c e n t r i c i t y r a t i o of j o u r n a l r e l a t i v e t o individual arcs. A s f a r as t h e a u t h o r s a r e aware t h e g r o o v i n g s t r a t e g y proposed i n t h e p r e s e n t p a p e r , a l t h o u g h s i m p l e h a s n o t appeared i n the open literature. It provides a compromise between s t a b i l i t y and s y n c h r o n o u s d a m p i n g c a p a c i t y a n d a t t h e same t i m e i s v e r y e a s y t o m a n u f a c t u r e and insensitive t o m a n u f a c t u r i n g e r r o r s , u n l i k e many o t h e r d e s i g n s (10). 2.
BEARING DESIGN
ROTATION
f
LOAD
I-II
-l
FIG. 1.
BEARING GEOMETRY
348 F o r t h e purpose of t h i s paper w e w i l l d e a l w i t h a s i n g l e c i r c u l a r a r c b e a r i n g a s shown i n Fig. 1. The o i l i n l e t and o u t l e t grooves a r e on t h e h o r i z o n t a l c e n t r e l i n e . A c e n t r a l c i r c u m f e r e n t i a l groove e x t e n d s from t h e o u t l e t s l o t , upstream through an a n g l e 8 which can be v a r i e d . The o n l y o t h e r d e s i g n p a r a m e t e r i s t h e r a t i o of s l o t w i d t h t o b e a r i n g length. To s i m p l i f y c a l c u l a t i o n s t h e o u t l e t s l o t i s h e l d a t atmospheric p r e s s u r e although t h i s i s not c r i t i c a l . Experimental work i s compared w i t h theoretical predictions for a 355 mm d i a m e t e r b e a r i n g of L / D r a t i o 0.8 and clearance r a t i s 0.0015. Groove arc8 c o n s i d e r e d a r e 0 ( p l a i n c i r c u l a r ) , 47O, 60 and 360' ( f u l l c i r c u m f e r e n t i a l groove) w i t h a groove width b e a r i n g l e n g t h r a t i o of 0.18. The o i l i n l e t t e m p e r a t u r e i s h e l d a t 3OoC t h e o i l used being THB 32 (ISO).
I
LOAD kN 300
+
E X P E RI ME N T
S P E E D 850 RPM
200
-
3.
RESULTS 100
Bearing s t a t i c and dynamic p r o p e r t i e s have been c a l c u l a t e d u s i n g a 2 d i m e n s i o n a l thermo-hydrodynamic s o l u t i o n of t h e Reynolds equation. The t r e a t m e n t i s a d i a b a t i c , i . e . no h e a t l o s s by conduction i s assumed, a l t h o u g h o i l i s allowed t o r e c i r c u l a t e o v e r t h e unloaded a r c o f t h e b e a r i n g , t o mix w i t h the o i l supplied a t the i n l e t s l o t . The upstream boundary is assumed t o occur a t t h e o i l i n l e t s l o t , a l t h o u g h i t would have been more c o r r e c t t o a l l o w f o r o i l r e c i r c u l a t i o n a n d b o u n d a r y r e f o r m a t i o n u p s t r e a m of t h e i n l e t location. T e s t i n g was c a r r i e d o u t on a f u l l s c a l e b e a r i n g i n t h e GEC b e a r i n g r i g Steady a t S t a f f o r d d e s c r i b e d f u l l y i n (11). loading was applied pneumatically and o s c i l l a t i n g l o a d s by means of e l e c t r o hydraulic exciters. Only a few w h i t e metal t e m p e r a t u r e s were measured f o r m o n i t o r i n g purposes. Measurements of c i r c u m f e r e n t i a l p r e s s u r e p r o f i l e and f i l m t h i c k n e s s were t a k e n using probes f i x e d i n t h e j o u r n a l . P r e s s u r e measurements were t a k e n a t t h e c e n t r e l i n e s and four other axial p o s i t i o n s . T e s t s on 47' and 60' grooved b e a r i n g s were conducted a t low s p e e d s and o v e r a v e r t i c a l l o a d r a n g e of 20 kN t o 260 kN. Theoretical r e s u l t s wereoobtained f o r t h e s e b e a r i n g s and a l s o a 360 grooved d e s i g n o p e r a t i n g o v e r a speed range of 500 t o 3000 rpm. A l l r e s u l t s w i l l use t h e f o l l o w i n g legend: O0 47O
E CCE N T RI CT T Y R A T I O (
0.2
FIG. 2.
0.4
0.6
0.8
1.0
LOAD v ECCENTRICITY RATIO.
o"",
------I_
-._.-._._ 4.
60' 360'
EVALUATION OF DESIGN
Most of t h e r e s u l t s p r e s e n t e d , r e l a t e t o t h e o r e t i c a l models of b e a r i n g s . Experim e n t a l r e s u l t s quoted a r e confined t o l o a d v a r i a t i o n s a t low speed (850 rpm) a l t h o u g h t h e r e would have been no problem i n t e s t i n g a t s p e e d s up t o 4000 rpm. On l a r g e b e a r i n g s h o w e v e r , i t h a s b e e n shown t h a t a s s p e e d i n c r e a s e s , so do t h e r m o - e l a s t i c effects (12). The shape of t h e o i l f i l m p r o f i l e i s changed s i g n i f i c a n t l y a t h i g h s p e e d s and t h i s h a s a profound e f f e c t on t h e dynamic coefficients.
0
FIG. 3.
JOURNAL LOCI.
F i g . 2 s h o w s t h e v a r i a t i o n of l o a d w i t h e c c e n t r i c i t y r a t i o f o r groove a n g l e s of o O , 47O, 60° amd 3 60'. The l o c i of t h e j o u r n a l c e n t r e s o v e r t h e same l o a d range a r e p l o t t e d i n f i g . 3. Fig. 2 i n d i c a t e s good agreement and t h e l o c i of f i g . 3 a l s o a g r e e a t e c c e n t r i c i t y r a t i o s of 0.6 and l e s s . At greater eccentricity there is some divergence i n t h a t the theory over estimates t h e a t t i t u d e angle. Neve r t h e l e s s t h e g e n e r a l e f f e c t of p a r t i a l grooving i s d e m o n s t r a t e d as i n c r e a s i n g t h e a t t i t u d e
349 a n g l e f o r any given bearing load. Thiz i n c r . e a s e is more p r o n o u n c e d f o r t h e 60 I t w i l l a l s o be s e e n t h a t a s groove angle. e c c e n t r i c i t y r a t i o i n c r e a s e s t h e e f f e c t of t h e p a r t i a l g r o o v i n g on a t t i t u d e a n g l e , decreases.
A k e y t o t h e r e a s o n why p a r t i a l g r o o v i n g s h o u l d a f f e c t t h e a t t i t u d e a n g l e i n a manner which d i f f e r s w i t h e c c e n t r i c i t y r a t i o i s provided by the measured pressure d i s t r i b u t i o n s of f i g s . 4 a t o 4d. Consider f i r s t t h e ungrooved b e a r i n g of f i g . 4a which r e f e r s t o a f a i r l y l i g h t l o a d of 100 kN c o r r e s p o n d i n g t o a n e c c e n t r i c i t y r a t i o of 0.6. The c a v i t a t i o n boundaroy i s l a r g e l y The e f f e c t a x i a l and o c c u r s a t around 165 a 60' partial groove upon this of Here t h e d i s t r i b u t i o n i s shown i n f i g . 4b. g r o o v e c u t s i n t o t h e c a v i t a t i o n boundary and modifies the pressure p r o f i l e s i g n i f i c a n t l y . I n o r d e r t o support a v e r t i c a l load the a t t i t u d e a n g l e o f t h e j o u r n a l h a s had t o i n c r e a s e t o r a i s e the pressure i n the v i c i n i t y of t h e groove t o compensate f o r t h e reduced a r e a . F i g . 4c shows t h e ungrooved b e a r i n g a t a l o a d of 260 kN c o r r e s p o n d i n g t o a n I t w i l l be e c c e n t r i c i t y o f a r o u n d 0.8. r e m a r k e d t h a t t h e c a v i t a t i o n boundary i s a t The i n t r o d u c t i o n o f a 60' a r o u n d 150'. groove m o d i f i e s t h i s c a v i t a t i o n boundary a l i t t l e b u t much less t h a n i n t h e p r e v i o u s , low e c c e n t r i c i t y c a s e . T h i s e x p l a i n s why t h e e f f e c t of g r o o v i n g on a t t i t u d e a n g l e is small at high eccentricities. This d i f f e r e n t i a l e f f e c t of p a r t i a l g r o o v i n g on t h e low e c c e n t r i c i t y c o n d i t i o n (low l o a d , h i g h s p e e d ) on t h e one hand, and t h e h i g h eccentricity condition (high load, low speed) on the other, permits partial g r o o v i n g t o be used a s a n o p t i m i s i n g technique. It i s p o s s i b l e f o r t h e bearing t o b e h a v e as a grooved b e a r i n g a t h i g h s p e e d s o r low l o a d s and a s a n ungrooved b e a r i n g a t low s p e e d s and h i g h l o a d s . Dynamic predictions are based on linearized stiffness and' damping coefficients defined according to the f o l l o w i n g equation.
.
00
900
FIG. 4a.
0'
180°
GROOVE, LOAD 100 kN.
L
900
FIG. 4b.
FIG.
4c.
60'
0'
180'
GROOVE, LOAD 100 kN.
GROOVE, LOAD 260 kN.
w h e r e E and B a r e d i m e n s i o n l e s s , ( x , y ) and (Fx,F ) a r e i n c r e m e n t a l d i s p l a c e m e n t and force' v e c t o r s r e s p e c t i v e l y , C i s r a d i a l c l e a r a n c e and W is a p p l i e d b e a r i n g l o a d . I f t h e s t i f f n e s s m a t r i x i s i n v e r t e d t o form a f l e x i b i l i t y m a t r i x , c o n s i d e r two elements of and Rx Mitchell e t t h i s matrix viz. R a 1 (13) have shown r x a t t h e s B two v a l u e s can a l s o be d e r i v e d from t h e l o c u s and t h u s used a s a check on e x p e r i m e n t a l s t a t i c and I t c a n f u r t h e r be dynamic r e s u l t s (13). shown t h a t when t h e s l o p e of t h e l o c u s i s p o s i t i v e t h e n Rx i s a l s o p o s i t i v e . There is a c o n n e c t i o n getween t h e change of s i g n and t h e o n s e t of i n s t a b i l i t y , so t h a t of R t h e R g h e r t h e speed ( o r lower t h e l o a d ) a t w h i c h t h i s s i g n change o c c u r s , t h e g r e a t e r the stability. On t h i s b a s i s f i g . 3 indicates that the partially grooved b e a r i n g s are l i k e l y t o be more s t a b l e t h a n t h e ungrooved b e a r i n g . T h i s , however, is o n l y a g u i d e and h e n c e f o r t h t h e d y n a m i c
.
90"
FIG. 4d. FIGS. 4. A N D 60'
60'
GROOVE, LOAD 260 kN.
EXPERIMENTAL ISORARS (kPa) FOR 0' GROOVES AT LOAP' O F 100 kN A N D 260 kN, SPEED 850 RPM.
18O0
350 c o e f f i c i e n t s w i l l be used t o compare t h e m e r i t s of v a r i o u s b e a r i n g s . Examples of t h e s e c o e f f i c i e n t s a r e p l o t t e d a g a i n s t speed i n f i g s . 5 and 6 f o r t h e f o u r b e a r i n g s u n d e r consideration.
d.Q
0.6
LOAD 70 kN STABLE
0.4
0.2
0 7.
FIG.
EYY
EYX
STABILITY CHART.
-2-
/-=
&
-3.
-
w'*\
7 -4
-
/ /
r
/'
./.-3..
r
-.'
The s i m p l e s t way of d e m o n s t r a t i n g t h e i n t r i n s i c s t a b i l i z i n g c h a r a c t e r of a b e a r i n g i s by m e a n s o f t h e s t a b i l i t y c h a r t s o f f i g . 7. A l i g h t l o a d of 70 kN i s used t o e n a b l e a comparison of a l l f o u r b e a r i n g d e s i g n s t o be made. The c h a r t s are o b t a i n e d by c o n s i d e r i n g v a l u e s o f t h e p a r a m e t e r (W / n ) a t a g i v e n r u n n i n g speed which r e s u l t s i n a r e a l e i g e n v a l u e of t h e f o l l o w i n g m a t r i x .
I
7
I
+ is n
E
+i%B XY
L
I
1
Exx
0
Bxx YX
E
+ i e B xy n xx
E
+ i S B n YY
YY
I
d
where n is t h e j o u r n a l r u n n i n g speed and o i s t h e non-synchronous f r e q u e n c y a t which t h e journal orbits. In effect t h i s is the c o n d i t i o n f o r t h e b e a r i n g t o a c t as a p u r e spring. U n s t a b l e c o m b i n a t i o n s of speed and f r e q u e n c y r a t i o a p p e a r t o t h e r i g h t of t h e It i s c l e a r t h e r e f o r e t h a t t h e boundaries. c i r c u l a r b e a r i n g is v e r y u n s t a b l e and t h a t t h e s t a b i l i t y o f t h e p a r t i a l l y grooved b e a r i n g s l i e s between t h i s b e a r i n g and t h a t of a fully circumf e r e n t i a l l y grooved bearing. The b e a r i n g w i t h a groove a r c of 60' i s m a r g i n a l l y more s t a b l e t h a n that w i t h a 47' a r c a t l o w running speeds. A t the higher speeds, t h e r e i s l i t t l e difference i n t h e s t a b i l i t y of t h e t w o d e s i g n s . The reason f o r t h i s i s t h a t a t high a t t i t u d e a n g l e s both grooves l i e i n t h e t h i n f i l m i . e . t h e h i g h p r e s s u r e zone. The f a c t t h a t groove extends s l i g h t l y f u r t h e r t h e 60' u p s t r e a m i n t o t h e low p r e s s u r e zone h a s l i t t l e e f f e c t on t h e r e s u l t a n t o i l f i l m force. Although t h e p a r t i a l l y grooved b e a r i n g s a r e less s t a b l e t h a n i n t h e f u l l y grooved 8 and 9 d e m o n s t r a t e t h e i r case, figs. s u p e r i o r i t y i n damping o u t r e s p o n s e due t o u n b a l a n c e a t system "criticals". FIG. 6. DAMPING COEFFICIENTS. LOAD 165 kN.
351 T h e r o t o r v i b r a t i o n s shown c o r r e s p o n d t o a common u n b a l a n c e f o r a l l c a s e s a n d , as might b e e x p e c t e d , t h e s t i f f r o t o r s y s t e m is much l e s s s e n s i t i v e than the f l e x i b l e rotor It i s common knowledge t h a t a p l a i n system. circular profile results in the best s y n c h r o n o u s damping p r o p e r t i e s i n hydrodynamic b e a r i n g s . The r e l a t i v e f i g u r e s i n t a b l e 1 show t h a t t h e p a r t i a l l y g r o o v e d b e a r i n g s approach t h e c i r c u l a r i n t h i s respect.
AMPLITUDE (mm)
0.5
MINIMUM SYSTEM DAMPING (%) 0
0.5 850
FIG. 8.
8.75 900 SHAFT SPEED RPM
925
RESPONSE OF 15 Hz ROTOR. 0.4
0.15 0.3
0.10
0.2
0.1
0.05
C
o ! 1600
FIG. 9.
I
I
i.700
1800
500
FIG. These curves r e l a t e t o t h e response of a s i m p l e ( J e f f c o t t ) r o t o r w i t h c e n t r e mass of The 3 3 t o n n e s u n b a l a n c e d by 0 . 3 3 k g m. r o t o r s are s i m i l a r i n mass and s t i f f n e s s t o certain practical generator/turbine rotors and a r e d e s i g n a t e d as " f l e x i b l e " and " s t i f f " respectively. This r e f e r s t o t h e i r natural f r e q u e n c i e s on r i g i d s u p p o r t s of 15 Hz and 30 Hz. It is p o s s i b l e t o c o n c e i v e of a v e r y much s t i f f e r r o t o r b u t t h i s would n o t p o s e a s t a b i l i t y problem from t h e p o i n t of view of classical oil whirl and is therefore i r r e l e v a n t i n the present context. Table 1 summarises t h e r e l a t i v e s e n s i t i v i t y of t h e v a r i o u s r o t o r b e a r i n g combinations.
Bearing
O0 47O 60' 360'
F l e x i b l e Rotor S t i f f Rotor Amplitude Normalised Amplitude Normalised (mm) Responses (mm) Responses 0.32 0.39 0.44 0.99
1 .o 1.22 1.38 3.09
TABLE 1.
0.075 0.078 0.09 0.185
1 .o 1.04 1.2 2.47
lo00
1;OO
2000
2500
SHAFT SPEED RPM
SHAFT SPEED RPM RESPONSE OF 30 Hz ROTOR. 10.
SYSTEM DAMPING OF 7.5 Hz ROTOR.
Earlier remarks on stability have concentrated on intrinsic bearing It i s useful a l s o t o judge properties. For b e a r i n g s t a b i l i t y i n a system context. t h i s purpose a very f l e x i b l e J e f f c o t t r o t o r h a s been chosen. The r i g i d l y s u p p o r t e d n a t u r a l f r e q u e n c y of t h i s r o t o r i s 7.5 Hz a n d t h e c e n t r a l mass h a s b e e n c h o s e n a s 14 tonnes i n o r d e r t o keep t h e b e a r i n g load low. I n t h i s v e r y s e n s i t i v e environment, t h e c i r c u l a r b e a r i n g is shown t o be t o t a l l y i n a d e q u a t e and t h e f u l l y grooved (360') b e a r i n g h i g h l y s t a b l e , a l b e i t w i t h v e r y low s y s t e m damping. The p a r t i a l l y grooved b e a r i n g s p r o v i d e h i g h e r s y s t e m damping a t l o w e r s p e e d s t h a n t h e 360' grooved b e a r i n g , b u t a t higher speeds t h e opposite is true. Nevertheless both designs represent a g r e a t improvement o v e r t h e p l a i n c i r c u l a r p r o f i l e bearing. The b e a r i n g s a r e n o t t h e o n l y s o u r c e o f d e s t a b i l i z a t i o n i n r o t a t i n g systems, an i m p o r t a n t mechanism b e i n g t h a t of r o t a r y d a m p i n g (14). The e f f e c t of r o t a r y damping is t o reduce t h e speed of o n s e t of i n s t a b i l i t y i n r e l a t i o n t o t h e lowest system - loosely called the f i r s t frequency c r i t i c a l speed. S m i t h ( 1 5 ) f i r s t showed t h a t bearing anisotropy reduces i n s t a b i l i t y
352 d u e t o r o t a r y damping, and i t is a l s o w e l l known t h a t b e a r i n g flexibility can be b e n e f i c i a l . Both of t h e s e p r o p e r t i e s c a n b e i n c r e a s e d by s u i t a b l e p a r t i a l g r o o v i n g .
10
MINIMUM SYSTEM DAMPING
(%I SPEED
3000 R P M 8
6.
T h e s t a b i l i z i n g p r o p e r t i e s of a b e a r i n g can be a l t e r e d e i t h e r by c h a n g i n g speed o r l o a d . Thus f a r we h a v e c o n s i d e r e d o n l y speed changes but in a practical situation, c i r c u m s t a n c e s of m i s a l i g n m e n t o c c u r i n which l o a d i s reduced on p a r t i c u l a r b e a r i n g s . F i g . 12 shows a s t a b i l i t y c h a r t d e r i v e d b o t h f r o m t h e o r e t i c a l and e x p e r i m e n t dynamic Here t h e i n d e p e n d e n t v a r i a b l e coefficients. is load. I t i s shown how i n t o l e r a n t t h e u n g r o o v e d b e a r i n g is of r e d u c t i o n s i n l o a d and t h e c o n s i d e r a b l e improvement r e s u l t i n g from p a r t i a l g r o o v i n g . Comparison between theoretical and experimentally derived r e s u l t s show t h e t h e o r y t o be s l i g h t l y conservative. The i n s e n s i t i v i t y of t h e r e s u l t s t o t h e p a r t i a l g r o o v e a n g l e is once again demonstrated and i n f e r s t h a t t h e s e will be i n s e n s i t i v e t o small bearings changes i n load angle.
4 mm
LOAD 165 k N
0.15-
2 0.1 .
_ _ _ -----
0.
--.-.-
0
% ROTARY D A M P I N G
FIG.
11.
SPEED
STABILITY WITH ROTARY DAMPING.
Fig. 11 shows minimum s y s t e m damping p l o t t e d a g a i n s t r o t a r y damping f o r v a r i o u s b e a r i n g d e s i g n s , u s i n g t h e 3 0 Hz J e f f c o t t r o t o r p r e v i o u s l y mentioned. J u d g i n g t h e merit of a b e a r i n g by t h e l e v e l of r o t a r y damping t h e system will tolerate before becoming u n s t a b l e ( z e r o s y s t e m damping), i t w i l l be s e e n t h a t u n l i k e t h e s i t u a t i o n shown in f i g . 10, t h e f u l l y grooved b e a r i n g i s i n f e r i o r t o 0 t h e 47 p a r t i a l g r o o v e d e s i g n .
w/n 0.4
0.2
FIG.
LOAD k N
FIG.
12.
STABILITY CHART (LOAD).
M I N I M U M O I L FILM THICKNESS.
P e r h a p s t h e most i m p o r t a n t c o n s i d e r a t i o n i n t h e d e s i g n of o i l l u b r i c a t e d b e a r i n g s i s t h a t of minimum oil f i l m t h i c k n e s s . This d e c i d e s t h e t o l e r a n c e of t h e b e a r i n g t o a n g u l a r m i s a l i g n m e n t r e s u l t i n g from a s s e m b l y It a l s o e r r o r s o r shaft deflections. oil influences the wear arising from c o n t a m i n a t i o n a s w e l l as l o c a l o v e r h e a t i n g effects. I n d e e d any b e a r i n g c o u l d b e made s t a b l e , provided t h e j o u r n a l e c c e n t r i c i t y c o u l d be increased without l i m i t . The need t o m a i n t a i n a n a d e q u a t e minimum o i l f i l m a t a appropriate running speeds provides constraint. F i g . 13 compares t h e minimum f i l m thickness a t various speeds f o r a l l d e s i g n s w i t h t h e recommendation of E.S.D.U. (16). It w i l l be s e e n t h a t t h e most s t a b l e , f u l l y g r o o v e d b e a r i n g , is c l o s e s t t o t h e l i m i t i n g v a l u e a t 3000 rpm, w h e r e a s t h e partially grooved bearings, although a d e q u a t e l y s t a b l e , h a v e a much l a r g e r margin of s a f e t y .
5. 0
13.
RPM
DISCUSSION
T h e b e n e f i t s of p a r t i a l g r o o v i n g have been p r e s e n t e d and t h e e x a m i n a t i o n l i m i t e d t o a s i n g l e load bearing arc. The g r o o v e i n e f f e c t , is a method o f m o d i f y i n g t h e j o u r n a l l o c u s by i n c r e a s i n g t h e a t t i t u d e a n g l e . T h i s is e f f e c t e d by u s i n g t h e g r o o v e t o
353 change t h e c a v i t a t i o n boundary a t low i.e. high speeds. eccentricity ratios B e c a u s e of t h e l i m i t e d g r o o v e a n g l e t h e lower speed b e h a v i o u r , when t h e j o u r n a l operates at high eccentricities, is unaffected. Consequently i n f l e x i b l e r o t o r a p p l i c a t i o n s where lower c r i t i c a l s p e e d s a r e well below t h e maximum r u n n i n g s p e e d , a l l t h e a d v a n t a g e s of an ungrooved c i r c u l a r p r o f i l e b e a r i n g are o b t a i n e d . I n t h e o r y , t h e j o u r n a l l o c u s c o u l d be e q u a l l y modified by s h a p i n g t h e b o r e o f t h e bearing. T h i s m i g h t w e l l be a d i f f i c u l t manufacturing process to carry out accurately. The p r i n c i p l e , however, of is s e l e c t i v e d e r a t i n g by any method, certainly less complicated than using p a r a s i t i c loading t o a d j u s t t h e j o u r n a l locus. The d e s i g n s p r e s e n t e d i n t h e p a p e r would not be much u s e i n c i r c u m s t a n c e s where t h e j o u r n a l was f o r c e d t o r u n a t v e r y low e c c e n t r i c i t i e s o r even i n t h e t o p h a l f of t h e b e a r i n g due t o s e v e r e misalignment w i t h contiguous rotors. I n t h i s case t h e p o s s i b i l i t y of a m u l t i - a r c d e s i g n based on p a r t i a l l y grooved components c o u l d w e l l be worth f u r t h e r i n v e s t i g a t i o n . The work p r e s e n t e d i n t h i s p a p e r is largely based on relatively simple theoretical considerations. Over t h e p a s t s i x y e a r s , however, a c o n s i d e r a b l e amount of e m p i r i c a l d a t a h a s ,been d e r i v e d from t e s t s c a r r i e d o u t on t h e S t a f f o r d b e a r i n g r i g . O p e r a t i o n a l e x p e r i e n c e w i t h such b e a r i n g s h a s confirmed t h a t t h e y c a n be used w i t h advantage i n a p p l i c a t i o n s i n v o l v i n g h i g h speed f l e x i b l e r o t o r s . 6.
b)
Morton, P.G. 'The i n f l u e n c e of coupled asymmetric b e a r i n g s on t h e motion of a massive f l e x i b l e r o t o r ' Proc. I n s t n . Mech. Engrs. 1967-8, 182,P t . 1 . Newkirk, B.L. and T a y l o r , H.D. ' S h a f t w h i p p i n g due t o o i l a c t i o n i n j o u r n a l bearings', G e n e r a l E l e c t r i c Review NO. 8 , 559-568. 1925,
s,
G l i e n e c k e , J. ' E x p e r i m e n t a l i n v e s t i g a t i o n of t h e s t i f f n e s s and damping c o e f f i c i e n t s of t u r b i n e b e a r i n g s and their application to stability p r e d i c t i o n ' , Proc. I n s t n . Mech. Engrs. 1966-67, 181,P t . 3B, 116-129. Akkok, M. and E t t l e s , C.M. McC. 'The e f f e c t of g r o o v i n g and bore shape on t h e s t a b i l i t y of j o u r n a l b e a r i n g s ' , ASLE P r e p r i n t No. 79-AM-6D-3. and H e a l y , S.P. 'AntiPope, A.W. vibration journal bearings (An e x p e r i m e n t a l s a g a ) , Proc. I n s t n . Mech. Engrs. 1966-67, 181,P t . 3B. A. 'Principles of Cameron, Lubrication', 1966 (Longman Group, London), 341-381.
A l l a i r e , P.E. and Nicholas, J.C., Lewis, D.H. ' S t i f f n e s s and damping c o e f f i c i e n t s f o r f i n i t e length s t e p j o u r n a l b e a r i n g s ' , ASLE P r e p r i n t No. 79-AM-6D-1, 1979.
CONCLUSIONS
A s t u d y h a s b e e n made o f t h e e f f e c t o f p a r t i a l circumferential grooving a t t h e downstream end of t h e l o a d b e a r i n g oil f i l m The of a c i r c u l a r p r o f i l e j o u r n a l bearing. d e s i g n i s s i m p l e and e a s y t o m a n u f a c t u r e accurately and posseses the following attributes:
a)
References
T h i s d e s i g n h a s a good a n t i o i l w h i r l e f f e c t o n f l e x i b l e r o t o r systems.
Where such s y s t e m s a r e s u b j e c t e d t o r o t a r y damping, t h e s t a b i l i z i n g e f f e c t is s u p e r i o r even t o t h a t of f u l l circumferential grooving. By r e a s o n of i t s a n i s o t r o p i c s t i f f n e s s t h e b e a r i n g is v e r y much b e t t e r t h a n , s a y , t i l t i n g pad designs i n t h i s respect.
c)
The synchronous damping p r o v i d e d a t h i g h e c c e n t r i c i t y r a t i o s is s u p e r i o r t o t h a t of many o t h e r a n t i - w h i r l d e s i g n s .
d)
Another a d v a n t a g e of the partially grooved b e a r i n g i s t h a t good s t a b i l i t y c a n be s e c u r e d w i t h o u t e x c e s s i v e l y t h i n minimum o i l f i l m s .
7.
ACKNOWLEDGEMENTS
The a u t h o r s g r a t e f u l l y acknowledge the c o m p u t a t i o n a l work o f D r . P.S. Keogh and t h e e x p e r i m e n t a l r e s u l t s p r o v i d e d by Messrs. J.E. Brown and G.S. Khera.
M.E., Flack, R.D. and Leader, A l l a i r e , P.E. ' E x p e r i m e n t a l s t u d y of three journal bearings with a flexible r o t o r ' , ASLE P r e p r i n t No. 79-AM-6D-2.
Kramer, shafts forces', 2 0 , NO.
E . ' S e l f e x c i t e d v i b r a t i o n of as a r e s u l t of transverse Brennstoff-Warne-Kraft, 1968, 7 , 307-312.
'The M a r t i n , F.A. and Ruddy, A.V. e f f e c t of m a n u f a c t u r i n g t o l e r a n c e s on the stability of profile bore b e a r i n g s ' , Paper C273184, Proc. I n s t n . Mech. Engrs. Conf. on V i b r a t i o n s i n R o t a t i n g Machinery, York, 1984. P. G. ' The experimental Morton, e v a l u a t i o n of f l u i d f i l m b e a r i n g s ' , P r o c . Machinery V i b r a t i o n M o n i t o r i n g and A n a l y s i s Meeting, N e w O r l e a n s , L o u i s i a n a , 1985, 147-158.
Morton, P.G. and Keogh, P.S. 'Thermoelastic influences i n journal b e a r i n g l u b r i c a t i o n ' , Proc. R. SOC. A 1986, 403, 111-134. M i t c h e l l , J.R., H o l m e s , R. and Van H. ' Experiment a1 Ballegooyen, d e t e r m i n a t i o n of bearing oil-film s t i f f n e s s ' , Proc. I n s t n . Mech. Engrs. 1965-6, , & l ( P t . 3K), 90.
354
( 1 4 ) Kimball, A . L . ' I n t e r n a l f r i c t i o n theory of s h a f t w h i r l i n g ' , P h i l . Mag. 1925, V O l . 49.
D.M. 'The motion of rotor ( 1 5 ) Smith, c a r r i e d by a f l e x i b l e s h a f t i n f l e x i b l e bearings' Proc. R. S O C . A 1933, 142, 92. ( 1 6 ) Engineering Sciences Data Unit 'Calculation methods for steadily loaded pressure fed hydrodynamic journal b e a r i n g s , I n s t n . Mech. Engrs. 1967, Item No. 66023.
355
Paper Xl(iii)
Theoretical and experimental orbits of a dynamically loaded hydrodynamic journaI bearing R.W. Jakeman and D.W. Parkins
T h i s paper g i v e s a comparison o f t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s o f a d y n a m i c a l l y loaded The j o u r n a l b e a r i n g h a v i n g a p r e s s u r i s e d o i l s u p p l y t o a c e n t r a l 3600 c i r c u m f e r e n t i a l groove. r e s u l t s o f two t h e o r e t i c a l analyses a r e presented: Methods A and B. Method B, r e f e r r e d t o as t h e computed v e l o c i t y Reaction Method, f e a t u r e s o i l f i l m f o r c e p r e d i c t i o n by means o f p r e c o e f f i c i e n t s , t h u s f a c i l i t a t i n g q u i c k e r computation. S a t i s f a c t o r y c o r r e l a t i o n o f t h e experimental r e s u l t s w i t h t h e p r e d i c t i o n s o f b o t h t h e o r e t i c a l methods i s shown. Comparisions a r e made f o r t h r e e examples i n c l u d i n q d i f f e r e n t r e l a t i v e phase and a m p l i t u d e o f t h e e x c i t a t i o n components a t b o t h once and t w i c e r o t a t i o n a l frequency. INTRODUCTION 1. 1.1 Notation
-
L i n e a r i s e d o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s have been commonly used t o model t h e i n f l u e n c e o f hydrodynamic j o u r n a l b e a r i n g s upon t h e l a t e r a l v i b r a t i o n c h a r a c t e r i s t i c s o f v a r i o u s s h a f t i n g systems. These c o e f f i c i e n t s a r e s u b j e c t t o a h i g h degree o f n o n - l i n e a r i t y which may lead to substantial errors, particularly with respect to amplitude p r e d i c t i o n , i n s i t u a t i o n s where s i g n i f i c a n t dynamic l o a d i n g i s encountered. I n more extreme cases o f dynamic loading, such as c r a n k s h a f t bearings, n o n - l i n e a r i t y renders t h e use o f a s i n g l e s e t o f displacement and A velocity coefficients t o t a l l y impractical. t i m e s t e p p i n g j o u r n a l o r b i t a n a l y s i s i s used i n these s i t u a t i o n s . Journal o r b i t analysis is inherently heavy on computing time, p a r t i c u l a r l y w i t h t h e more r i g o r o u s t y p e s o f a n a l y s i s , where o i l f i l m c h a r a c t e r i s t i c s must be computed a t each t i m e step. A c o n s i d e r a b l e r e d u c t i o n i n computing t i m e can be gained b y t h e use o f e i t h e r an approximate s o l u t i o n o f the oil film pressure distribution or pre-computed o i l f i l m data. The o b j e c t i v e o f t h e work r e p o r t e d i n t h i s paper was t o compare t h e r e s u l t s o f two journal orbit prediction methods with experimental data obtained from a t e s t r i g ( 1 ) . T h e o r e t i c a l Method A i s o f t h e r i g o r o u s film pressure type thus u s i n g numerical s o l u t i o n s a t each t i m e s t e p (2), w h i l s t Method B, r e f e r r e d t o as t h e R e a c t i o n Method, achieves a f a s t o r b i t s o l u t i o n b y t h e use o f pre-computed v e l o c i t y c o e f f i c i e n t s . Method A has been p r e v i o u s l y d e s c r i b e d i n r e f e r e n c e ( 3 ) , and t h e development o f t h e o i l f i l m f o r c e e q u a t i o n s upon which t h e R e a c t i o n Method i s based i s o u t l i n e d i n r e f e r e n c e ( 4 ) . Alignment between t h e j o u r n a l and b e a r i n g was m a i n t a i n e d f o r a l l c o n d i t i o n s covered i n t h i s paper, and t h e b e a r i n g f e a t u r e d a p r e s s u r i s e d o i l supply t o a c e n t r a l l y positioned f u l l circumferential groove. Both theoretical methods took account o f j o u r n a l i n e r t i a l f o r c e s , and Method A had an o p t i o n a l f a c i l i t y f o r modelling o i l f i l m h i s t o r y .
AXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r s m a l l displacement p e r t u r b a t i o n s . BXX,etc. L i n e a r i s e d o i l f i l m c o e f f i c i e n t s f o r small v e l o c i t y perturbations. Btt, Brt,
Brr
Separate wedge and squeeze a c t i o n velocity coefficents
B r r t , B t r t I n t e r a c t i v e wedge and squeeze a c t i o n velocity coefficients. C
Radial clearance
Fr, F t
Radial,
Fex,Fey
Horizontal,
v e r t i c a l external forces.
F x , FY
Horizontal,
v e r t i c a l o i l f i l m forces.
j, i
Circumferential, p o s i t i o n reference
m
J o u r n a l mass
qn ( j , i )
N e t t o i l volume f l o w r a t e i n t o element j, i.
R
Radial journal v e l o c i t y
T
Dynamic c y c l e t i m e
t
Time f r o m s t a r t o f dynamic c y c l e and a t the s t a r t o f time step At.
Ve(j,i)
Total
volume
Vo ( j , i )
Volume
of
x, y
Horiziontal, displacement*
i,
Horizontal, vertical journal velocity*
tangential
oil
f i l m forces.
axial
e 1ement
*
oil
of in
element element
v e r t ica 1
j,
i.
j,
i.
journal
356
ic', 4;
Horizontal, vertical acceleration* Time s t e p increment
At
B
journal
Angular velocity of about b e a r i n g a x i s . *
journal
60
Equivalent journal*
velocity
E
Eccentricity
W
Journal a n g u l a r own a x i s .
*
angular
r a t i o = (x2 velocity
+
axis of 2.2
about
its
Suf f ixes : h,j,o
b e a r i n g housing, j o u r n a l , o i l f i l m .
P
perturbation etc.
max
maximum p e r m i t t e d value.
S
i n i t i a l l y e s t i m a t e d value. denotes c o n d i t i o n s a t t + A t , s u f f i x denotes c o n d i t i o n s a t t.
A
to
Axx,
compute
no
Prefix: A
denotes t h e change i n any parameter over A t e.g. Ax = xA -x
2.
BRIEF REVIEW OF PREVIOUS WORK
2.1
Factors relevant Analysis
to
Journal
Orbit
There a r e t h r e e main f a c t o r s p e r t a i n i n g to t h e j o u r n a l o r b i t a n a l y s i s methods p u b l i s h e d t o date. The v a r i o u s o p t i o n s w i t h i n t h e s e a r e o u t 1 i n e d below: (1)
(2)
Bear inq e l a s t ic it y : a.
B e a r i n g assumed t o be r i g i d .
b.
Taken i n t o account b y i n t e r a c t i v e film pressure solution of distribution and corresponding b e a r i n g e l a s t i c deformation.
P r e v i o u s Work
Y ~ ) ~ / ~ / C
R e f e r e r s t o t h e normal s i t u a t i o n o f a "fixed" bearing. I n t h e experimental t e s t r i g t h e s e parameters r e f e r t o t h e b e a r i n g housing s i n c e t h e j o u r n a l i s "fixed".
used
(3)
O i l f i l m force derivation: a.
S o l u t i o n o f Reynolds e q u a t i o n b y s h o r t b e a r i n g approximation.
b.
Numerical f i l m p r e s s u r e s o l u t i o n f o r bearings o f f i n i t e length.
c.
Use o f pre-computed o r measured o i l f i l m properties t o f a c i l i t a t e a f a s t o r b i t solution.
d.
O i l f i l m h i s t o r y modelling.
Journal mass: a.
I n e r t i a l f o r c e s assumed t o be negligible in relation to e x t e r n a l and o i l f i l m f o r c e s , therefore journal velocity components a r e d e r i v e d t o produce o i l f i l m f o r c e s equal t o t h e e x t e r n a l f o r c e s a t each step.
b.
I n e r t i a l f o r c e s n o t neglected.
One o f t h e b e s t known f a s t s o l u t i o n s i s t h e M o b i l i t y Method o f Booker(5) which f e a t u r e s option (lc). The M o b i l i t y data, upon which t h i s method depends, was o r i g i n a l l y d e r i v e d by t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( l a ) , and was consequently o f l e s s e r accuracy t h a n more recent numerical solutions. Finite bearing solutions or experimental measurements may a l s o be used t o produce M o b i l i t y data, t h e r e b y s u b s t a n t i a l l y improving accuracy. T h i s method was designed f o r s i t u a t i o n s where j o u r n a l i n e r t i a l f o r c e s c o u l d be n e g l e c t e d (2a) and t h e r i g i d b e a r i n g assumption (3a), and i s t h e o r e t i c a l l y l i m i t e d t o b e a r i n g s h a v i n g c i r c u m f e r e n t i a l symmetry. The a n a l y s i s b y Holmes and Craven ( 6 ) i s one o f t h e few t o have taken account o f j o u r n a l i n e r t i a l f o r c e s (2b), t h e i r work b e i n g based on t h e s h o r t b e a r i n q approximation ( l a ) , and applied t o a r i g i d bearinq (3a). O i l f i l m h i s t o r y m o d e l l i n g ( I d ) has a l s o r e c e i v e d v e r y l i t t l e a t t e n t i o n , t h e paper by Jones ( 7 ) g i v i n q a good account o f t h i s , b u t with the limitations o f neglecting i n e r t i a l f o r c e s (2a) and t h e r i g i d b e a r i n g assumption (3-3)
L i t t l e work has been c a r r i e d o u t on t h e modelling o f e l a s t i c i t y i n a dynamically loaded b e a r i n g (3b) due t o t h e excessive computing t i m e i n v o l v e d . The paper b y LaBouff and Booker ( 8 ) i s an example o f t h i s , and used a f i n i t e b e a r i n q s o l u t i o n ( l b ) and n e g l e c t e d i n e r t i a l f o r c e s (2a). F a n t i n o e t a1 ( 9 ) a t t a i n e d a more a c c e p t a b l e computing t i m e by u s i n g t h e s h o r t b e a r i n g a p p r o x i m a t i o n ( 1 a), b u t w i t h a consequent l o s s o f accuracy. Goenka and Oh (10) used t h e b a s i c methods o f b o t h ( 8 ) and ( 9 ) , b u t w i t h v a r i o u s r e f i n e m e n t s t o improve b o t h accuracy and computing time. 2.3
Relation o f P r e v i o u s Work
Methods
A
and
B
to
I n r e l a t i o n t o the foregoing analysis option c a t e g o r i e s , i t may be n o t e d t h a t t h e o r e t i c a l Method A i n t h i s paper used a numerical f i n i t e bearing s o l u t i o n (lb), with an o p t i o n a l f a c i l i t y f o r o i l f i l m h i s t o r y modelling (Id). Journal inertial f o r c e s were taken into consideration (2b), b u t t h e b e a r i n g was assumed t o be r i g i d ( 3 a ) . Method A i s therefore closely comparable to the (7), and a theoretical work by Jones comparison w i t h r e s u l t s t h e r e f r o m u s i n g t h e i n t e r m a i n c r a n k s h a f t b e a r i n g o f a 1.8 l i t r e 4 - s t r o k e c y c l e p e t r o l engine as a t e s t case, was g i v e n i n r e f e r e n c e ( 3 ) . The i n c l u s i o n o f inertial f o r c e s was t h e main d i f f e r e n c e between t h e above analyses, Method A h e r e i n and t h a t by Jones ( 7 ) . I n t h i s respect
357
Method A i s comparable Holmes and Craven ( 6 ) .
to
the
analysis
Method B d i f f e r e d from Method A, in that pre-computed v e l o c i t y c o e f f i c i e n t s were used i n order t o obtain a f a s t o r b i t s o l u t i o n (lc). The c o e f f i c i e n t s were d e r i v e d by a numerical f i n i t e b e a r i n g s o l u t i o n ( l b ) , b u t t h i s method negated t h e p o s s i b i l i t y o f o i l f i l m h i s t o r y m o d e l l i n g ( I d ) , f o r which no f a s t A particular s o l u t i o n i s known t o e x i s t . f e a t u r e o f Method B i s t h a t t h e v e l o c i t y coefficients used take account of the interaction o f squeeze and wedge action r e s u l t i n g f r o m t h e presence o f c a v i t a t i o n , and the a s s o c i a t e d non-1 i n e a r behaviour. In u t i l i s i n g pre-computed c o e f f i c i e n t s , Method 8 may be compared w i t h B o o k e r ' s M o b i l i t y Method (5), b u t d i f f e r s i n t h a t i t r e a d i l y a l l o w s j o u r n a l i n e r t i a l f o r c e s t o be taken i n t o account. The M o b i l i t y Method may appear t o be s i m p l e r t h a n Method B i n t h a t o n l y two parameters a r e r e q u i r e d , namely t h e M o b i l i t y Number and t h e a n g l e o f t h e squeeze p a t h r e l a t i v e t o t h e load vector. However, t h e s e two parameters are functions of both e c c e n t r i c i t y r a t i o and a t t i t u d e angle, even f o r a c i r c u m f e r e n t i a1 1y symmetri c a l b e a r i n g Method B r e q u i r e s f i v e v e l o c i t y c o e f f i c i e n t s , b u t f o r t h e c i r c u m f e r e n t i a l l y symmetrical b e a r i n g t h e s e are f u n c t i o n s o f e c c e n t r i c i t y r a t i o only. The t o t a l amount o f pre-computed d a t a r e q u i r e d by Method B i s t h e r e f o r e s u b s t a n t i a l l y less than f o r t h e M o b i l i t y Method. I n a d d i t i o n , Method B may be extended t o cover n o n - c i r c u m f e r e n t i a l l y symmetrical b e a r i n g s b y computing t h e five velocity c o e f f i c i e n t s as f u n c t i o n s o f e c c e n t r i c i t y r a t i o and a t t i t u d e angle.
.
3.
EXPERIMENTAL METHOD
3.1
Design o f T e s t R i q
Turnbuckle
by
F i g u r e 1 shows t h e apparatus on which t h e experimental orbits were obtained. The r o t a t i n g s h a f t i s supported a t e i t h e r end i n r o l l i n g element " s l a v e " bearings, w h i l s t t h e t e s t b e a r i n g i s mounted i n a " f l o a t i n g " housing. I n c o n t r a s t t o t h e normal p r a c t i c a l i t was therefore t h e bearing situation, housing orbits relative to the "fixed" j o u r n a l , r a t h e r t h a n j o u r n a l o r b i t s , t h a t were measured e x p e r i m e n t a l l y . Steady f o r c e s were applied separately or together in both h o r i z o n t a l and v e r t i c a l d i r e c t i o n s d i r e c t l y t o t h e t e s t b e a r i n g housing t h r o u g h t e n s i o n e d wires. R e l a t i v e displacement between t e s t b e a r i n g and journal was measured by f o u r pairs of non-contacting i n d u c t i v e transducers located on each s i d e o f t h e b e a r i n g i n t h e h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . T h i s arrangement permitted calculations o f displacement a t e i t h e r b e a r i n g end o r t h e a x i a l c e n t r e plane. The t e n s i o n e d w i r e s were a t t a c h e d t o f i x e d p o i n t s l o c a t e d a t a d i s t a n c e many t i m e s ( a p p r o x i m a t e l y 20,OOO:l) greater than t h e maximum possible housing motion. This prevented housing displacement f r o m a l t e r i n g t h e d i r e c t i o n o f t h e steady f o r c e s . Stiffness of t h e s p r i n g elements i n each l o a d i n g system was made small compared t o t h a t o f t h e o i l film. This meant that test housing displacement d i d n o t a l t e r t h e magnitude o f
Steady force measurement
Electro magnetic Steady force measurement
J4,I:'
q.>
Electro magnetic vibrator-
Fig.1 Schematic Arrangement of Test Rig Loading System t h e steady f o r c e s . F i g u r e 1 shows t h a t t h e horizontal and vertical steady loading arrangements each have i n t e r m e d i a t e p u l l e y s between t h e steady f o r c e gauge and t h e t e s t b e a r i n g housing. These comprised wheels, supported by low f r i c t i o n r o l l i n q element bearings, which a l l o w e d t h e t e s t b e a r i n g housing freedom t o r o t a t e around two m u t u a l l y p e r p e n d i c u l a r t r a n s v e r s e axes w h i l s t under a l a r g e steady force. Freedom around t h e s e axes allows the bearing t o a l i g n i t s e l f w i t h the j o u r n a l l o n g i t u d i n a l axis. Moreover, this l o a d i n g arrangement e l i m i n a t e d any c o n s t r a i n t around t h e b e a r i n g c e n t r e l i n e . Hence any t o r q u e e x e r t e d b y t h e o i l f i l m was r e s i s t e d by a separate t o r q u e r e s t r a i n i n g l i n k .
A magnetic sensor i n d i c a t e d s h a f t o r i e n t a t i o n and provided a pulse for an accurate r o t a t i o n a l speed i n d i c a t o r . Dynamic b e a r i n g f o r c e s Fex, Fey, measured by p i e z o gauges, c o u l d be a p p l i e d t o t h e bearing housing either vertically or h o r i z o n t a l l y o r t o g e t h e r w i t h any r e l a t i v e phase and magnitude by t h e two e l e c t r o magnetic v i b r a t o r s . Signals f o r these e l e c t r o magnetic v i b r a t o r s and t h e i r power a m p l i f i e r s were c r e a t e d by a sinewave generator d r i v e n from t h e t e s t shaft. The v i b r a t o r connectors were designed t o impose n e g l i g i b l e c o n s t r a i n t on t h e t e s t b e a r i n g housing, t h i s c o n d i t i o n b e i n g v e r i f i e d f o r each experiment. W i t h t h e steady f o r c e o n l y a p p l i e d t o t h e t e s t bearing, plus the torque r e s t r a i n t , the housing remains f r e e t o move a small a x i a l distance along t h e shaft. This feature checked whether full convienientl y hydrodynamic c o n d i t i o n s had been e s t a b l i s h e d . However, when dynamic loads were a p p l i e d i t was found t h a t o n l y a m i c r o s c o p i c misalignment thereof was sufficient to cause an
358
unacceptably l a r g e l o n q i t u d i n a l v i b r a t i o n and torsional o s c i 1l a t i o n about an axis perpendicular t o j o u r n a l centre l i n e . To obviate t h i s , l o c a t i n q wires p a r a l l e l t o t h e b e a r i n q c e n t r e l i n e were i n t r o d u c e d f o r t h e dynamic t e s t s . They a l l o w e d t h e b e a r i n g t o move t r a n s v e r s e l y and remain p a r a l l e l t o t h e s h a f t w h i l s t p r e v e n t i n q any misalignment o r It was shown t h a t these w i r e s a x i a l motion. t r a n s m i t t e d no s t a t i c o r dynamic f o r c e s t o t h e test bearing housing in any direction perpendicular t o t h e bearing centre l i n e .
housing acceleration were recorded. Immediately a f t e r each dynamic l o a d i n g t e s t F were smoothly reduced t o f o r c e s Fex, time z e r o and an e X o r i g i n " displacement h i s t o r y recorded. T h i s accounted f o r e f f e c t s such as s m a l l o u t o f balance f o r c e s and journal runout. J o u r n a l c e n t r e l o c a t i o n was t h e n checked. Displacements due t o dynamic f o r c e s a l o n e were subsequently o b t a i n e d by s u b t r a c t i n g t h e "orgin" ordinates from those at corresponding cycle times in the i m m e d i a t e l y p r e c e d i n g d y n a m i c a l l y loaded t e s t .
3.2
4.
METHOD A: THEORETICAL JOURNAL ORB1T ANALYSI S
4.1
Introduction
T e s t R i q Equations o f M o t i o n
Equations o f m o t i o n f o r are: Fex F x = mh 'X'h + mo yo Fey
-
Fy = mh vh
+
mo
t h e housinq and o i l
yo
[11
where Fx, Fy a r e t h e o i l f i l m forces a c t i n g on t h e j o u r n a l . These f o r c e s a r e f u n c t i o n s o f t h e r e l a t i v e housing t o j o u r n a l displacements x, y, (measured d i r e c t l y by t r a n s d u c e r s ) , and r e l a t i v e v e l o c i t i e s iC, j o b t a i n e d by numerical d i f f e r e n t i a t i o n o f t h e measured x, y t i m e h i s t o r y . Accelerometers attached to the housing j;h w i t h Xh, Yh obtained measured yh, b y double n u m e r i c a l i n t e g r a t i o n . The j o u r n a l displacements w i t h r e s p e c t t o a f i x e d p o s i t i o n i n space were o b t a i n e d from:
[21 I t was
shown
moYo s 0.01
t h a t i f -3 c X j / X h 4 +3 t h e n mhyh, and s i m i l a r l y for the
y direction.
A l l experimental d a t a r e p o r t e d i n t h i s paper were found t o meet t h i s c o n d i t i o n . Morton (11) a l s o notes t h a t t h e o i l f i l m t r a n s v e r s e inertial forces may be neglected. The equations of motion may therefore be s imp1 if ied t o :
[31 3.3
T e s t B e a r i n g Data
The f o l l o w i n g d a t a d e f i n e t h e r e l e v a n t t e s t b e a r i n g dimensions and o p e r a t i n g c o n d i t i o n s used: 63.5 mm. 2x9.3 mm. lands 0.0836 mm 5.08 mm x 3600 1180. RPM. 0.0517 MPa (gauge) -0.175 MPa (gauge) 0.0186 Pa.s
Journal diameter B e a r i n g Length D iamet r a1 c 1ear ance O i l groove Journal speed O i l supply pressure C a v i t a t i o n pressure Effective viscosity 3.4
T e s t Procedure
A t each steady l o a d
-
speed combination, t e s t r i g temperatures were s t a b i l i s e d and t h e d a t a obtained f o r t h e determination o f a t t i t u d e a n g l e and e c c e n t r i c i t y . Data r e p o r t e d i n t h i s paper were a l l o b t a i n e d a t a s i n g l e v a l u e o f s t e a d y e c c e n t r i c i t y r a t i o and a t t i t u d e angle. Time histories of the bearing housing h o r i z o n t a l and v e r t i c a l displacements r e l a t i v e t o t h e j o u r n a l , e x t e r n a l dynamic f o r c e s and
-
RIGOROUS
T h i s method i s based on t h e p r e d i c t i o n o f j o u r n a l displacement and v e l o c i t y components a t t h e end o f each t i m e s t e p by means o f displacement and velocity coefficients A full computed f o r t h e c u r r e n t c o n d i t i o n s . d e s c r i p t i o n o f t h i s method i s g i v e n i n I n r e l a t i o n t o Method A the, reference (3). t h e description "rigorous", e s s e n t i a l l y r e f e r s t o t h e use o f a n u m e r i c a l s o l u t i o n o f t h e f i l m p r e s s u r e d i s t r i b u t i o n ( 2 ) a t each o r b i t step. T h i s t y p e o f s o l u t i o n can accommodate f i n i t e l e n g t h t o diameter r a t i o s and p r e s s u r i s e d o i l feed f e a t u r e s . I n e v i t a b l y t h e term "rigorous" i s r e l a t i v e , and t h e most s i g n i f i c a n t a p p r o x i m a t i o n o f t h i s method i s c o n s i d e r e d t o be t h e r i g i d b e a r i n g assumption. As i n d i c a t e d i n t h e r e v i e w o f p r e v i o u s work, m o d e l l i n g b e a r i n g e l a s t i c i t y a t p r e s e n t r e s u l t s i n e x c e s s i v e computing t i m e u n l e s s approximate f i l m p r e s s u r e s o l u t i o n s a r e used. Bearing elasticity may be quite a p p r e c i a b l e i n some p r a c t i c a l . a p p l i c a t i o n s , n o t a b l y c o n n e c t i n g r o d bearings, b u t t h e t e s t b e a r i n g used t o o b t a i n t h e e x p e r i m e n t a l o r b i t presented i n t h i s paper, was c o n t a i n e d i n a substantial housing. Differences in the e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s due t o bearing e l a s t i c i t y are t h e r e f o r e u n l i k e l y t o be s e r i o u s i n t h i s i n s t a n c e .
4.2
C a v i t a t i o n Model
A c a v i t a t i o n model which took account o f f l o w continuity, whilst assuming a constant c a v i t a t i o n pressure, was used i n Method A. D e t a i l s o f t h i s model a r e a l s o g i v e n i n No account i s t a k e n o f t h e reference ( 2 ) . n e g a t i v e p r e s s u r e s p i k e preceeding t h e r u p t u r e boundary which has been r e p o r t e d i n s e v e r a l e x p e r i m e n t a l s t u d i e s , b u t no p r a c t i c a l system f o r m o d e l l i n g t h i s f e a t u r e i s known t o e x i s t a t present. The method whereby c o n t i n u i t y i s s a t i s f i e d w i t h i n t h e c a v i t a t i o n zone i s s i m p l e and easy t o a p p l y w i t h i n a r e l a x a t i o n s o l u t i o n Only t h e o f t h e f i l m pressure d i s t r i b u t i o n . c a v i t a t i o n p r e s s u r e has t o be s p e c i f i e d , no assumptions o r i n i t i a l e s t i m a t i o n s f o r t h e l o c a t i o n o f t h e c a v i t a t i o n zone boundaries, o r t h e p r e s s u r e g r a d i e n t s a t t h e s e boundaries, a r e necessary. Furthermore, t h i s method i s eminently suitable to oil film history m o d e l l i n g , w h i c h may be d e f i n e d as t h e s t e p b y s t e p m o n i t o r i n g and u p d a t i n g o f t h e e x t e n t o f cavitation zones and the volumetric d i s t r i b u t i o n o f o i l w i t h i n them t h r o u g h o u t t h e journal orbit.
359 O i l F i l m H i s t o r y Model
4.3
changes by assuming t h a t a c c e l e r a t i o n v a r i e s l i n e a r l y w i t h time during t h i s i n t e r v a l :
A detailed description o f the o i l f i l m history model i s given i n r e f e r e n c e ( 3 ) and t h e f o l l o w i n q n o t e s o u t l i n e t h e main f e a t u r e s : The o i l f i l m i s d i v i d e d i n t o r e c t a n g u l a r elements, f o r t h e purpose o f s o l v i n g t h e f i l m pressure d i s t r i b u t i o n b y c o n s i d e r a t i o n o f f l o w continuity (2). O i l f i l m h i s t o r y modelling i s based on t h e premise t h a t i n a d y n a m i c a l l y loaded bearing, elements s u b j e c t t o c a v i t a t i o n do n o t have t o s a t i s f y f l o w c o n t i n u i t y , s i n c e t h e y may be f i l l i n g o r emptying a t any g i v e n time. D u r i n g each o r b i t t i m e s t e p t h e n e t t f l o w r a t e o f o i l t o each element i s computed, t h i s b e i n g used t o update t h e volume o f o i l w i t h i n each element a t t h e t i m e s t e p end: VoA(j,i) = Vo ( j , i )
+
qn ( j , i ) .
At
[4 1
T r a n s f e r o f a c a v i t a t i n g element t o a f u l l f i l m element occurs when e q u a t i o n [ 4 ] p r e d i c t s an o i l volume equal t o o r exceeding t h e The element volume: Vo(j,i) 3 VeJj,i). r e v e r s e t r a n s f e r may o c c u r d u r i n g t h e f i l m pressure relaxation process, when a s u b - c a v i t a t i o n p r e s s u r e i s computed f o r a f u l l film element. By means of the above processes, c a v i t a t i o n zones may expand o r c o n t r a c t i n any d i r e c t i o n a c c o r d i n g t o t h e p r e v a i l i n g c o n d i t i o n s as t h e dynamic c y c l e proceeds. 4.4
O r b i t Time Step S o l u t i o n
Since journal mass inertial f o r c e s were included, t h e o r b i t t i m e s t e p procedure was based on t h e s o l u t i o n o f t h e e q u a t i o n s o f m o t i o n f o r t h e mean c o n d i t i o n s d u r i n g each o r b i t step: (FexA-
FxA +
Fe,
- F,
)/ 2 = rn A i / A t
(FWA-FyA+ Fey- F,)/2
151
= m Ai/At
A t t h e s t a r t o f each t i m e s t e p t h e j o u r n a l displacement and v e l o c i t y components (x,y,i?,j) w i l l b e known, and t h e c o r r e s p o n d i n g o i l f i l m forces Fx, Fy can thus be computed. E x t e r n a l f o r c e components Fex, a t the end s t a r t p o i n t and FexA , FeyA hte': p o i n t can be i n t e r p o l a t e d f r o m t h e s p e c i f i e d e x t e r n a l l o a d c y c l e data. There remains 6 unknowns XA , YA , ~ C A , ?A , FXA ., FYA corresponding t o t h e end p o i n t . An a d d i t i o n a l f o u r e q u a t i o n s are t h e r e f o r e r e q u i r e d i n o r d e r t o obtain a solution. Two f u r t h e r e q u a t i o n s are p r o v i d e d by u s i n g o i l f i l m displacement and v e l o c i t y c o e f f i c i e n t s t o r e l a t e t h e o i l f i l m f o r c e changes w i t h t h e c o r r e s p o n d i n g displacement and v e l o c i t y changes d u r i n g t h e t i m e step. The displacement and v e l o c i t y c o e f f i c i e n t s a r e computed f o r t h e c o n d i t i o n s c o r r e s p o n d i n g t o t h e s t a r t p o i n t , and i t i s assumed t h a t t h e s e v a l u e s do n o t v a r y s i g n i f i c a n t l y over t h e t i m e step: +B,,Ai
[71
Ax+ A,, Ay + B,, A i + &,,A9
[81
AF, = & A x
AF, = A,,
+ A,,Ay
+B,A~
The remaining two e q u a t i o n s r e q u i r e d a r e o b t a i n e d b y r e l a t i n g t h e displacement changes during A t w i t h t h e corresponding v e l o c i t y
A X = ( 2 i + i AA)t / 3
+ i(AtI2/6
[91
[lo1
by= (29 + ~ ~ ) A 1 / 3 + ~ ( b t ) ~ / 6
The d is p l acement and v e l o c i t y c o e f f i c i e n t s used i n e q u a t i o n s [ 5 ] and [ 6 ] were computed a t each o r b i t s t e p p o i n t b y t h e a p p l i c a t i o n o f displacement and v e l o c i t y p e r t u r b a t i o n s and film p r e s s u r e s o l u t i o n s t o determine t h e A critical c o r r e s p o n d i n g o i l f i l m forces. f e a t u r e o f t h e t i m e s t e p s o l u t i o n was t h e minimising of errors arising from the n o n - l i n e a r i t y of t h e s e c o e f f i c i e n t s . T h i s was achieved i n two ways: F i r s t l y , t h e d u r a t i o n of each t i m e s t e p ( A t ) was computed t o m a i n t a i n t h e s t e p displacement and v e l o c i t y changes w i t h i n c e r t a i n maximum values. These maximum changes were computed f r o m e m p i r i c a l A( ) = K1 K~.E functions o f t h e form where K1 and K2 a r e c o n s t a n t s . Secondly, t h e procedure made i n i t i a l e s t i m a t e s o f t h e s t e p and v e l o c i t y changes, these d is p l acement v a l u e s t h e n b e i n g used as t h e p e r t u r b a t i o n s t o compute t h e c o e f f i c i e n t s . Full details o f t h i s method a r e g i v e n i n r e f e r e n c e ( 3 ) .
-
5.
THEORETICAL METHOD 6: METHOD
5.1
Introduction
THE
REACTION
T h i s method achieves a s u b s t a n t i a l l y f a s t e r o r b i t a n a l y s i s by t h e use o f pre-computed velocity coefficients. The name "Reaction Method" was chosen since the velocity c o e f f i c i e n t s enable t h e t o t a l o i l f i l m f o r c e r e a c t i o n t o be e s t i m a t e d f o r any combination o f j o u r n a l v e l o c i t y and p o s i t i o n w i t h i n t h e b e a r i n g clearance. Both squeeze and wedge actions are included, together w i t h the i n t e r a c t i o n between them due t o c a v i t a t i o n . 5.2
O i l F i l m Force Equations
The o i l f i l m f o r c e equations, which f o r m t h e b a s i s o f t h e R e a c t i o n Method, a r e expressed i n p o l a r c o - o r d i n a t e terms: F t = Btt6o + B t r t
k6o
F r = Brt0o + B r r R + B r r t
.. RBo
[I11
c 121
The development of these equations is d e s c r i b e d i n d e t a i l i n r e f e r e n c e (4), and t h e p r i n c i p a l f e a t u r e s a r e as f o l l o w s : a)
The f o r c e components Fr, F t are t h e t o t a l o i l f i l m f o r c e s and n o t changes i n f o r c e f r o m an e q u i l i b r i u m osition. T h i s i s f a c i l i t a t e d by b e i n g t h e t o t a l e f f e c t i v e wedge v e l o c i t y since it incorporates both t h e angular v e l o c i t y o f t h e j o u r n a l about i t s own a x i s ( w ) and t h e angular v e l o c i t y o f t h e j o u r c a l a x i s about t h e b e a r i n g a x i s (8) i.e. 0,=0-W2 (assuming a stationary bearing).
lo
b)
The
velocity
coefficients
are
computed f o r a range o f p a r t i c u l a r v a l u e s of e c c e n t r i c i t y r a t i o and interpolation (linear or logarithmic) between the adjacent values is carried out for any given eccentricity ratio. C o n s i d e r a t i o n o f t h e dimensionless forms f o r t h e v e l o c i t y c o e f f i c i e n t s i s f u l l y d e t a i l e d i n reference (4). This indicated t h a t t h e v a l i d i t y o f dimensionless v e l o c i t y c o e f f i c i e n t s i s r e s t r i c t e d t o given values o f dimensionless s u p o l v and c a v i t a t i o n pressures, wbictj in turn are and t h e p r o d u c t funcJions o f R, 8, R . 8, according t o t h e type o f coefficient
.
Predicted o r b i t s using c o e f f i c i e n t s d e r i v e d b y R p e r t u r b a t i o n amplitudes o f 0.6, 1.2 and 1.8 mm/s d i d n o t It indicate significant differences. is therefore evident that the predicted o r b i t s are not unduly sensitive to the perturbation a m p l i t u d e f r o m which t h e v e l o c i t y coefficients a r e obtained. This means t h a t t h e p r e d i c t i o n accuracy when u s i n g dimensionless velocity c o e f f i c i e n t data i s not c r i t i c a l l y dependant on satisfying the s i m i 1a r i t y r e q u i r e m e n t s w i t h r e s p e c t to dimensionless supply and c a v i t a t i o n pressures. Reference ( 4 ) i n d i c a t e d t h a t t h e F r R curves for 8, # ? are = 0 asymptotic t o t h e c u r y e f o r €lo as t h e magnitude o f R i n c r e a s e d i n both positive and negative directions. A c c o r d i n g l y , when u s i n g t h e l i n e a r i s e d e q u a t i o n s [ll] and [12], Fr i s n o t a l l o w e d t o f a j l below t h e v a l u e c o r r e s p o n d i n g t o 8, = 0; i.e. F r E r r R.
-
The determination of velocity c o e f f i c i e n t s by t h e a p p l i c a t i o n o f v e l o c i t y perturbations i s a quasi dynamic s o l u t i o n i n t h a t i t i g n o r e s t h e dependence on p r e v i o u s c o n d i t i o n s I n o t h e r words t h e i n the o i l f i l m . R e a c t i o n Method does n o t t a k e account history, and s h o u l d of oil film t h e r e f o r e be used w i t h c a u t i o n i n s i t u a t i o n s where t h i s f a c t o r may be s i g n if ic a n t . F a s t O r b i t Time S t e p p i n g Procedure orbit time steppinq procedure was v i r t u a l l y i d e n t i c a l t o t h a t used f o r Method A. A Cartezian co-ordinate system was retained with displacement and velocity c o e f f i c i e n t s i n C a r t e z i a n terms computed by [ll] and [12] with means o f equations appropriate Polar Cartezian transformation. The most i m p o r t a n t d i f f e r e n c e i n t h e procedure concerned t h e d e t e r m i n a t i o n o f t h e t i m e s t e p duration A t . As shown i n r e f e r e n c e ( 3 ) , A t f o r each t i m e s t e p was determined t o ensure t h a t t h e changes i n the components of displacement and v e l o c i t y d u r i n g t h e s t e p were w i t h i n p r e s c r i b e d l i m i t s which were f u n c t i o n s o f eccentricity ratio. T h i s was necessary i n
-
o r d e r t o m a i n t a i n an a c c e p t a b l e accuracy o f t h e p r e d i c t e d of o i l f i l m f o r c e components a t t h e t i m e s t e p end when u s i n g l i n e a r i s e d displacement and v e l o c i t y c o e f f i c i e n t s . At was found b y p r o g r e s s i v e r e d u c t i o n o f t r i a l values u n t i l a l l t h e above l i m i t s were satisfied. I n Method A t h e i n i t i a l t r i a l v a l u e o f A t was a r b i t r a r i l y s e t equal t o about 1.5% o f t h e c y c l e t i m e and reduced by 1% f o r each successive t r i a l . T h i s approach ensured t h a t A t was w i t h i n 1% o f t h e maximum A t p e r m i t t e d b y t h e increment l i m i t c o n s t r a i n t s . The c o r r e s p o n d i n g t i m e r e q u i r e d t o compute A t was a small p a r t of t h e t o t a l computing time. W i t h t h e R e a c t i o n Method, t h e computing t i m e r e q u i r e d t o e s t a b l i s h A t became s i g n i f i c a n t . It was clearly necessary to find a satisfactory compromise between the c o n f l i c t i n g requirements t o maximise A t w i t h i n t h e g i v e n c o n s t r a i n t s , and y e t m i n i m i s e t h e computing t i m e r e q u i r e d t o determine t h i s value. After several t r i a l s , t h e optimum s o l u t i o n f o r t h e t e s t c o n d i t i o n s covered i n t h i s paper was found t o be as f o l l o w s : The i n i t i a l t r i a l A t was s e t a t 30% g r e a t e r t h a n t h e v a l u e f o r t h e p r e v i o u s t i m e step, and t h i s v a l u e was t h e n reduced by 10% f o r each successive t r i a l .
6.
D I S C U S S I O N OF RESULTS
Journal o r b i t s predicted by both t h e o r e t i c a l methods and those measured with the e x p e r i m e n t a l t e s t r i g a r e presented i n F i q u r e s The c o r r e s p o n d i n g e x t e r n a l f o r c e 2 t o 4. c y c l e d a t a i s a l s o g i v e n i n p o l a r form. E x t e r n a l f o r c e c y c l e f r e q u e n c i e s a r e a t once ( T e s t c o n d i t i o n s 1 and 3, F i g u r e s 2.and4.) and t w i c e ( T e s t c o n d i t i o n 2, F i g u r e 3) t h e j o u r n a l rotational frequency. The form of the external force cycle i s similar f o r t e s t c o n d i t i o n s 1 and 2 and r e s u l t s i n o b l i q u e o r b i t s . However, t h e e x t e r n a l f o r c e c y c l e f o r t e s t c o n d i t i o n 3 i s d i f f e r e n t and r e s u l t s i n a substantially horizontal orbit. All three t e s t c o n d i t i o n s show good agreement between t h e o r b i t s p r e d i c t e d by b o t h t h e o r e t i c a l methods. The d i f f e r e n c e s between t h e r e s u l t s o f t h e two t h e o r e t i c a l methods a r e m a i n l y due t o t h e approximations i n t r o d u c e d when f i t t i n q t h e r e l a t i v e l y s i m p l e e q u a t i o n s [ll] and [12], used i n Method B y t o p r e d i c t e d o i l f i l m f o r c e j o u r n a l v e l o c i t y data. Agreement between t h e e x p e r i m e n t a l t h o s e p r e d i c t e d by b o t h t h e o r e t i c a l However, g e n e r a l 1y s a t i s f a c t o r y . two r e g i o n s i n which s i g n i f i c a n t between the experimental and o r b i t s a r e apparent.
o r b i t s and methods i s there are differences theoretical
The l a r g e s t apparent d i s c r e c a n c y was i n t e s t c o n d i t i o n 3 where F i g u r e 4. shows t h a t b o t h t h e o r e t i c a l methods have s u b s t a n t i a l l y g r e a t e r excursions i n t h e d i r e c t i o n against r o t a t i o n . I n approachinq t h e o r b i t e x t r e m i t y a t t / T h 0.15 a s t r o n g e r o i l f i l m wedge a c t i o n w i l l be generated b y t h e a n t i c l o c k w i s e movement o f t h e j o u r n a l c e n t r e ( n e g a t i v e j). T h i s r e g i o n a l s o c o i n c i d e s w i t h n e g a t i v e R. These two e f f e c t s combine t o cause a much s t r o n g e r tendency t o c a v i t a t e i n t h e area t o t h e downstream s i d e o f t h e minimum f i l m t h i c k n e s s p o s i t i o n . It i s therefore postulated that the difference i n t h e e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s i n
36 1
t h i s r e g i o n , may be due t o t h e c a v i t a t i o n pressure b e i n g s i g n i f i c a n t l y lower t h a n t h a t used f o r t h e p r e d i c t i o n o f t h i s dynamic situation. The c a v i t a t i o n p r e s s u r e used t o compute t h e t h e o r e t i c a l o r b i t s , was based on t h e v a l u e d e r i v e d by P a r k i n s ( 1 ) t o y i e l d agreement between the experimental and predicted equilibruim positions. Sensitivity o f t h e o r b i t s t o such d i f f e r e n c e s i n t h e c a v i t a t i o n p r e s s u r e would be i n c r e a s e d by t h e
Dynamic Force Cycle
80
Experimental -o-Theoretical method A .. * .. . -t) ineorericai metnoa
----
I,
Test Condition 1 External Force Data and Journal Orbit Fig 2
Experimental Theoretica I method A
low f o r c e s i n t h e c o r r e s p o n d i n g p a r t o f t h e dynamic f o r c e c y c l e . Hume and Holmes ( 1 2 ) have a l s o i n d i c a t e d t h a t t h e c o r r e l a t i o n between e x p e r i m e n t a l and t h e o r e t i c a l o r b i t s o f i s significantly a squeeze f i l m bearing, dependent upon t h e p r o v i s i o n f o r s u b s t a n t i a l l y sub-atmospheric c a v i t a t i o n pressures i n t h e t h e o r e t i c a l model. I n c o n t r a s t w i t h t h e above o b s e r v a t i o n s f o r t e s t c o n d i t i o n 3, t h e o r b i t a l movement a g a i n s t the direction o f rotation i n t e s t condition 1 ( F i g u r e 2) corresponds t o t h e b u i l d up t o maximum load. This r e s u l t s i n a mainly p o s i t i v e 6 and hence much l e s s c a v i t a t i o n t h a n t h a t experienced i n t h e corresponding p a r t o f The reduced t h e o r b i t f o r t e s t c o n d i t i o n 3. extent o f cavitation, combined w i t h t h e p r o x i m i t y t o maximum 1oad, would r e n d e r o i 1 f i l m f o r c e p r e d i c t i o n e r r o r s associated w i t h cavitation insignificant i n t h i s situation. These c o n d i t i o n s a l s o a p p l y t o t e s t c o n d i t i o n 2 ( F i g u r e 3), and e x p l a i n s why t h e s i g n i f i c a n t differences in the experimental and t h e o r e t i c a l e x t e n t o f movement a g a i n s t t h e d i r e c t i o n o f r o t a t i o n i n t e s t c o n d i t i o n 3 does n o t occur i n t h e other t e s t conditions. The second i m p o r t a n t d i f f e r e n c e between t h e t h e o r e t i c a l and e x p e r i m e n t a l o r b i t s , i s t h e s i g n i f i c a n t l y greater e c c e n t r i c i t y r a t i o o f t h e e x p e r i m e n t a l o r b i t s i n t h e v i c i n i t y o f q~ = 200 i n F i g u r e s 2 and 3, and s i m i l a r l y a t w = 400 i n F i g u r e 4. I n a l l three t e s t conditions, the location of the above d i s c r e p a n c y corresponded t o t h e r e g i o n o f maximum t o t a l load. Since t h e t h e o r e t i c a l 'models assumed a r i g i d b e a r i n g and j o u r n a l , e l a s t i c d i s t o r t i o n i s t h e most l i k e l y cause o f t h e l a r g e r experimental e c c e n t r i c i t y r a t i o s a t t h e more h i g h l y loaded p a r t s o f t h e dynamic cycle.
7 v
c -Dynamic Force Cycle
Steadv
Measured static equilibrium
-2p7 Theoretical method A Theoretical method 8
Fig 3
Test Cmdition 2 Force Data and Journal Orbit
-----
------
Test Condition 3 External Force Data and Journal Orbit Fig 1
362
Theoretical journal o r b i t s f o r t e s t condition 1 ( F i g u r e 2) were computed b o t h w i t h and w i t h o u t t h e o i l f i l m h i s t o r y model u s i n g The d i f f e r e n c e s were n e g l i g i b l e . Method A. T h i s r e s u l t was c o n s i d e r e d t o be due t o t h e combination o f a small o r b i t i n r e l a t i o n t o t h e c l e a r a n c e c i r c l e , and t o t h e e f f i c i e n t supply of oil provided by the full c i r c u m f e r e n t i a l groove.
7.
(3)
JAKEMAN R. W. "The I n f 1uence of C a v i t a t i o n on t h e n o n - l i n e a r i t y o f Velocity Coefficients in a Hydrodynamic J o u r n a l Bearing." To b e Lyon T r i b o l . Symp. p u b l i s h e d : Leeds Leeds Sept. 1986.
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CONCLU SI ON S
T h i s paper compares measured j o u r n a l o r b i t s w i t h t h e p r e d i c t i o n s o f two t h e o r e t i c a l methods. The t e s t c o n d i t i o n s used covered d i f f e r e n t forms o f e x c i t a t i o n a t b o t h once and t w i c e r o t a t i o n a l frequency.
(5)
Good agreement was shown between t h e r e s u l t s o f b o t h t h e o r e t i c a l methods, any d i f f e r e n c e b e i n g l a r g e l y due t o t h e approximations i n t r o d u c e d i n Method B i n o r d e r t o achieve f a s t e r computation. Method 6, introduces simple equations f o r t h e t o t a l o i l f i l m f o r c e components, u s i n g a new t y p e o f v e l o c i t y coefficient. An e q u i v a l e n t angular v e l o c i t y i s used which combines r o t a t i o n o f t h e j o u r n a l about i t s a x i s w i t h r o t a t i o n o f t h e j o u r n a l a x i s about t h e b e a r i n g a x i s , t h u s f a c i l i t a t i n g application to both steady and dynamic situations. The computation t i m e f o r t h i s Method was reduced b y a f a c t o r o f o v e r 300 r e l a t i v e t o t h a t f o r Method A.
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JONES G.J. " C r a n k s h a f t Bearings: O i l Film History" 9 t h Leeds-Lyon T r i b o l Symp.Sept.1982 pp 83 - 88. LA BOUFF G.A. and BOOKER J.F. "Dynamically Loaded Journal Bearings: A F i n i t e Element Treatment f o r R i g i d and Elastic S u r f aces". A.S.M.E./A.S.L.E. Conf. Oct. 1984 A.S.M.E. Paper 84 TRIB 11.
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(9) Generally good agreement between the experimental and theoretical orbits was attained. The main differences were considered t o be r e l a t e d t h e i n f l u e n c e o f c a v i t a t i o n a t low l o a d and b e a r i n g e l a s t i c i t y a t h i g h load.
REFERENCES PARKINS D.W. "Theoretical and Experimental Determination of the Dynamic Characteristics of a Hydrodynamic Journal Bearing". A.S.M.E. J n l . Lub. Tech. Vol. 101. A p r i l 1979 pp 129 139.
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(21
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FANTINO B., FRENE J. and GODET M. "Dynamic Behaviour o f an E l a s t i c Theoretical Connecting Rod B e a r i n g -. S t u d i e s i n Study". SAE/SP-539 Engine Bearings and Lubrication, Paper No. 83037, Feb. 1983 pp 23 32.
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GOENKA P.K. and OH K.P. "An Optimum A Connecting Rod Design Study Lubrication Viewpoint". A.S.L.E./A.S.M.E T r i bology Conference. A t l a n t a October 1985. A.S.M.E. P r e p r i n t No. 85 Trib 50.
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ACKNOLEDGEMENTS
The a u t h o r s would l i k e t o thank t h e Head o f the College of Manufacturing, Cranfield I n s t i t u t e o f Technology and t h e Committee o f L l o y d ' s R e g i s t e r o f Shipping f o r p e r m i s s i o n t o p u b l i s h t h i s paper. They would a l s o l i k e t o thank t h e i r many c o l l e a g u e s a t b o t h C r a n f i e l d I n s t i t u t e o f Technology and L l o y d ' s R e g i s t e r o f Shipping f o r t h e h e l p and c o o p e r a t i o n throughout t h e work.
(1)
BOOKER J.F. "Dynamically loaded Journal Bearings: Nume r ica 1 A p p l i c a t i o n o f t h e M o b i l i t y Method". A.S.M.E. J n l . Lub. Tech. Jan 1971. pp 168 - 176. and CRAVEN A.H. "The HOLMES R. I n f l u e n c e o f C r a n k s h a f t and Flywheel mass on t h e Performance o f Enqine Main B e a r i n g s " 1.Mech.E T r i b o l . Conv 1971 Paper C63/71 pp 80 85.
The e f f e c t of o i l f i l m h i s t o r y was f o u n d t o be n e g l i g i b l e i n t h e case analysed.
8.
JAKEMAN R.W. "Journal O r b i t Analysis t a k i n g account o f O i l F i l m H i s t o r y and J o u r n a l Mass". Proc. o f Conf: Numerical Methods i n Laminar and T u r b u l e n t Flow. Swansea. J u l y 1985 pp 199 - 210.
JAKEMAN R.W. "A n u m e r i c a l a n a l y s i s method based on f l o w c o n t i n u i t y f o r hydrodynamic journal bearinqs". T r i b o l . I n t . Vol. 17. No. 6. Dec. 1984 pp 325 - 333.
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MORTON, P.G. "Measurement o f t h e Dynamic C h a r a c t e r i s t i c s o f a Large A.S.M.E. Sleeve B e a r i n g " . Trans. Jnl. Lub. Tech. Jan. 1971. pp. 143 150.
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HUME, B and HOLMES, R. "The Role o f Sub-Atmospheric F i l m Pressures i n t h e V i b r a t i o n Performance o f Squeeze F i l m Bearings". I. Mech. E. J n l and Mech. No. 5. 1978. Eng. Science. Vol. 20. p . 283.
363
Paper Xl(iv)
The effect of dynamic deformation on dynamic properties and stability of cylindrical journal bearings Z. Zhang, 0.Mao and H. Xu
Reynolds e q u a t i o n and d e f o r m a t i o n e q u a t i o n a r e d e r i v e d i n complex form f o r t h e dynamic increment of o i l f i l m p r e s s u r e and b e a r i n g s u r f a c e dynamic d e f o r m a t i o n . The complex a m p l i t u d e s o f dynamic p r e s s u r e and d e f o r m a t i o n d i f f e r e n t i a t e d w i t h r e s p e c t t o j o u r n a l motions a r e solved by numerical methods w i t h i t e r a t i o n s between them, and t h e w h i r l r a t i o a t s t a b i l i t y t h r e s h o l d i s found by f u r t h e r i t e r a t i o n s , wherefrom t h e dynamic p r o p e r t i e s r e l e v a n t t o s t a b i l i t y t h r e s h o l d a r e c a l c u l a t e d and from them t h e s t a b i l i t y c h a r a c t e r i s t i c s . I t i s concluded t h a t t h e e f f e c t o f dynamic d e f o r m a t i o n on b e a r i n g dynamic p r o p e r t i e s and s t a b i l i t y is u n n e g l i g i b l e for s o f t e r b e a r i n g m a t e r i a l s . I t i s a l s o shown t h a t t h e two c r o s s damping c o e f f i c i e n t s d i f f e r from each o t h e r , i n s t e a d o f b e i n g e q u a l a s p r e d i c t e d by t h e t h e o r y f o r s t a t i o n a r y c o n t o u r s .
1 I N TRODUC TI ON
S t a t i c p r o p e r t i e s o f j o u r n a l b e a r i n g s u n d e r EHL c o n d i t i o n s have been i n v e s t i g a t e d by v a r i o u s r e s e a r c h e r s i n t h e l a s t twenty y e a r s ( 1 ) , ( 2 ) . But dynamic p r o p e r t i e s have mostly been c a l c u l a t e d t a k i n g i n t o c o n s i d e r a t i o n o n l y of s t a t i c deformation of t h e b e a r i n g (2-4). L i n e a r d e f o r m a t i o n model was used i n ( 5 ) t o account f o r t h e e f f e c t o f dynamic d e f o r m a t i o n on dynamic p r o p e r t i e s , but i t s e f f e c t on s t a b i l i t y t h r e s h o l d i s s t i l l t o be i n v e s t i g a t e d , and more e l a b o r a t e d e f o r m a t i o n model might a l s o be a p p l i e d . These a r e a t t e m p t e d i n this p a p e r . 1.1
Nondimensional p r e s s u r e 2 P=Pq / (2pwo 1
P
O i l f i l m pressure
T
Nondimensional time o r a n g l e of r o t a t i o n T=tw 0
t
Time
z
Axial c o o r d i n a t e measured from middle p l a n e of b e a r i n g width
€0
A
Eccentricity r a t i o a t s t a t i c equilibrium Nondimensional a x i a l c o o r d i n a t e A=z/ (1/2 )
Notation Nondimensional damping c o e f f i c i e n t
B
P
00
Attitude angle a t s t a t i c equilibrium
1
1u
Dynamic v i s c o s i t y o f l u b r i c a n t
b
Damping c o e f f i c i e n t
'p
d
Bearing diameter
Angular c o o r d i n a t e , measured from maximum c l e a r a n c e
Hadial clearance
ICI
C
Clearance r a t i o 9=c/ (d/2 1 R a t i o o f e x c i t a t i o n frequency t o r u n n i n g frequency
B=$/
-E
$1
Nondimensional e l a s t i c i t y modulus
n
E=E$/(2pd0) E
Young's modulus
H
Nondimensional f i l m t h i c k n e s s H=h/c
h
Film thi c h e s s
K
Nondimensional s t i f f n e s s c o e f f i c i e n t 3 K=k$ / ( p ~ , l )
k
Stiffness coefficient
1
B e a r i n g width
Nondimensional c r i t i c a l mass o f r i g i d Mcr rotor 3 Mcr=mcraot# /(pl m
cr
C r i t i c a l mass o f r i g i d r o t o r
n -w/cl, w a0
Excitation frequency Running frequency
2 THEORY The c h a r a c t e r i s t i c motion o f t h e j o u r n a l c e n t e r on s t a b i l i t y t h r e s h o l d i s a s t a t i o n a r y w h i r l on a c l o s e d e l l i p t i c a l o r b i t around t h e s t a t i c e q u i l i b r i u m p o s i t i o n of t h e j o u r n a l c e n t e r , with a d e f i n i t e whirl r a t i o expressing t h e r a t i o o f w h i r l frequency t o r u n n i n g frequency. I n o r d e r t o be a b l e t o p r e d i c t t h e t h r e s h o l d s p e e d , t h e e i g h t dynamic c o e f f i c i e n t s of the o i l f i l m corresponding t o such a c h a r a c t e r i s t i c motion should f i r s t be
364 c a l c u l a t e d , wherefore a l l t h e b a s i c r e l a t i o n s h i p s c o n c e r n i n g t h e dynamic f i l m p r e s s u r e and t h e dynamic d e f o r m a t i o n o f t h e b e a r i n g s h o u l d be d e r i v e d r e l e v a n t t o s u c h a s t a t e o f m o t i o n . The n o n d i m e n s i o n a l form o f R e y n o l d s e q u a t i o n may be w r i t t e n as:
Fig.1 shows t h e j o u r n a l a t i t s s t a t i c equilibrium position r e l a t i v e t o the bearing.
deformation :
,X1
HA = . ! ~ U ~ ~) lf (~f Jl , % y l 1 d T , d h , ( 8 ) A The i n c l u s i o n o f t h e dynamic d e f o r m a t i o n Hs i n e q u a t i o n ( 6 ) makes t h e e x p r e s s i o n more complete t h a n i f o n l y s t a t i c d e f o r m a t i o n i s taken i n t o consideration. The i n s t a n t a n e o u s v a l u e o f t h e r e s u l t a n t f i l m p r e s s u r e is t h e n e x p r e s s e d as: P = P
iflT + q 5 = P o + Q e
(9)
where Po d e n o t e s t h e s t a t i c component o f t h e film pressure. S u b s t i t u t i n g e q u a t i o n s ( 6 ) and ( 9 ) i n t o e q u a t i o n ( 1 ) * s e p a r a t i n g t h e time dependent p a r t from t h e s t a t i o n a r y o n e , and n e g l e c t i n g small terms o f h i g h e r o r d e r , we o b t a i n
and
Reyn(U) = -9-
PI aHo
Ho
-+ a7
aM
3%
+ 6inM
-
THO[
3M (Hm
F i g . 1 End view O f b e a r i n g and j o u r n a l . where the. o p e ' r a t o r Heyn( ) d e n o t e s I f the journal i s excited i n t o a simple harmonic m o t i o n o f small a m p l i t u d e , t h e i n s t a n t a n e o u s e c c e n t r i c i t y r a t i o and a t t i t u d e a n g l e may be e x p r e s s e d r e s p e c t i v e l y as: inT L = E + A&einT and 0 = 0 + b e e (2) 0
u h e r e A & and d e a r e complex a m p l i t u d e s of e c c e n t r i c i t y r a t i o and a t t i t u d e a n g l e r e s p e c t i v e l y . The c o r r e s p o n d i n g dynamic i n c r e m e n t o f f i l m t h i c k n e s s c o n s i s t s o f two components, one o f which is t h a t c a u s e d by t h e v a r i a t i o n of t h e geometric p o s i t i o n of t h e journal center: inT H = ( ~ € c o s c p+ t , b e s i n c p ) e (31 d t h e o t h e r is t h e dynamic d e f o r m a t i o n c a u s e d by t h e dynamic i n c r e m e n t o f f i l m p r e s s u r e : (4) Hg = JJ Q,(Cpl A, ) f(cP*h,'Pl, h l ) dTldh1 A where f (q,h,v1h l ) is t h e f l e x i b i l i t y c o e f f i 9 c i e n t expressing t h e r a d i a l deformation a t p o i n t (?,A) caused by u n i t p r e s s u r e a t P o i n t (cpl A l 1: Q (T1 X 1 ) is t h e dynamic
and M = bEcoscp + EoAOsincp + Ha
For each e c c e n t r i c i t y r a t i o , simultaneous e q u a t i o n s ( 7 ) and ( 1 0 ) a r e s o l v e d t o o b t a i n t h e s t a t i c f i l m thickness H s t a t i c pressure P 0,
and a t t i t u d e a n g l e
O0
I n order t o calculate
t h e dynamic c o e f f i c i e n t s o f t h e o i l f i l m , e q u a t i o n s ( 1 1 ) and ( 8 )a r e f i r s t p a r t i a l l y differentiated with respect t o perturbation A&, t o o b t a i n
+ 6in(coscq + H A L )
-
3Ho{ [ H o ( - s i n y + a-)HAL
acp
9
9
i n c r e m e n t o f f i l m p r e s s u r e a t p o i n t ('pl h l ) ; 9
A d e n o t e s t h e p r e s s u r e domain. Under s m a l l
s i m p l e harmonic v i b r a t i o n , Us c a n a l s o b e e x p r e s s e d i n s i m p l e harmonic form: Qs = Ue i n T
(5)
The i n s t a n t a n e o u s v a l u e o f f i l m t h i c k n e s s may t h e r e f o r e be e x p r e s s e d as: H = H
o
= H
+ H
d
+Hs
+ (AEcoscp+ EoAesincp + H,)e inT
(6) where Ho i s t h e s t a t i c f i l m t h i c k n e s s i n c l u d i n g the s t a t i c deformation of the bearing:
The complex d i s t r i b u t i o n Q& i s s o l v e d from t h e s i m u l t a n e o u s e q u a t i o n s ( 1 2 ) and ( 1 3 ) by iterations. Stiffness coefficients K Kee and
ee ,
damping c o e f f i c i e n t s B
a r e then
ee, c a l c u l a t e d by i n t e g r a t i o n s : -~~U,COS'f
d'f d h = K e e + inBee
A
-jJU,sincpdcp
d h = KBe
+
iflBBe
(14)
A and H, i s t h e complex a m p l i t u d e of t h e dynamic
S i m i l a r l y , p a r t i a 1 d i f f e r e n t i a t i o n s with
365 o b t a i n e d , and they can be used t o p r e d i c t t h e s t a b i l i t y t h r e s h o l d speed.
r e s p e c t t o A9 a r e made t o g e t
+ 6in(sincp + HA@ ) -3Ho[
( Ho(coscp + 3% 7) Ep
I n t h e c a l c u l a t i o n s , the o i l f i l m domain is d i s c r e t i e e d , t h e Reynolds e q u a t i o n s a r e s u b s t i t u t e d by systems of f i n i t e d i f f e r e n c e e q u a t i o n s and solved by SOH method. The f l e x i b i l i t y c o e f f i c i e n t f (cp,h,c~, ,A1 ) i s o b t a i n e d i n d i s c r e t e form by a p p l y i n g SAP5 w i t h subsequent r e d u c t i o n . I t e r a t i o n s between t h e Reynolds e q u a t i o n s and t h e d e f o r m a t i o n c a l c u l a t i o n s a r e performed t o o b t a i n t h e simultaneous s o l u t i o n s f o r o i l film pressure ( o r i t s d e r i v a t i v e s ) and b e a r i n g deformation (or its derivatives).
3 RESULTS OF CALCULATIONS The complex d i s t r i b u t i o n Q, i s solved from e q u a t i o n s ( 1 5 ) and ( 1 6 ) . S t i f f n e s s e s Kee and dampings B d
,
9
BOB a r e c a l c u l a t e d by
Km
integrations:
The dynamic c o e f f i c i e n t s a r e t h e n transformed i n t o C a r t e s i a n c o o r d i n a t e s t o obtain
C a l c u l a t i o n s have been made f o r a c y l i n d r i c a l x=200 and b e a r i n g w i t h l/d=0.6, € =0.1-0.95, 20 ( c o r r e s p o n d i n g approximately t o bronze and nylon r e s p e c t i v e l y ) , Y o i s s o n ' s r a t i o V = O . 7 , anti r a t i o o f o u t e r t o i n n e r d i a m e t e r o f t h e bush D/d=1.6. The b e a r i n g bush is supposed t o be u n i f o r m l y and r i g i d l y supported on i t s outer surface. R e s u l t s have been o b t a i n e d b o t h when o n l y s t a t i c d e f o r m a t i o n is considered and when dynamic d e f o r m a t i o n i s a l s o i n c l u d e d . The r e s u l t s i n t h e f i r s t c a s e correspond w e l l w i t h ( 4 ) . The r e s u l t s of t h e two c a s e s d i f f e r s i g n i f i c a n t l y from e a c h o t h e r f o r t h e s o f t e r m a t e r i a l (E=20), they a r e t h e r e f o r e shown h e r e . Fig.2 shows t h e comparison of t h e f o u r s t i f f n e s s c o e f f i c i e n t s . Fig.? shows t h a t of t h e f o u r damping c o e f f i c i e n t s , where i t may be seen
P r i n c i p a l l y , t h e dynamic c o e f f i c i e n t s w i l l depend on t h e e x c i t a t i o n frequency r a t i o n . The procedure of c a l c u l a t i o n of t h e s t a b i l i t y threshold speed should t h e r e f o r e be a l i t t l e d i f f e r e n t from t h a t f o r nondeformable b e a r i n g s . A value of fl should f i r s t be e s t i m a t e d , t h e e i g h t dynamic c o e f f i c i e n t s a r e t h e n c a l c u l a t e d , from which t h e e q u i v a l e n t s t i f f n e s s c o e f f i c i e n t K and w h i r l r a t i o Yst a t s t a b i l i t y t h r e s h o l d eq a r e obtained ( 6 ) :
25.1
15.8 10.0 6 -7 4.0
2.5 1.6
0.8
From them, t h e nondimensional c r i t i c a l mass M
cr r e p r e s e n t i n g a l s o t h e nondimensional t h r e s h o l d speed of a r i g i d r o t o r is o b t a i n e d :
0.6
0.4 0.2
I f the calculated value of whirl r a t i o does n o t c o i n c i d e w i t h t h e e s t i m a t e d e x c i t a t i o n r a t i o , a new e s t i m a t i o n should be made and t h e c a l c u l a t i o n r e p e a t e d , u n t i l c o i n c i d e n c e is obtained t o a s u f f i c i e n t a c c u r a c y . I t h a s been seen i n t h e c a l c u l a t e d c a s e s t h a t t h e v a l u e s of t h e dynamic c o e f f i c i e n t s v a r y b u t l i t t l e with e x c i t a t i o n r a t i o between 0.5 and 1 , s o t h a t t h e above s a i d procedure converges r a p i d l y . The v a l u e s of t h e e i g h t dynamic c o e f f i c i e n t s , t h e e q u i v a l e n t s t i f f n e s s and whirl r a t i o a t s t a b i l i t y threshold a r e thus
0
-0.2
-0.4 -0.6 -0.8
Fig.2 Nona imensional s t i f f n e s s coefficients. s t a t i c and dynamic d e f o r m a t i o n s t a t i c deformation only
_----
366
v a l u e s o f t h e two c r o s s dampings determined e x p e r i m e n t a l l y under c e r t a i n c i r c u m s t a n c e s . Fig.4 shows t h a t of t h e e q u i v a l e n t t h e w h i r l r a t i o YSt stiffness coefficient K eq * and t h e nondimensional c r i t i c a l mass M of a cr r i g i d r o t o r . I t may be s e e n t h a t s i g n i f i c a n t u n d e r e s t i m a t i o n o f s t a b i l i t y t h r e s h o l d speed may r e s u l t i f t h e e f f e c t o f dynamic deformat i o n i s neglected, e s p e c i a l l y i f the material i s s o f t and t h e e c c e n t r i c i t y r a t i o i s l a r g e .
4 CONCLUSIONS ( 1 ) The e f f e c t o f dynamic d e f o r m a t i o n o f b e a r i n g on dynamic p r o p e r t i e s and s t a b i l i t y t h r e s h o l d speed is u n n e g l i g i b l e f o r s o f t e r bearing materials. ( 2 ) The d e v i a t i o n s o f t h e damping c o e f f i c i e n t s from n e g l e c t i n g dynamic d e f o r m a t i o n i s more pronounced t h a n t h a t o f t h e s t i f f n e s s coefficients. ( 3 ) Cross damping c o e f f i c i e n t s d i f f e r from e a c h o t h e r when dynamic d e f o r m a t i o n i s t a k e n i n t o consideration. ACKNOWLEDGEMENT The a u t h o r s g r a t e f u l l y acknowledge t h e a s s i s t a n c e of Mr. Yu L i i n f o r m u l a t i o n o f t h e computer programs and his v a l u a b l e a d v i c e s . References Fig.3 Nondimensional damping coefficients. s t a t i c and dynamic d e f o r m a t i o n s t a t i c deformation only
--_--
Fig.4 Nondimensional e q u i v a l e n t s t i f f n e s s , threshold whirl r a t i o and c r i t i c a l mass of r i g i d r o t o r . s t a t i c and dynamic d e f o r m a t i o n ----- s t a t i c d e f o r m a t i o n o n l y
-
t h a t t h e d e v i a t i o n s a r e c o n s i d e r a b l e . I t is noteworthy t h a t t h e two c r o s s dampings d i f f e r from each o t h e r s i g n i f i c a n t l y when dynamic deformation i s taken i n t o c o n s i d e r a t i o n , i n contrast t o the t h e o r e t i c a l predictions f o r s t a t i o n a r y b e a r i n g c o n t o u r s ( 7 ) , which might w e l l be one of' t h e c h i e f c a u s e s o f t h e unequal
Higginson, G.R. 'The t h e o r e t i c a l e f f e c t s of e l a s t i c d e f o r m a t i o n o f t h e b e a r i n g l i n e r on j o u r n a l b e a r i n g performance', PrOc. Symposium on Elastohydrodynamic L u b r i c a t i o n , Inst.mech.Engrs.,l965, 180, P a r t 3B, 71-38. Oh, K.P. and Hueber, K.H. ' S o l u t i o n of t h e elastohydrodynamic f i n i t e j o u r n a l b e a r i n g problem', Trans. ASME, J. Lub. Techn., V01.95, 1973, 342-352. J a i n , S.C., S i n h a s e n , R. and Singh, D.Y. ' A s t u d y of elastohydrodynamic l u b r i c a t i o n o f a c e n t r a l l y loaded 120' a r c p a r t i a l b e a r i n g i n d i f f e r e n t flow r e g i m e s ' , Proc, I n s t . Mech. Engrs,, Vo1.197, P a r t C , 1983, 97- 108. Mao, Q., Han, D,C. and G l i e n i c k e , J . ' S t a b i l i t a e t s e i g e n s c h a f t e n von G l e i t l a g e r n b e i Beruecksichtigung d e r L a g e r e l a s t i z i t a e t ' , Konstruktion 35 (1987) H.2, S -45-52 'The i n f l u e n c e of b e a r i n g N i l s s o n , L.R.K. f l e x i b i l i t y on t h e dynamic performance o f r a d i a l o i l f i l m b e a r i n g s ' , Proc. F i f t h Leeds-Lyon Symposium on Tribology, 1978, Paper I X ( i ) , 311-319. Zhang, Z. 'Theory o f l u b r i c a t i o n o f hydrodynamic b e a r i n g s ' , Feb. 1979, J i a o t o n g U n i v e r s i t y , Shanghai ( i n Chinese ). Shang, L. ' A m a t r i x method f o r computing t h e dynamic c h a r a c t e r i s t i c c o e f f i c i e n t s of hydrodynamic j o u r n a l b e a r i n g s ' , J o u r n a l o f Z h e j i a n g U n i v e r s i t y , No.3, V01.18, Sept. 1984, 126-135 ( i n Chinese).
SESSION XI1 BIO-TRIBOLOGY Chairman: Professor K. Aho PAPER Xll(i)
Development of transient elastohydrodynamic models for synovial joint lubrication
PAPER Xll(ii)
An analysis of micro-elasto-hydrodynamic lubrication in synovial joints considering cyclic loading and entraining velocities
PAPER Xll(iii) Lubricating film formation in knee prostheses under walking conditions
This Page Intentionally Left Blank
369
Paper Xll(i)
Development of transient elastohydrodynamic models for synovial joint lubrication T.J. Smith and J.B. Medley
Numerical analysis was developed for the transient elastohydrodynamic lubrication of human synovial joints which included the local squeeze velocity. The human ankle joint was selected for specific study and it was considered representative of all synovial joints. Initially, steady state analysis was performed and a formula for minimum film thickness was developed. The full transient analysis was applied to a bearing configuration which was similar to the human ankle joint. An approximate plane inclined surface model was shown to provide a good estimate for the minimum film thickness when compared to the full transient analysis. It was applied to a bearing configuration which accurately represented the human ankle joint and a reasonably constant minimum film thickness of about 1.0 rn was predicted for the walking cycle. 1
INTRODUCTION
The roleof transient elastohydrodynamic lubrication in the tribology of human synovial joints has been considered previously by Medley et a1 (1). The human ankle joint was examined but the findings were considered to apply, in a general fashion, to all of the weight bearing synovial joints. A rather extensive numerical analysis was performed but local squeeze velocity was not included in the formulation. Arguments were presented for the application of a model in which the bearing profile was approximated as a plane inclined surface. This plane inclined surface model predicted the variation of the minimum film thickness during the walking cycle. In the present study, the details of the changes in film profile with time were sought and thus the local squeeze velocity was included in the formulation. This aspect of transient elastohydrodynamic lubrication for surfaces of low elastic modulus had not been studied in sufficient detail by previous investigators, although some speculation was made by Medley and Dowson (2) and by Hooke (3). Furthermore, such detail was considered relevant to the recent work of Dowson and Jin ( 4 ) who extended the modelling of synovial joint lubrication to include microelastohydrodynamic lubrication. The transient elastohydrodynamic lubrication of compliant surface layers had been examined in considerable detail by Smith ( 5 ) . The specific cases examined by Smith were not as heavily loaded as human synovial joints. However, steady state analysis was performed by Smith for heavily loaded contacts which were similar to human synovial joints and some of these results were presented by Smith and Medley ( 6 ) . In the present study, the work of Smith (5) was used as a foundation and further numerical analysis for transient elastohydrodynamic lubrication was developed and applied to a specific case that had features which were similar to human synovial joints. The results of the full transient analysis were compared to those predicted by the plane inclined surface model.
1.1
Notation
F'
load per unit width film thickness
h hO
rigid body film thickness
k
deformation coefficient
P R
reduced radius
pressure
t
time
t
period
P
entrainment velocity
U
X
spacial co-ordinate
X
cavitation boundary
e
X.
inlet boundary
rl
viscosity
j
superscript indicating progression in time
i
subscript indicating spacial progression except when it refers to the inlet boundary
2
A REPRESENTATION OF THE HUMAN ANKLE JOINT
Medley et a1 (1) presented an equivalent bearing geometry for the human ankle joint based on measurements given in Medley et a1 (7). The same configuration was proposed f o r the present study (Fig. 1) with a reduced radius of 0.35 m. The deformation of cartilage was represented as a linearly elastic constrained column with deflection directly proportional to the local pressure (Fig. 2).
370
I
c F'
3.0
+
z z w W
20.0
!c
_--
Stouffer et al (1977) Murray et al (1964
7
Fig. 1 An equivalent bearing representing the human ankle joint.
0.0
t
stance phase 0'5
HEEL STRIKE
t
I.o swing phase
TOE OFF
TIME ( s ) I
E,u
Fig. 3 The relative angular displacement between talus and tibia during the walking cycle
d I
--_
Stauffer et al (1977) Seireg 01 Arvikar (1975)
k
k =
t
******-*..
4.0
a d o p t e d i n the p r e s e n t study
d (l-l*z)- v D
Fig. 2
The adopted deformation model for cartilage.
The column model was considered an appropriate approximation of cartilage deformation for the present lubrication analysis. A value of 16 MPa was chosen for the elastic modulus, Poisson's ratio was specified as 0.4 and the viscosity was set at 0.01 Pa s . The relative angular displacement between the talus and the tibia was measured by Murray et a1 ( 8 ) for the entire walking cycle and by Stauffer et a1 (9) for only the stance phase (Fig. 3 ) . The values measured by Murray et a1 were used to estimate entrainment velocities in subsequent lubrication analysis by assuming a centre of rotation of 20.8 nun and that the talar surface remained stationary. The load divided by body weight was estimated by Seireg and Arvikar (10) for the entire walking cycle and by Stauffer et a1 (9) for the stance phase only. A value of 704 N was specified for body weight from ( 7 ) and the loads during walking were calculated (Fig. 4 ) . The estimates from Seireg and Arvikar were used with a value of 26 mm for the ankle width from ( 7 ) to calculate load per unit width in subsequent lubrication analysis. It was necessary to apply
a
0
i
OD
.)
stance phaseOm5
HEEL STRIKE
1
1.0
swinq phase
TOE OFF
TIME ( s 1 Fig. 4
The load during the walking cycle.
a higher load in the swing phase for the present modelling as shown in Fig. 4 .
371
The lubricant flow was represented by the Reynolds equation,
a { H 3 ap) ax ax
a ax
12 { U
=
s aH) aT
U = fl(T)
where
n
+ -XL 2
H = H0
+ P
with boundary conditions, -ap = p = 0 @
ax
@
p = o
x=xE
X = X 1
The load capacity was given by
xE W
=
f2(T)
=
1
P dX
Both fl(T) and f2(T) were functions derived from the data presented in Figs. 3 and 4 . 3
NUMERICAL ANALYSIS
position algorithm (11) was then used in the inner loop and a secant method was applied in the outer loop. The marching procedure in the inner loop had to approach bracketing the root in a constrained manner because if Ho was too small, the library routine for solving the integrated Reynolds equation would fail or negative pressures, which were not physically realistic, would occur in the first few steps of the solution. In either case, Ho was increased automatically and the solution was attempted again. This constraint on the values of Ho for which a solution could be obtained slowed the iterative procedure but solutions were obtained eventually in all cases. The calculated film thickness changed rapidly near the cavitation boundary and thus the first step was subdivided into up to 100 small steps to allow the calculation of precise values for the minimum film thickness. The iteration between film pressure and elastic deflection fields, involved in direct elastohydrodynamic solutions, was avoided and the non-linearity of the integrated Reynolds equation was handled within the library routine (IMSL-DGEAR). Reliability checks were performed by varying the allowable local truncation error within DGEAR and the number of discrete pressure values used in the integration for the load. The resulting solutions had smooth pressure distributions and film profiles (Fig. 5).
3.1 Steady state solutions
u = 5.994x 10-l~
w = 7,507
The Reynolds equation was integrated once and the pressure gradient boundary condition was applied to yield
Px lo3
Hx06
-- 3.0 with boundary conditions,
P = O
@
P = O
@
x = xE x = xI
The numerical solution of the equation was achieved by using a library routine which selected the order and step size automatically and was designed for stiff differential equations (IMSL-DGEAR). The routine was applied in the following iterative scheme:
r
I
6.0
4.0
2.0
0.0
2.0
4.0
6.0
x x lo2
Fig. 5 A typical steady state solution.
SPECIFY
x~
SPEC IFY H o SOLVE FOR P USING
IMSL-DGEAR
L
ITERATE FOR P=O @ X = X l ITERATE FOR W
The load capacity was obtained using library routines which integrated a natural cubic spline representation of the pressure distribution (IMSL- ICSICU and DCSQDU) The two iteration loops involved marching procedures at a fixed step size until the roots (the required Ho and X ) were bracketed. A false
E
The developed solution procedure, with some variations in the iteration scheme, was applied by Medley (12) and by Smith (5). The results were used to derive a formula for the minimum film thickness. This derivation involved applying the concepts developed by Johnson (13) to the present bearing configuration. The required expression for the maximum dry contact stress was taken from Bentall and Johnson (14) and the minimum film thickness for rigid cylindrical contacts was taken from Dowson et a1 (15) along with the maximum film pressure for rigid cylindrical contacts. Fourty-three data points from both Medley and Smith were plotted and a formula was derived by performing a least squares curve fit (Fig. 6 ) . Using the solution procedure developed for the present study, a minimum film
372
h = 1.66+ 0.752 g, std. error = 0.3 1500- 9, = 4 3 --c2 5 8 3
L
0
1.C
500 -
-
doto from numerical solutions
0
0
0
1000
500
9,
1500 0'
2000
w 516
3000
2500
~
3.300U"2
Fig. 6 A formula for minimum film thickness under steady state conditions. thickness was found for a contact with a g value The formula was applied and prezicted a minimum film thickness within 0.075% of the calculated value. of 11587.
3.2 Transient solutions The developed procedure for steady state conditions was very successful and solutions were obtained for both light and heavily loaded contacts. It seemed logical to extend this procedure to a transient analysis. As before, the Reynolds equation was integrated to yield
aP
12
+ P) + s(x-xE)
dH
n
o
X
+ S
1
ap
dX}
xE local squeeze velocity The squeeze velocity was represented implicitly by a first order backward difference. Thus, a discretized form of the equation was derived as follows:
+
S
[(X-XE)
In the representation of the local squeeze velocity, an embedded Tr-lpezoidal rule was employed for the pressure values from the current time step. The accuracy of this representation was influenced by the spacing of the discrete pressure distribution which was not controlled within the DGEAR library routine. Thus, reliability checks had to be performed by varying the level of discretization. The level of discretization also influenced the accuracy of the calculated load. The reliability of the part of the solution which was controlled by DGEAR had checks made by varying the allowable local truncation error within DGEAR. The pressure values from the previous time step were represented by a natural cubic spline (IMSL-ICSICU), then integrated (IMSL-DCSQDU) and the values of the interal at various X-locations were themselves represented as a natural cubic spline (ISML-ICSICU). Thus, the values of the definite integral involving the pressure values from the previous time step were available to the function subroutine of DGEAR at any X-location using a cubic spline interpolation (IMSL-ICSEVU). At each time step, the overall iterative scheme and load integration was identical to that used in the steady state analysis. Also, the first spacial step was subdivided as in the steady state analysis. An outer loop was constructed to allow marching in time. Final reliability checks involved the size of the time step. The additional complexity of the transient formulation led to much greater computational effort. In addition, a much more serous problem occurred. In some cases, under-flow errors developed within DGEAR and in other cases iterations for Ho continued until all 18 digits were determined but the inlet pressure was not close to zero. Based on the results of Smith (5) and a trial and error procedure with the present transient formulation, it was found that severe numerical problems occurred when
gE > 60. U with gE calculated using the instantaneous values of U and W. Using this discovery as a guide, the values specified to represent the human ankle joint were modified so that full transient solutions could be obtained (Table 1).
(Ho- Hi-') Table 1 The two cases considered in the present study. Ankle Joint
C
Modified Case
.
EMBEDDED TRAPEZOIDAL RULE
X J
xE I
CUBIC SPLINE
20. 2.0
16.
2.4 0.4 0.35 0.01 0.28 0.03 1.o
0.4
- 3.73 - 39.
0.1 0.015 0.37 -0.40 54. - 100. 2.0
373
The resulting modified case for the transient analysis was only somewhat similar to the case which represented the human ankle joint. The time step was set to 0.02 and the variation in film thickness with time was calculated (Fig. 7). Later, a time step of 0.01 was used and the resulting value of Hmin at a T of 0.06 differed by 0.41% from the value when the time step was 0.02
I
PX1o3 H x i O 5 p:----
H :-
.
v)
0 c
X
3 -5.0
-2.0
XXlO
5.0
2.0
00
2
w -0 c
X
Fig. 8 Transient film profiles and pressure distributions.
3
Table 2 Comparison of the minimum film thickness from the full transient analysis and those from the plane inclined surface model.
v)
s X
.-C
E
I
I
00
0.1
T
Hmin Full Transient
Fig. 7 The full transient solutions for minimum film thickness. The surface profiles slowly flattened as the entrainment velocity decreased and the length of the contact zone was reduced as the load decreased (Fig. 8 ) . The propagation of thinner films from the inlet zone, which had been suggested as a possibility by Hooke ( 3 ) , did not occur in this case, Also, the phenomenon was not found in the less heavily loaded contacts examined by Smith ( 5 ) .
3.3 The plane inclined surface model The plane inclined surface model involved the specification of a plane inclined surface bearing with a length equal to the dry contact length ( 1 ) ( 2 ) . The inclination was determined by equating the minimum film thickness of the plane inclined surface bearing to that obtained from the formula given in Fig. 6 for the instantaneous load and entrainment velocity. The length and inclination changed immediately at each instant in time and a single ordinary differential equation described the variation of film thickness with time ( 2 ) . The test case for the transient analysis was solved using the plane inclined surface model. The resulting predictions for minimum film thickness were remarkably close to those obtained with the full transient analysis (Table 2 ) .
0.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
1.508 1.508 1.504 1.493 1.474 1.447 1.412 1.369 1.321
x 105 Plane Inclined 1.508 1.515 1.514 1.504 1.484 1.454 1.415 1.366 1.310
A difference of 0.8% occurred at a T of 0.16. The results obtained with the full transient formation and given in Table 2 took a total of 1.45 CPU hours, The results obtained with the plane inclined surface model took a total of about 10 CPU seconds. Since the plane inclined surface model had performed reasonably well, when compared to the full transient analysis, it was applied to the case which represented the human ankle joint (Table 1 ) . The plane inclined surface model did not work, in its present form, at very low low loads and thus the loads during the swing phase had to be increased as shown in Fig. 4 . A cubic spline interpolation with periodic end conditions (IMSL-ICSPLN) was used to represent the load versus time function and a cubic spline evaluation routine (IMSL-ICSEVU) made load values available at any instant in time. The displacement function measured by Murray et a1 (8) and shown in Fig. 3 was
314
represented by a cubic spline interpolation with periodic end conditions (IMSL-ICSPLN). The first derivative of the function was then evaluated (IMSL-DCSEVU) and entrainment velocity was calculated using u = 1.815~10 -4
2
where 8 is the angular displacement.
The variation of entrainment velocity was represented by ICSPLN and interpolated values were supplied by ICSEVU in the same manner as the load function. The resulting variation in minimum film thickness for the converged cycle w'as reasonably constant at just less than 1.0 pm (Fig. 9 ) . However, for the specified values representing the ankle joint, the dry contact length exceeded the length of the talus for 0.255 4
T
6 0.530
This apparent physical inconsistency could be eliminated by decreasing the reduced radius o r increasing the elastic modulus of cartilage. One could also consider that lubricant entrainment did not occur in this portion of the walking cycle and squeeze action alone would act to prevent significant decreases in the film thickness. Further pursuit of this issue was considered beyond the scope of the present study.
4
CONCLUSIONS
.The steady state analysis was successful in predicting minimum film thickness for both lightly and heavily loaded contacts. A general formula for minimum film thickness was developed. 0 The full transient analysis provided solutions
for a modified case which was similar to the human ankle joint. Thinner films did not propagate from the inlet zone in these results. .The plane inclined surface model gave results remarkably close to the full transient solutions and was successfully applied to the case representing the human ankle joint.
6
Hminx 10
3 20
'.O
O
I
0.9pm
t
1
I 0.5
0.0
t
T
I 1.0
Fig. 9 The predicted minimum film thickness from the plane inclined surface model f o r the human ankle joint during the walking cycle.
(3)
HOOKE, C.J. 'Discussion', ASLE Trans.,
(4)
DOWSON, D. and JIN, Z.M. 'Microelastohydrodynamic lubrication of synovial joints', Engng Med., 1986, Is,2 , 63-65. SMITH, T.J. 'Numerical modelling of the transient elastohydrodynamic lubrication of compliant surface layers', MASc Thesis, Dept. of Mech. Eng., University of Waterloo, 1985. SMITH, T.J. and MEDLEY, J.B. 'Elastohydrodynamic lubrication models to aid in the development of compliant surfaced components for hemiarthroplasty', 6th Annual Meeting of the Canadian Biomaterials Society, 1985,. Kingston, Ontario, 44-45. MEDLEY, J.B., DOWSON, D. and WRIGHT, V. 'Surface geometry of the human ankle jo:*it', Engng Med., 1983, 12, 1 , 35-41. MURRAY, M.P., DROUGHT, A.B. and KORY, R.C. 'Walking patterns of normal men', J.Bone 335-360. Jt. S u r g . , 1964, STAUFFER, R.N., CHAO, E.Y.S. and BREWSTER, R.C. 'Force and motion analysis of the normal. diseased and orosthetic ankle joint', Clin. Orthop., 1977, 127,
1984,
(5)
7, 3 ,
249-250.
m,
189-196.
SEIREG, A. and ARVIKAR, R.J. 'The 5
ACKNOWLEDGEMENT
Financial support was provided by the Natural Science and Engineering Research Council of Canada.
References (1)
MEDLEY, J.B., WWSON, D. and WRIGHT, V.
(2)
'Transient elastohydrodynamic lubrication models for the human ankle joint', Engng Med., 1984, 13,3 , 137-151. MEDLEY, J.B. and DOWSON, D. 'Lubrication of elastic-isoviscous line contacts subject to cvclic time-varvine loads and entrainment velocities', ASiE Trans., 1984, 27, 3 , 243-251.
prediction of musculature load charing and joint forces in the lower extremities during walking', J. Biomechs, 1975, 8, 89-102.
CONTE, S.D. and DEBOOR, C. 'Elementary numerical analysis', 3 r d Edition 1980 (McGraw-Hill, New York). MEDLEY, J.B. 'The lubrication of normal human ankle joints', PhD Thesis, Dept. of Mech. Eng., The University of Leeds, 1981. JOHNSON, K.L. 'Regimes of elastohydrodynamic lubrication', J. Mech. Engng Sci., 1970, 1 2 , 9-16.
BENTALL, R.H. and JOHNSON, K.L. 'An elastic strip in plane rolling contact' Int. J. Mech. Sci., 1968, lo, 637-663. DOWSON, D., MARKHO, D.H. and JONES, D.A. 'The lubrication of lightly loaded cylinders in combined rolling, sliding, and normal motion', J. Lubr. Techn., 1976,
98,
4 , 509-516.
375
Paper Xll(ii)
An analysis of micro-elasto-hydrodynamic lubrication in synovial joints considering cyclic loading and entraining velocities D. Dowson and Z.M. Jin SYNOPSIS Attention has heen drawn to the powerful role of micro-elasto-hydrodynamic action in the lubrication The findings from this of synovial joints under quaisi-static conditions in a previous paper (7). earlier analysis did much t o reconcile the conflicting indications of experimental and theoretical studies of synovial joint lubrication extending over many years. Experimental investigations increasingly suggested that natural synovial joints experienced fluid-film lubrication, whereas the theories failed to confirm the formation of such films. It has now become necessary to extend the micro-elastohydrodynamic analysis to conditions more representative of physiological motion and loading and this is the subject of the present paper. The results of this extended analysis confirm that, even under the dynamic conditions representative of the walking cycle, the effective film thickness remains remarkably constant and that micro-elastohydrodynamic action remains as a powerful and highly beneficial action in synovial joints. A new feature of the solutions for cyclic conditions is the partial re-emergence of the surface rugosities at instants when entraining action is very small and squeeze-film action predominates. This is evident in the vicinity of the central dimple which is known to form under elastohydrodynamic squeeze film action between smooth surfaces. The current study has provided further confirmation of the major role played by micro-elastohydrodynamic lubrication in synovial joints. The model adopted is still relatively simple and it may be necessary to give further attention to lubricant rheology in order to bring theory and experiment fully into accord in relation to joint friction. However, the imposition of fully dynamic conditions has not impaired the essential indications of quasi-static micro-elastohydrodynamic analysis of synovial joints reported earlier. 1. INTRODUCTION
During the past two or three decades sharp contradictions have emerged between the indications from theoretical and experimental studies of synovial joint lubrication. Most experimental investigations have revealed fluid film lubrication characteristics in synovial joints under conditions of simulated level walking, whereas hydrodynamic and even elastohydrodynamic analyses of increasing sophistication have failed to predict film thicknesses of sufficient magnitude relative to the surface roughness of articular cartilage to support this view. If the elastohydrodynamic film thickness expressions developed in recent years for both point and line contacts and steady loading of semi-infinite elastic solids are applied to the major joints in the lower limb, it can he shown that the minimum film thickness in such joints is likely to he of the order of 1 pm. Dowson (1,2) estimated minimum film thicknesses under quasi-static conditions in the hip and the knee of 1.3 pm and 1.25 pm respectively. Higginson ( 3 ) developed a Grubin-Ertel type solution under pure squeeze-film motion and predicted minimum film thicknesses of 5 pm in the hip and 1 Um in the knee after a loading period of 0.5s. However, some doubt remains about the initial
film thickness to be adopted in such calculations and Dowson ( 2 ) estimated the film thickness at the end of the lightly loaded swing pl e to be only 3 . 4 pm. It ts well known that the surface of articular cartilage is by no means smooth and the roughness average (Ra) has been estimated to lie in the range 2 to 5 Hm (Gardner et a1 ( 4 ) ; Sayles et a1 (5)). It is customary to relate theoretical predictions of film thickness (h) developed on the basis of perfectly smooth surfaces, to the sum of the surface roughnesses (R ) of the opposing bearing surfaces when Zssessing the mode of lubrication in a bearing. It thus appears that smooth surface film thickness predictions in excess of 5 to 1 0 V m are required if the theoretical predictions are to be consistent with the experimental evidence of fluid film lubrication in synovial joints. None of the theoretical approaches outlined above considered adequately the cyclic nature of physiological loading and motion in synovial joints. Medley et a1 ( 6 ) included such effects in their studies of ankle joint lubrication and they found that a reasonable approximation to the instantaneous film thickness at any instant in the loading cycle could he obtained by means of an extended
376 Grubin-Ertel type of solution. They also noted that the dynamic conditions experienced by the ankle joint in steady walking had but a modest effect upon the variation of lubricant film thickness throughout the cycle.
t
Time.
X
Coordinate in entraining direction.
X X
Dowson and Jin recently (7,8) introduced the concept of micro-elasto-hydrodynamic lubrication in synovial joints. They found that the rugosities on a single sided roughness model of a synovial joint could readily generate local perturbations to the overall pressure distribution which were themselves capable of smoothing out to an impressive extent the initially rough surfaces within the load bearing conjunction under quasi-static conditions. The results of this preliminary analysis indicated that the articular cartilage might be effectively smoothed out in the loaded conjunction and that it might, after all, be possible to contemplate fluid-film lubrication in synovial joints as a result of micro-elasto-hydrodynamic action. These initial quasi-static analyses have done much to bring theory and experiment into accord in the field of synovial joint lubrication, but it became necessary to extend the study to transient conditions representative of the walking cycle. This was the objective of the investigation reported in this paper. 1.1
Semi-dry contact zone width.
a
Compliance for pressure perturbation generated by surface roughness.
cS
Compliance for smooth surfaced cartilage layer.
E
Modulus of elasticity.
R
Radius of equivalent cylinder adjacent to rigid plane.
Ra
Amplitude of surface roughness profile
Sliding velocity of lower (plane) surface (= twice entraining velocity).
6
Elastic deformation
6S
Elastic deformation of smooth surface layer of thickness (d).
r'
Elastic deformation of rough (wavy) surface profile of amplitude (ao).
rl
Lubricant viscosity.
h,
Wave length of surface roughness profile
d
Thickness of elastic layer.
h
Film thickness.
ho
Minimum film thickness of undeformed smooth cylinder near a rigid plane.
*
1
=
0
Phase angle of surface roughness profile . a
Central film thickness
'Average' central film thickness
P
Pressure.
MODEL OF SYNOVIAL JOINT
The general features of synovial joints are shown in Figure 1. The bearing material is articular cartilage bonded to subchondral bone and the lubricant is synovial fluid.
ARTICULAR
SYNOVIAL FLUID
Pressure on 'Smooth' cartilage layer.
'ten
Central pressure.
pd
Pressure due to dry contact between an elastic layer and a rigid plane.
*
- Q]
(]PO).
) of film Film thickness at the point (x cav rupture due to cavitation.
'ma,
cos[y
-b).
hcav
pS
[F- 0 )
Poisson's ratio.
2
Inlet location ( x
cos
V
- 01
b
hcen
Roughness average.
"0
0
hcen
Location of film rupture point due to cav cavitation.
. a
a
c o s F
Inlet location.
r'
Notation
a
1
Maximum pressure.
P
Amplitude of sinusoidal pressure profile applied to semi-infinite elastic solid.
UP)
Perturbation pressure generated by surface roughness.
Figure 1
Synovial Joint
It is both possible and convenient to represent the ankle joint by a cylindernear-a-
311
plane configuration for the present analysis. Furthermore, the two layers of cartilage will be portrayed by a single, douhle thickness layer of 'soft' o r low elastic modulus material mounted on the cylindrical surface, leaving the opposing member as a rigid, plane surface as depicted in Figure 2. W
1 I
The cosine term represents the modification to the parabolic film associated with an undeformed transverse roughness profile and phase of amplitude (a ), wavelength (),I angle ( 0 > , relagive to the origin of coordinates. It is convenient to represent the total elastic deformation ( 6 ) at any location by the deformation which would occur with smooth hut lubricated cartilage (6 ) together with the additional and relatively'very small deformation associated with the rough, wavy surface profile (6,) in micro-elastohydrodynamic luhrication analysis. hence;
-
-b t
ho t
h
u0-
Figure 2
b t X
Equivalent Bearing Model (single sided roughness) for Micro-elastohydrodynamic Analysis.
In this single-sided roughness model of a synovial joint the rugosities will be represented by a transverse, sinusoidal waveform as described by Dowson and Jin (7,8). It will further be assumed that synovial fluid can be treated as a Newtonian fluid and the articular cartilage as an ideal elastic (Hookean) solid. The property values adopted will be based upon the detailed study reported by Medley et a1 ( 6 ) . 3 TRANSIENT MICRO-ELASTO-HYDRODYNAMIC LUBRICATION ANALYSIS
The basic formulation of the governing equations for the transient micro-elasto-hydrodynamic lubrication problem is developed from the outline presented by Dowson and Jin (7,8) for quasi-static conditions. The integrated Reynolds equation for an isoviscous, incompressible fluid can he written in the form,
-
where the sliding velocity (U of the lower, plane surface is itself a functPon of time. Medley et a1 ( 6 ) assumed a sinusoidal form for (u ) , but in the present study the data prgsented by Murray et a1 (9) for motion within the ankle joint in level walking was used directly to predict the entraining velocity throughout the cycle. The instantaneous film thickness at any location (x) within the joint is given by,
where ( C ) and (C: ) represent the cartilage elastic co;pliancesrto be associated with the smooth surface hydrodynamic pressure (p ) and the perturbation pressure (A p) generat% by the surface roughness. Furthehore, the overall cartilage deformation under lubricated conditions is s o close to that of dry contact under physiological loading that the analytical, parabolic contact stress expression (p,) for a thin layer of soft material on a rigid hacking can be used, not only to provide good estimates of the overall conjunction geometry, but also to provide a reasonable approximation to (ps>. hence,
The 'contact stresses', contact zone widths and deformations for a smooth surfaced layer of elastic material in contact with a rigid plane are sensitive to both Poisson's ratio ( v ) and the ratio of layer thickness to semi-contact zone width (d/a) (see Johnson, l o ) . If the latter ratio is large compared with unity, the effect of the substrate i s small and the solutions approach the Hertzian conditions associated with the well known semi-elliptical normal stress distribution. F o r plane strain and a small value of (d/a) the normal stress adopts a parabolic distribution (see Johnson, 101, providing that the Poisson's ratio is not close to 0.5. F o r lubricated conjunctions formed between a thin, soft layer of elastic material and a rigid plane, the elastic deformation is very similar to the paraholic profile for dry contact, with the typical build-up of pressure on the inlet side associated with lubricant entrainment superimposed. It is therefore convenient to consider plane strain and to determine the local deformation associated with the local pressure on the basis of a constrained column model (Medley et a1 6 ) in lubrication analysis. For a local pressure (p), a soft layer of thickness (d) having a modulus of elasticity (E) and a Poisson's ratio ( v ) , the normal deflection is thus given by,
It can readily be shown that small perturbations to the pressure profile developed
378
by rough surfaces will completely flatten the initial roughness on the cartilage due to m i c r w e l a s t o - h y d r o d y n a m i c action. Since the elastic deformation ( 6 ) is proportional to the product of pressure (p) and layer thickness (d) and the roughness amplitudes (a jar, but a few microns compared with layer thi%knesses ( d ) of a few millimet es, pressure perturbations of the of the smooth surface pressure are order of 10 adequate to achieve full flattening of the initially rough surfaces. However, for the relatively small wavelength roughnesses ( A ) compared with the layer thickness (d), the elastic deformations associated with these pressure perturbations need to be determined on the basis of a semi-infinite solid to permit interactions between adjacent protuberances and pressure profiles to be considered. A sinusoidal surface deformation in an Initially smooth semi-infinite solid is associated with a sinusoidal normal surface stress (see Johnson, 10) p 3 9 8 ) , and the two are related as follows.
-5
shown f o r a value of (p*/E) of 0.87 x and a P o i s s o n ’ s ratio of 0.4 in Figure 3 . It should be remembered, however, that the elastic deformations (&.) associated with the qessure perturbations (Ap) are only ahout (10 ) times the bulk deformation of the smooth cartilage layer (s,), and hence little error is encountered if ( C ) is used rather than (Cr) i n many circumstanceZ.
It will therefore be assumed that the local rough surface deformation (6 ) associated with the pressure perturbation ( A F ) in the micro-elastohydrodynamic 1ubri:ation analysis takes the form,
0 0.1
1
10
100
Wavelength Layer Thickness Figure 3 The compliance factors ( C ) and (C ) to be adopted in the analysis (see eluation ( 5 ) ) are thus; r
and
“7
c = b r
E
It should he noted that the plane strain column model (layer) and small wave length rough surface (semi-infinite) model yield identical deflections when subjected to unit pressures when,
k]
is particularly sensitive to The ratio ( v ) as the latte quantity approaches 0.5, and for the value of 0.4 adopted in this analysis it equals 1.75. The constrained column model thus provides a reasonable estimate of the elastic deformation when the wavelength of the initial sinusoidal roughness is a few times greater than the layer thickness, but the small wave length roughness model has to be adopted for wavelengths less than the layer thickness. A comparison between the predictions of elastic deflections from these useful, limiting models and a full finite element solution for a layer of material having a sinusoidal wavy surface is
Predictions of Elastic Deformation Amplitude €or a Sinusoidal Surface Pressure Distribution
For a fixed load and entraining velocity the film thickness at a given ( x ) location is constant and thusah = 0. However, for the dynamic conditionakncountered by synovial joints in the lower limb in level walking, where both load and entraining velocities vary with time, ah# 0 . The local normal velocity can he detegined from equation (2) by noting that, (10)
The first term represents the rigid body normal velocity and the second term the variation in normal velocity associated with changes in the local elastic deformation. The latter term often has a relatively small influence upon the overall soluticn to the Reynolds equation (see Dowson, Ruddy and Economou (11) and Medley et a1 ( 6 ) ) and yet it introducesa major increase in the numerical effort required to obtain a solution. It is particularly important in cyclic problems of the kind considered here to select an effective normal velocity for the bearing surfaces if local effects are not to be considered. In the present problem it is convenient to use the variation of a mean film thickness based upon conditions in the centre of the contact but corrected for roughness and local deformation of the wavy surface. This can be illustrated
379
by rewriting ( 2 ) in terms of the central film thickness (hcen) at (FO) and introducing the expression for elastic deformation ( 3 ) .
+ Cr tap),
(11)
lubrication of synovial joints. Medley et a1 ( 6 ) considered a realistic loading cycle in their study of ankle joint lubrication, but represented the entraining velocity by a simple cosine function. In the present analysis we use the data presented hy Murray et a1 ( 9 ) for displacement, o r motion, and the load distribution adopted by Medley et a1 ( 6 ) to represent the ankle joint in steady walking as shown in Figure 4 .
-1 00
A representative normal velocity is then taken as, _al =
.-90 --80
'Zen
_
at
at
This approximation is particularly important at times in the cycle when the entraining velocity is small and the behaviour of the lubricating film is governed essentially by squeeze-film action. At other times, when combined entraining and squeeze film action is considered the surface waves are effectively smoothed out everywhere by micro-elastohydrodynamic lubrication action such that the terms in the square brackets in equation ( 1 2 ) are almost self cancelling. In such circumstances,
0
0.1
OQ
Ttme
Figure 4
~
-
at
ahcen
(14)
at
The significance of this approach d l 1 be discussed later. If the average normal velocity derived from ( 1 3 ) is introduced into ( I ) , the Reynolds equation can be integrated to yield;
(8)
Loading Pattern and Entraining Veloctties in Walking Cycle (Ankle Joint). Load - - - - Entraining Velocity.
* ahcen
O B W 0 6 0.6 0.7 0-8 0.9 1 0
The procedure adopted for the integration of equation (15) follows the method outlined earlier by the authors (7,8) for quasi-static conditions. Integration procedes upstream from an initially selected outlet o r cavitation location (X ) related to (h en) and an iterative p%edure is adopte8 until the pressure curve returns to ze o to within a at the specified tolerance Pinlet/Pmax < inlet location.
I
-
where subscript (cav) refers to conditions at the point of film rupture due to cavitation. At this location it is assumed that the Reynolds cavitation boundary condition applied at all times. (16)
At inlet the pressure is assumed to build up from ambient (zero) from a location (-b) where the film thickness is large compared with the minimum film thickness in the conjunction. Thus, p 4
=
0
when
x
i
=
-b
(17
1
NUMERICAL PROCEDURE
The essential problem is to solve the integrated Reynolds equation (15) throughout the complete walking cycle for selected loading and motion cycles. In earlier papers (7,8) we considered quasi-static conditions in our analysis of micro-elasto-hydrodynamic
The numerical Algorithm Group (NAG) routine DOZERF hased upon the Gear method was used to integrate ( 1 5 ) in a manner similar to that described hy Medley et a1 ( 6 ) . A bisection technique was used f o r both inlet pressure and load iterations, with the latter integration being achieved by the method of Gill and Miller (12) using NAG DOlGAF. The tolerance on central pressure,- efined as I (p:en - P~;~)/(P~~,) was 10
9
I,
-
The iteration consisted of five loops as shown in Figure 5. The initial specifications of x ,h and pce were based upon soluffhs 8Fained unzer quasi-static conditions, but as mentioned earlier the dry contact solution f o r normal stress (equation ( 4 ) ) gave a good initial estimate of p The total cycle time of 1 second was subdi88ed into 6 4 time intervals and at each time step the iterative procedure progressed until the tolerance on load capac ty satisfied the condition I IAWI/W < 10- I. After each cycle the loadings and velocities shown in Figure 4 were re-applied and a satisfactory, converged cyclic film thickness profile was generally achieved after about ten complete cycles, corresponding to some lo3 C.P.U. seconds on the Amdahl computer at the University of Leeds.
.
1
380
The tolerance on central film thickness adopted in this procys was 1(6hc n)(t=o)/(hcen (t=o))l
.
I INPUT LOAD AND SPEED AT TIME (t)
1
SPECIFY CAVITATION BOUNDARY POSITION
I SPECIFY CENTRAL FILN
I
VELOSITY rc h cen(t)-h cen(t-bt)
I
6t
SPECIFY CENTRAL PRESSURE VALUE
J
PRESSURE
ITERATE FOR CENTRAL PRESSURE
I
=
when x = f b
0
Solutions have been obtained for the cyclic variation of film thickness and pressure under conditions representative of the ankle joint in level walking. A single-sided roughness model of the form depicted in Figure 2 has been adopted and results are presented for an undeformed sinusoidal roughness profile of amplitude 1 and wavelength 1 mm. Detailed consideration of the influence of wave amplitude and roughness upon film thickness under quasi-static conditions has been reported elsewhere by the authors (8). The geometrical and property values shown in Table 1 have been adopted unless alternative values are quoted.
= 2.4 =
1
ITERATE FOR LOAD
_______Lr__j TIME INCREMENT
I
ITERATE UNTIL CYCLIC STEADY STATE OCCURS
Figure 5
Flow Chart for Micro-Elastohydrodynamic Lubricatton Analysis of Synovial Joints.
The iterative procedure outlined above does not readily converge when the entraining velocities are small. Medley et a1 (6) reported similar problems in their analysis of smooth
mm
0.35 m
E
=
16 MPa
v
= 0.4
17 = 0.01 Pas
X=lmm TABLE 1
'CHANGE CAVITATION BOUNDARY
(17
5 RESULTS
d
SOLVE REYNOLD'SEQUATIO FOR PRESSURE
I
p
The overall computational time was therefore reduced in the present study by switching from the full (entraining + squeezefilm) solution of equation (15) to a consideration of squeeze-film action alone whenever the entraining velocity was very small. The errors in film thickness prediction resulting from this more rapid prediction of cyclic behaviour were checked and found to be less than 1 percent for smooth surfaced cartilage layers.
SPECIFY CENTRAL FILM THICKNESS AT TIME
I I
surface layers of cartilage in the ankle joint In pure squeeze-film motion the boundary conditions (16) and (17) are no longer appropriate, since symmetry applies and the requirements are simply,
GEOMETRICAL AND PROPERTY VALUES
A representative set of solutions for the pressure profiles, film shapes and deformed surface roughness profiles for times of Os, 0.109s, 0.391s, 0.766s and 0.797s in the one second walking cycle is presented in Figures 6-10 respectively. The reasons for selecting these time intervals become clear when the loading and entraining velocity cycles shown in Figure 4 are consulted. The first set of solutions (t=o) shown in Figure 6 depict conditions at a high entraining velocity and relatively low load, whereas those shown in Figure 7 (t = 0.109s) exhibit the characteristics of an elasto-hydrodynamic solution for a lower entraining velocity and higher load. After (0.391s) the entraining velocity is almost zero whilst the load is close to its maximum value and the set of results shown in Figure 8 thus reflects a pure squeeze-film situation. Conditions return t o
381
Amp l l t u d e
(pm)
Wavelength
(mm)
= 1. 00 = 1.00
(pm) =
Wavelength
(mm)
1.00
= 1.00
Pressure d l s t r l b u t l o n
Pressure d l s t r l b u t i o n
:I
Amplitude
-
3
(D
a a
x (mm)
F I Lm p r o f I l e
- , . - " 4 l 4
-10
.a
-5
4
. I
a
0
*
8
.
5
*
;o
11
I,
x (mm) Deformed waves
w
(N/m) = 0. 15077E 05 Time
(9)
( Sliding
Figure 6
u (m/s) =
0.424E-01
= 0.0000
8 Squeeze 1
P r e s s u r e D i s t r i b u t i o n , F i l m Shape and Deformed S u r f a c e Roughness P r o f i l e (t = 0 s )
combined e n t r a i n i n g and s q u e e z e - f i l m a c t i o n a f t e r ( 0 . 7 6 6 s ) as shown i n F i g u r e 9 , s i n c e t h e entraining velocity retains a s i g n i f i c a n t value w h i l e t h e l o a d i s s t i l l small. F i g u r e 10 shows t h a t squeeze-film a c t i o n e s s e n t i a l l y dominates t h e f i f t e e n p e r c e n t of t h e w a l k i n g c y c l e e n c o u n t e r e d w i t h low e n t r a i n i n g v e l o c i t i e s a n d r e l a t i v e l y low l o a d s j u s t h e f o r e (0.8s).
w
(N/m) = 0.32764E 05 u (m/s) = 0.287E-01 T i m e (9) = 0. 1094 ( S l l d l n g 8 Squeeze 1
Figure 7
P r e s s u r e D i s t r i b u t i o n , F i l m Shape and Deformed S u r f a c e Roughness Profile. ( t = 0.1094s)
The c y c l i c v a r i a t i o n of p r e d i c t e d c e n t r a l and minimum f i l m t h i c k n e s s e s a r e shown i n F i g u r e 11 and 1 2 . t h e c o r r e s p o n d i n g q u a s i - s t a t i c s o l u t i o n s , which n e g l e c t s q u e e z e - f i l m a c t i o n , d e r i v e d by t h e p r o c e d u r e o u t l i n e d by Dowson and J i n ( 8 ) a r e superimposed upon t h e s e F i g u r e s t o i l l u s t r a t e t h e r e m a r k a b l e
382
Amplitude (pm) = 1.00 Uavelength (mm) = 1.00 Pressure d l s t r l b u t l o n b-
.*.
4
.I..
-
-
a n
a n
cL
n I
.
L . : . *
.
*
I
4
.
;
-10
:
4 !
4
4
i,
-5
;
.
'
a
*
.
:
:
Yo
5
:
\
O
U
.
.C
z
z
3
a*.
.*- 2 LC-
:'
x (mm)
x (mm)
FI l m p r o f I le
FI l m prof1 le
I
:
:
:
. u - m . n 4
-10
;
.
:
4
-5
: A
! a
:
i,
'
:
8
:
:
I
:
5
:
(
Ib
m
9 . . u .
u
; m .
; M
: 4
-10
: 4
-5
4
! ': I:
: 1
i,
.:
U: *
5
x (mm)
x (mm) Deformed. wave8
Deformed waves
tl
I
-.+ -2
: D :U :
10
't
jl:
I
*--3 ~~
w
(N/m) = 0.85341E 05
Figure 8
u ( m / d = 0. 164E-02 Time (a) = 0.3906 ( Pure squeeze )
Pressure Distribution, Film Shape and Deformed Surface Roughness Profile. (t = 0 . 3 9 0 6 s )
u (N/m) = 0.81 729E 04
u ( m / d = 0.208E-01 Ttme (a) = 0.7656 ( S l l d t n g 8 Squeeze 1
Figure 9
Pressure Distrihution, Film Shape and Deformed Surface Roughness Profile. (t = 0 . 7 6 5 6 s )
extent to which normal motion of the cartilage surface preserves an essentially steady film thickness similar to that generated at times of substantial entraining action. Comparisons between the predicted cyclic variations of central and minimum film thicknesses for smooth (curve 1 ) and initially
rough (curves 2-1 and 2-2) cartilage surfaces are shown in Figures 13 and 14. The sliding velocity in the ankle varies from about 2 x 10-3 m/s to 5 x m/s throughout the walking cycle and since the predicted film thicknesses are remarkably constant at ahout the synovial fluid will be subjected to shear rates
383
Pm) = 1.00 Amplttude Wave length (mm) = 1.00 Pressure dtstrtbut ton
I
bS
4
3
2
I , U
. .';
.
*
~
4
4
;
:
4
4
-5
-10
Ft l m
I : ; : : : Y
U
.
-10
1
4
4
-5
!
;
:
*
1
8
:': *
f
*
5
0
.
: .
f
U
Figure 11
Cyclic Variation of Predicted Central Film Thickness (hcen). (O quasi-static analysis) ( a o = 1 p m X = I mm)
Figure 12
Cyclic Variation of Predicted Minimum Film Thickness (h i n > (0 quasi-static analysis'? (ao = 1 l.rm h = 1 mm)
10 x (mm)
proft l e
: : ! : : : : : : : 4
4
6
#
*
a
5
' A
m
u
x (mm)
Deformed waves
I3
*
2
were unable to locate any evidence of the viscosity of synovial fluid at even higher shear rates, but s nce it cannot he less than that of water ( l o d 4 Pas), it was decfded to repeat all the previous film thickness -2 calculations for a viscosity of 0.5 x 10 Pas. The results are represented by traces 2-2 in Figures 13 and 1 4 . w
(N/m) = 0.94094E 04 u ( m / d = 0.102E-01 Ttme (a) = 0.7969 ( Pure squeeze 1
Figure 10
Pressure Distribution, Film Shape and Deformed Surface Roughness Profile. (t
=
0.7969s)
4 in the range 2 x lo3 to 5 x 10 reciprocal seconds. Cooke et a1 (13) have shown that the viscosity of synovial fluid is strongly affec ed by shear rate, but that It falls t abou lo-' Pas at shear rates in the range 10' - 10' reciprocal seconds. It was for this reason that this value was adopted in the present study. We
It is interesting to note that although the present micro-elasto-hydrodynamic theory of synovial joint 1.ubrication yields film thickness predictions which accord with the experimental indications of fluid-film lubrication, the corresponding coefficients of friction are much smaller than the experimental values. A possible explanation is that the analysis takes no account of the apparent increase in surface viscosity experienced by very thin films. In their initial analysis of 'boosted lubrication' Dowson et a1 ( 1 4 ) assumed that the effective viscosity of the lubricant would increase as the film thickness hecame smaller, according to the relationship,
384
The film thickness predictions have been recalculated according to this boosted lubrication analysis, for a value of (F) of 187.5 pm adopted earlier by Dowson et a1 (14). The results are recorded as curves 2-3 in Figures 13 and 14. The corresponding maximum coefficient of friction was 0.004, a value which is just within, but at the lower limit of, the range recorded by Swanson (15).
I
1.51
0
+ 0.2
04
0.6
0.8
1.0
0
Ttme (el Figure 15
Comparisons of Central Film Thickness for 'Pure Squeese - - -' and 'Squeeze + Entraining -' Motion during one Cycle.
-
6
0.5..
2-2'
Figure 13
Central Film Thickness Predictions for the Walking Cycle. 1 - Smooth cartilage 2 - Rough cartilage (a = 1 um h = 1 mm) 2-9 = 0.01 Pas 2-2 Q = 0.005 Pas 2-3 q boosted lubrication.
DISCUSSION
The pressure curves presented in Figures 6-10 confirm the predictions reported earlier by the authors (8) for quasi-static conditions, that very small pressure perturbations on the smooth cartilage pressure profiles are sufficient to produce a major flattening of the initially rough (wavy) cartilage surface in the loaded conjunction. This means that the predicted film thicknesses of about 1 urn are no longer small compared with the effective composite surface roughness and that fluid-film lubrication can confidently he anticipated in the model of a synovial joint considered in the present analysis. The dominant rdle of microelasto-hydrodynamic lubrication in achieving this situation is evident in the representations of the deformed waves shown in Figures 6-10. A major flattening of the elastic surface waves occurs as they ripple through the loaded conjunction and then spring back to their initial size.
3 0 1 25
-m
5
2-3
It is also evident that pressure profiles preserve a form consistent with the dry predictions of normal stress cycle.
Figure 14
Minimum Film Thickness Predictions for the Walking Cycle. 1 - Smooth cartilage. 2 - Rough cartilage (a = 1 pm h = 1 mm) 2-P Q = 0.01 Pas. 2-2 rl = 0.005 Pas. 2-3 boosted lubrication
It is interesting to compare the predictions of film thickness throughout the cycle based upon combined entraining and squeeze-film action with those obtained if squeeze-film motion alone i s considered. This is shown in Figure 15 for central film thickness for the same initial film thicknesses at (t=o).
the hydrodynamic near parabolic contact throughout the
When entraining motion plays a significant role in the fluid-film forming process the characteristic wedge-shaped film profile noted by Medley et a1 (6) and the authors (7,R) emerges as shown in Figures 6 , 7 and 9 . However, when the entraining action is small and squeeze-film motion dominates the film behaviour an interesting and previously unknown characteristic of the film profile emerges. It can be seen from Figures 8 and 10 that relatively small pressure perturbations are generated by micro-elasto-hydrodynamic action in the middle of the symmetrical film profile and that this allows the asperities to recover a substantial proportion of their initial size. This is particularly evident in the troughs of the waves, rather than on the peaks and this gives rise to a central dimple with a decaying roughness profile superimposed upon the cartilage on either side of the central region.
385 The emergence of this central dimple and the partially relaxed surface waves is the main reason for the use of equation ( 1 3 ) as the effective uniform squeeze-film velocity in the conjunction If he squeeze film velocity is equated to I!$cen] alone, the neglect of local squeeze fil cti n on either side of the central dimple cannot be justified. However, by assuming an average conjunction squeeze-film velocity, some error can be anticipated in the predictions of film shape in the central (dimple) region. This question is currently receiving further consideration. The powerful role of squeeze-film action in maintaining a fairly constant film thickness throughout the dynamic loading and entraining velocity cycles is evident, particularly in relation to the central film thickness, from the comparisons between the full and quasi-static predictions in Figures 11 and 12. The mean values of both central and minimum film thickness are well predicted by the quasi-static solutions, but the cyclic fluctuations are greatly reduced when account is taken of the normal motion between the cartilage surfaces. The more erratic nature of the minimum film thickness trace shown in Figure 12, compared with that for central film thickness shown in Figure 11, is attributable mainly to the changes in film shape, relative to the centre of the conjunction, under dynamic conditions, but partly to the use of squeeze-film solutions alone whenever the entraining velocities are small. The minimum film thickness is seen to range from about 0.6 um to 1.1 Urn and the central film thickness from about 1.0 pm to 1.2 um throughout the cycle. The introduction of surface roughness into the analysis has but a minor effect upon the prediction of central film thickness throughout the cycle, as shown in Figure 13, but microelasto-hydrodynamic action largely flattens the surface waves. A somewhat larger effect is evident when minimum film thickness is considered (Figure 14) and this is attributable to the partial recovery of the surface waves in the centre of the conjunction when entraining action is neglected and pure squeeze-film action is assumed to govern the film behaviour. The reduction in predicted film thickness evident in Figures 13 and 14 for the synovial fluid with a viscosity of only 50 percent of that considered throughout most of the analysis is neither dramatic nor unexpected. Even with this very small viscosity of 0.005 Pas microelasto-hydrodynamic action would be expected to ensure fluid-film lubrication in the present model. The coefficients of friction for these micro-elasto-hydrodynamic models are still small compared with experimental values. If allowance is made for an increase in the effective viscosity as the films become small, according t o the methods adopted earlier in boosted lubrication analysis, the maximum coefficient of friction rises to about 0.004. It appears that a more complete analysis of friction in synovial joints must await further clarification of the rheological hehaviour of synovial fluid in thin films at very high shear rates.
If squeeze-film action alone is considered during the cycle, the film thickness will decrease throughout the period considered, as shown by the broken line in Figure 15. Clearly, repeated cycles of loading would lead to a gradual reduction in film thickness with time, unless it be assumed that the surfaces are separated instantaneously by exactly the appropriate distance to ensure a renewed start at the inttial value of film thickness at the end of each period. The comparisons between combined (entraining and squeeze-film) and pure squeezefilm predictions shown in Figure 15 are based upon the assumptions of equal initial film thicknesses at the start of each cycle. The agreement is good throughout much of the stance phase, but the beneficial action of entraining motion, particularly in the swing phase, is clearly evident. Clearly, combined entraining and squeeze-film actions need to be considered in a satisfactory, repeating, cyclic prediction of film thickness. Furthermore, a difficulty arises in pure squeeze-film calculations in specifying the initial film thickness as noted by Higginson (3) and Dowson (2). Throughout this analysis it has been assumed that the two layers of articular cartilage which form a synovial joint can be represented with sufficient accuracy by a single layer of cartilage of equivalent thickness on the upper, curved component in the model shown in Figure 2. Furthermore, a single-sided roughness model has been adopted and it is well known from analyses of the lubrication of rough surfaces that the benefits of micro-elasto-hydrodynamic action are most evident in such representations. The current predictions are therefore presented as the most beneficial indications of micro-elasto-hydrodynamic lubrication in synovial joints, whilst more complete representations of the bearing are contemplated.
7 CONCLUSIONS In the present study we have extended our previous (Dowson and Jin, 7,U) quasi-static analysis of micro-elasto-hydrodynamic lubrication action in synovial joints to take account of combined entraining and squeeze-film motion in a dynamically loaded cycle representative of conditions encountered in the human ankle joint. The following conclusions have been reached. (i)
Micro-elasto-hydrodynamic lubrication action effectively smooths out the initial wavy surface roughnesses, in the single sided transverse roughness model considered, throughout the dynamic loading and entraining velocity cycles encountered in steady walking. This further confirms the major contribution to synovial joint lubrication of microelasto-hydrodynamic action.
(ii)
Squeeze-film action greatly restrains the film thickness excursions encountered in the walking cycle, compared with those predicted for entraining action (quasi-static) alone.
The film pressures are very similar to the parabolic profiles of normal stress distribution for an elastic layer on a rigid backing throughout the walking cycle. Very small pre sure perturbatims (typically lo-' times the 'smooth' surface pressures) are sufficient to effect a major flattening of the initial wavy surface throughout the conjunction for the conditions considered.
(8)
DOWSON, D. and JIN, Z.M. (19871, 'MicroElastohydrodynamic Action in the Lubricatiofi of Synovial Joints', to be puhlished.
(9)
MURRAY, M.P., DROUGHT, A.R. and KORY, R.C. (19641, 'Walking Patterns of Normal Men', Journal of Bone and Joint Surgery, 46A, pp 335-360. (1985), 'Contact Mechanics', Cambridge University Press.
( 1 0 ) JOHNSON, K.L.
In pure squeeze-film motion a central dimple emerges in the soft layer, with a pronounced hut partial recovery of the initial wavy surface profile.
(11) DOWSON, D.,'RUDDY, B.L. and ECONOMOIJ, P.N. (1983), 'The Elastohydrodynamic
The introduction of surface roughness into the bearing model has but a small effect upon film thickness prediction under cyclic conditions, but microelasto-hydrodynamic action effectively smooths out the surface ripples and ensures a fluid-film lubrication mechanism.
(12) GILL, P.E.
The analysis has shown that it is essential to consider both entraining and squeeze-film motion in a full analysis of synovial joint lubrication under physiological conditions. References DOWSON, D. (1981), 'Lubrication of Joints (A) Natural Joints', Chapter 13 in 'An Introduction to the Biomechanics of Joints and Joint Replacements', Edited by Dowson, D. and Wright, V., Mechanical Engineering Publications, pp 120-123. DOWSON, D. (1983), 'The Lubrication of Synovial Joints', Proceedings of the Ninth Canadian Congress of Applied Mechanics, Saskatoon, pp 15-31. HIGGINSON, G.R. (1978), 'Elastohydrodynamic Lubrication in Human Joints', Engineering in Medicine, 7, pp 35-41. GARDNER, D.L., O'CONNOR, P. and OATES, K., (1981), 'Low Temperature Scanning Electron Microscopy of Dog and Guinea-pig Hyaline Articular Cartilage', J. Anat., 132, No. 2, pp 267-282. SAYLES, R.S., THOMAS, T.R., SANDERSON, J., HASLOCK, I. and UNSWORTH, A. (19791, 'Measurement of the Surface Micro-Geometry of Articular Cartilage', J. Biomechs., 12, pp 257-267. MEDLEY, J.B., DOWSON, D. and WRIGHT, V. (1984), 'Transient Elastohydrodynamic Lubrication Models for the Human Ankle Joint', Engineering in Medicine, 13, pp 137-151.
DOWSON, D., and JIN, Z.M. (1986), 'MicroElastohydrodynamic Lubrication of Synovial Joints', Engineering in Medicine, 15, pp 63-65.
Lubrication of Piston Rings', Proc. R. Soc. A., 386, pp 409-430. and MILLER, G.F. (19721, 'An Algorithm for the Integration of Unequally Spaced Data', Computer Journal, 15, pp 80-R 3. ( 1 3 ) COOKE, A.F., DOWSON, D. and WRIGHT, V. (1978), 'The Rheology of Synovial Fluid
and Some Potential Synthetic Lubricants for Degenerate Synovial Joints', Engineering in Medicine, 7, pp 66-72. (14) DOWSON, D., UNSWORTH, A. and WRIGHT, V. (1970), 'Analysis of 'Boosted Lubrication'
in Human Joints', Journal of Mechanical Engineering Sciences, 12, No. 5, pp 364-369. (1979), 'Friction, Wear and Lubrication' in 'Adult Articular Cartilage', Ed. by Freeman, M.A.R., Pitman Medical, London, pp 415-460.
(15) SWANSON, S.A.V.,
387
Paper Xll(iii)
Lubricating film formation in knee prostheses under walking conditions T. Murakami and N. Ohtsuki
To investigate the lubricating film formation in knee prostheses under walking condition, a knee joint siniulator is used to simulate the tibial axis load and flexion-extension motion. The effect of swinging motion and load variation on the fluid film formation has been examined by measuring the contact electric resistance between conductive femoral and tibial components. It is shown that the elastohydrodynamic film formation in a knee prosthesis with an elastomeric tibial component is considerably excellent compared with the prosthesis with a polyethylene tibial component. It is also noted in the former that the film thickness changes depending on the walking phase, and becomes minimum at the peak load just prior to toe off during walking motion.
1 INTRODUCTION
2 EXPERIMENTAL
Natural synovial joint is an ideal joint or bearing which has low friction, high mobility accompanied with stability, high load-carrying capacity and long life. Therefore, the lubrication mechanism in natural joints has attracted the interest of many researchers. Reynolds (1) suggested that the fluid film formation in natural joints may be mainly due to the squeeze action. It is known from recent investigations (2, 3 , 4 ) that the transient elastohydrodynamic film formation and supplemental lubrication mechanism such as micro-elastohydrodynamic lubrication, weeping, biphasic, boosted or boundary lubrication are capable of providing sufficient protection to the articular cartilages in natural joints. On the contrary, there is just a little information on the actual lubricating film formation in total replacement joints. For artificial hip joint as spherical bearing composed of high density o r ultrahigh molecular weight polyethylene and anticorrosive metal or bioceramics, considerable fluid film is elastohydrodynamic likely to be formed by squeeze action under the normal walking condition (5). Sasada et al. (6) pointed out that a total hip prosthesis containing polyvinyl alcohol gel as an artificial articular cartilage exhibits quite similar frictional behaviour to natural joint. However, in many kinds of knee prostheses with low geometrical congruity, the fluid film formation seems to be insufficient to prevent the direct contact between femoral and tibial components. In the present study, a knee joint simulator is used to simulate the tibial axis load and flexion-extension motion under walking condition. The lubrication condition o r fluid film formation in knee prosthesis models is evaluated by measuring the electric contact resistance between the femoral metallic component and tibial conductive polymeric o r elastomeric component under the constant applied voltage in a knee joint simulator.
2.1 Knee joint simulator The mechanical simulation of movement and load conditions of the knee joint is attained by the knee joint simulator shown in Fig.1. The general view of the simulator is shown in Fig. 2. The tibial axis load[7 in Fig. 13 and flexion-extension motion [ 4 ] for steady walking were fed to this simulator. Their model patterns ( 6 , 7) are In the present test, the depicted in Fig. 3 .
AnteriorPosterior axis
MedialLateral ax i
7
(a)
I
Tibia1 axis
Fig. 1 Mechanical simulation of walking condition for knee joint by knee joint simulator
Fig. 2 General view of knee joint simulator
388
-
Heel strike
3
Seven kinds of silicone oils (dimethylpolysiloxan) different in viscosity are used ( kinematic viscosity:l.O, 2 . 0 , 10, 30, 100, 1000 and 10000 mm2/s at 25OC). Oil names are designated as S-1, S-2, S-10, S-30, S-100, S-1000 and S-10000 respectively.
Toe off
I
2.0
W
m
0 d
m ;'
2.3 Experimental procedure
1.0
m
The metallic upper femoral specimen cemented to an epoxy resin bone model was fastened to the femoral driving frame of the simulator, while the lower tibial one was fixed to the lower specimen holder with an oil bath as exhibitted in Fig. 5.
d
m n .rl
..+
-
0
-
Percent o f c y c l e c (Swing p h a s e )
(Stance phase)
Fig. 3 Model patterns for tibial axis load and flexion-extension motion during walking tibial axis load was slightly simplified as shown in Figs. 7 and 8. This time-dependent load was applied by the personal computor controlled hydraulic system. The flexion-extension motion was achieved by means of a cam driven rack-and-pinion system. The periodic time of the walking cycle is 2.0 s , which is a little long compared with usual walking. Maxima of sliding speed during the stance and swing phase are about 0.06 and 0.18 m/s respectively. The other free linear axial motions [3, 51 and free rotations [ 2 , 61 about the tibial and anteriorposterior axes are accepted by using low friction dry bearings and linear ball bearings. In this study, the linear movements [ 3 , 51 were restricted within narrow limits. The detailed mechanism has been described elsewhere (9). 2.2 Materials and lubricants
Two kinds of knee prosthesis models were tested. One type is the Kodama-Yamamoto Mark-II Knee Prosthesis (10) consisted of a femoral COP steel component and an ultrahigh molecular weight polyethylene (UHMWPE) tibial component as shown in Fig. 4 , the latter of which was coated with platinum by the sputtering method. Surface roughness of metallic femoral component is 0.07 pm (rms) The molecular weight M and elastic modulus E of UHMWiE are as follows: M > .2 x 10 , E = 550 MPa.
Fig. 5 Test section of knee joint simulator In order to examine the fluid film formation between femoral and tibial specimens, the separation voltage is measured under the applied voltage of 100 mV. Electric circuit and the relationship between the degree of separation and electric contact resistance are shown in Fig. 6. The degree of separation is defined as the ratio of measured voltage to the applied voltage(100 mV). That is, Degree of separation = 1 : Perfect separation (full fluid film formation). Degree of separation = 0 : Direct intimate contact. In the tests using the conductive silicone rubber, which has electric resistance of about 50 R , it is corrected so that the voltage derived from the bulk resistance of non-swollen conductive rubber corresponds to the degree of separation of 0. The effect of swelling phenomena on the measured voltage is described later.
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2 Prosthesis
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Fig. 6 Electric circuit for measuring electric contact resistance and relationship between degree of separation and electric resistance
Kodama-Yamamoto Mark-II Knee Prosthesis
Another is the combination of the same femoral component and conductive silicone rubber tibial flat specimen( E = 9.1 MF'a, 3 nun in thickness) supported by acrylic flat plate. The latter model corresponds to the knee prosthesis with compliant artificial cartilage.
The simulator tests were carried out under the following two conditions at room temperature. (1) 'NO SWING' condition : Normal approach and separation without swinging motion under the walking load condition. ( 2 ) 'SWING' condition : Walking condition with swinging motion.
389
For each oil, the 'NO SWING' test was followed by the 'SWING' test. In the tests for the conductive rubber, the elastomeric specimens were exchanged for different oils to avoid the influence of the swelling by different oil on fluid film formation.
3 RESULTS 3.1 Knee prosthesis with a polyethylene tibial comDonent
In the 'NO SWING' test of the knee prosthesis with a polyethylene tibial component, i.e., Kodama-Yamamoto Mark-II Knee Prosthesis, the degree of separation is zero even for high viscous oils during all walking cycle. This means fluid film has been practically squeezed out. In the 'SWING' test, slight fluid film formation is observed immediately after the heel strike for high viscosity oil in Fig. 7. During the swing phase, fluid film recovers to perfect separation only for no tibial loading period. P
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Percent of cycle Fig. 7 Tibia1 axis load and degree of separation for knee prosthesis with a polyethylene tibial component during the fifth walking cycle.
3.2 Knee prosthesis with an elastomeric tibial component Typical results of the loading pattern and degree of separation during the fifth walking cycle are shown in Fig. 8 with oil S-100. The degree of separation for this oil is considered to correspond to the fluid film formation, because the change in electric resistance of rubber due to immersion in this lubricant is inconsiderable as mentioned later. The slight difference between the loading patterns is attributable to the phase lag in hydraulic control system. In the 'NO SWING' test, the degree of separation changes, depending on the load variation. I n the load increasing process, such as ab, cd or ef, it increases until the maximum value at i, j or k, and then decreases to the minimum value at b', d' o r f'. In the following load decreasing stage such as bc, de or fa, it first recovers to higher level by unloading, and thereafter decreases almost exponentially by squeeze out action. In general, at the peak load just before toe off, oil film becomes minimum during one cycle. In the 'SWING' test, i.e., under usual walking condition, fluid film formation is enhanced by the swinging motion as shown in Fig. 8. During the stance phase, the degree of separation decreases slightly and rapidly drops to the minimum value at the peak load before toe off, and then recovers to the perfect separation level during the swing phase. In both tests, the minima and maxima during one cycle are observed at A and B near toe off as indicated in Fig. 8 . The variation of these values with the repetition of walking cycle in 35 cycles is shown in Fig. 9. The swinging motion is effective for fluid film formation irrespective of oil viscosity, In the 'NO SWING' test, the fluid film is squeezed out by the repetition of loading cycle, and the decreasing rate depends on the lubricant viscosity. On the other hand, in the 'SWING' test, the fluid film formation except S-10 is enhanced by the repetition of walking cycle, which is caused by the improvement of the geometrical conformity due to the creep deformation of the elastomer.
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Fig. 9 The variation of minima and maxima of degree of separation at A and B with repetition of walking cycle
4 DISCUSSION From the experimental results, it was found that the elastohydrodynamic film formation during the stance phase is little expected except immediately after heel strike for the existing total knee replacement with a polyethylene tibial component which has low geometrical congruity. For improvement of fluid film formation, it is necessary to design the knee prosthesis to have higher geometrical congruity or to be possessed of compliant artificial cartilage and adequate lubricant with high lubricating performance. In the following, the effect of load variation, swinging motion and viscosity of lubricant on the fluid film formation for the knee prosthesis with an elastomeric tibial component is discussed.
4.1 Effect of load variation and swinging motion on fluid film formation For the lubrication condition in joints during walking motion, the effect of load variation and swinging motion on fluid film formation should be taken into consideration. Even in the 'NO-SWING' test without the swinging motion, fluid film formation considerably varies depending on the load fluctuation as shown in Fig. 8 . The change in fluid film thickness from the increasing tendency to decreasing one with an increase in load is attributable to the temporary thickening of the minimum film thickness due to profile change accompanied with enlargement of contact area, followed by the thinning of fluid film caused by the squeeze out action, as pointed out for step load condition by Ikeuchi et a1 (11). During one cycle, the film thickness attains the minimum value generally at peak load just before toe off. In swinging or reciprocating motion, the mutual relationship between wedge and squeeze film actions becomes important for fluid film formation. In the former report on the elastohydrodynamic film formation in reciprocating motion
under constant load (12), one of the authors pointed out that for the shorter strokes than the critical length, the hydrodynamic film is no more maintained essentially in a stable state, but finally results in breakdown as indicated by the inverse theory (13). For shorter stroke condition under constant load, where the recovery of fluid film thickness by wedge action is too little to avoid the asperity contact, there is the transition from the initial temporary elastohydrodynamic condition to the non-hydrodynamic one caused by hydrodynamic instability. The critical stroke length for this simulator test is not decided at the present stage, because there is no numerical solution available for the swinging motion of this knee prosthesis model under variational load condition. However, it is considered that the stroke should be longer than the contact width of tibial component in the moving direction; i.e., the length of total contact area of femoral component during one cycle should be at least two times of tibial one to ensure the entrainment of sufficient lubricant. In the present experimental results, the swinging motion is very effective for fluid film formation. In this test under almost pure sliding contact condition, the sliding stroke for the swing phase is longer than contact width (about 20 mm) in the sliding direction at the highest load, and the load during the swing phase is light, which enhance fluid film formation. In consequence, sufficient elastohydrodynamic film is formed with the aid of squeeze film action during the stance phase with lubricants of the appropriate viscosity. It is necessary that time-dependent variation of fluid film formation and frictional behaviour in the present knee prosthesis model should be estimated by the analytical study on transient elastohydrodynamic lubrication (3,14). The effect of rolling motion on fluid film formation during walking motion should be also examined. On the basis of the experimental data on frictional characteristics of natural hip joints, Unsworth et al. (15) proposed the model
391
Film profile
Toe off
Heel
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Fig. 10 Lubrication modes in natural synovial joint under walking condition (modified from Ref. 15) of lubrication modes in natural synovial joints under steady walking condition as depicted in Fig. 10. It is ascertained from the present results in Fig. 8 that the fluid film thickness slightly decreases during the stance phase and is reduced to a minimum thickness at the peak load just before toe off, and then recovers to the previous thickness during the swing phase. 4.2
Effect of lubricant viscosity and behaviour on fluid film formation
swelling
Typical examples of the degree of separation in the fifth cycle of the 'SWING' tests for different oils S-1, S-10 and s-100 are shown in Fig. 11. It is noted that the minimum values of the degree of separation during one cycle are
observed at the peak load just prior to toe off for all oils. It is further noticed that S-1 of the lowest viscosity shows higher separation than S-10, although fluid film formation with S100 is superior to other oils. Minimum and maximum values of degree of separation at A and B in the vicinity of toe off in the fifth cycle of the 'NO SWING' and 'SWING' tests are plotted against lubricant viscosity in Fig.12. In both tests, the degree of separation increases with an increase of viscosity for higher viscosity oils, but it increases with a decrease of viscosity in the lower viscosity range. Therefore, the minimum separation is observed for oil viscosity of about 10 mm2/s. For higher viscosity range, an increase of viscosity enhances the fluid film formation by the viscous effect in the elastohydrodynamic lubrication. In contrast, it is considered for low viscosity oils that the penetrative behaviour of lubricants into conductive rubber have a significant effect on the fluid film formation or the change of bulk electric resistance of rubber. Experimental results on changes with time in the electric resistance and volume of elastomeric sDecimen in the immersion swelling test at 21 O k are shown in Fig. 13, where -volume change ratio is defined as Vs/Vi (Vs:volume after swellinn, Vi:initial volume;30 mm x 10 mm -. x 3.0 mm). It is confirmed the swelling behaviour is suppressed with an increase of viscosity. Accordingly, the degree of se aration for oil S-100(kinematic viscosity:100 mm / s at 25OC) corresponds to the fluid film formation in the
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Effect of lubricant viscosity on degree of separation in the fifth cycle of 'SWING' tests
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5 CONCLUSIONS The lubricating film formation in knee prostheses under walking condition has been investigated by measuring the contact electric resistance between femoral and tibial components in a knee joint simulator. In a knee prosthesis with an elastomeric tibial component, elastohydrodynamic film is considerably formed even during the loading stance phase, whereas in a prosthesis with a polyethylene tibial component, slight fluid film formation is observed only immediately after heel strike except during the swing phase. It is also noticed in the former that the film thickness decreases during the stance phase and generally becomes minimum at the peak load just before toe off, and recovers to the previous level during the swing.phase. 6
ACKNOWLEDGEMENT
The authors wish to thank Prof. Y. Yamamoto of Department of Mechanical Engineering at Kyushu University for valuable discussion and Messrs. J. Inasaka, H. Matsui and S. Kimoto for their help with the experiments. The Kodama-Yamamoto Mark-IS Knee Prosthesis was supplied by Mizuho Ika Co. Ltd., Japan. References (1)
REYNOLDS, 0. 'On the theory of lubrication and its application to Mr.Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil', Phil, Trans. Roy. SOC. London, 1886,
177, 157-234. DOWSON, D. 'Modes of lubrication in human joints', Proc'. Instn. Mech. Engrs. 1966-67, 181, Pt3J, 45-54 (3) MEDLEY, J.B., WWSON, D. and WRIGHT, V. 'Transient elastohydrodynamic lubrication models for the human ankle joint', Engng. Med. 1984, 13, No.3, 137-151. (4) GQWSON, D. and JIN, ZHONG-MIN. 'Microelastohydrodynamic lubrication of synovial joints', Engng. Med. 1986, 15, No.2, 63-65. (5) SASADA, T. and MABUCHI, K. 'Elastohydrodynamic lubripation of total hip prostheses', Proc. JSLE Int. Tribology Conf. 1985, (2)
949-954. (6) SASADA, T.,. TAKAHASHI, M., WATAKABE, M., MABUCHI, K., TSUKAMOTO, Y. and NANBU, M. 'Frictional behavior of a total hip prosthesis containing artificial articular cartilage', J. Japan. SOC. Biomaterials (in Japanese), 1985, 3, No.3, 151-157. (7) MORRISON, J.B. 'Bio-engineering analysis of force actions transmitted by the knee joint', Bio-Med. Eng., 1968, 3, No.4, 164170. (8) DOWSON, D., JOBBINS, B., O'KELLY, J. and WRIGHT, V., 'A knee joint simulator', Ch.7, 'Evaluation of artificial hip joints', 1977 (Biological Engineering Society, U.K.), 7990. (9) MURAKAMI, T., OHTSUKI, N. and MATSUI, H. 'The evaluation of lubricating film formation in knee prostheses in a knee joint simulator', Trans. JSME, (to be published in Japanese). (10) YAMAMOTO, S. 'Total knee replacement with the Kodama-Yamamoto knee prosthesis', Clin. Orthp., 1979, 145, 60-67. (11) IKEUCHI, K. and MORI, H. 'Squeeze film on compliant surface under step load (2nd Report, Spherical thruster)., Bulletin of JSME, 1984, 27, N0.231, 2030-2035. (12) HIRANO, F. and MURAKAMI, T. 'Photoelastic study of elastohydrodynamic contact condition in reciprocating motion', Proc. 7th Int. Conf. on Fluid Sealing (BHRA), 1975, 51-70. (13) HIRANO, F. and KANETA, M. 'Theoretical investigation of friction and sealing characteristics of flexible seals for reciprocating motion', Proc. 5th Int. Conf. on Fluid Sealing (BHRA), 1971, 17-32. (14) MEDLEY, J.B. and DOWSON, D. 'Lubrication of elastic-isoviscous line contacts subject to cyclic time-varying loads and entrainment velocities', ASLE Trans., 1984, 27, No.3, 243-251 (15) UNSWORTH, A,, DOWSON, D. and WRIGHT, V. 'Some new evidence on human joint lubrication', Ann.rheum. Dis., 1975, 34, 277-285. (16) MCCUTCHEN, C.W. 'Physiological lubrication', Proc. Instn. Mech. Engrs., 1966-67, 181, Pt3J, 55-62. (17) MANSOUR, J.M. and MOW, M.C. 'On the natural lubrication of synovial joints:normal and degenerate', Trans. Am. SOC. Mech. Engrs. J. Lub. Tech. 1977, 99, 163-173. (18) MOORE, D.F. Principles and Applications of Tribology, 1975, 147, Pergamon Press.
SESSION Xlll SUPERLAMINAR FLOW IN BEARINGS Chairman: Professor
H.Blok
PAPER Xlll(i)
A review of superlaminar flow in journal bearings
PAPER Xlll(ii)
Frictional losses in turbulent flow between rotating concentric cylinders
PAPER Xlll(iii) Turbulence and inertia effects in finite width stepped thrust bearings PAPER Xlll(iv) A theory of non-Newtonian turbulent fluid films and its application to bearings
This Page Intentionally Left Blank
Paper Xlll(i)
A review of superlaminar flow in journal bearings F.R. Mobbs
T h e r e h a v e r e c e n t l y been r a p i d advances i n o u r understanding of Taylor-Couette flow between a r o t a t i n g i n n e r c y l i n d e r and a c o n c e n t r i c s t a t i o n a r y o u t e r c y l i n d e r . %is p r o g r e s s is reviewed and its i m p l i c a t i o n s i n the case o f e c c e n t r i c c y l i n d e r s or j o u r n a l bearings considered.
1 INTROWCTION The flow between c o n c e n t r i c r o t a t i n g c y l i n d e r s h a s becane the focus o f c o n s i d e r a b l e a t t e n t i o n i n the b e l i e f that it o f f e r s many advantages f o r the s t u d y o f t r a n s i t i o n f r a n laminar to t u r b u l e n t flow. T h i s is g e n e r a l l y r e f e r r e d to as Taylor-Couette flow, but the term superlaminar flow is used by tribolqists to d e s c r i b e flows a t v a l u e s o f Taylor n m k r T = 2 bi2 R , d 3 / ( R + R 2 ) y 2 ( w h e r e $]is t h e r o t a t i o n a l speed of! the i n n e r c y l i n d e r , R~ the inner c y l i n d e r r a d i u s , the o u t e r c y l l n d e r the f l u i d kinematic r a d i u s , d the gap, and v i s c o s i t y ) which exceed the c r i t i c a l v a l u e Tc corresponding to the formation of Taylor vortices (Fig. 1).
>
1
T a y l o r v o r t i c e s , amplitude modulated wavy T a y l o r v o r t e x f l o w , and f i n a l l y c h a o t i c or t u r b u l e n t flow. However, r e c e n t experimental work s u g g e s t s t h a t t h e t r a n s i t i o n process depends c r i t i c a l l y on both the annulus aspect r a t i o , r = 1/d, and t h e v o r t e x c e l l a x i a l l e n g t h / g a p ratio, ~ / dand that many d i f f e r e n t r o u t e s to t u r b u l e n c e are p o s s i b l e . The case o f e c c e n t r i c c y l i n d e r s , w i t h its a p p l i c a t i o n to j o u r n a l b e a r i n g s , h a s n o t been as e x t e n s i v e l y i n v e s t i g a t e d , b u t r e c e n t r e s u l t s for concentric cylinders have imprtant i m p l i c a t i o n s f o r t h e e c c e n t r i c c y l i n d e r case. 2 SUB-CRITICAL VORTICES The c r i t i c a l Taylor number T f o r the o n s e t of T a y l o r v o r t i c e s can be pred?cted by examining the s t a b i l i t y o f mall amplitude d i s t u r b a n c e s when superimposed on t h e b a s i c Couette flow. The u s e o f t h i s l i n e a r s t a b i l i t y a n a l y s i s f o r c o n c e n t r i c c y l i n d e r s h a s been e x t e n s i v e l y reviewed by Chandrasekhar (1) and S t u a r t ( 2 ) . A l l such a n a l y s e s assme t h a t t h e c y l i n d e r s are i n f i n i t e l y long. I n a d d i t i o n to T they p r e d i c t an i n i t i a l Taylor v o r t e x cell' a x i a l length, A E X p e b e n t a l l y determined values of T are i n gccd agreement with l i n e a r s t a b i l i t y &eory p r e d i c t i o n s d e s p i t e end e f f e c t s . However, even i f t h e annulus is long, end e f f e c t s g i v e rise to m e problems. T h e p r e s e n c e o f weak v o r t e x m o t i o n s a t T a y l o r nunbers less than T have been r e p o r t e d by many i n v e s t i g a t o r s a n d s a v e been d e s c r i b e d i n d e t a i l by Jackson, Robati and Mobbs ( 3 ) , f o r b o t h c o n c e n t r i c and e c c e n t r i c cylinders. That t h e s e motions are due to end e f f e c t s h a s been shown by numerical solutions of the Navier-Stokes e q u a t i o n s f o r the f l w i n a f i n i t e l e n g t h annulus. S o l u t i o n s have been o b t a i n e d by A l z i a r y d e Roquefort and G r i l l a u d ( 4 ) f o r c o n c e n t r i c c y l i n d e r s w i t h end plates r o t a t i n g w i t h the i n n e r c y l i n d e r , P r e s t o n (5) f o r c o n c e n t r i c c y l i n d e r s w i t h f i x e d end p l a t e s and El-Dujaily ( 6 ) f o r c o n c e n t r i c and e c c e n t r i c c y l i n d e r s w i t h f i x e d end p l a t e s . I n a l l cases a s i n g l e v o r t e x pair, occupying the f u l l annulus l e n g t h , is Eound to b e p r e s e n t a t v a n i s h i n g l y mall Taylor ncmber of v o r t e x ( F i g . 2 ) . As T i n c r e a s e s , the n-r p a i r s is i n c r e a s e d b y t h e f o r m a t i o n o f new v o r t e x pairs a t the c e n t r e o f the annulus ( F i g s
.
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A n n u l u s Mid-length Fig. 2 Sub-critical vortex formation between an inner r o t a t i n g c y l i n d e r and a concentric s t a t i o n a r y outer c y l i n d e r w i t h s t a t i o n a r y end plates. T/T = 0.01, r a d i u s ratio = 0.9, aspect ratio = &, 2 cells values of stream function :v1 = -0.00200, v 2 = - 0.00020, I#) = -0.00002 O.C. = outer c y l i n d e r , 1.C. = i n n e r cylinder The gradual strengthening of these s u b c r i t i c a l v o r t i c e s as T increases results i n a departure f r a n l i n e a r i t y i n the t o r q u e s p e e d r e l a t i o n s h i p (Mobbs and Ozcgan ( 7 ) ) (Fig. 5 ) . However, t h e y d o n o t a p p e a r t o d i s t o r t t h e basic Couette flow s u f f i c i e n t l y to i n v a l i d a t e the i n f i n i t e c y l i n d e r linear s t a b i l i t y theory p r e d i c t i o n s of T , although they do tend to m t h the s h a 6 increase i n t o r q u e s p e e d g r a d i e n t due to Taylor vortex onset. Taylor vortex o n s e t is marked by a sudden increase i n vortex c i r c u l a t i o n , s t a r t i n g w i t h t h e already more vigorous end cells (see Figs 3 , 4 ) and extending r a p i d l y inwards towards the annulus c e n t r e l i n e . When the c y l i n d e r s are e c c e n t r i c the sub-critical v o r t i c e s d o n o t e x t e n d t o t h e o u t e r cylinder near the widest gap. Taylor vortex o n s e t occurs simultaneously a l l round the circunference and is marked by an extension o f the v o r t i c e s across to the outer c y l i n d e r n e a r the widest gap (Castle and Mobbs ( 8 ) ) .
A
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d
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'I!/Tc= 0.1, 4 cells values of stream function := -0.008233,
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.
T h e v a r i a t i o n o f T, w i t h e c c e n t r i c i t y
ratio has been predicted by Eagles, S t u a r t and D i prima ( 9 ) using l i n e a r s t a b i l i t y theory and confinned experimentally by ozcgan and Mobbs
(10) and many others. 3.
TAYLORVORTEXFLCkJ
Linear s t a b i l i t y theory f o r the i n f i n i t e c y l i n d e r case predicts the appearance of T a y l o r v o r t i c e s w i t h cell s i z e Kogehan and D i prima (11) have analysed t%e s t a b i l i t y of Taylor vortex flow f o r mall values of "4' and have shown that there e x i s t s a continu& o f s o l u t i o n s that are stable covering a range o f x centred approximately on x I n a f i n i t e length apparaEus the nunber o f p o s s i b l e flows is constrained by t h e n e c e s s i t y o f f i t t i n g a n i n t e g e r number o f v o r t e x cells i n t o a given annulus length. I n an experiment w i t h an apparatus having a r a d i u s 0.6 and annulus aspect ratio 12.61, Benjanin and Mullin (12) obtained a m u l t i p l i c i t y of Taylor vortex flows with the number of cells r a w i n g f r a n 8 to 1 8 ,
.
.
397
End
(including odd numbers). For l a r g e b u t f i n i t e aspect r a t i o s , t h e y a r g u e t h a t t h e number of d i s c r e t e cellular modes w i l l i n c r e a s e u n t i l i n t h e l i m i t the continuun o f a l t e r n a t i v e c e l l u l a r flows allowed f o r i n the i n f i n i t e model is f i l l e d . I n practice cell s i z e may vary over the l e n g t h of the annulus s i n c e it is observed t h a t s u r f a c e imperfections tend to 'attract' a cell boundary. The major importance of these observations lies i n the influence which the value of h a s been found t o e x e r t on t h e subsequent developnent of the flow a t higher T.
Plate +
c
+
WAVY TAYLOR WFTEX FLOW
4.
When t h e Taylor nunber is increased to a value T , travelling azimuthal waves apyar d p e r i m p s e d on the Taylor vortices, r e s u l t i n g i n a reduction i n the slop of the torque-speed c h a r a c t e r i s t i c (see Fig. 5 ) . mrenzen, P f i s t e r and Mullin (13) have reported experiments carried o u t a t a r a d i u s r a t i o 0.507 i n which t h e y found Tw t o be extremely s e n s i t i v e to cell size. ?heir r e s u l t s have been confirmed by Sharif (14) using r a d i u s ratio 0.497. The v a r i a t i o n o f T T- with cell aspect ratio X/d is shown i n F g.'6.
U
0
OA
0
0
O A O A
n
d
2501 200
Fig. 4.
As f o r Fig. 3. = 0.5, 6 cells
T/T
of stream function
va&s
= -0.023266,
v2
:-
= -0.011633,
v3
=
"-0.006980 = 0.0032566, v 5 = 0.0016283, q 6 v4-0. 000074 O.C. = o u t e r c y l i n d e r , I.C. = Inner Cylinder
W
800
3
u L
0
+ 00
i
150
1
1001
0
0.2
0.4 0.6 0.8 1.0 A/d
1.2
Fig. 6 Influence of axial cell length X on the Radius wavy vortex onset Taylor nunber, T r a t i o = 0.497, aspect r a t i o ran& 17.39 19.13
.
0
0
cu
200 400 6 00 5 p e e d , r pv./rni n.
Fig. 5 Measured o u t e r c y l i n d e r torque. Radius ratio = 0.9, e c c e n t r i c i t y ratio = 0.1. Torque u n i t s a r e a r b i t r a r y . A- Taylor vortex onset. B - o n s e t o f vortex waves.
-
S i n c e i n the case of e c c e n t r i c c y l i n d e r s x d o e s not vary around t h e azimuth, the cell a s p e c t r a t i o w i l l so v a r y and it m i g h t b e a n t i c i p a t e d t h a t vortex waves w i l l not appear simultaneously around t h e annulus. ?his azimuthal v a r i a t i o n of T was observed sane years ago by Jones ( 1 5 ) . #is results (Fig. 7) show waves to appear f i r s t 90 deg. downstream of t h e narrowest gap. mrenzen e t al. (13) also measured the frequency of t h e wave oscillations a t onset using laser-Doppler anemanetry. Maintaining 1 2 vortex cells they v a r i e d A/d by varying r
.
398
r e s u l t s f o r two d i f f e r e n t i n i t i a l c e l l s i z e s
are shown i n Figs. 9 and 10. By r e f e r e n c e to Fig. 8 , the m u l t i p l e spectral frequencies
1-51 1.&
(apart fran harmonics and difference attributed to the frequencies) can be s i m u l taneous presence o f d i f f e r e n t t y p e s of wavy mode and/or d i f f e r e n t c e l l u l a r modes. T h i s s u g g e s t s t h a t multi-frequency spectra are d i r e c t l y related to end e f f e c t s .
1
p""1
X
= 0.0
U
&
ry
0
0.2
0.4
0.6
I, F.O.
0.61
e
Fig. 7 C i r c m f e r e n t i a l v a r i a t i o n of the ratio o f wavy v o r t e x o n s e t Taylor nunber, T to the critical Taylor number a t z e r o eccenTr_icity, 0 , with e c c e n t r i c i t y ratio,,. x, 0 = 0 , +,e 0 ~ c ~ O o , O , O = 180°, A.0 = 270 e measured frCm the narrowest g a p i n the direction of rotation.
.
X
= 0.2
T h e y o b s e r v e d t h r e e d i f f e r e n t t y p e s of v o r t e x waves depending on X/d each o f which had its own c h a r a c t e r i s t i c frequency range as follows :I n t h e aspect r a t i o r a n g e r = 10.04 11.55, t h e outward flow cell boundaries oscillated. This they called t h e jet mcde The relative frequency ( frequency/cyl i n d e r speed) ranged from 0.51 - 0.46 w i t h a wave nunber m = 1. 2. I n t h e aspect ratio range 11.16 - 15.06, t h e inward flow cell boundaries oscillated w i t h a r e l a t i v e frequency o f 0.77 (m = 4 ) or 0.94 (m = 5 ) . 3. I n t h e aspect ratio range 14.21 - 16.87, both boundaries oscillated w i t h a r e l a t i v e f r e q u e n c y r a n g e 0 . 0 7 5 - 0.09 and wave nunber m = 1. T h i s they called t h e wavy mode. S i m i l a r o n s e t f r e q u e n c i e s have been m e a s u r e d by S h a r i f ( 1 4 ) u s i n g h o t f i l m probes w i t h c y l i n d e r r a d i u s ratio 0.497. The aspect ratio w a s v a r i e d and t h e cell number also v a r i e d a t each aspect ratio. A typical set o f results is shown i n Fig. 8. AS i n Lorenzen e t a l ' s measurements frequency f a l l s with t h e j e t mode i n c r e a s i n g cell s i z e while t h e wavy mode frequency rises. The range o f x/d o v e r which t h e inward j e t o s c i l l a t i o n f r e q u e n c i e s were p r e s e n t diminished w i t h T h i s mode increasing aspect ratio. appears t o v a n i s h i f r is s u f f i c i e n t l y large. Fenstermacher , Swinney and Gollub ( 1 6 ) ( r a d i u s r a t i o 0.875) were t h e f i r s t to report t h e appearance o f a secod inccnnnensurate f r e q u e n c y i n wavy v o r t e x flow as T increased. ?his second frequency r e s u l t e d i n amplitude Later Gorman, w i t h modulation o f t h e waves. and Swinney ( 1 7 ) obtained spectra w i t h t h r e e and f o u r frequencies. S h a r i f ( 1 4 ) also obtained multi-frequency spectra i n h i s experiments w i t h r a d i u s ratio 0.497. A large nunber o f spectra were taken ?he spectral f r e q u e n c i e s o v e r a range o f T. whose p e r exceeded a p r e s e l e c t e d percentage o f t h e power o f t h e dominant s p e c t r a l frequency were then plotted a g a i n s t T / T ~ . Typical 1.
.
1 0
b4.M.
#
0.2
0.L
0.6
vd
0.8 1.0
1.2
Fig. 8 I n f l u e n c e of a x i a l cell l e n g t h on relative wavy vortex onset frequency ( frequency/cylinder speed) R a d i u s r a t i o = 0.497, aspect ratio range 17.39-19.13. I.F.O. = Inward flow o s c i l l a t i o n , J.M. = Jet Mode, W.M. = wavy mode.
J.MJ
.c
1 *J.M.,18
"...-.-?
.-----.si-.-m,L
J.M.,l4~elI s
......."..-.., c..e l l s .............. -. t ells,, ................ .. . . _ ... '---::,I.F.O.,IL
*
I.
.I..
1. :
Fig. 9 V a r i a t i o n of r e l a t i v e f r e q u e n c i e s with Taylor n m k r ratio T/T Radius r a t i o = 0,497, i n i t i a l c e l l numb&= 16. A s p e c t ratio = 1 4 , c u t o f f f a c t o r ( f r a c t i o n o f power i n the h i g h e s t spectral peak) = 0.1 I.F.O. = Inward flow o s c i l l a t i o n J.M. = Jet mode.
.
~f multiple f r e q u e n c i e s are a s s o c i a t e d w i t h m u l t i p l e cellular modes then it may be expected t h a t i n c r e a s i n g t h e annulus aspect r a t i o w i l l l e a d to more spectral frequencies. Frequency T h i s appears to be t h e case. s p e c t r a obtained by Haji Isnail (18) from an a p p a r a t u s o f r a d i u s ratio 0.9 and aspect r a t i o 1 0 0 , show that s i n g l e frequency and multiple
399
frequency waves occur over a l t e r n a t i n g ranges of T/T (Fig. 11). Typical frequency spectra Sre shown i n Figs. 12 and 1 3 f o r s i n g l e and multiple frequency regimes, respectively. Many of the frequencies i n Fig. 13 l i e close together, a r e s u l t t h a t might have been anticipated s i n c e i n a long annulus modes whose cell nunbers d i f f e r only by a snall amount w i l l have mall d i f f e r e n c e s i n h / d a n d t h e r e f o r e i n wave frequency.
'9
3, 0.8 EL
ZI
"C 2 0)
3 U Q)
L
- 1
d
Q)
....,a
.->
0 L 8 1'2 16 Relative frequency
I.F. 0.J 2 c e I I s
....-...".
............ A M . J 2 c e Ils
Fig. 12 Frequency spectrun with a s i n g l e frequency and its f i r s t two harmonics. Radius ratio = 0.9, aspect ratio = 100, T/T = 4.72
200
T/
400 Tc
C
6 00
Fig. 10 V a r i a t i o n of r e l a t i v e frequencies w i t h Taylor number ratio T/T Radius ratio = 0.497, E n i t i a l cell number = 1 2 , aspect ratio = 12.2, c u t o f f f a c t o r = 0.1 I.F.O. = Inward flow o s c i l l a t i o n , J.M. Jet mcde, W.M. = Wavy mode.
.
0 1 2 3 4 Relative frequency I 1
I
1
I
I I I t l I
10
I
1
1
1
50
Fig. 1 3 Frequency spectrun containing multiple frequencies. Radius ratio = 0.9, aspect ratio = loo, T / T ~= 43.9 5.
Fig. 11 Ranges of T/T over which s i n g l e and multiple frequency s&tra occur. m i u s ratio = 0.9, aspect ratio = 100. S = Single frequency, M = multiple frequencies For e c c e n t r i c cylinders, no systematic wave frequency measurements have been made, although O'Brien and Mobbs (19) have c m n t e d on the azimuthal dependence of the flow. Since the s i t u a t i o n is h/d v a r i e s around t h e azimuth f a r mre canplex than i n the concentric cylinder case.
nmBULENCE
Landau ( 2 0 ) proposed t h a t t r a n s i t i o n f r a n laminar to turbulent flow could take place by a sequence of flow i n s t a b i l i t i e s each of i h i c h would introduce a n& frequency i n t o the spectrum. However, Ruelle and Takens (21) argued t h a t n o w l i n e a r i n t e r a c t i o n s would produce chaos a f t e r a small number of i n s t a b i l i t i e s and Newhouse, Ruelle and Takens ( 2 2 ) predicted that chaos would follow the introduction of a second spectral frequency. N e i t h e r o f t h e s e t w o models a p p e a r s t o be validated by the experimental results of section 4 , s i n c e t h e multiple frequencies observed do not appear to result f r a n successive wave i n s t a b i l i t i e s , b u t f r a n m u l t i p l e cellular modes. I t should be noted, h a e v e r , t h a t it is impossible to t e l l whether or n o t sane canplex spectra (e.g. Fig. 1 3 ) contain an e l e n e n t of weak background turbulence.
400
Fenstenacher et al. (16) stated t h a t t h e i r flow became c h a o t i c or t u r b u l e n t a t T/T = 450, accompanied by t h e d i s a p p e a r a n c e 0% s h a r p spectral peaks, b u t w r k i n g with t h e same r a d i u s ratio. Walden and m n n e l l y ( 2 3 ) observed t h e reemergence of a s h a r p spectral peak i n t h e range T/Tc = 784 to 1296. Barcilon, Brindley, Lessen ti Mobbs (24) ( r a d i u s r a t i o 0.908 , r = 6 5 ) reported t h e disappearance o f cell boundary waves a t T/T = 445, b u t w a v i n e s s c o n t i n u e d i n t h e inte%ior o f t h e v o r t e x cells. Haji Hassan (25) showed t h e corresponding spectra to c o n t a i n two f r e q u e n c i e s of high power, i n a d d i t i o n to a complex of weak frequencies. The two frequencies remained up to T/T = 2500 and then disappeared. A s i n g l e frequegcy re-merged i n t h e range T/T = 6000 to 60,000, t h e l i m i t of t h e experimenfs. A d i s t i n c t i v e Taylor v o r t e x structure remained throughout. B a r c i l o n e t a l . (24) observed t h a t t h e disappearance o f boundary waves was followed by t h e appearance o f a r e g u l a r p a t t e r n o f streaks superimposed on the Taylor v o r t i c e s which were believed t o be due to Gortler v o r t i c e s g e n e r a t e d i n t h e boundary layers fonned on t h e two c y l i n d e r s . Experiments by Townsend ( 2 6 ) a t r a d i u s ratio 0.666, r = 24.15, suggest that t h e regular Taylor v o r x s t r u c t u r e breaks d a m a t T / T ~ ,o f o r d e r 1 0 T h i s is l i n k e d t o changes i n t h e boundary l a y e r s . As T i n c r e a s e s t h e i n f l u e n c e o f f l o w c u r v a t u r e is r e d u c e d close to t h e c y l i n d e r s and the Gortler v o r t i c e s are r e p l a c e d by e d d i e s s i m i l a r t o t h o s e i n p l a n e flows. I t would appear, t h e r e f o r e , t h a t t u r b u l e n c e o f t h e plane flow type o n l y occurs a t T a y l o r numbers f a r beyond t h e r a n g e o f practical l u b r i c a t i o n i n t e r e s t ( u p to T / T ~ of order 200).
F.
6. CCNCLUSION Much f u r t h e r w r k is required to extend t h e p r e s e n t knowledge o f superlaminar flow between to the eccentric concentric cyliniders Indications cylinder/journal bearing case. are t h a t w i t h i n t h e range of Taylor n&rs encountered i n practice the flow c o n s i s t s p r i m a r i l y of Taylor v o r t i c e s w i t h superimposed t r a v e l l i n g waves and any p o s s i b l e turbulence is
weak. S i n c e a n n u l u s aspect ratios i n j o u r n a l bearings are large, extremely canplex wave motions are to be anticipated. References CHANDWEKHAR, S. "HYDRODYNAMIC AND Hydranagnetic S t a b i l i t y " . Oxford Clarendon Press , 1961. S'IUAWT, J.T. i n RDSENHEAD, L(editor1 "Laminar Boundary Layers", Oxford Clarendon P r e s s , 1963. B. and MOBBS, F.R. JACKSCN, P.A., =TI, "Secondary flows between eccentric r o t a t i n g c y l i n d e r s a t s u b - c r i t i c a l Taylor nunbers. 'I ALZIARY d e WEFORT, T. and GRILLAUD, D. "Canputation on Taylor v o r t e x flow by a t r a n s i e n t implicit method." Canputers and Fluids, 1978 , 6, 259-269. PREsn=pJ, W.S. "A s t u d y o f t h e sub-critical and wavy v o r t e x regimes i n t h e flow between c o n c e n t r i c r o t a t i n g c y l i n d e r s . " PhD. Thesis. Department of Mechanical Engineering , U n i v e r s i t y o f Leeds, 1979.
EL-DUJAILY,
M.J. "End effects on and Taylor vortex flow between concentric and eccentric cylinders." PhD. Thesis. Department o f Mechanical Engineering , u n i v e r s i t y of weds, 1983. MOBBS, F.R. and OZOGAN, M.S. "Study Of s u b - c r i t i c a l Taylor votex flow between e c c e n t r i c r o t a t i n g c y l i n d e r s by torque measurements and v i s u a l Observations." I n t e r n a t i o n a l Journal o f Heat and F l u i d Flow, 1984, VOl. 5 , nO.4, 251-253. CASrzE, P. and MOBBS, F.R. "Hydrodynamic s t a b i l i t y o f t h e flow between e c c e n t r i c Proc. I n s t n . Mech r o t a t i n g cylinders." Engrs , 1968, 182 I 41-52. EAGLES, P.M., SIUART, J.T. and D I PRIMA, R.C., ##The e f f e c t o f e c c e n t r i c i t y on torque and load i n Taylor vortex flow." J . F l u i d Mech. 1978 , v o l . 8 7 , p a r t 2 , 209-231. M.S. and M)BBs, F.R., ( 1 0 ) OZOGAN, "Superlaminar flow between e c c e n t r i c r o t a t i n g c y l i n d e r s a t small c l e a r a n c e ratios." Am. Soc. Mech. Engrs. Energy Conservation through Fluid Film Lubrication Technology: Frontiers i n Research and Design. 1979, 181-189. (11) KOGELMAN, S. and D I PRIMA, R.C. "Stability of spatially periodic s u p e r c r i t i c a l flows i n hydrodynamics." Phys. F l u i d s 1970, 13, 1-11. (12) BENJAMIN, T.B. and MULLIN, T. "Notes on t h e m u l t i p l i c i t y of flows i n the Taylor e x p e r i m e n t . J. F l u i d Mech. 1982, 1 2 1 , 219-230. ( 1 3 ) LORENZEN, A. PFISTER, G. and ElLTLLIN, T. " ~ n de f f e c t s on the t r a n s i t i o n to time dependent motion in the Taylor experiment." Phys. F l u i d s , 1983, 2 6 ( 1 ) , 10-1 3. ( 1 4 ) SHARIF, N.S. "An experimental s t u d y o f the f l u i d dynamics o f axisymnetric and disturbances in non-ax isymnet r i c Taylor-Conette flow." PhD. Thesis. Mech ~ n g .Dept., u n i v e r s i t y of ~ e e d s ,1986. (15) JONES, C.D. "A s t u d y of secondary flow and turbulence between e c c e n t r i c r o t a t i n g cylinders." PhD. Thesis Mech. Eng. Dept., U n i v e r s i t y o f Leeds. 1973. SWINNEY, H.L. and (16) FENSTERMACHER, P.R., GOD, J.P. "Dynamic i n s t a b i l i t y and t h e t r a n s i t i o n to c h a o t i c Taylor v o r t e x flow." J. F l u i d Mech. 1979, 94, 103-128. (17) GORMAN, M. , REITH, L.A. , and SWINNEY, H.L. "Modulation p a t t e r n s , multiple f r e q u e n c i e s and o t h e r phenanena in c i r c u l a r C o u e t t e flow" i n ' N o n l i n e a r Dynamics" edit by HELLEMAN, R. Annals o f t h e New York Acadamy o f Sci. 1980. (18) HAJI ISMAIL, A.W.B. "An i n v e s t i g a t i o n o f flow using torque Taylor vortex measurements and h o t f i l m aneanetry." PhD. T h e s i s , Dept. Mech. Eng. U n i v e r s i t y o f Leeds. 1982. and MOBBS, F.R."The ( 1 9 ) O'BRIEN, K.T. e v o l u t i o n of t u r b u l e n c e between e c c e n t r i c r o t a t i n g cylinders." 2nd Leeds-Lyon Symposiun on Tribology. 1975, 51-56. L. "on the problem of (20) m u , C.R. Acad. Sci. U.R.S.S. turbulence." 1944, 44, 311-315. ( 2 1 ) RUELLE, D. and TAKENS, F. "On the n a t u r e o f t u r b u l e n c e .'I Commun, Math. Phys. 1971, 20, 167-192.
sub-critical
.
40 1
(22) NEMlCUSE, J., RUELLE D., and TAKENS, F. "Occurence of s t r a n g e wian A a t ractors near quasi-periodic flows on 4 , m > 3 1 1 . C m u n . Math. Phys. 1978, 64, 35-40. R. w. and DONNELLY, (23) W E N , R.J."Re-emergent o r d e r of c h a o t i c circular C o u e t t e flow." Phys. Rev. L e t t . 1979, 42(5) , 301-463. ( 2 4 ) BARCIm, A., BRINDLEY, J., LESSEN, M. and m38BS, F.R. "Marginal i n s t a b i l i t y i n Taylor-Couette flows a t very high Taylor J. Fluid Mech. 1979, 94, nunber." 453-463. HASSAN, M. "An experimental (25) W I i n v e s t i g a t i o n of Taylor vortex waves extending to very high Taylor nunbers." PhD. Thesis. Dept. Mech. Eng. University of ~ e e d s ,1982. "Axisymnetrical Couette (26) TaWNSEND, A.A. Fluid flow a t large Taylor nunber." Mech. 1984, 1 4 4 , 329-362.
J.
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Paper Xlll(ii)
Frictional losses in turbulent flow between rotating concentric cylinders C.G. Floyd
The f r i c t i o n a l losses a r i s i n g in t h e t u r b u l e n t flow between a r o t a t i n g c y l i n d e r a n d a s t a t i o n a r y concentric o u t e r c y l i n d e r have been s t u d i e d experimentally. Tests have been c a r r i e d o u t u s i n g c i r c u l a r c y l i n d e r s a n d o t h e r geometries i n c l u d i n g c i r c u m f e r e n t i a l g r o o v e elements a n d r a d i a l disc It has been f o u n d t h a t t h e f r i c t i o n a l losses c a n b e a c c u r a t e l y p r e d i c t e d by t h e simple elements. addition o f t h e effects o f t h e r o t a t i n g s u r f a c e elements, a n d by assuming t h a t t h e dependence o f the s k i n f r i c t i o n coefficient on Reynolds number i s t h e same for a l l p o i n t s o n t h e r o t o r surface. 1
INTRODUCTION
A t v e r y high Reynolds numbers, t y p i c a l l y greater t h a n l o 6 , t h e flow o f a fluid between a rotating cylinder and a stationary concentric outer c y l i n d e r becomes fully t u r b u l e n t , as t h e superlaminar T a y l o r v o r t e x flow s t r u c t u r e breaks down. T u r b u l e n t flows o f t h i s t y p e occur in t h e a n n u l a r spaces a r o u n d t h e s h a f t s o f c e n t r i f u g a l pumps, in t h e a n n u l i between r o t o r s a n d s t a t o r s o n high speed e l e c t r i c motors, a n d c a n also o c c u r in l a r g e high speed journal bearings. T h e f r i c t i o n a l losses in s u c h systems d u e t o t h e s h e a r i n g o f t h e fluid f i l m a r e s i g n i f i c a n t a n d c a n g i v e r i s e t o excessive localised heat generation. In a l a r g e b o i l e r feed pump, f o r example, heat generation levels o f t h e o r d e r o f 40 k W have been r e c o r d e d in the a n n u l u s a r o u n d t h e mechanical seal i n s t a l l ation. Turbomachinery components s u c h as mechanical seals a r e sensitive t o fluid temperature, a n d t h e r e i s t h e r e f o r e a clear need f o r an accurate heat generation p r e d i c t i o n method f o r s u c h components t o p e r m i t d e s i g n optimisation. Considerable p u b l i s h e d w o r k i s available o n t h e subject o f superlaminar flow between r o t a t i n g c y l i n d e r s but l i t t l e f r i c t i o n a l loss data i s available, a n d t h e n n o t a t Reynolds numbers g r e a t e r t h a n 106.5. F u r t h e r m o r e t h e published w o r k i s limited t o e i t h e r p l a i n c i r c u l a r c y l i n d e r s o r t o discs, a n d t h e r e i s n o data available f o r more complex shapes i n c o r p o r a t i n g both c y l i n d r i c a l s u r f a c e a n d r a d i a l d i s c elements such as make up t h e m a j o r i t y o f t u r b o m a c h i n e r y components. T h e w o r k p r e s e n t e d in t h i s p a p e r i s a s t u d y o f these more complex s h a f t a n d h o u s i n g geometries, a n d i s a n attempt t o determine t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat generation a n d t o p r o v i d e empirical guidelines f o r p r e d i c t i n g t h e magnitude o f t h e heat generation. T h i s s t u d y i s p a r t o f a wider i n v e s t i g a t i o n i n t o flow problems in turbomachine r y a n n u l i (1).
Nomenclature
1.1
C
T o r q u e e x e r t e d o n t h e r o t o r by t h e fluid
R
Radius o f t h e r o t o r
I
Axial length o f the rotor
r
Local r a d i u s o f a p o i n t o f i n t e r e s t
t
Width o f a n n u l a r g a p between t h e r o t o r a n d t h e stator
v
Local v e l o c i t y a t a p o i n t o f r a d i u s r
v
Kinematic v i s c o s i t y o f t h e fluid
p
D e n s i t y o f t h e fluid
T
Shear s t r e s s o n t h e r o t a t i n g surface
w
Angular velocity o f t h e r o t o r
Non-dimensional Coefficients F Non-dimensional t o r q u e F = Re
R
C /p v 2 R
Rotor Reynolds number ReR = w R 2 / v
Cf S k i n f r i c t i o n c o e f f i c i e n t CF = o r local coefficient 2
cf
‘I / 0 . 5
pw2R
= T 10.5 pw2 r 2
EX PER IMENTAL A R RAN C EMEN T
A p u r p o s e built t e s t rig was developed f o r t h i s work, a n d a schematic arrangement o f t h e r o t o r a n d h o u s i n g assembly i s i l l u s t r a t e d in F i g u r e 1. As c a n b e seen f r o m t h e f i g u r e , t h e r o t o r s h a f t a n d h o u s i n g assembly were so designed t h a t d i f f e r e n t r o t o r s a n d housings c o u l d b e f i t t e d , t o g i v e a wide r a n g e o f a x i a l lengths, s u r f a c e finishes a n d r o t o r t o h o u s i n g g a p sizes. Representative t e s t a n n u l i a r e i l l u s t r a t e d in F i g u r e 2.
404
Fluid Annulus
water by a n e x t e r n a l heat exchanger, a n d t h e water was p r e s s u r i s e d t o between 2 a n d 8 b a r t o p r e v e n t c a v i t a t i o n a n d a i r entrainment t h r o u g h t h e seals from o c c u r r i n g . One o f t h e major problems in developing t h e experimental equipment was t h e d e s i g n o f suitable seals between t h e r o t o r a n d t h e housing. These seals were r e q u i r e d t o operate o v e r a r a n g e o f speeds a n d fluid p r e s s u r e s but w i t h minimal leakage a n d w i t h t h e minimum possible f r i c t i o n a l losses. T h e f i n a l arrangement u s e d consisted o f a c a r b o n seal ring running against a t u n g s t e n c a r b i d e seal ring w i t h t h e axial c l o s i n g f o r c e p r o v i d e d by compressed a i r . T h e c l o s i n g f o r c e between t h e t w o seal r i n g s c o u l d t h e r e f o r e b e regulat e d during t e s t s t o c o n t r o l t h e leakage.
Shaft
T h e use o f stainless steel f o r t h e r o t o r a n d h o u s i n g components a n d t h e use o f water as t h e w o r k i n g fluid were d i c t a t e d by t h e d e s i r e t o r e p r e s e n t t h e conditions in a n actual pump f o r w h i c h some experimental data T h e size o f t h e equipment were available. was also f i x e d f o r t h e same reason. T h e possibilities o f u s i n g a scale model o r o f u s i n g a l t e r n a t i v e f l u i d s were considered, but as t h e importance o f t h e v a r i o u s parameters a f f e c t i n g heat g e n e r a t i o n were n o t k n o w n it was t h o u g h t desirable t o eliminate a n y scale effects.
Figure 1 T e s t riq arrangement
I
/,
I'
J
A
rotor seal
T h e p r i n c i p a l parameters measured were:
seal
(i)
S h a f t speed a n d torque, measured by a s h a f t mounted optical s l i t t e d d i s c system, p r o v i d i n g f i l t e r e d analogue voltage o u t p u t s . F l u i d i n l e t a n d o u t l e t temperatures, measured by p l a t i n u m resistance thermometers. F l u i d pressures, measured by analogue p r e s s u r e gauges.
(ii)
F l u i d c i r c u l a t i o n flow rate, measured by a p a i r o f Rotameter u n i t s . Seal leakage flow rates, measured by t h e time t a k e n t o fill a g r a d u a t e d container
.
0
(iii) Fiqure 2 T y p i c a l a n n u l a r t e s t qeometries T h e r o t o r a n d h o u s i n g components were manufactured f r o m stainless steel, a n d t y p i c a l r o t o r r a d i i were in t h e r a n g e 107mm to 12lmm. T h e s h a f t was d r i v e n by a 75 k W motor t h r o u g h a b e l t d r i v e system t o g i v e a rotational speed r a n g e o f 2100 r p m t o 9400 rpm. T h e o p e r a t i n g fluid was water, a n d t h i s was c i r c u l a t e d t h r o u g h t h e t e s t cell a t a r a t e o f between 15 a n d 50 l i t r e s / m i n u t e by a n e x t e r n a l pump. Heat was e x t r a c t e d from t h e
S u b s i d i a r y measurements o f t h e compressed a i r s u p p l y p r e s s u r e s t o t h e seals a n d t h e b e a r i n g temperatures were also made. T h e heat g e n e r a t i o n in t h e system was determined by t h e s h a f t speed a n d t h e t o r q u e measurements. Determination o f t h e heat generation from t h e r i s e in fluid temperature a n d t h e fluid c i r c u l a t i o n f l o w r a t e g a v e o n l y a b o u t 709, o f t h e power loss, o w i n g t o t h e additional heat t r a n s f e r i n t o t h e b o d y of t h e housing.
3
CYLINDER TESTS
F o r fully t u r b u l e n t flow w i t h Reynolds numbers g r e a t e r t h a n lo6, t h e flow between t h e r o t o r a n d t h e h o u s i n g has been shown by Pai ( 2 ) a n d Ustemenko ( 3 ) t o consist o f t w o thin s u r f a c e shear l a y e r s a n d a c e n t r a l flow r e g i o n
405 over 80% o f th e gap f o r which v r i s a constant. The s k i n f r i c t i o n coefficient o n t h e r o t o r surface, a n d hence t h e heat generation ,is dependent o n th e velocity g r a d i e n t in t h e It w i l l t h e r e f o r e b e o n l y surface shear layer. weakly dependent o n t h e Reynolds number a n d on the housing geometry. It also follows t h a t the flow in th e a nnular g a p i s essentially tw o dimensional. T h e influence o f t h e helical flow a r i sin g from t h e c i r c u l a t i o n of t h e fluid by t h e external pump was considered t o b e small. The axial Reynolds number was less t h a n 0.1% o f the rotational Reynolds numbers a n d t h e work o f Yamada (4) a n d o t h e r s suggests t h a t for these conditions helical flow effects can b e neglected. T h e r e s u l t s obtained in t h i s s t u d y showed no dependence o n fluid c i r c u l a t i o n rate. Sig nificant t h r e e dimensional e n d e f f e c t s did occur, however, b o t h d u e t o t h e additional surfaces o f t h e seals a n d t h e c y l i nder ends, a n d d u e t o t h r e e dimensional flow e ffects in t h e flow caused by these additional surfaces. T h e t o r q u e d u e t o these e n d effects comprised up t o 50% o f t h e total t o r q u e measured, a n d in p a r t i c u l a r t h e f r i c t i o n a t t h e seals was high. T h e t o r q u e d u e t o all t h e e n d effects could n o t b e measured separately fro m the t o r q u e d u e t o t h e cylinders, a n d it was t h e r efore necessary t o t e s t t w o c y l i n d e r s of d i f f e r e n t axial lengths, t o eliminate t h e e n d effects by s u b t r a c t i n g t h e two results. Problems h ave been encountered w i t h t h i s technique in t h e past (5) because o f t h e size o f t h e e n d effects, a n d so f o u r d i f f e r e n t c y l i n d e r l e n g t h s were tested, t o p e r m i t t h e s u b traction calculations t o be c r o s s checked. T h e shortest axial l e n g t h chosen was 40mm. and t h i s l e n g t h was also chosen f o r most o f the t e s t i n g o f more complex r o t o r a n d stator geometries. T h e o t h e r l e n g t h s chosen f o r t h e c y l i nder te sts were 65mm, 80mm a n d 120mm. It was assumed t h a t f o r a g i v e n r o t o r and housing geometry t h e s k i n f r i c t i o n coefficient i s related t o t h e Reynolds number b y a power law relationship CF = aReRb, where t h e s k i n f r i c t i o n coefficient i s g i v e n by CF = G / ( n p I R 4 w 2 ) .
However, t h e s k i n f r i c t i o n coefficient cannot b e determined d i r e c t l y from t h e experimental data, as t h e total t o r q u e measured ( G T), includes a component d u e t o e n d e f f e c t s ( G E ) as well as t h e c y l i n d e r surface component ( G C). T h e r e f o r e t h e s k i n f r i c t i o n coefficient is g i v e n as
where Ic i s t h e actual l e n g t h o f c y l i n d e r s u r face, a n d IE i s a n equivalent e x t r a l e n g t h o f c y l i n d e r surface t o allow f o r t h e e n d effects. T o analyse t h e data, a non-dimensional t o r q ue co efficient F was introduced, where: F = (GC + GE)/pu2R
F can t h e n b e d e f i n e d in terms o f Reynolds n u m b e r by :
a n d thence f r o m a series o f t e s t s w i t h diffe r e n t c y l i n d e r l e n g t h s , a, b a n d IE can b e found.
F has n o p h y s i c a l significance, but was chosen as t h e non-dimensional c o e f f i c i e n t t o g i v e a significant gradient o n a plot o f log F against l o g ReR t o f a c i l i t a t e t h e data processing. I m p l i c i t in t h i s a p p r o a c h was t h e assumption t h a t t h e e n d e f f e c t s c a n b e q u a n t i f i e d by a n a d d i t i o n a l l e n g t h o f c y l i n d r i c a l s u r f a c e ( IE) F o r t h i s assumption t o b e valid, a l l t h e e n d e f f e c t s must v a r y w i t h R e y n o l d s Number in a similar fashion t o t h e c y l i n d e r t o r q u e . The additional torque due t o the e n d effects can be s p l i t i n t o t w o components; the torque d u e t o t h e seals, a n d t h e t o r q u e d u e t o t h e end surfaces o f the cylinders. The torque d u e t o t h e b e a r i n g s c a n b e i g n o r e d as d r y r u n s o f t h e t e s t rig showed t h a t t h e b e a r i n g t o r q u e was n e g l i g i b l e .
.
T o assess t h e t o r q u e d u e t o t h e seals, it was assumed t h a t t h e flow a t t h e seal i n t e r f a c e was t u r b u l e n t , w i t h a seal Reynolds Number o f t h e o r d e r o f 105. T h e seal Reynolds Number was c a l c u l a t e d f r o m t h e s l i d i n g v e l o c i t y a n d a n estimate o f t h e i n t e r f a c e f i l m t h i c k n e s s based o n t h e measured leakage a n d p r e s s u r e differential. The c i r c u m f e r e n t i a l flow a t t h e seal i n t e r f a c e i s t h e n comparable to t h e flow between m o v i n g p a r a l l e l plates, f o r w h i c h t h e dependence o f s k i n f r i c t i o n o n Reynolds n u m b e r i s similar t o t h a t f o r c o n c e n t r i c r o t a t i n g c y l i n d e r s ( 6 ) . It i s t h e r e f o r e reasonable t o assume t h a t t h e t o r q u e d u e t o t h e seals can b e c o n s i d e r e d t o b e equal t o the torque due t o a n additional length o f c y l i n d e r surface. T h e t o r q u e d u e t o t h e e n d surfaces o f t h e r o t a t i n g c y l i n d e r c o u l d not b e d e t e r m i n e d theoretically. However, as t h e flow in t h e annular gap i s a basically irrotational turbulent flow a t a high R e y n o l d s number, t h e t u r b u l e n t b o u n d a r y l a y e r s o n t h e e n d surfaces o f t h e a n n u l a r g a p w i l l b e similar t o those o n t h e From t h i s i t can be c y l i n d e r surfaces. assumed t h a t t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r w i l l also b e similar. It i s t h e r e f o r e reasonable t o assume t h a t a l l t h e e n d e f f e c t s w i l l v a r y w i t h Reynolds n u m b e r in a similar fashion t o t h e c y l i n d e r torque. T h e n f o r each a n n u l a r g a p w i d t h t h e r e s u l t s for a l l t h e a x i a l l e n g t h s c a n b e p l o t t e d in t e r m s o f l o g F a g a i n s t l o g ReR a n d processed simultaneously t o g i v e a least squares r e g r e s s i o n fit o f f o u r p a r a l l e l l i n e s t o t h e f o u r sets o f experimental data.
T h e r e g r e s s i o n a n a l y s i s was weighted t o t a k e account o f t h e estimated a c c u r a c y o f t h e measurements. F i g u r e 3 shows a plot of t h e data p o i n t s f o r a 5mm g a p f o r t h e t w o
406 It can b e seen t h a t t h e c o r r e l a t i o n between t h e data a n d t h e f i t t e d lines i s good, a n d in f a c t t h e c o r r e l a t i o n c o e f f i c i e n t s were t y p i c a l l y g r e a t e r t h a n 0 . 9 9 . T h e r e g r e s s i o n analysis d e t e r m i n e d n o t o n l y t h e slope o f t h e line, but also t h e m a g n i t u d e o f t h e e n d e f f e c t s in t e r m s o f the equivalent e x t r a length o f cylindrical surface. T h e e q u i v a l e n t e x t r a l e n g t h s were calculated t o b e 5 8 , 75 a n d 75mm f o r a n n u l a r g a p w i d t h s o f 5 , 10 a n d 20mm r e s p e c t i v e l y . The reduced magnitude o f the end effects f o r t h e 5mm g a p was n o t d u e t o experimental e r r o r , as r e p e a t t e s t s showed t h a t t h e a c c u r a c y o f d e t e r m i n a t i o n o f t h e e n d l e n g t h s was It i s t h e r e f o r e consida p p r o x i m a t e l y f 3mm. e r e d t h a t it was a r e s u l t o f some t h r e e dimensional flow e f f e c t o v e r t h e c y l i n d e r e n d surfaces.
extreme axial lengths o f 40 a n d 120mm togeth e r w i t h t h e f i t t e d lines. T h e r e s u l t s for t h e i n t e r mediate lengths have been omitted f o r c l a r i t y .
12 .o
I
I = 120
log
11.5
11
Al l o w i n g f o r t h e calculated i n f l u e n c e o f t h e e n d effects, t h e r e l a t i o n s h i p s between s k i n f r i c t i o n c o e f f i c i e n t a n d Reynolds n u m b e r were determined f o r the different gap widths.
.o
These were f o u n d t o be:
10.5 6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
log ReR
T h e r e appears t o b e o n l y a s l i g h t dependence o n t h e g a p r a t i o ( t / R ) , a n d t h i s is in agreement w i t h t h e r e s u l t s o f o t h e r workers. T h i s c a n b e seen in F i g u r e 4 w h i c h is a plot o f other published results and the present results.
Fiqure 3 T o r q u e aqainst Reynolds number f o r two a xial lenqths
1.3
1.2
1.1
1 .o
l-
i
P t/R
o
0.15-0.4
A
0.07-0.15
V
0.04-0.07
0
0.02-0.04
5.6
1
5.8
Fiqure 4
6.0
I
6.2
I
f
I
6.4
6.6
6.8
1
7.0 l o g ReR
Selected p u b l i s h e d data compared w i t h mean lines f r o m p r e s e n t r e s u l t s f o r cylinders.
Published r e s u l t s from B i l q e n ( 7 ) . G o r l a n d ( 8 ) & T a y l o r ( 9 )
7.2
407
It would appear from F i g u r e 4 t h a t t h e s k i n f r i c tion coefficient i n i t i a l l y decreases w i t h decreasing g ap r a t i o but t h e n begins t o increase as t he g ap r a t i o f u r t h e r decreases. T h e minimum s k i n f r i c t i o n coefficient o c c u r s when t h e g a p r a t i o i s in th e approximate r a n g e o f 0 . 1 t o 0 . 2 . A more precise statement o n t h e e f f e c t o f g a p r a t i o o n s k i n f r i c t i o n coefficient i s n o t possible, as the scatter o f experimental data p o i n t s i s greater t h a n a n y g a p r a t i o effect.
4
assumption was based o n t h e f a c t t h a t t h e s k i n f r i c t i o n c o e f f i c i e n t data f o r complete discs p u b l i s h e d by o t h e r w o r k e r s ( 1 0 , 1 1 , 1 2 ) showed good agreement w i t h t h e p r e s e n t r e s u l t s f o r c y l i n d r i c a l surfaces, as shown in F i g u r e 6 .
COMPLEX GEOMETRY EFFECTS Influence o f Housinq Geometry
4.1
A series o f te sts were c a r r i e d o u t u s i n g p l a i n c y l i nd rical r o t o r s but w i t h changes t o t h e housing geometry. Two d i f f e r e n t housing geometry effects were tested. T h e f i r s t o f these was a change t o t h e i n l e t geometry from t h a t shown in Figure 2 ( i ) t o t h a t shown in F i g u r e 2 ( i i ) . Figure 5 shows t h e data f o r b o t h geometries plotted as log F against log ReR f o r a n 80mm axial l e n g t h c y l i n d e r . It can b e seen t h a t changing t h e i n l e t geometry h a d n o discernab l e effect o n t h e r o t o r torque.
12.0
.9
.7 .6 .8
+
6.3
-1
+
F i+
+
closed inle t
Q
open i n l e t
+
F I
6.4
6.5
1
6.6
6.8
6.9
7.0
7.1
The variation o f the total torque for the 121mm r o t o r w i t h r o t o r Reynolds n u m b e r c o u l d t h e n b e p r e d i c t e d by i n t e g r a t i n g t h e local s k i n f r i c t i o n c o e f f i c i e n t o v e r t h e complete r o t o r s u r face. T h e e n d e f f e c t s were i n c l u d e d as a n e x t r a axial length o f cylinder. The predicted total t o r q u e a n d t h e experimental data a r e p l o t t e d in F i g u r e 7 a n d show good agreement.
10.5 1
6.7
published radial disc results
0
6.3
6.6
Comparison o f c y l i n d e r r e s u l t s against
0
1
6.5
Fiqure 6
8
11.0
6.4
log ReR
++ 11.5
Present c y l i n d e r r e s u l t s lppen (10) Ketola & McGrew ( 1 1 ) D a i l y & Nece ( 1 2 )
1
.5
1
-
I
I
1
6.7
6.8
6.9
1
7.0
1
7.1
12.0
Effect o f doubling
-
log ReR Figure 5 Influence o f i n l e t qeometry o n t o r q u e 11.5T h e second housing change i n v o l v e d t h e i n t r o d u c t i o n o f a circumferential groove, as shown in F i g u r e 2 ( i i ) . Again t h i s h a d n o discernable effect o n t h e r o t o r torque. T h i s i s n o t s u r p r i s i n g as one would expect t h e r o t o r t o r q u e to depend o n t h e velocity g r a d i e n t in t h e shear layer o n t h e ro tor, a n d t h i s will b e l i t t l e affect e d by h ousing changes. 4.2
11.0
-
10.5
I
Radial Elements
The n e x t series o f tests i n v o l v e d t h e i n t r o d u c t i o n o f r a d i a l elements o n t o t h e r o t o r . T w o r o t o r s were used, w i t h r a d i i o f 116mm a n d 1 2 l m m , t o g i v e radial surface elements a t t h e c y l i n d e r ends. It was assumed t h a t t h e relations h i p between local s k i n f r i c t i o n coefficient a n d local Reynolds number f o r b o t h disc a n d c y l i n d e r elements was t h e same as t h a t between the s k i n f r i c t i o n coefficient a n d r o t o r Reynolds This number f o r t h e c y l i n d r i c a l surfaces.
6.3
1
,
6.4
I
I
6.5
6.6
I
6.7
1
6.8
I
1
6.9 7.0 log ReR
Fiqure 7 Pr e d i c t e d a n d measured t o r q u e f o r a r o t o r w i t h r a d i a l s u r f a c e elements
1
7.1
408
The proportion o f the total torque due to the r a d i a l surfaces i s small, t y p i c a l l y a r o u n d 15%, but t h e estimate o f t h e t o t a l t o r q u e i s s t i l l sensitive t o l a r g e e r r o r s in t h e estimate o f t h e F i g u r e 7 shows a r e radial surface torque. calculated p r e d i c t i o n assuming t h a t t h e local s k i n f r i c t i o n coefficient for the radial surfaces was double t h e value f o r t h e c y l i n d r i c a l surfaces. It can b e seen t h a t t h e p r e d i c t i o n d i f f e r s s i g n i f i c a n t l y f r o m t h e experimental data. It can be concluded t h a t , f o r t h i s t y p e o f geometry where t h e p r o p o r t i o n o f t o t a l t o r q u e d u e t o t h e r a d i a l surfaces i s small, t a k i n g t h e local r a d i a l s u r f a c e s k i n f r i c t i o n c o e f f i c i e n t t o b e equal t o t h e c y l i n d e r s u r f a c e s k i n f r i c t i o n coefficient i s a v a l i d a p p r o a c h t o p r e d i c t i n g t h e total torque.
T o t e s t f o r i n t e r a c t i o n s between t h e c y l i n d r i c a l a n d r a d i a l s u r f a c e elements, a stepped r o t o r was tested. T h i s r o t o r was o f 40mm a x i a l l e n g t h , w i t h lOmm a t a r a d i u s o f 107mm a n d 30mm a t a r a d i u s o f 116mm. T h e t o r q u e f o r t h e r o t o r was p r e d i c t e d b y t h e i n t e g r a t i o n o f t h e local s k i n f r i c t i o n c o e f f i c i e n t across t h e r o t o r surface, as before, a n d t h e p r e d i c t i o n was again f o u n d t o a g r e e v e r y closely w i t h t h e experimental data, w i t h n o detectable interaction effects. T h e above r e s u l t s show t h a t t h e t o r q u e f o r a n y complex r o t o r a n d h o u s i n g geometry w i t h i n t h e r a n g e t e s t e d can b e p r e d i c t e d by t h e simple a d d i t i o n o f t h e components o f t o r q u e d u e t o t h e r o t a t i n g s u r f a c e elements, t o g e t h e r w i t h a n e x t r a component o f t o r q u e t o allow f o r e n d effects. The influence o f the housing geometry o n t h e t o t a l t o r q u e o f t h e system i s minor a n d can be neglected. T h i s i s most c l e a r l y shown by t h e r e s u l t s f o r t h e system shown in F i g u r e 2 ( i i i ) w h i c h has b o t h a stepped r o t o r a n d a c i r c u m f e r e n t i a l g r o o v e in t h e housing. F i g u r e 8 shows v e r y close agreement between t h e measured t o r q u e r e s u l t s a n d t h e p r e d i c t e d values.
11.5
4
5
GENERAL A P P L I C A B I L I T Y
T o a p p l y these r e s u l t s g e n e r a l l y , it i s necessary t o know w h e t h e r a l l o f t h e e n d e f f e c t s a r e d u e t o t e s t rig effects, o r whether t h e y a r e in p a r t caused by flow e f f e c t s a r o u n d t h e edges o f t h e r o t a t i n g s u r f a c e . It i s clear t h a t a l l t h e e n d e f f e c t s a r e n o t caused b y t h e seal f r i c t i o n a n d c y l i n d e r e n d A 40mm l o n g r o t o r w i t h a 5mm s u r f a c e s alone. a n n u l a r g a p has a n e n d e f f e c t e q u i v a l e n t l e n g t h o f 58mm. A p p r o x i m a t e l y h a l f o f t h i s can b e e x p l a i n e d as d u e t o t h e r a d i a l s u r f a c e s a t t h e sides o f t h e r o t o r a n d t h e l i k e l y losses in t h e t h i c k fluid films o n t h e seal faces. T h e remaining e n d e f f e c t s a r e p r o b a b l y d u e t o t h r e e dimensional flow e f f e c t s w h i c h may be u n i q u e t o t h e t e s t r i g o r may b e g e n e r a l l y applicable
.
T h e o n l y way t o find o u t i s t o compare t h e t o r q u e p r e d i c t e d f o r a complex geometry w i t h measured t e s t r e s u l t s o b t a i n e d o n a completely d i f f e r e n t t e s t rig. T h i s comparison was made, u s i n g r e s u l t s f o r a piece o f t u r b o m a c h i n e r y made up o f t w o sections similar t o F i g u r e 2 ( i i i ) a n d a l o n g p l a i n c y l i n d e r o f 400mm a x i a l l e n g t h . It was f o u n d t h a t t h e p r e d i c t e d t o r q u e , a g r e e d w i t h t h e t e s t data t o w i t h i n 10% i f a l l t h e e n d e f f e c t s were assumed t o b e d u e t o seal losses a n d t e s t r i g e f f e c t s . However, t h i s level o f agreement between p r e d i c t i o n a n d experimental data i s n o t c o n c l u s i v e as t h e l o n g p l a i n c y l i n d e r compone n t swamped a n y i n f l u e n c e o f t h e e n d e f f e c t s T h e r e remains, o n the total torque. t h e r e f o r e , some e n d e f f e c t s w h i c h a r e a f u n c t i o n o f u n k n o w n e f f e c t s in t h e flow o v e r t h e r o t a t i n g surface, a n d it i s s t i l l n o t clear if t h e y a r e u n i q u e t o t h e t e s t r i g o r a r e g e n e r a l l y applicable. It i s t h e r e f o r e concluded t h a t t h e following g u i d e l i n e s w i l l p r o v i d e a reasonable a n d c o n s e r v a t i v e estimate o f t h e t o r q u e o n t h e r o t o r o f a piece o f t u r b o m a c h i n e r y .
(1)
O n l y t h e r o t a t i n g s u r f a c e s h o u l d be considered.
(2)
A mean v a l u e o f s k i n f r i c t i o n c o e f f i c i e n t should be applied to b o t h the radial and t h e c y l i n d r i c a l surfaces. T h e value Re-Oe2 s u g g e s t e d i s Cf = 3.16. w h e r e Re i s e i t h e r a f u n c t i o n o f R f o r t h e c y l i n d r i c a l surface, o r a v a r i a b l e f u n c t i o n o f r f o r t h e r a d i a l surface. These numerical values a r e mean values o f those d e t e r m i n e d f o r t h e d i f f e r e n t g a p r a t i o s a n d h a v e been t e s t e d a n d f o u n d t o g i v e good c o r r e l a t i o n .
(3)
It i s c o n s i d e r e d t h a t a n additional length o f cylindrical surface should be a d d e d as a n estimate o f t h e u n k n o w n effects, p r i n c i p a l l y d u e t o i n t e r f e r e n c e e f f e c t s adjacent t o t h e r o t o r c o r n e r s . F o r r o t o r s w i t h t w o major changes in r a d i u s a l o n g t h e i r a x i a l lengths, s u c h as those tested, a n additional l e n g t h o f 35mm a t a r a d i u s o f 107mm i s appropriate. For r o t o r s w i t h a
11.0
10.5
6.3 6 . 4 Fiqure 8
6.5
6.6
6.7
6.8
6.9
7.0 7.1 log ReR
Comparison of p r e d i c t e d a n d measured r e s u l t s f o r a stepped r o t o r a n d a q r o o v e d h o u s i n q
409
(4)
greater number o f step changes in radius then it is likely t h a t t h i s l e n g t h should be increased.
(9)
TAYLOR, G 1 . " F l u i d f r i c t i o n between r o t a t i n g c y l i n d e r s 1-torque measurements" Proc. Roy. SOC. Ser A., 1936, 157 p546
In addition, f o r the test rig used f o r the present work, it i s necessary t o add a f u r t h e r l e n g t h o f 35mm o f c y l i n d r i c a l s u r face t o allow f o r t h e f r i c t i o n due t o the seals and adjacent radial surfaces.
(10)
IPPEN, A T "Influence o f viscosity on c e n t r i f u g a l pump performance". T r a n s ASME, 1946, 68, p 823
(11)
KETOLA, H N McCrew, J M "Pressure, frictional resistance a n d flow characteristics o f t h e p a r t i a l l y wetted r o t a t i n g d i s k " Trans. ASME. Ser. F., 1968, 90, p 295 -
(12)
DAILY, J W, NECE, R E "Chamber dimension effects o n induced flow and frictional resistance o f enclosed r o t a t i n g disks". Trans. ASME. Ser D., 1969, 82, p 217
It is considered t h a t t h e e r r o r in t h e estimate o f torque obtained u s i n g the above procedure is unlikely t o be greater than 20%, f o r r o t o r s w i t h gap ratios in t h e range 0.05 t o 0.2, and w i t h similar ratios o f radial t o c y l i n d r i c a l r o t o r surface areas t o those o f t h e r o t o r s tested. 6
ACKNOWLEDGEMENTS
The author acknowledges t h e assistance o f the SERC and o f t h e T.I. Research Laboratories who sponsored t h i s work u n d e r a CASE Project. The author also wishes t o acknowledge the s u p p o r t a n d advice g i v e n by t h e s t a f f o f the Department o f Aeronautical Engineering at the U n i v e r s i t y o f Bristol.
References FLOYD, C G "A S t u d y o n frictional losses o f enclosed r o t o r s a t high Reynolds numbers" Ph. D. Thesis, U n i v e r s i t y o f Bristol, September 1982. PAI, S I " T u r b u l e n t flow between r o t a t i n g cylinders" NACA Tech. Note 892, 1943. USTIMENKO, B P e t al " T u r b u l e n t t r a n s f e r in r o t a r y flows o f an incompressible fluid". F l u i d Mech. Soviet Research, 1972, p 121
1,
YAMADA, Y "Torque resistance Of a flow between r o t a t i n g coaxial c y l i n d e r s h a v i n g axial flow". Bull. JSME, 1962, 5. p 635 SULMONT, P. BOURGET, P L "Mesure experimentale du couple de frottement d'un c y l i n d r e t o u r n a n t dans un c y l i n d r e f i x e p o u r des faible e n t r e f e r s e t des g r a n d s nombres du Reynolds". Annales de Mechanique, Ecole Nat. de Mech., Nantes, 1969. ROBERTSON, J M. "On t u r b u l e n t plane Couette flow", 6th Mid West Conference o n F l u i d Mech., Univ. o f Texas 1959. BILGEN, E, BOULOS, R "Functional dependence o f t o r q u e coefficient o f coaxial c y l i n d e r s o n gap w i d t h a n d Reynolds numbers". Trans. ASME Ser. I . 1973 95, p 122.
-
GORLAND, S H e t al "Experimental windage losses f o r close clearance r o t a t i n g c y l i n d e r s in t h e t u r b u l e n t flow regime". NASA TM X-52851, 1970
This Page Intentionally Left Blank
411
Paper Xlll(iii)
Turbulence and inertia effects in finite width stepped thrust bearings A.K. Tieu
The inertia effect in hydrodynamic thrust bearings are considered here with finite width stepped oil film profiles. In the analysis, the momentum and continuity equations are integrated across the oil film thickness, and combined to produce the modified Reynolds equation. The oil film pressure are obtained from the Reynolds equation by the finite element method, and corrected by a finite difference scheme. I . INTRODUCTION 1.1 Notations
In thin film hydrodynamic lubrication theory, the importance of inertia forces in the fluid film depends on the magnitude where U of the effective Reynolds number Re* ( Re* = is the slider velocity, Y kinematic viscosity, h film thickness and L bearing length). The inertia effects are significant when Re* exceeds unity. In normal bearing applications, the effective Reynolds number is so low such that the inertia effect is justifiably ignored in the analysis. There are increasing number of applications at high speeds and low viscosity fluids, where Re* is larger than unity. Moreover, cases with unsteady load or where bearing geometries exhibiting abrupt change in film thickness can give rise to large inertia forces. Thus load capacity of hydrodynamic bearings should be evaluated with inertia taken into account at high Reynolds number.
constants coefficient of inertia effects bearing width coefficient of pressure correction fri t'onal force on rotor surface Ef, Z - d T film thickness minimum film thickness step height inertia terms turbulent coefficients bearing length weighting function shape function local pressure,corrected pressure
9
The infinitely long Rayleigh step bearing is of fundamental importance to the theory of hydrodynamic lubrication, as it has the highest load capacity. As shown by Putre (1) for step bearing with no side leakage operating at Reynolds number up to 1000, there is a pressure drop across the step which increases the bearing load capacity . Up to now there are not many published results on finite width stepped thrust bearings. Other computational work such as those by Kennedy et a1 (2), King and Taylor (3), Launder and Leschnizer (4) , Constantinescu and Galetuse (5,6) dealt with a tapered oil film profile in journal or thrust bearings. Tieu (7) considered stepped slider bearing. Upwind finite difference method were used in (3,4), whilst finite element method was used in (2,6,7). It was reported in (6,7) that some convergence problems were encountered with the finite element scheme. For unsteady lubrication films, Sestieri and Piva (8) found that the inertial terms start to contribute significantly of the order 1 or higher. to the pressure for R e = The inverse problem to determine the influence of inertia for Reynolds number R e * = 0.1 was considered by Malvano and Vatta (9). Pasquantonio and Sala (10) evaluated the performance of an infinitely long slider bearing with combined effects of temperature, inertia and turbulence. In this paper, the inertia and turbulence effect are included in the analysis of finite width bearing with a stepped film profile. This involves the solution of a set of coupled, non-linear differential equation for three primitive variables Um , Vm and p, which are mean velocities and pressure respectively . They are solved in an iterative scheme by the finite element method, and results are obtained for cases of high ratios. Reynolds number and large
2
mean bearing pressure U, h,,,& Y
U, h 2 Y L
slider velocity in x-direction mean film velocity in x-direction, predicted and corrected value mean film velocity in y-direction, predicted and corrected value P h2
pu,L
cc P 7
coordinates in direction of sliding, along width, and across film height upwind constants coefficients for laminar or turbulent flow viscosity density shear stress
2. THEORY
The flow within the finite width stepped slider bearing in Fig.1 is governed by three-dimensional continuity and momentum equations. Since the film thickness h is in the order of times the bearing length , the pressure can be considered constant in the z-direction.
412 For finite width rectangular bearing with zero slider velocity normal to the direction of motion, the governing integral equations, momentum and continuity, are given in ( 5 ) as following :
this paper. The modified Reynolds equation has the following form :
subject to a prescribed pressure around the bearing boundary.
where T ~ ,7x,h ~ , are shear stresses in the x-direction on the slider and bearing surfaces respectively
At high Reynolds number, a fraction of the velocity head is converted to a pressure head at inlet . This ram pressure at inlet can be considered in the present simulation by prescribing the appropriate velocity head in the numerical scheme. Equation ( 4 ) is similar to the standard non-inertial Reynolds equation in lubrication, except for the last two terms on the right hand side, which are inertia terms. The flow in the x and y direction are obtained from :
This paper adopted the commonly accepted assumption (3,4,5) that the velocity profiles is not affected by inertia effects. The velocity distribution then take the usual parabolic form, which allows the evaluation of the above integrals in the inertia and shear stress terms. The integrals were reduced to the following forms: I x x = Q U L h + PU:h - YUmUoh I x y = (QUm -
I,, Tx,o
7x,h =
Y’U0)Vmh (3)
UO - -) 2
7 y . o - 7y,h =
=
V . { h- v3p
h + 6pu2m ddx
kr
- h V + p- Ih)2 2 kr
= 0
subject to :
PkY
TVrn
For laminar flow where R e .C 1000 Q
In short, the modified Reynolds equation can be written as :
= aVLh
h
3. GALERKIN METHOD AND FINITE ELEMENT FORMULATION
=
12
1.2,p = 0 . 1 3 3 , ~= 0 . 2 , ~ = ’ 0.1,6 = 0.133,kx = k,
For turbulent flow where R e 2 1000
+ 0.0136Re0.90 12 + 0.0043Re0.96
i. p = p o = Prescribed pressure on boundary S1
ii. qn = - n . [ $ V p + S2, n is normal unit vector
gI]
on boundary
kx = 12 k, =
Applying the Galerkin method, equation (6) becomes :
[v.($vp
from the mixing length theory ( 5 )
- hV
2 Q
= 0,p = 0.885Re-0.367,y =
=
0,6 = 1.95Re-0.43
The system of equation (1) for steady-state condition contains three unknowns Urn,V, and p, and it can be solved by one of the following methods. One method is to introduce stream function, which is solved together with pressure ( 3 ) . The second method is to solve for Urn,Vm and p in their primitive form (4). Both these methods made use of upwind finite difference technique. The third method employs Galerkin’s method in a finite element formulation to solve the single modified Reynolds equation, which is obtained by introducing explicit expressions of Urn,V, into the continuity equation (2,5). An appeal of the finite element method is its ease of implementing varying grid size close to the step. Similar technique is adopted in
+ Ph2 --I)LidA
=
0
kr
(7)
Integrating equation (7)by parts and using appropriate boundary conditions, the integral expression can be written as :
The shape functions has the standard form Ni = ai + bjx + ciy for linear triangles. The pressure, film thickness, velocity and inertia terms are linear combination of the nodal values. P = C N i p i ; h = CNihi ; I, = C N i I x i e t c...
(9)
In the upwind Galerkin method, the weighting function Li = Ni + Fi , Fi is parabolic function of Ni , and they can be found in Appendix 1 . Substitution of expression (9) into equation (8), and standard finite element procedure such as
413 integration of the individual terms and assembly of matrices can be found in reference (12). The weighting function Lj can be chosen to be the same as the shape function Niin the standard Galerkin’s method.
4. COMPUTATION
B/L=l,L=O.l06m,p = 4 - 4 2 ~ l O - ~ P a . s , + mrn
Table 1 and Friction Factor
Load Capacity
The following basic parameters were used in the computation
hsrep
The results is now applied to the finite width stepped bearing with hsrep m i n = 0 - 2 . The increase in load capacity and friction factor are shown in Table 1.
=
0.0056,
from 0 to 2.0.
The bearing pressure can be obtained initially from equation (4) with I, and I, = 0, which is then used to determine U,,,, Vm from equation ( 5 ) . The new pressure p is then computed from equation (4) with new I, and I, different from zero. The procedure is iterated until the final pressure converge to within a specified tolerance. This was used in reference (7) for tapered as well as stepped slider . The model compared very well with others (I ,3,4,6). The computational method adopted in (7) encountered convergence problem for stepped or tapered oil film profile at 2 0.003 and the Reynolds number Re exceeding 1000. It confirms similar problems reported by Constantinescu (6). This paper produces results for higher and Reynolds number than those in reference (7).
9
9
In reference (7), the finite element grid in the computation was varied
i. in grid spacings in the x- and/or y- direction ,
Case
Re,,,
W
F
1 2 3 4
2000 6000 9000 18000 54000
0.240 0.469 0.640 1.137 3.000
0.975 1.183 1.332 1.740 3.330
5
2
=
F
hsiep 0.0056, h , = 1.0
The pressure distribution from inlet to outlet of case 2, Table 1 are shown in Fig. 3 for a square step thrust bearing Due to inertia, the inlet pressure before the step increases above the non-inertial pressure, but decreases after the step. The drop in pressure at the step is not as sharp as is the case of infinite stepped thrust bearing. The pressure contours and distribution are shown in Figs. 4 and 5 . The increase in load and friction at high Reynolds numbers are shown in Fig. 6, where it can be seen that the increase in friction factor F is not as much as that of the load. The load capacity and friction for Re = 3870, based on
%
ii. in the number of grids from 9 x 9 (128 triangles) to 17 x 17 (512 triangles),
the outlet film thickness, and from 0.5 to 2.0 are shown in Table 2. The inertial and non-inertial solution are shown in Fig. 7.
iii. in the orientation of the triangles within the grid.
Table 2
These affected 5-6070 change in mean bearing pressure, but they did not improve the convergence of the numerical scheme. Results from the 17x17 grid in Fig.2 will be discussed from now on.
Effect of Step Heights on Load ani’ Friction hsrep
h,
-
W
F
0.330 0.469 0.496 0.485
0.995 1.183 1.357 1.582
The upwind Galerkin’s method was then used . This was found to produced some improvement to the results, such as delaying the divergence by 2 or 3 iterations. But satisfactory convergence could not be achieved.
0.5*
The method in reference (4) to correct the pressure field for the effect of inertia was found to give reasonable convergence. In this case, the pressure is obtained by the standard , V,,,are calcuGalerkin’s method. The mean velocities U,,, lated from non-inertial pressures. They d o not satisfy the continuity equation, and therefore result in residual mass sources . These residuals together with the LHS of equation (4) produce the pressure correction p’, which is then used to compute the velocity corrections u’ , v’, as shown .in Appendix 2. The process is repeated until the pressure fields do not change by 0.3% . In some cases, oscillations of the pressures behind the step occurred. This was smoothed partly by the upwind Galerkin’s method, but it could not be completely removed.
‘Re,,, = 6OOO
It was found that the standard Galerkin’s method and the evaluation of the mass source term by the central difference technique gave stable results and they were adopted in this paper.
1.0 1.5 2.0
It can be seen from Fig. 7 that compared with noninertial results, inertia produces larger change in load with lower step height, whilst higher increase in friction is found at larger step height.
5. CONCLUSIONS The inertia effects is significant for finite width step thrust bearing when the Reynolds number exceeds 1OOO. The bearing pressure and friction increases with Reynolds number although the friction does not increase as much as pressure. The increase in pressure and friction is higher than those
414
for tapered film profile. For finite width stepped bearing, there is some pressure drop at the step, and it is not as steep as for the case of infinitely wide bearing . The inertia effect produces larger increase in load and smaller change in friction at lower step height.
6. ACKNOWLEDGEMENT The author acknowledges the financial grant from the University of Wollongong for this project.
APPENDIX 1
References 1.
2.
3.
4.
PUTRE,H.A. ’ Computer Solution of Unsteady Navier Stokes Equations for An Infinite Hydro-dynamic Step Bearing’ NASA TN D5682 1970 KENNEDY, F . E., CONSTANTINESCU, V. N. and GALETUSE, S. ’ A Numerical Method for Studying Inertia Effects in Thin Film lubrication’ Proc. of the 1975 Symposium (Leeds-Lyon) Super Laminar Flow in Bearings, Inst. Mech Eng Publication London 1977 pp174-180 KING,K.F. and TAYLOR,C.M. ’An Estimation of The Effects of Fluid Inertia on The Performance of the Plane Inclined Slider Thrust Bearing with Particular Regard to turbulent Lubrication ’ ASME Journal of Lubrication Technology, Vol 99 No I 1977 pp129-135 LAUNDER B.E. and LESCHZINER,M. ’Flow in Finite Width, Thrust Bearings Including Inertial Effects I. Laminar Flow’ ASME Journal of Lubrication Technology, July 1978 VOI 100 ~330-338
WEIGHTING FUNCTIONS FOR LINEAR TRIANGLES The weighting functions of the Upwind Galerkin method is given by :
with the following properties : i. ii. ...
111.
c:=, F~ = o F;(Ni = 0) = 0 I Fi(N; = 0) I = 3 a; Nk N;(i # j # k )
From reference (1 I), the weighting functions for linear triangle are given as following : F1 = 3 ( f f 2 N 3 N 1 - a 3 N 2 N l ) F2 = 3 (a3 N I N2
5.
6.
7.
8.
9.
CONSTANTINESCU, V. N. and GALETUSE, S ’ On The Possibilities of Improving the Accuracy of The Evaluation of Inertia Forces in Laminar and Turbulent Films’, ASME Journal of Lubrication Technology, Vol 97, NO 1 1974 ~ ~ 6 9 - 7 9 CONSTANTINESCU,V.N. and GALETUSE,S. ’Operating Characteristics of Journal Bearings in Turbulent Inertial Flow’ ASME Journal of Lubrication Technology, April 1982 VOI 104 ~173-179 TlEU A.K. ’Inertia Effects In Finite Width Stepped Hydrodynamic Thrust Bearing’ Proc. Computational Techniques and Applications Conference, Melbourne Australia, August 1985, Eds J . Noye and R.May SESTIER1,A. and PIVA,R. ’ The Influence of Fluid Inertia in Unsteady Lubrication Films’ Trans. ASME, J.Lubrication Technology, April 1982, Vol 104 ,pp180186. MALVAN0,R. and VATTA,F. ‘The Influence of Fluid Inertia in Steady Laminar Lubrication’ Trans. ASME, J. of Lubrication Technology, Jan. 1983 Vol 105,pp7783
10.
DiPASQUANTONIO F. and SCALA R. ’Influence of the Thermal Field on The Resistance Law in The Turbulent Bearing Lubrication Theory’ Trans. ASME, J.Tribology, Vol 106 July 1984, pp368-374.
11.
HUYAKORN P.S. ’Solution of Steady State, Convective Transport Equation Using An Upwind Finite Element Scheme’ Appl . Math . Modelling, March 1977, VOI I , ~ ~ 1 8 7 - 1 9 5
12.
HUEBNER,K.H ’The Finite Element Methods for Engineers’ John Wiley and Sons, New-York 1975
(2)
-
(YI
N3 N2)
(3)
F3 = 3 ( C Y ~N2 N3 - a2 Nl N3)
2‘
The sign of C Y ~depends on v = (Vi + 5 ). li; where li, is direction vector,and V; , V, velocity vectors at nodes i and j respectively. a > Oif v > O
-= O i f
a
v
APPENDIX 2 CORRECTIONS FOR VELOCITY AND PRESSURE The pressure and mean velocities are given by : Pm
=F+P’ -
urn = urn+
u’
-
vm = v, + v’ where on the right hand side, the first terms are the predicted values, and the second terms the corrected values It was shown in reference (4) that the corrected values u’ and v’ can be approximated by the following :
v’p =
D V @‘s - P ’ P ) I V
“ P
where A p includes inertia effects, and subscripts S,E,P are the south,east and centre of the upwind difference grid.
415
Fig. 1 Finite Width Stepped Thrust Bearing
- - - non-inertial
-inertial
Fig. 4 Pressure Contours of Finite Width Stepped Bearing with Inertia Same Data as Fig. 3
Fig. 2 Finite Element Mesh
-
r
P
1.0
0.5
Fig. 3 Pressure From Inlet To Outlet
Re,,,
(b) inertial pressure
hsrrp
= 6000,T=1.0
Fig. 5 Pressure Distribution of Finite Width Stepped Thrust Bearing
416
W
IF
0.6
1.5
0.1
0.2 lo
~
~
50 ~
~
1
0
0 Fig. 6 Percentage Increase in Load and Friction with Reynolds Number for
%
=
1
1.o
~
0.5
1.0
1.5
2.0
sbdh Fig. 7 Effect of Step Height on Load and Friction
417
Paper Xlll(iv)
A theory of non-Newtonian turbulent fluid films and its application to bearings J.F. Pierre and R. Boudet
A modified k - E model f o r d e t e r m i n a t i o n of turbulent non-newtonian s t r e s s e s has been used to provide velocity profiles in Poiseuille-Couette flow. Generalized functions a r e found f o r t h e nonnewtonian turbulent Liscosity c o e f f i c i e n t s Cx a n d CZ , which lead us to formulation of t h e generalized
Reynold equation which is valid both for newtonian and non-newtonian fluids, in laminar o r non-laminar flows. Similarly, approached functions f o r t h e s t r e s s e s n e a r t h e wall a r e obtained. T h e Reynolds equation is then resolved f o r t h e case of f i n i t e plain bearings in turbulent regime. The results confirm t h e interest of shearthinning fluids regarding t h e reduction of power losses. I INTRODUCTION The i n c r e a s e of r o t a t i o n speeds of shafts, t o g e t h e r with t h e use of very low kinemat i c viscosity fluids as lubricants, have, for a l r e a d y a long time, m a d e i t necessary to a c q u i r e a theoritical knowledge of turbulent newtonian flows in bearings. Several approaches h a v e succeeded in rnodelizing newtonian turbulent bearings [I]. S o m e of t h e m o s t used a r e t h e law of t h e wall [2], t h e P r a n d t l mixing length model [3], t h e bulk flow theory [41, t h e energy model of t u r b u l e n c e [51. They h a v e q u i t e well converged to t h e formulation of a single generalization of t h e laminar Reynolds equation introducing turbulent viscosity coeffic i e n t s CX and GZ. Nowadays, t h e a i m e d technological improvements of machines lead to f u r t h e r demands regarding bearings, which should provide higher maximum rotation speed, t o g e t h e r with increased s t a t i c and d y n a m i c p e r f o r m a n c e s o v e r a wide r a n g e of r o t a t i o n speeds.
1.1 N o t a t i o n Pressure Space coordinates Laminar s h e a r s t r e s s Shear s t r e s s at t h e wall Shear r a t e Velocity c o m p o n e n t s Velocity of t h e mobile wall o r s h a f t s u r f a c e speed Film thickness Radial c l e a r a n c e Laminar non-newtonian viscosity Turbulent non-newtonian viscosity
Usual lubricants d o not m e e t t h i s new demand because of t h e very high. power losses, which no longer allow speeds to increase.
Non-newtonian exponent Fluid density
Therefore, w e have to use non-classical lubricants, which a r e pseudoplastic fluids, t h e power losses reducing properties of which h a v e been experimentally verified by Nicolas [ 6 ] in 1979 using C u a r and Polyox in plain bearings. Turbulent flows of non newtonian fluids have been studied e a r l i e r in smooth round t u b e s by Dodge [7] and Bogue [S]. These studies w e r e not a b l e to e n t a i l t h e development of a single theory for turbulent flows of non-newtonian fluids. Our approach, based o n a n e n e r g e t i c a l description of t u r b u l e n c e and on a power law model for t h e c o n s t i t u t i v e equation of fluids, leads us to t h e formulation of a t h e o r y generalizing newtonian and laminar cases [9].
K i n e t i c e n e r g y of turbulence Dissipation r a t e of turbulence Non-newtonian Reynolds number equals to v (1-n) P 2 THEORY \c/
=(-\
T h e behavior of shearthinning fluids is r a t h e r well described by t h e power law model which a l s o allows simple comparisons with t h e newtonian case for n = 1. The behavior of shearthickening fluids is obtained f o r n > 1. I t gives t h e shear s t r e s s T as a function of t h e shear r a t e 6 :
418
(1
1
T
,
where
~
.n
V
m =
non-newtonian viscosity
n =
non-newtonian exponent o r index.
The turbulent m o v e m e n t f o r such fluids satisfies t h e equations of motion obtained by 0. Reynolds, with t h e usual hypothesis in thin films. They a r e :
6~
6
-=-6 x
6 y
6,
-=
[
Txy-
k and E , which a r e respectively t h e kinetic e n e r g y and t h e dissipation r a t e of turbulence, a r e given by t h e following system, w h e r e t h e nonnewtonian influence a p p e a r s only in t h e velocity gradient term :
1
p u'v'
0
6~
62
by
6 -
p U ' V ' and - p v'w'are t h e Reynolds
stresses. They c a n b e approached by t h e k model for t h e newtonian case [ l o ] :
-pu'v'=vT
-
6Y
E
[(P 33 +
6u -
6~ (3)
f l , f 2 , f p a r e functions introduced to m a k e t h e model valid f o r t h e laminar sublayer n e a r t h e wall :
For power law fluids, and f o r homogeneity reasons with laminar shear stress, w e suppose t h e y c a n b e w r i t t e n as :
(4)
{
f 2 = 1-e -PV'W'
Rt
*
= vT(-)n
pkY
I
vT=
'h
with R k = p m T = p Cp fp
k2
E
m
and
Rt
=-
Pk2 m~
419 Assuming m T
is
known as a function of
3-1 R e s u l t s
y, t h e equation of motion c a n b e i n t e g r a t e d o n c e to yield to t h e velocity derivatives. A Taylor development is t h e n necessary to linearize t h e s e expressions, which c a n then b e i n t e g r a t e d o n c e again t o lead to :
0.10E.03
0. IOE'04
0.lOE.O
1
Fig.] C a l c u l a t e d GX for t h e newtonian case (n = 1) and d i f f e r e n t pressure g r a d i e n t s c o m p a r e d to t h e approached function of t h e turbulent new t o n ran t h e o r y with
I(y
Writing t h e equation of continuity, then q u i t e e a s y to o b t a i n t h e generalized Reynolds equation:
it
is
V
6h
Where Gx and GZ a r e t h e turbulent non-newtonian viscosity co.2fficients and depend on 1 a n d J. 3 PKOCEDURE AND RESULTS
Details of t h e m e t h o d of resolution have been described in [9] and [ I I]. Considering t h e Poiseuille-Couette flow, t u r b u l e n c e equations a r e resolved by a half implicit method, consisting into impliciting t h e t e r m s which will increase t h e diagonal dominance of t h e matrix. Then I and J a r e e v a l u a t e d and velocity profiles a r e obtained from equations (6). These s t e p s a r e i t e r a t e d until a convergence c r i t e r i o n is satisfied. Together with velocity profiles, t h e c o e f f i c i e n t s GX a n d and t h e s h e a r s t r e s s at t h e wall T,, a r e GZ calculated.
Fig.2 Calculated GZ f o r t h e newtonian case (n = 1 ) and d i f f e r e n t pressure gradients compared to t h e approached function of t h e turbulent newtonian t h e o r y
C a l c u l a t e d Gx and GZ f o r d i f f e r e n t non-newtonian indexes a r e plotted versus nonnewtonian Reynolds number in logarithmic coordinates - W e h a v e been a b l e to approach t h e c a l c u l a t e d c u r v e s by linear functions of Ln (Ren). The s a m e is done f o r shear s t r e s s e s at t h e wall.
Fig.3 C a l c u l a t e d GX f o r n = 0.9 and linear approximation obtained
420
below : n
I
1/G,
12
+ 0.0136 R e
0.90 0.80
0.9
12 + 0.0045 R e n
0.8
12 + 0.0026 R e n
0.7
12 + 0.0015 R e n
0.6
0.64 1 2 + O.OOp9 R e n 1 2 + 0.0007 R e n 0.59
0.5 n 1
0.74 0.69
l/GZ
12
t
0.0043 R e
0.96 0.87
0.9
1 2 + 0.0022 R e n
Fig.4 Calculated GZ for n = 0.9
0.8
12
and linear approximation obtained
0.7
12 + 0.0011 R e n
0.6
I 2 + 0.0008 R e n
0.67
0.5
12 + 0.0006 R e n
0.61
+ 0.0016 R e n 0.79 0.73
Shear s t r e s s e s at t h e walls
T
can
w r i t t e n as follows : Tw=f--
introduces
6,
h
6,
2
+
To
T h e turbulent newtonian t h e o r y pc as defined below +
6,
h
6x
V
-
TW=----+PC
2
h
Similarly, w e c a n define non-newtonian case :
p c for t h e
Fig.5 Calculated GX for n = 0.8 6x
a n d linear approximation obtained
Calculated figures 7 to 12
2 lo/a ,, nd
k a r e shown in
Fig.6 C a l c u l a t e d Gz f o r n = 0.8 I
I
and linear approximation obtained G
X
a n d Gz will now b e approximated,
f o r t h e d i f f e r e n t non-newtonian indexes n to t h e respective approached functions found and given
Fig.7 C a l c u l a t e d
p c f o r t h e newtonian case
(n = 1) c o m p a r e d to t h e approached function of t h e newtonian turbulent t h e o r y
0 . IOE.03
Pc-
I
0.lOE.OZ
O.IOE*OI
,
,
,
. , , . .. ,
0.lOE.OC
Fig.8 Calculated
,
. . . . , . ,. 0 . IOE.07
,
,
. . . , ... O.lOE+OI
pc for n = 0.9 and linear
Fig.11 Calculated lo/,, for n = 0.6 and linear approximation obtained
approximation obtained
Fig.9 Calculated
p c for n = 0.6
Fig. 12 Calculated newtonian indexes
lo/,,
lo/,,
for different non-
The approached functions found for pc, and for different nonand
newtonian indexes a r e given below :
n
1
-
1.o
0 . IOE'Oh
0.
IOL.07
Fig.10 Calculated lo/,, for n = 0.9 and linear approximation obtained
4
0.9
5.7 x
Ren
0.8
2.5 x
Ren
0.7
1.8 x
Ren
0.6
2.0 x
Ren
0.5
3.1 x
Ren
n 1.o
0.9 0.8 0.7 0.6
'1.39 1.16
0.84 0.55 0.39
PC 1 + 0.0012 R e
0.94
0.79 I + 0.0012 Ren 0.54 I + 0.008 Ren 0.4 1 + 0.024 Ren
-
422
3.2 Application to plain bearings
The functions a r e used for GX and Cz n :0,9
in t h e above mentioned Reynolds equation, which IS solved by a variationnal method proposed f i r s t by Bayada [ 121. The boundary conditions a r e t h o s e of Reynolds, which a r e still valid f o r turbulent newtonian flows, as established in [13]. The friction t o r q u e is c a l c u l a t e d using t h e previously given approached functions of t h e shear s t r e s s at t h e wall.
n :0.8
W e h a v e done simulation runs for a bearing in turbulent r e g i m e defined as follows : S h a f t radius : 0.05 m Radial C l e a r a n c e :
3 0,l
m
0,2
0,3
0.4
0,s
0,0
0.7
0,s
0,9
E
Fig.15 Non-newtonian influence upon t h e flow r a t e
Rotation speed : 19.000 rpm
h
z
Bearing with : 0.1 m Fluid viscosity : 0.003 PI
v
E
'
%
30-
Y c 0 I .
C
b
aJ 3 30000..
i
.-0
25000-
x
U
20000U
x
=
f o r - E = 0,3
25 ..
0 .+ c ;. 20.-
m
5
n: I
15000..
n:0,9
L 15..
n:O,9
n:0,8
R o t a t i o n speed (rpm) Fig.16 Influence of t h e r o t a t i o n speed upon t h e f r i c t i o n torque. Comparison b e t w e e n newtonian and non-newtonian case Fig.13 Non newtonian influence upon t h e hydrodynamic w e a r
4. DECUSSION
h
z
A
For n = 1 c a l c u l a t e d GX, GZ and
t
pc
a r e in good a g r e e m e n t with t h e well-known f o r m u l a s of t h e newtonian turbulent theory. Non-newtonian GX a n d GZ depend slightly on t h e pressure gradient as in t h e newtonian case. The influence of t h e pressure gradient on p c b e c o m e s i m p o r t a n t as t h e non-newtonian index n decreases.
A
n-0'9
0.1
0,2
0,3
0.4
0,s
0.6
Fig.14 Non-newtonian influence upon t h e friction t o r q u e
0,7
0,8
0,9
C
Thus, as soon as n b e c o m e s smaller t h a n 0.7, i t is n o m o r e possible for t h e c o m p u t e d Ln ( p,) to b e adjusted to linear model, w h a t e v e r t h e pressure gradient is. This is why, a n d use t h e s e w e had to a p p r o x i m a t e approached functions to c a l c u l a t e t h e friction t o r q u e for small non-newtonian indexes. Solving t h e Reynolds equation and using t h e approached functions provides us with t h e c h a r a c t e r i s t i c s of plain bearings lubricated by newtonian o r non newtonian fluids in turbulent regime.
423
The figures presented show t h e interest of non-newtonian pseudoplastic fluids regarding t h e reduction of t h e friction torque. For instance, the friction torque in a bearing decreases tenfolds when using a non-newtonian lubricant of n = 0.8 instead of a conventionnal newtonian one, and this for a n identical wear capability. 5. CONCLUSION
This study does not mean to solve every problem risen by non-newtonian turbulent lubrication, about which many questions remain unsolved - Nevertheless, i t is hoped to provide here a basis for debating and further researching in order t o improve t h e theory proposed here. The first s t e p needed to go ahead is undoubtedly a comparison to experimental results, so as to appraise t h e grade of precision obtained with t h e theory described in this paper. Unfortunately, it was not found possible to achieve this goal using Nicolas's experimental data [6], since n and m, which vary according to the preparation method of t h e fluid, were not available. References (I)
VINAY KUMAR "Plain hydrodynamic bearings in t h e turbulent regime - a critical review", WEAR, Vol 72, nol, 1981, p 13.28.
(2)
ELROD H.G and NG C.W. "A theory for turbulent fluid films and i t s application to bearings" Trans. A.S.M.E., Journal of Lubrication Technology, Vol 89, n03, p.346-362, 1967.
(3)
CONTANTINESCU V.N., "On turbulent Lubrication", Proc. Inst. Mech. Eng., Vol 173, n"38, p 881-899, 1959.
(4)
HIRS. "A bulk flow theory for turbulence in Lubricant films", 3. of Lub. Tech., Vol 95, n02, p 137-146, 4/73.
HO M.K. and VOHR J.H., "Application of energy model of turbulence to calculation of Lubricant flows", J. of Lub. Tech., Vol 96, nol, p 95-102, 1974. (5)
NICOLAS D., "Les rkgimes non Laminaires e n lubrification, rdduction du f r o t t e m e n t par addition d e polym&es", T h k e Docteur-Ingdnieur, Universitd Claude BERNARD, Lyon, 17 Septembre 1979. (6)
(7)
DODGE D.W., "Turbulent flow of nonnewtonian fluids in smooth round tubes", Thesis presented to t h e University of Delaware, at Newark, Del., in 1958, in partial fulfillment of t h e requirements for t h e degree of Doctor of Philosophy. (8)
BOGUE D.C., "Velocity profiles in turbulent non-newtonian pipe flow", Thesis presented to t h e University of Delaware, at Newark, Del., in 1960, in partial fullfillment of t h e requirements for t h e degree of Doctor of Philosophy. PIERRE J.F., "Ecoulements turbulents d e fluides non-newtoniens e n films minces", T h k e d e Doctorat, Ecole Nationale Supkrieure d'Arts et Mdtiers, Paris, 13 Mars 1986.
(9)
(10) LAM C.G.K. and BREMHORST K.,
"A modified from of t h e K model for predicting wall turbulence", Trans. of t h e A.S.M.E., J. of fluid Engng, Vol 103, Sept 81, p 456-460. (11) PIERRE J.F. et BOUDET R., "Turbulent flow of non-newtonian fluid in thin films",
Proc. of t h e 4th Conference on Numerical methods in laminar and turbulent flow, Swansea, 9-12 July 1985, pages 235-243. ( 1 2) BAYADA G., "Indaquations variationnelles
elliptiques aux conditions aux limites pbriodiques. Application i I'dquation d e Reynolds". Doctorat d e specialit&, Universitd C. BERNARD, Lyon, 1972. (13) VINAY KUMAR, "The Reynolds boundary conditions : should they be used during turbulent hydrodynamic lubrication", WEAR, VOI 65, 1981, p 295-306
This Page Intentionally Left Blank
SESSION XIV BEARING ANALYSIS Chairman: Professor H. Marsh
PAPER XIV(i)
A new numerical technique for the analysis of lubricating films. Part I: Incompressible, isoviscous lubricant
PAPER XIV(ii)
The boundary element method in lubrication analysis
PAPER XIV(iii) Thermohydrodynamic analysis for laminar lubricating films PAPER XIV(iv) The lubrication of elliptical contacts with spin
This Page Intentionally Left Blank
427
Paper XIV(i)
A new numerical technique for the analysis of lubricating films. Part I: Incompressible, isoviscous lubricant C.H.T. Pan, A. Perlman and W. Li
This paper describes a new algorithm, which produces high resolution results, for the numerical computation of pressure and flux profiles in a lubricating fluid-film. The new method avoids the use of elementary functions to approximate the pressure field. Natural functions are generated by performing a directional integration of the pressure-flux relationship along a mesh line in the discretized domain. Pressure profiles which accurately depict details can be produced with a coarse computational mesh. Illustrative examples are given for a flat slider and a journal bearing. E
eccentricity ratio of journal bearing
A high resolution of film pressure is often
e
azimuthal coordinate
desired in engineering studies. Critical issues may be related to the determination of load capacity, the estimation of lubricant heating, the calculation of minimum surface separation, the analysis of elastohydrodynamic deformation, and the description of film rupture. Adaptations of the conventional Finite Difference Method (F.D.M.) and the Finite Element Method (F.E.M.) are common practices in studies of hydrodynamic lubrication (1,Z). Use of a fine computational mesh is usually mandatory if good numerical accuracy is desired. The proposed new method is capable of producing very accurate numerical results without being dependent on the use of fine computational meshes. By performing directional integration of the relationship between pressure gradient and flux locally, a high resolution algorithm is derived and given the name Local Partial Differential Equation Method (L.P.D.E.M.).
-
1
INTRODUCTION
Q
dimensionless film flux vector
'PX
x-component of dimensionless film flux
z
dimensionless time
2
THEORY
Conventional F.D.M. and F.E.M. are dependent on the use of elementary functions which may not be most suitable to describe the details of the desired fields. In the derivation of the new algorithm, emphasis is placed on a practical procedure to compute the numerical relationship between the pressure field and the flux components according to the theory of lubrication, which is presented as: - aH Divergence Statement: div Q t - = 0 (1) az
1.1 Notation Flux Formula:
5
=
-H3 grad P t % ( 2 )
A, B
constants of local flux field
CsD
constants of continuous flux field
H
dimensionless film thickness
,I
mth moment integral of the nth reciprocal power of H
LID
length/diameter ratio of journal bearing
2 . 1 Directional Integration of the Local P.D.E.
P
dimensionless film pressure
V
unit vector along the sliding direction
"X
directional cosine between x and sliding
X
local coordinate in arbitrary direction; also, abscissa of Cartesian system
Y
ordinate of Cartesian system
2
axial coordinate
It is presumed that a resonable mesh size can be selected according to the knowledge of the field property of the squeeze term in Equation (1) and the geometrical parameter of the bearing (e.g. the lengthldiameter ratio of a journal bearing), so that the flux components are smoothly varying in the discretized domain. The primary approximation to be imposed is that the flux components are adequately represented by truncated Taylor expansions. Consider an arbitrarily oriented mesh line, on which the distance x is measured from a local
-
v
Where is the unit vector aimed along the sliding direction. The flux formula used here is retricted to an isoviscous film. If temperature variation is to be considered, then factors to account for velocity distortion due to viscosity variation would have to be added (3).
428
origin. The linear Taylor expansion of flux is: cpx=A+Bx
(3)
where A and B are local constants. The proposed pressure profile generating equation in the x-direction is obtained by eliminating 'px between Equations ( 2 ) and (3): dP = Vx(1/H2) - A (l/H3) - B (x/H3) (4) dx Since A and B are local constants, Equation ( 4 ) can be integrated in a closed form:
-
Computation of flux and pressure profiles between adjacent mesh nodes. Algebraic details of the first step depend on the discretization scheme. In the following, the orthogonal rectangular mesh system will be used to obtain illustrative examples. (4)
3.1 Pressure and Fluxes at Mesh Nodes Consider a local rectangular domain centered at the intersection of two mutually perpendicular mesh lines. A central approximation of the divergence operator may be derived for the uniform rectangular mesh by using spatial derivatives of the flux fields given by Equation (3). Thus, in coordinates (x,y)
Subscript "k" refers to the local origin situated at the k-th mesh point and the functions ,I are the exact integral from the local origin: Im(x)
=
(xm/Hn) dx ik
(6)
Accurate numerical treatment of these integrals is an essential part of the new algorithm in its practical implementation. For the theoretical film profile of an eccentric journal bearing, it is possible to apply the method of Sommerfeld to calculate their precise values (4). However, to make available a more versatile procedure, which may be applied to any sectionally smooth film profile, it is proposed that the mesh interval be divided into four subintervals, and the approximation of a polygonal periphery to span the subintervals be assumed. These integrals can be thus replaced by simple algebraic expressions as shown in the APPENDIX. This approximation is consistent with the discretization scheme. The local constants are linked to the discretized P-field through the "exact finite integrals".
where "c" denotes values at the central point or the local origin. Upon substitution of Equation (7) for Bx and By, a discretized approximation of Equation (1) is established, involving the local origin and its four adjacent points. The five point algorithm presented above can be written for all internal mesh nodes to form a complete, linear, non-homogeneous, algebraic system to define the discretized P-field. The system is blockwise tridiagonal, and can be readily solved by standard efficient matrix methods. Upon solving the discretized P-field, the local constants, A and B, can be calculated according to Equations (7) at all internal mesh nodes. At a boundary node, Equation (7) can only be used to find B along the boundary. However, Equations (1) and (8) can then be used to find B across. Afterwards, Equation (5) is applied toward the interior to obtain an equation, which is used for the calculation of A across. The set of values of A at all mesh nodes is the discretized cpx-field. 3.2 Computation of Profiles
2.2 Calculation of Local Constants The local constants, A and B, are determined by the increments of film pressure from the local origin to either adjacent mesh node. Thus,
+[:
- +
I13
I13-*13 - +
I03
I03-IO3 -I03
Equation (3) does not assure continuity between Ak and Ak+l-Bk+l(fbX). Consequently, there is a corresponding amguity in the pressure profile as given by Equation (5) and two distinct values of dP/dx exist at every internal mesh node. In order to construct a smooth pressure profile, it is necessary to replace Equation ( 3 ) by
A
'pcorr=
r
+
C x
+ D.x2
(9)
1
where superscript "+" or "-" defines the upper limit of integration of Equation (6) to be k+l. 3 COMPUTATION PROCEDURE The required computation procedure of the lubricating film involves the following steps: ( 1 ) Computation of the pressure field at discretized mesh nodes. (2) Computation of flux components at internal mesh nodes. ( 3 ) Computation of the cross flux at boundary nodes.
The coefficients C and D are selected to ensure continuity of P and of 'pcorr. That is,
+ 102vx
+ +
- 103Ak
-
113ck
+
- 123Dk
(11)
123 is to be computed by the general procedure previously described for other.,I The pressure at a point between mesh nodes can now be calculated as
(12)
429
4 EXAMPLES 4.1 Flat Slider
A preliminary trial of the new algorithm was applied to a flat inclined slider. This served as as a convenient model problem since its geometry naturally suggests the use of the formulas given in the APPENDIX. The flat slider considered has the following parameters: Length/Width Ratio = 1.0 Inlet/Exit Gap Ratio = 9.0 Calculations were made with a coarse mesh ( 6 ~ 6 ) ~ z and also with a fine mesh (18x18). An encouraging experience from this example Figure 1. Film Pressure of Finite Journal Brg is the relative insensitivity of the result on 16x8 L.P.D.E.M., L/D=l, ~=0.9 the mesh size. Since all nodes of the coarse mesh are repeated in the fine mesh, one can compare the computed pressure values at these nodes directly. The ratios of the computed film pressure using the coarse mesh to that using the fine mesh at all common mesh nodes are shown in Table 1. Indices "i" and "j" respectively mark nodes along and across sliding. The entrance of of the slider film is at (i=O) and the centerline is at (j=3). The biggest discrepancy is a modest 5.9%. It is of interest to note that the coarse mesh calculation onsistently yields a lower value of film pressure than that from the fine mesh calculation. The residual inaccuracy in film pressure exhibits a second order trend. Figure 2. Azimuthal Flux of Finite Journal Brg 16x8 L.P.D.E.M. , L/D=l, ~ 0 . 9 1.000 1.000 1.000 1.000 Table 1.
1.000 0.972 0.974 0.974
1.000 0.973 0.972 0.971
1.000 0.970 0.969 0.969
1.000 0.960 0.965 0.967
1.000 0.941 0.962 0.968
1.000 1.000 1.000 1.000
Mesh Size Sensitivity of L.P.D.E.M. Square Flat Slider, 9:l Gap Ratio
4.2 Finite Length Journal Bearing The journal bearing of finite length was used as another model to evaluate the new method for two dimensional calculations. Specific examples are for ~ 0 . 9 0 and L/D=l. Mesh sizes ranging from 8x8 to 16x8 were used. Results calculated with the 16x8 mesh using the L.P.D.E.M. are shown in Figures 1, 2, and 3. Results from coarser mesh calculations were substantially similar. Direct comparison of the centerline pressure profile is shown in Figure 4 between 8x8 and 16x8 computations. The profile of the 8x8 computation is remarkably accurate, not only for the level of the peak, but also for its overall shape. It is especially noteworthy that the highest mesh node film pressure of the 8x8 calcuation is only onethird of the peak pressure. The journal bearing was also calculated by the standard F.D.M. using various meshes. There is much more dependence on mesh size of both the level and the profile shape of film pressure. An interesting tell-tale symptom of computational pathology is the rather bizzare shape of the azimutahl flux as shown in Figure 5. An abrupt dip is seen on either side of the location of the minimum gap. Some improvement is gained by mesh refinement, but the anomalous tendency remains evident even in the 16x8 calculation. There is no trace of similar difficulty in L.P.D.E.M. results.
Figure 3. Axial Flux of Finite Journal Bearing 16x8 L.P.D.E.M., L/D=l, ~=0.9 a
'A
4
a
16x8
R I 0
t
I -4
-
Figure 4. Centerline Pressure, Finite Journal Brg, L.P.D.E.M., L/D 1, E = 0.9 Thus, numerical evidence strongly suggests that a coarse mesh calculation by L.P.D.E.M. can achieve a level of precision which is very close to the corresponding asymptotic limit. Using a conventional F.D.M., similar accuracy can be attained only by a combination of mesh refinement and extrapolation.
430
7 ACKNOWLEDGEMENT Partial support from the Digital Equipment Corporation is gratefully acknowledged. References /
z
(a) 8x8 Mesh
(b)
16x8 Mesh
Figure 5. Azimuthal Flux of Finite Journal Brg Calculated by F.D.M., L/D=l, ~ 0 . 9 5 CONCLUSIONS A new method for the numerical analysis of an incompressible lubricating film has been demonstrated. Based on the illustrative example of a flat slider of finite width, and a comparative study against the F.D.M., using a finite length journal bearing as the model problem, L.P.D.E.M. is seen to possess three important relative advantages. The same relative advantages are believed to hold against the F.E.M. since the same fundamental issues are involved. They are: (1) Accurate results can be obtained with a very coarse computational mesh. (2) Accurate flux fields are obtained. ( 3 ) Intermesh profiles can be generated.
(1) Raimondi, A.A. and Boyd, J. 'A Solution for for the Finite Journal Bearing and Its Application to Analysis and Design', Trans. ASLE, 1958, 1, 159-209. (2) Reddi, M.M. 'Finite Element Solution of the Incompressible Lubrication Problem', Trans. ASME. ser. F, 1969, 9 l , 524-533. ( 3 ) Dowson, D. and Hudson, J.D. 'Thermo-hydrodynamic analysis of the infinite sliderbearing, I. The plane-inclined sliderbearing', Instn. Mech. Engrs., Proc. of the Lubr. and Wear Group, 1964, 34-44. (4) Somerfeld, A. 'Zur hydrodynamischen Theorie der Schmiermittelreibung', Z. Math. Phys., 1904, 50, 97-155.
APPENDIX There are four combinations of (m,n) needed of the integral ,I as defined by Equation (6). The closed form algebraic expressions of them based on the subdivided secant approximation of the film thickness profile are derived as follows. The secant approximation of the film thickness profiles allows one to write dx = (6x/6H) dH
6x and 6H are increments of the local coordinate and the film thickness of the secant segments. It can be shown that, with the lower limit of integration placed at the vertex x of the secant segment, integrated formulas of Eqaution (6) are
6 PROSPECTS OF FURTHER DEVELOPMENTS Although present examples are based on the field equations which are restricted to films of uniform viscosity, generalization to treat the effects of cross-film viscosity variation is a matter of rounding out details. The same can be said about extending L.P.D.E.M. to be used with non-rectangular mesh setups. The high resolution capability of the new algorithm opens up the possibility of performing fluid film analysis on small computers. In all likelihood, a 10x10 mesh would be very adequate for most problems. Since accurate pressure gradient is inherent in the high resolution result, L.P.D.E.M. will enhance theoretical studies of film rupture in its various forms. Because the flux fields can be accurately calculated with ease by L.P.D.E.M., it will be a powerful adjunct to numerical studies of thermohydrodynamic problems, where convective heat transfer is one of the dominating factor.
(A.1)
where
431
Paper XIV(ii)
The boundary element method in lubrication analysis D.B. Ingham, J.A. Ritchie and C.M. Taylor
A Boundary Element Method (B.E.M.) t e c h n i q u e h a s b e e n a p p l i e d t o e f f e c t a s o l u t i o n t o t h e problem of low Reynolds number f l o w a t t h e i n l e t t o a t h r u s t p a d b e a r i n g . By t h e i n t r o d u c t i o n of a s u i t a b l e t h e Biharmonic E q u a t i o n , V 4 z = 0 , w a s o b t a i n e d a n d s o l v e d i n t h e i n l e t r e g i o n . stream function, An a n a l y t i c a l a s y m p t o t i c p e r t u r b a t i o n s o l u t i o n was o b t a i n e d t o improve t h e a c c u r a c y of t h e s o l u t i o n . R e s u l t s a r e p r e s e n t e d i n t e r m s of s t r e a m l i n e s of c o n s t a n t v a l u e a n d a l s o v e l o c i t y p r o f i l e s . I n a d d i t i o n t h e hydrodynamic p r e s s u r e g e n e r a t e d a t t h e r u n n e r s u r f a c e i s c a l c u l a t e d , showing a d i s c r e p ancy between t h e a m b i e n t ( z e r o g a u g e ) p r e s s u r e a n d t h e p r e s s u r e a t t h e l e a d i n g edge of t h e p a d .
5,
1
WTRODUCTION
The Boundary Element Method (B.E.M.) h a s e n j o y e d o n l y a l i m i t e d a p p l i c a t i o n i n s t u d i e s of f l u i d f i l m l u b r i c a t i o n . Khader ( 1 ) a n d Ingham a n d Kelmanson ( 2 ) h a v e p r e s e n t e d i n v e s t i g a t i o n s of i t s u s e a n d i d e n t i f i e d t h e a b i l i t y of t h e method t o h a n d l e complex g e o m e t r i e s a n d t h e a s s o c i a t e d boundary c o n d i t i o n s . Khader d e m o n s t r a t e d t h e a p p l i c a t i o n t h r o u g h t h e a n a l y s i s of a hydrodynamic s l i d e r b e a r i n g a n d t h e s q u e e z e f i l m b e t ween two c i r c u l a r p l a t e s , whilst Ingham a n d Kelmanson a d d r e s s e d t h e s t e a d y f l o w of a n incompr e s s i b l e v i s c o u s f l u i d between a n i n n e r r o t a t i n g c i r c u l a r c y l i n d e r and an o u t e r s t a t i o n a r y s l e e v e of a r b i t r a r y c r o s s s e c t i o n . The l a t t e r geometry i s r e p r e s e n t a t i v e of t h a t of p l a i n j o u r n a l b e a r i n g s , i n c l u d i n g n o n - c i r c u l a r b e a r i n g s , which a r e w i d e l y u s e d i n t h e s u p p r e s s i o n of s h a f t v i b r a t i o n s . The p r e s e n t p a p e r i s d i r e c t e d t o w a r d s t h e f u r t h e r development of t h e B.E.M. i n l u b r i c a t i o n a n a l y s i s and d e a l s i n p a r t i c u l a r with t h e f l o w a t entry t o a f l u i d f i l m thrust bearing.
p e r m i t t i n g s u b s t a n t i a l energy d i s s i p a t i o n i n a s m a l l s p a c e . A n a l y s i s a n d d e s i g n of such b e a r ings has a t t r a c t e d considerable attention. More c o m p r e h e n s i v e s t u d i e s i n v o l v e n o t o n l y i n v e s t i g a t i o n of t h e hydrodynamics of t h e l u b r i c a n t f i l m , b u t a l s o t h e g e n e r a t i o n of e n e r g y i n t h e f i l m and i t s d i s s i p a t i o n t o t h e l u b r i c a n t , In a d d i t i o n boundary s o l i d s and t h e environment. t h e thermal and e l a s t i c d i s t o r t i o n s of t h r u s t bearing pads a r e frequently c r u c i a l i n determining t h e o p e r a t i o n a l c h a r a c t e r i s t i c s . The d e s i g n p r o c e s s i s t h u s a p a r t i c u l a r l y complex one w i t h a number o f d i f f i c u l t a n a l y t i c a l a n d practical aspects.
The s t r a t e g y of t h e B.E.M. is t o transform the governing equations a n a l y t i c a l l y i n t o integr a l e q u a t i o n s v a l i d on t h e boundary of t h e r e g i o n of i n t e r e s t . T h i s a n a l y t i c a l s t e p r e d u c e s t h e dimension of t h e s o l u t i o n domain a n d h e n c e t h e c o m p l e x i t y of t h e problem. In addition t h i s t r a n s f o r m a t i o n c a n a v o i d t h e d i f f i c u l t i e s encount e r e d w ith unusual o r i n t r i c a t e geometries t h a t t h e more e s t a b l i s h e d f i n i t e d i f f e r e n c e a n d f i n i t e element approximation techniques f r e q u e n t l y f a c e . Once t h e a n a l y t i c a l f o r m u l a t i o n h a s b e e n a c h i e v e d numerical techniques can b e a p p l i e d t o o b t a i n a solution.
the order t h e s t u d y of i n l e t f l o w e f f e c t s h a s had t o b e d i v o r c e d f r o m t h a t o f t h e main l o a d b e a r i n g f i l m . A common a i m of t h e s t u d y of t h e i n l e t r e g i o n h a s b e e n t o c a l c u l a t e a v a l u e of t h e f l u i d p r e s s u r e a t i n l e t t o a pad a n d t o u s e t h i s a s a boundary c o n d i t i o n i n t h e lubricating film analysis. S t u d i e s of t h e i n l e t r e g i o n t o a t h r u s t pad h a v e v a r i e d from s i m p l e c o n s i d e r a t i o n s i n c o r p o r a t i n g a dynamic ( o r Bernoulli) pressure with a loss c o e f f i c i e n t ( 4 ) , t o a n a l y s e s i n c o r p o r a t i n g boundary l a y e r assessment with d i f f e r i n g d e g r e e s of complexity ( 5 , 6 , 7 , 8).
is thus an analytical-numerical t e c h n i q u e f o r t h e s o l u t i o n of b o u n d a r y v a l u e p r o b l e m s . The f o r m u l a t i o n of t h e i n t e g r a l e q u a t i o n s i s u s u a l l y achieved u s i n g Greens Theorems a n d G r e e n s F u n c t i o n s . F o r more d e t a i l s of t h e b a s i c t h e o r y a n d a n up-to-date a c c o u n t of Brebbia e t a 1 ( 3 ) g i v e a n e x c e l l e n t t h e B.E.M., review.
Here t h e i n l e t r e g i o n t o a f l u i d f i l m t h r u s t bearing w i l l be analysed using the B.E.M., t h e f i r s t t i m e t h a t such a n a p p r o a c h h a s b e e n a d o p t e d . The p r e s e n t f o r m u l a t i o n of t h e method l i m i t s a p p l i c a t i o n t o c o n d i t i o n s of slow viscous flow. This i s c l e a r l y a r e s t r i c t i o n , however, t h e p r i m a r y p u r p o s e of t h e s t u d y r e p o r t e d was t h e i m p l e m e n t a t i o n of t h e B.E.M. a n a l y s i s t o t h e problem and t h e i d e n t i f i c a t i o n of t h e a t t e n d a n t a d v a n t a g e s . Enhancement of the technique t o cover i n e r t i a l flow s i t u a t i o n s i s on6 o f a number o f i m p o r t a n t developments for the future.
The B.E.M.
O i l l u b r i c a t e d t h r u s t pad b e a r i n g s a r e w i d e l y u s e d i n a v a r i e t y o f m a c h i n e s where t h e r e i s a r e q u i r e m e n t t o accommodate a n a x i a l l o a d . They o f f e r h i g h r e l i a b i l i t y a n d l o n g l i f e whilst
There h a v e been few s t u d i e s of t h e p r e s s u r e head g e n e r a t e d a t t h e i n l e t t o t h r u s t b e a r i n g p a d s a n d s u c h c o n s i d e r a t i o n s h a v e been l i m i t e d t o two d i m e n s i o n a l f l o w ( i n f i n i t e pad w i d t h ) s i t u a t i o n s . Because t h e a s p e c t r a t i o ( f i l m h e i g h t / f i l m l e n g t h ) of l u b r i c a t i n g f i l m s i s of
432 In employing t h e B.E.M. t o t h e problem i n h a n d , a d i m e n s i o n l e s s stream f u n c t i o n - v o r t i c i t y f o r m u l a t i o n h a s been used. R e s u l t s a r e presented f o r constant stream function values i n t h e v i c i n i t y of t h e i n l e t t o a t h r u s t p a d enabl i n g a v i s u a l i s a t i o n of t h e f l o w p a t t e r n . A d e t e r m i n a t i o n of t h e p r e s s u r e v a r i a t i o n on t h e moving s u r f a c e ( o r r u n n e r ) of t h e b e a r i n g i n t h e The non-zero i n l e t region is a l s o undertaken. v a l u e of p r e s s u r e a t t h e nominal i n l e t t o t h e pad i s d e t a i l e d w i t h c o n s i d e r a t i o n of b o t h P o i s e u i l l e a n d C o u e t t e dominated f l o w s .
V e l o c i t y component i n y - d i r e c t i o n
V
V
V e l o c i t y component i n y - d i r e c t i o n , d i m e n s i o n l e s s , v = v/U
X
Coordinate a x i s along flow
X
Coordinate axis-along flow, dimensionless, x = x/h
Y
C o o r d i n a t e a x i s p e r p e n d i c u l a r t o Ox
-
Coordinate axis-perpendicular dimensionless, y = y/ho
Y 1.1
t o Ox,
Notation 6..
Kronecker d e l t a
General p o i n t s i n t h e dimensionless (x,y) plane
S(5-L)
Dirac d e l t a function
Boundary c o n t o u r l i n e
n(a)
Space f u n c t i o n
Boundary e l e m e n t i
e
Cylindrical angle coordinate
Film thickness
u0
Fluid viscosity
Z i l m thickness, dimensionless, h = h/ho
PO
F i l m t h i c k n e s s between b e a r i n g p l a t e s
- -
13
@
X,+ Square m a t r i c e s
-
Number of b o u n d a r y e l e m e n t s
'i w
Fluid pressure
v4
Eluid pressure, dimensionless, P = Pho/(uoUo) F l u i d flow r a t e per u n i t width F_luid f l o w r a t e , dimens i o n 1 e s s , Q = Q/(hoUo) M i d p o i n t of e l e m e n t i D i s t a n c e between R = Ia-bI
a and
Fundamental s o l u t i o n Stream f u n c t i o n s , d i m e n s i o n l e s s Approximate v a l u e of
on e l e m e n t i
V o r t i c i t y , dimensionless,
=
V2G
B iharmon i c o p e r a t o r , d imens i o n l e s s ,
v4
= VZ(V2)
Subscripts
i/j
Boundary e l e m e n t i / j
x/y
Derivative with respect t o p o s i t i v e xly axis
Superscript
b,
dimensionless,
I
8-
Reynolds number, Re =
Fluid density
-
PoUoho
D e r i v a t i v e w i t h r e s p e c t t o outward u n i t normal
~
UO
2
Cylindrical r a d i a l coordinate, d imen s i o n l e s s S u r f a c e e n c l o s e d by c o n t o u r C
THEORY
When a p p l y i n g t h e Boundary Element Method (B.E.M.) t o a p r o b l e m t h e s t e p s i n v o l v e d may b e c a t e g o r i s e d i n a g e n e r a l form. These a r e surmnarised below:-
Speed of moving p l a t e (1)
The g o v e r n i n g e q u a t i o n s o f t h e problem a r e t r a n s f o r m e d i n t o i n t e g r a l e q u a t i o n s which a r e v a l i d on t h e b o u n d a r y of t h e s o l u t i o n domain.
(2)
The b o u n d a r y i s d i s c r e t i s e d i n t o e l e m e n t s a n d t h e s o l u t i o n i s a p p r o x i m a t e d on t h e s e elements
R e f e r e n c e s p e e d of moving p l a t e Speed of moving p l a t e , . d i m e n s i o n l e s s ,
i = u/uo
Velocity vector i n Cartesian coordinates, u = (U,V,O) Velocity vector i n iimensionless c a r t e s i a n c o o r d i n a t e s , != (;,c,O) V e l o c i t y component i n x - d i r e c t i o n V e l o c i t y component i n x - d i r e c t i o n , d i m e n s i o n l e s s , u = u/Uo
.
(3)
Using t h e e q u a t i o n s o b t a i n e d i n s t e p s ( 1 ) a n d ( 2 ) i n c o n j u n c t i o n w i t h t h e boundary v a l u e s g i v e n by t h e o r i g i n a l p r o b l e m a s y s t e m of a l g e b r a i c e q u a t i o n s i s o b t a i n e d which c a n b e s o l v e d t o g i v e a c o m p l e t e s o l u t i o n on t h e b o u n d a r y .
433 Using t h e s o l u t i o n on t h e boundary and t h e transformation derived in s t e p (1) the s o l u t i o n a t any p o i n t i n t h e o r i g i n a l problem domain c a n b e found.
(4)
These s t e p s a r e i l l u s t r a t e d by a p p l y i n g t h e B.E.M. t o t h e problem d e s c r i b e d by low Reynolds number f l u i d flow. By assuming t h e f l o w of a v i s c o u s , i n c o m p r e s s i b l e f l u i d a t low Reynolds number t h e Navier-Stokes e q u a t i o n s r e d u c e t o , i n non-d imen s iona 1 form,
yp
(1)
= V'U
-
J (G(V';)
'-($) 'v'T+v'+ =
-
The p r e s s u r e , p , and t h e v e l o c i t i e s , u a n d v, have been n o n - d i m e n s i o n a l i s e d w i t h r e s p e c t t o t h e r e f e r e n c e v a l u e s of t h e f i l m t h i c k n e s s , ho , t h e speed of t h e moving p l a t e , Uo,
where S(5-L) i s t h e t h r e e d i m e n s i o n a l g i r a c d e l t a f u n c t i o n . With t h i s c h o i c e f o r Q e q u a t i o n (9) r e d u c e s t o : -
C
and t h e c o n t i n u i t y e q u a t i o n c a n b e w r i t t e n a s
-
5,
So f a r , i n e q u a t i o n ( 9 ) h a s been a n e n t i r e l y a r b i t s a r y f u n c t i o n . To make u s e of e q u a t i o n ( 9 ) , 4, i s chosen t o s a t i s f y t h e f o l l o w i n g equation
and t h e v i s c o s i t y ,
p - i n t h e f o l l o w i n g manner
-
q(5) = 271
5
E
S-C
q(a)
5
E
C and C c o n t i n u o u s a t a
=
'TI
~ ( 5= ) ' i n t e r n a l
-
-a -;-- ; ,
* =-- ; ax
aY
-
(41
which i s d e f i n e d by
and t h e v o r t i c i t y , w,
v4;
=
0
(b)
To t r a n s f o r m e q u a t i o n ( 6 ) i n t o a n i n t e g r a l e q u a t i o n form, u s e i s made of a form of a G r e e n ' s Theorem, namely [($ (0';) J
' -($)
'V';)
ds=
($V45 0'yV';)ds
(7)
C
where ( > 'i n d i c a t e s d i f f e r e l t i a t i o n w i t h r e s p e c t t o t h e outward u n i t n o r m a l , v,, s a t i s f i e s e q u a t i o n (6) a n d , is an e n t i r e l y a r b i t r a r y function. This-equation i s d e r i v e d by s u b s t i t u t i o n of for i n Greens Second Theorem. The f u n c t i o n s , $, a n d , can b e transposed t o give
+,
+
angle'
aE
C and C d i s c o n t i n u o u s
at a. The i m p l i c a t i o n of e q u a t i o n s (10) and (11) t o g e t h e r i s t h a t , i f a s o l u t i o n t o e q u a t i o n (10) e x i s t s and t h e v a l u e s o f , I#, and i t s d e r i v a t i v e s a r e known a t a i l p o i n t s on t h e boundary, t h e n t h e v a l u e of , I), anywhere i n S c a n b e c a l c u l a t e d by e v a l u a t i n g t h e i n t e g r a l i n e q u a t i o n ( 1 1 ) . Many s o l u t i o n s t o e q u a t i o n (10) e x i s t b u t o n l y t h e non-homogeneous s o l u t i o n i s r e q u i r e d . T h i s s o l u t i o n i s known as t h e Fundamental S o l u t i o n , which, f o r e q u a t i o n (10) i s
Q(R) The i n t r o d u c t i o n of t h e s t r e a m f u n c t i o n i s such t h a t the continuity equation (2) i s automatically s a t i s f i e d . E q u a t i o n ( 1 ) now r e d u c e s t o t h e Biharmonic E q u a t i o n ,
(11)
rl(a)G(a>
where q(5) i s a s p a c e f u n c t i o n which t a k e s t h e f o l lowing v a l u e s :
(3) In o r d e r t o a p p l y t h e B.E.M. t o e q u a t i o n s ( 1 ) and ( 2 ) it i s c o n v e n i e n t - t o i n t r o d u c e a non-dimensiona 1 s t r e a m f u n c t i o n , 9 , which i s d e f i n e d a s
'-(V'JI) ' 5 ) d s
();
where R =
=
1 b(R2 log
R
- Rz)
'( 12)
]a-k].
Thus i t o n l y remains t o f i n d a l l t h e v a l u e s of, ; , and i t s d e r i v a t i v e s on t h e boundary, C , i n o r d e r t o c a l c u l a t e , 9, anywhere. It i s a t t h i s p o i n t t h a t n u m e r i c a l t e c h n i q u e s a r e employed as a n a l y s i s r a r e l y a l l o w s f u r t h e r p r o g r e s s . A numerical s o l u t i o n can b e r e a c h e d by d i s c r e t i s i n g t h e boundary i n t o n e l e m e n t s . On t h e s e e l e m e n t s , v,, and i t s d e r i v a t i v e s , are approximated by s i m p l e f u n c t i o n s . Thus e q u a t i o n (11) c a n b e approximated by a series of n u m e r i c a l e q u a t i o n s and t h e f i r s t s t e p i n t h e t r a n s f o r m a t i o n i s t h e d i s c r e t i s a t i o n of
c,
V'5
5,
-
C
Subtraction gives
S
T h i s i n t u r n , by a p p r o x i m a t i o n o f , 9 , becomes
of e q u a t i o n (8) from e q u a t i o n (7)
I f 5 i s now chosen t o b e a p o i n t on e a c h boundary segment C. i n t u r n t h e n a series of n e q u a t i o n s is obtained:-
434
where
Mi j = i j ; ( q 1. c 3. I d s 'j Equation (15) c a n b e r e - e x p r e s s e d form a s
in matrix
Fig. 1.
where
N o t a t i o n f o r a n a l y t i c e v a l u a t i o n of i n t e g r a l s on C j
.
I f t h e c o o r d i n a t e s of t h e e n d p o i n t s of C The s o l u t i o n of e q u a t i o n (17) g i v e s t h e r e q u i r e d s o l u t i o n on t h e boundary so t h a t e q u a t i o n (14) can b e used t o f i n d t h e v a l u e of V, anywhere i n S. Having f o r m u l a t e d t h e problem u s i n g t h e B.E.M. two f e a t u r e s c a n b e r e c o g n i s e d : By u s i n g t h e B.E.M. t h e dimension of t h e problem h a s been reduced by one, which a u t o m a t i c a l l y r e d u c e s t h e c o m p l e x i t y of t h e problem.
(i)
The problem a s f o r m u l a t e d i s geometry independent a s no a s s u m p t i o n s a b o u t t h e shape of t h e boundary, C , have been made. T h i s means t h a t once t h e B.E.M. h a s been a p p l i e d t o a p a r t i c u l a r governing e q u a t i o n i t can b e e a s i l y a p p l i e d t o any problem f o r which t h i s e q u a t i o n i s r e l e v a n t , i . e . t h e a c t u a l shape of t h e boundary i s of l i t t l e importance when f o r m u l a t i n g a B.E.M. problem.
(ii)
I n t h e p a s t t h e i n t e g r a l s r e p r e s e n t e d by e q u a t i o n (16) have been performed by n u m e r i c a l t e c h n i q u e s . However, Ingham and Kelmanson ( 2 ) have shown t h a t t h e s e i n t e g r a l s c a n b e e v a l u a t e d a n a l y t i c a l l y . Using t h e n o t a t i o n shown i n F i g u r e ( 1 ) t h e s e i n t e g r a l s are:-
Jij = K..
13
Lij Mij
'
= 2 ( l o g II -log
1
=
=
1
19. (2K..-R 4 1 ij 1 1 - -{-[ 4 3
3
II )cosa+II l o g 2
3
9. -P.
2
3
+P. y s i n a 1
)sins
4 (k3-LlCOSa)' ( l o g 1 1 2 7 )
[Kij--P.2 + ( P . l c ~ s a )( 3l o g !?.l-~~]+(P.1~ina)2 4 3 3 1 -& y s i n a ] } 3 1
-
(18)
j
are
known, a s w e l l a s t h e c o o r d i n a t e s of 5 t h e n t h e f o u r matrices J , K , L , M c a n b e formed. So f a r t h e t e c h n i q u e s of t h e B.E.M. and t h e p h y s i c s of t h e problem have been e x p l a i n e d . To b e a b l e t o p r o c e e d , a geometry must now b e chosen and boundary c o n d i t i o n s have t o b e speci f i e d .
3
ANALYSIS OF THE INLET REGION TO A FLUID FILM THRUST BEARING
A f l u i d flow t h r u s t bearing usually c o n s i s t s of two f l a t s u r f a c e s h a v i n g r e l a t i v e motion and s e p a r a t e d by a narrow gap f i l l e d w i t h l u b r i c a n t . Normally one of t h e s u r f a c e s ( t h e pad) i s s t a t i o n a r y whilst t h e o t h e r ( t h e r u n n e r ) draws f l u i d i n t o t h e l u b r i c a t i n g f i l m which h a s a t h i c k n e s s of t h e o r d e r of 25 um. In o r d e r t o g e n e r a t e p r e s s u r e s hydrodynamically t h e f l u i d must, on a v e r a g e , b e c o n s t r a i n e d t o f l o w i n t o a c o n v e r g i n g gap. T h i s may b e a c h i e v e d by machining a t a p e r i n t o a f i x e d pad b u t more commonly t h e pad i s p r o v i d e d w i t h a pivot t o enable it t o t i l t as required. A l t e r n a t i v e l y t h e pad may c o n s i s t of two f l a t p o r t i o n s w i t h a s t e p between them ( t h e steppedp a r a l l e l t h r u s t b e a r i n g ) w i t h s u r f a c e motion d i r e c t e d from t h e t h i c k e r t o t h e t h i n n e r f i l m , In e i t h e r b u t t h i s i s a n u n u s u a l arrangement. case t h e i n l e t t o t h e b e a r i n g g e n e r a l l y h a s t h e geometry shown i n F i g u r e ( 2 1 , where any r e l a t i v e t i l t i n g of t h e p l a t e s does n o t show on t h e s c a l e used.
In t h i s c o n fi g u ra t i o n t h e bottom p l a t e o r r u n n e r ) i s c o n s i d e r e d t o b e moving w i t h a c o n s t a n t speed U i n t h e p o s i t i v e x - d i r e c t i o n . 0
T h i s movement, i n c o n j u n c t i o n w i t h a p r e s s u r e d i f f e r e n t i a l form f a r u p s t r e a m t o f a r downstream of t h e l e a d i n g edge of t h e pad, i n d u c e s the f l u i d i n the region x < 0 t o enter the
435
Y
STA TIONARY LEADING EDGE -
=
FLUID
PAD &
I
1 h,
I
THIN LUBRICATING
I
t
-
/ / / /////////////////////
0
RUNNER F i g . 2.
x
U,
Geometry of a t h r u s t pad b e a r i n g i n l e t .
channel (x > 0) e s t a b l i s h i n g a t h i n f i l m lubr i c a t i n g flow. This f l o w g e n e r a t e s a p r es s u re on t h e b e a r i n g pad and r u n n e r t o produce a l o a d I n t h e problem b e i n g carrying capability. considered t h e region b e f o r e t h e channel i s f u l l y f l o o d e d so t h a t t h e r e a r e no f r e e s u r f a c e s . A l s o , we t a k e t h e f l o w t o b e s t e a d y and b o t h t h e pad and t h e r u n n e r t o b e i n f i n i t e i n e x t e n t i n t h e z - d i r e c t i o n so t h a t t h e problem i s two-dimensional. With t h e assumption of a n i s o - v i s c o u s , incompr e s s i b l e f l u i d , e q u a t i o n s ( 1 ) and ( 2 ) now h o l d and t h e Biharmonic Equation ( 6 ) h a s t o b e s o l v e d i n t h e r e g i o n of t h e l e a d i n g edge of t h e pad. To s o I v e t h i s problem u s i n g t h e B . E . M . a s p e c i f i c a r e a e n c l o s e d by a known boundary h a s t o b e chosen. T h i s boundary w a s d e r i v e d from t h e geometry shown i n F i g u r e ( 2 ) by p l a c i n g a boundary downstream of t h e l e a d i n g e d g e , perpe n d i c u l a r t o t h e r u n n e r and t h e pad. A q u a r t e r c i r c l e boundary w a s t h e n p l a c e d i n t h e u p s t r e a m r e g i o n , l i n k i n g t h e pad t o t h e r u n n e r and creating a closed contour. This quarter-circle w a s chosen s i n c e i t w a s found t h a t t h e boundary c o n d i t i o n s u p s t r e a m of t h e l e a d i n g edge a r e best expressed in c y l i n d r i c a l p o l a r coordinates ( r , @ ) . A l l l e n g t h s were t h e n non-dimensionali s e d w i t h r e s p e c t t o ho, t h e d i s t a n c e between
t h e p l a t e s . T h i s new geometry i s shown i n F i g u r e ( 3 ) where t h e d i s t a n c e DE i s now 1 . The arrows i n F i g u r e ( 3 ) i n d i c a t e t h e p o s i t i v e d i r e c t i o n f o r t r a v e r s i n g t h e c o n t o u r ODEFGH.
Fig. 3 .
3.1
S o l u t i o n domain and boundary.
Boundary C o n d i t i o n s
To d e r i v e t h e boundary c o n d i t i o n s a p p l i c a b l e t o
t h e geometry shown i n F i g u r e ( 3 ) two p a r a m e t e r s had t o b e s p e c i f i e d . These were U , t h e speed of t h e moving p l a t e and Q t h e f l u i d f l o w r a t e The f l o w r a t e i s needed a s a p a r a a c r o s s DE. meter as it e n a b l e s t h e x - d i r e c t i o n u r e s s u r e d i f f e r e n t i a l , from f a r u p s t r e a m t o f a r downs t r e a m , t o b e i n c l u d e d i n t h e f o r m u l a t i o n of t h e problem. On HOD t h e p l a t e moves t o t h e r i g h t w i t h speed U , w h i l e EFG i s s t a t i o n a r y . In a d d i t i o n t h e r e i s a f l u i d f l o w r a t e of Q a c r o s s DE. T h i s f l o w r a t e must b e matched by a n o p p o s i t e f l o w r a t e of -Q a c r o s s GH s i n c e t h e b o u n d a r i e s HOD and EFG a r e s o l i d . The p a r a m e t e r s Q and U were non-dimensionalised u s i n g a r e f e r e n c e speed U 0’
a s shown i n e q u a t i o n (191, and t h i s a l l o w s t h e p h y s i c a l boundary c o n d i t i o n s t o b e e x p r e s s e d mathematically i n equation (20).
(19)
u = u/u -
J<,,o,
;(x ;co,;
>
=
U
0, )
= o
>
= o
)
(20)
J
G
I n terms of t h e s t r e a m f u n c t i o n , $, d e f i n e d i n s e c t i o n ( 2 ) t h e s e boundary c o n d i t i o n s become,
436
I n a d d i t i o n t o t h e boundary c o n d i t i o n s shown i n e q u a t i o n (21) t h e f a c t t h a t t h e b o u n d a r i e s HOD and EFG a r e s o l i d c a n b e e x p r e s s e d i n t e r m s of the stream function a s
where k, and k
2
a r e constants.
-
S i n c e , Q, i s
To d e r i v e t h e boundary c o n d i t i o n s on GH two f l o w c o n d i t i o n s had t o b e superimposed and t h e i r boundary c o n d i t i o n s combined. These two f l o w s a r e t h e f l o w due t o a p o i n t s i n k (which r e p r e s e n t s t h e f l u i d e n t e r i n g t h e c h a n n e l ) and t h e f l o w due t o a moving p l a t e i n a c o r n e r (which r e p r e s e n t s t h e f l o w induced by t h e p l a t e HOD a t l a r g e d i s t a c c e s from 0 ) . The a n a l y t i c a l e x p r e s s i o n f o r , 9 , on GH was d e r i v e d s e p a r a t e l y f o r each f l o w and t h e n t h e r e s u l t s were added together. The p o i n t s i n k f l o w w a s chosen t o r e p r e s e n t t h e f a c t t h a t f l u i d was e n t e r i n g t h e c h a n n e l t h r o u g h t h e gap OF and t h a t , i f a l a r g e enough s c a l e were c h o s e n , t h i s would resemble a p o i n t sink a t 0. S i m i l a r l y t h e moving p l a t e f l o w was chosen a s , w i t h o u t t h e p r e s e n c e of t h e c h a n n e l , t h i s i s t h e p a t t e r n of f l o w t h a t t h e f l u i d would a d o p t . These two f l o w p a t t e r n s a r e shown i n F i g u r e ( 4 ) . By s u p e r p o s i n g t h e s e two f l o w s t h e c o r r e c t f l o w p a t t e r n was o b t a i n e d .
o n l y dztermined t o w i t h i n a c o n s t a n t , k l c a n b e chosen t o e q u a l 0 . Then, u s i n g c o n t i n u i t y arguments t h e boundary c o n d i t i o n s shown i n e q u a t i o n s ( 2 1 ) and ( 2 2 ) a r e combined i n t o one s e t a s shown i n e q u a t i o n ( 2 3 )
(23)
Fig. 4.
where it s h o u l d b e n o t e d t h a t on HOD t h e outward u n i t normal i s (0,-1) so t h a t (24)
To d e r i v e t h e boundary c o n d i t i o n s on DE a p a r a b o l i c v e l o c i t y p r o f i l e was imposed i n addi t i o n t o t h e f l u i d flow r a t e c o n d i t i o n . This r e s u l t e d in t h e stream function, V y taking t h e f o l l o w i n g form on DE,
POINT SINK
CORNER FLOW
The two s e p a r a t e f l o w p a t t e r n s b e f o r e b e i n g combined.
Using t h e l i n e OG a s t h e r e f e r e n c e l i n e , 8=0, i n c y l i n d r i c a l p o l a r c o o r d i n a t e s , with 0 being t h e o r i g i n and a n g l e s b e i n g measured i n r a d i a n s and p o s i t i v e i n t h e a n t i - c l o c k w i s e d i r e c t i o n , t h e boundary c o n d i t i o n s on GH c a n b e found. F o l l o w i n g t h e example i n - B a t c h e l o r ( 9 ) t h e form of t h e stream f u n c t i o n , 9 , f o r t h e moving p l a t e flow i s
The v a l u e s of t h e c o n s t a n t s i n e q u a t i o n ( 2 8 ) a r e o b t a i n e d by n o t i n g t h a t on GF, I )' = 0 , and on HO, $I' = -U. The r e l e v a n t boundary c o n d i t i o n s f o r t h e moving p l a t e a r e t h u s b c d a r e c o n s t a n t s . The v a l u e s of 1 ' 1 ' 1' 1 t h e c o n s t a n t s i n e q u a t i o n (25) a r e found from e q u a t i o n (23) by c o n t i n u i t y arggments and g i v e t h e a n a l y t i c a l expression f o r , 9, as
where a
;=o, =
0,
This then gives
-
-
With t h i s e x p r e s s i o n f o r , q , t h e v o r t i c i t y , w , was evaluated a s
e = ~ , v r
;'=o
-
-
q ' = -U
e
(29) = n/2,
vr
431 The boundary c o n d i t i o n s f o r t h e p o i n t s i n k a r e given by
V , = ~ , V , ' = O V, = 0 ,
e=o,
e
9' = 0
ftr
(31) = n / 2 , ftr
The boundary c o n d i t i o n s i n e q u a t i o n (31) a r e most e a s i l y d e r i v e d by s u b t r a c t i n g t h e boundary c o n d i t i o n s shown i n e q u a t i o n (29) from t h e r e l e v a n t boundary c o n d i t i o n s l i s t e d i n e q u a t i o n ( 2 3 ) . With t h e boundary c o n d i t i o n s shown i n e q u a t i o n ( 3 1 ) , V , , i s found t o have t h e form
4
ASYMPTOTICS AND SINGULARITIES
When t h e boundary c o n d i t i o n s i n e q u a t i o n s ( 2 3 ) , ( 2 6 ) , (27) and (34) were d e r i v e d , t h e i m p l i c i t assumption was made t h a t b o t h DE and GH were s u f f i c i e n t l y d i s t a n t from 0 t o e n a b l e t h e derived expressions t o b e a c c u ra t e . Since the s e boundary c o n d i t i o n s have t o b e imposed a t a f i n i t e d i s t a n c e from 0 t h e r e w i l l always b e a r e s i d u a l e r r o r i n t h e a c c u r a c y of e q u a t i o n s ( 2 3 ) , ( 2 6 1 , ( 2 7 ) and (34). More a c c u r a t e e x p r e s s i o n s c a n b e o b t a i n e d by i n c l u d i n g a n a s y m p t o t i c p e r t u r b a t i o n s o l u t i o n which compensates f o r t h e f i n i t e d i s t a n c e s from 0 t o DE and GH. T h i s p e r t u r b a t i o n t a k e s t h e form
$(r,f3) = a 03+b 02+c B+d 3 3 3 3
(32)
=
u, =
and more s p e c i f i c a l l y (33)
-
By adding t h e two e x p r e s s i o n s f o r V , , a s shown in e q u a t i o n s (30) and ( 3 3 ) , t o g e t h e r t h e b o u n i a r y c o n d i t i o n on GH was d e r i v e d . The v o r t i c i t y , w, was a l s o c a l c u l a t e d and b o t h , ;, and w, are shown i n e q u a t i o n ( 3 4 ) .
(34)
With t h e boundary c o n d i t i o n s a s r e p r e s e n t e d by e q u a t i o n s (231, ( 2 6 ) , ( 2 7 ) , (34) and t h e problem f o r m u l a t e d a s shown i n e q u a t i o n (171, a system of n e q u a t i o n s i n 4n unknowns w i t h 2n boundary e q u a t i o n s h a s been o b t a i n e d . I n o r d e r t o d e r i v e a s o l v a b l e system of e q u a t i o n s i t w a s found n e c e s s a r y t o u s e e q u a t i o n (35) t o p r o v i d e a f u r t h e r n e q u a t i o n s t o g i v e 2n e q u a t i o n s i n 2n unknowns. v2;
$
= 0
(35)
This e q u a t i o n f o l l o w s d i r e c t l y from e q u a t i o n ( 6 ) and by u s i n g a s i m i l a r a n a l y s i s scheme t o t h a t a l r e a d y d e s c r i b e d , e q u a t i o n (35) c a n b e t r a n s formed i n t o e q u a t i o n (36) where t h e m a t r i c e s J and K a r e i d e n t i c a l t o t h o s e i n e q u a t i o n ( 1 6 ) .
Go + e-aXF(;) i,+ e-BxG(8)
on DE on GH
(39)
-
where, Go, i s g i v e n i n e x p r e s s i o n ( 2 6 ) and 9 , i n e x p r e s s i o n (34) and t h e c o n s t a n t s a , and t h e f u n c t i o n s F ( y ) , G(8) have t o b e found. This means t h a t V, a n d q 1 , a r e , i n e f f e c t ,
p e r t u r b e d by an a s y m p t o t i c s o l u t i o n which i s i n v e r s e - e x p o n e n t i a l l y dependent on t h e d i s t a n c e from 0. Thus, a t small d i s t a n c e s from 0, t h i s p e r t u r b a t i o n can be very s i g n i f i c a n t . To f i n d t h e forms of a , B , F@, G(B) i t i s n o t e d t h a t t h e stream f u n c t i o n , 9 , i n e q u a t i o n s (38) and (39) must s t i l l s a t i s f y e q u a t i o n ( 6 ) and t h a t t h e boundary c o n d i t i o n s i n (23) s t i l l h o l d . t h i e n a b l e s e q u a t i o n s (40) and (41) t o b e d e r i v e d Y t h t h e boundary condi t i o n s shown Fiv + 2aZF" + a 4 F
F(0) = F'(0) = F (
= o
It c a n b e s e e n t h a t e q u a t i o n s (40) and (41) can b e made i d e n t i c a l by s c a l i n g y i n e q u a t i o n (40) by a/2.
Solving equation (40) g i v e s
a = f sina F(y)=y{sinay-ay c o s a y + ( a c o t a - l ) y s i n a y }
J i + KW'
=
O_
(38)
(42) (43)
(36)
where y i s a c o n s t a n t . Equations (17) and ( 3 5 ) , t o g e t h e r w i t h t h e boundary c o n d i t i o n s i n e q u a t i o n s ( 2 3 ) , ( 2 6 ) , (27) and (34) were r e - e x p r e s s e d a s e q u a t i o n (37) which was s o l v e d u s i n g Crout r e d u c t i o n ( a s t h e m a t r i x N i s a v e r y dense m a t r i x ) (37) the Using t h e s o l u t i o n of ( 3 7 ) , t o g e t h e r wit! known boundary c o n d i t i o n s , t h e v a l u e of V, anywhere i n S + C c a n b e c a l c u l a t e d u s i n g e q u a t i o n (14).
Both a and y a r e found t o b e complex c o n s t a n t s and so a r e a l r e p r e s e n t a t i o n of e q u a t i o n (38) i s s o u g h t . By l o o k i n g f o r bounda r y c o n d i t i o n s of t h e form shown i n e q u a t i o n (44) t h e c o n s t a n t s i n e q u a t i o n (44) a r e t h e n a l l real and no complex number r e p r e s e n t a t i o n is required.
(44 1
438 E v a l u a t i n g t h e e x p r e s s i o n s i n e q u a t i o n s (44) u s i n g e q u a t i o n ( 3 8 ) f o r t h e f o r m of $J, a n d i t s d e r i v a t i v e s g i v e s t h e r e q u i r e d e x p r e s s i o n s f o r t h e cons t a n t s in equation (44). This then allows e q u a t i o n (44) t o b e used a s a replacement boundary c o n d i t i o n i n s t e a d of e q u a t i o n s ( 2 6 ) a n d ( 2 7 ) . A s i m i l a r process is followed t o o b t a i n replacement boundary c o n d i t i o n s f o r e q u a t i o n ( 3 4 ) .
d e v e l o p e d . An i n t e r e s t i n g f e a t u r e of t h e v e l o c i t y p r o f i l e s c a n b e seen a t t h e p o i n t ( c ) . T h i s p r o f i l e w a s c a l c u l a t e d a t t h e l e a d i n g edge a n d shows t h a t t h e f l u i d h a s a n a l m o s t f u l l y developed form b e f o r e i t h a s e n t e r e d t h e channel. It t h u s seems t h a t most of t h e a d j u s t m e n t i n t h e f l u i d flow occurs before the f l u i d enters the channel.
I n p r a c t i c e , however, i t w a s f o u n d t h a t . a n a s y m p t o t i c c o r r e c t i o n was u n n e c e s s a r y on DE a s the stream function very r a p i d l y a t t a i n s t h e form a s s p e c i f i e d i n e q u a t i o n ( 2 6 ) . Thus a n a s y m p t o t i c p e r t u r b a t i o n was o n l y u s e d on GH a n d n o t on DE.
Looking a t t h e s t r e a m l i n e d i a g r a m i n F i g u r e ( 5 ) i t i s seen t h a t t h e s t r e a m l i n e s s e t t l e down v e r y q u i c k l y a f t e r e n t e r i n g t h e c h a n n e l (which c o n f i r m s t h e r e s u l t s of t h e It i s a f e a t u r e of t h i s velocity profiles). problem t h a t t h e s t r e a m l i n e s r a p i d l y s e t t l e down i n t h e c h a n n e l f o r a l l v a l u e s of t h e p a r a meters studied.
A n o t h e r f e a t u r e i n t h e p r o b l e m i s o n e due e n t i r e l y t o t h e g e o m e t r y . The c o r n e r F i n F i g u r e ( 3 ) h a s a n a n g l e g r e a t e r t h a n 180" a n d t h i s automatically produces a s i n g u l a r i t y i n t h e f l u i d f l o w a t t h i s p o i n t . However, t h i s s i n g u l a r i t y can b e d e a l t with a n a l y t i c a l l y using t h e B.E.M. a n d t h u s c a u s e s no p a r t i c u l a r d i f f i c u l t y i n t h e f o r m u l a t i o n of t h e problem. In essence t h e s i n g u l a r i t y c a n b e d e a l t w i t h by f i n d i n g - a n a n a l y t i c a l s e r i e s e x p a n s i o n f o r t h e f o r m of n e a r t h e s i n g u l a r i t y . The s i n g u l a r p a r t of t h i s e x p a n s i o n i s t h e n s u b t r z c t e d t o l e a v e a nons i n g u l a r f u n c t i o n , s a y x. The p r o b l e m i s t h e n -s o l v e d i n t e r m s of t h e n o n - s i n g u l a r f u n c t i o n , x, a n d t h e s i n g u l a r t e r m s a r e a d d e d t o t h e f i n a l s o l u t i o n . T h i s a v o i d s any p r o b l e m s w i t h t h e s i n g u l a r i t y a n d smooth s o l u t i o n s c a n b e o b t a i n e d . F u r t h e r d e t a i l s c o n c e r n i n g t h e t r e a t m e n t of Biharmonic s i n g u l a r i t i e s may b e f o u n d i n Ingham a n d Kelmanson ( 2 ) .
+,
5
RESULTS
The r e s u l t s p r e s e n t e d were c a l c u l a t e d on t h e Amdahl 5860 d i g i t a l computer a t t h e U n i v e r s i t y of L e e d s . T h e r e a r e two p a r a m e t e r s i n t h e p r o b lem, t h e d i m e n s i o n l e s s f l u i d f l o w r a t e Q a n d t h e d i m e n s i o n l e s s p l a t e s p e e d U. Normally t h e n o r m a l i s e d p l a t e v e l o c i t y ( n . p . v . 1 was c h o s e n t o b e 1.0 a n d t h e n o r m a l i s e d f l u i d f l o w r a t e w a s v a r i e d . The e x c e p t i o n t o t h i s was t h a t when t h e p r e s s u r e on t h e moving p l a t e was c a l c u l a t e d a n n . p . v . of 0 . 0 was d e s i r e d , a s w e l l a s a n n . p . v . of 1.0, f o r a f i x e d f l o w r a t e . With t h e p r o b l e m a s s t a t e d t h e r e a r e t h r e e t y p e s of f l o w , a p r e s s u r e d r i v e n f l o w , a p l a t e d r i v e n f l o w a n d a m i x t u r e of t h e two. A p r e s s u r e d r i v e n f l o w i s t a k e n t o b e a f l o w d o m i n a t e d by P o i s e u i l l e f l o w i n t h e c h a n n e l ODEF b u t which i s n o t a pure P o i s e u i l l e flow, Conversely a p l a t e d r i v e n f l o w i s a f l o w d o m i n a t e d by C o u e t t e f l o w i n t h e c h a n n e l . Using t h e s e d e f i n i t i o n s , r e s u l t s a r e presented f o r both a p l a t e driven and a pressure driven flow. In addition t h e pressure g e n e r a t e d on t h e moving p l a t e i s a l s o p r e s e n t e d f o r a f i x e d f l o w r a t e Q a n d f o r v a r i o u s s p e e d s U. The r e s u l t s f o r a f l o w r a t e of 3 . 0 a n d a p l a t e s p e e d of 1.0 a r e shown i n F i g u r e ( 5 ) . The u p p e r -d i a g r a m shows l i n e s of c o n s t a n t s t r e a m f u n c t i o n , 9 , c a l c u l a t e d a t equal i n t e r v a l s between 0 and Q ( 3 . 0 i n t h i s e x a m p l e ) . The v e l o c i t y p r o f i l e s beneath a r e c a l c u l a t e d a t the s t a t i o n s ( a ) t o ( e l i n t h e x - d i r e c t i o n a n d t o a n o r m a l i s e d h e i g h t of 1.0 ( t h e f i l m t h i c k n e s s ) i n t h e y - d i r e c t i o n . This allows the p r o f i l e a t ( a ) t o be d i r e c t l y compared t o t h e p r o f i l e a t ( e ) . Examining t h e s e p r o f i l e s , i t can b e seen from t h e p r o f i l e a t (e) t h a t a p r e s s u r e d r i v e n f l o w h a s been
6
of 113 C h o o s i n g now a f l u i d f l o w r a t e a n d a s p e e d U of 1.0 t h e s i t u a t i o n shown i n F i g u r e ( 6 ) r e s u l t s . Again t h e s t r e a m l i n e s a n d v e l o c i t y p r o f i l e s a r e p r e s e n t e d i n t h e same manner a s F i g u r e ( 5 ) . From t h e v e l o c i t y p r o f i l e a t (e) it is seen t h a t f l u i d i s b e i n g dragged i n t o t h e c h a n n e l by t h e moving p l a t e a n d t h u s a p l a t e d r i v e n flow h a s been developed. I n t h i s e x a m p l e , however, t h e c h a n g e i n v e l o c i t y p r o f i l e s f r o m ( a ) t o ( e ) i s more g r a d u a l t h a n f o r t h e p r e s s u r e d r i v e n f l o w of F i g u r e ( 5 ) . By l o o k i n g a t t h e s t r e a m l i n e d i a g r a m of Figure ( 6 ) it can be seen t h a t the streamlines s e t t l e down v e r y r a p i d l y upon e n t e r i n g t h e channel b u t t h i s time they c l u s t e r towards the moving p l a t e ( r e f l e c t i n g t h e s t r o n g C o u e t t e f l o w ) . Of p e r h a p s more i n t e r e s t i s t h e s t r e a m l i n e which t e r m i n a t e s a t t h e l e a d i n g e d g e . T h i s s t r e a m l i n e h a s t h e v a l u e 1/3(Q) a n d i n d i c a t e s t h e p r e s e n c e of a s t a g n a t i o n p o i n t a t t h e c o m e r F. T h i s s t a g n a t i o n p o i n t i s c a u s e d by t h e s i n g u l a r i t y i n t h e f l o w due t o t h e c o r n e r being g r e a t e r t h a n 180". One of t h e f e a t u r e s o f i n t e r e s t i n t h i s p r o b l e m i s t h e p r e s s u r e on t h e moving p l a t e . T h i s p r e s s u r e i s shown i n F i g u r e ( 7 ) c a l c u l a t e d f o r v a r i o u s p l a t e speeds with a c o n s t a n t f l u i d f l o w r a t e . To c a l c u l a t e t h e p r e s s u r e on t h e p l a t e a b o u n d a r y c o n d i t i o n of p = 0.0 a t 0 w a s imposed a s a r e f e r e n c e p r e s s u r e . T h i s c o n d i t i o n i s f r e q u e n t l y i n c o r p o r a t e d by a s s u m i n g t h a t t h e p r e s s u r e a t i n l e t i s t h e same a s t h e a m b i e n t ( o r z e r o g a u g e ) p r e s s u r e . The r e s u l t s shown i n F i g u r e ( 7 ) s u g g e s t t h a t t h i s a s s u m p t i o n i s gene r a l l y false and t h a t t h e r e is a d i f f e r e n c e b e t w e e n t h e i n l e t p r e s s u r e ( p = 0) a n d t h e p r e s s u r e f a r upstream (considered as ambient). I n o r d e r t o o b t a i n a n i d e a of t h e s i z e of t h i s p r e s s u r e d i f f e r e n c e a t y p i c a l example was u s e d . Choosing r e a l i s t i c v a l u e s f o r h o' U0' P o , P o t h e p r e s s u r e p was c a l c u l a t e d f o r a c o r r e s p o n d i n g d i m e n s i o n l e s s p r e s s u r e of p = 12.01. The p r e s s u r e h e a d a c c o r d i n g t o B e r n o u l l i t h e o r y was The c h o s e n a l s o c a l c u l a t e d as a comparison. v a l u e s f o r h o , Uo, p o , p o a r e shown i n e q u a t i o n (45). ho = 25 x lo-'
m
uo
= 5
ms
po
= 0.05
Nsm
p o = 850
-1
(45)
-2
kgm-3
439 i s t h e p r e s s u r e a t t h e i n l e t of t h e t h r u s t pad bearing"
Rearranging e q u ation (3) g i v e s
PJJouo P=h0 which, u s i n g t h e v a l u e s i n e q u a t i o n (45) g i v e s
I n comparison, t h e B e r n o u l l i p r e s s u r e h e a d i s given by P
=
1P0U:,
References
Using t h e v a l u e s i n e q u a t i o n ( 4 5 ) t h i s g i v e s p = 0 . 5 x 850 x ( 5 ) *
=
4 11.06 x 10 INm-2
(49)
The R e y n o l d s number i n t h i s example i s a p p r o x i m a t e l y 2 a n d t h i s may go some way t o w a r d s e x p l a i n i n g t h e d i s c r e p a n c y b e t w e e n t h e two p r e s s u r e s . The two c a l c u l a t e d p r e s s u r e s a p p l y t o low a n d h i g h Reynolds number f l o w s r e s p e c t i v e l y so t h a t t h e ' r e a l ' p r e s s u r e p r o b a b l y l i e s b e t w e e n t h e two c a l c u l a t e d p r e s s u r e s . However t h e r e s u l t s show t h a t a d i s p a r i t y may a r i s e when u s i n g t h e B e r n o u l l i p r e s s u r e h e a d t o f i n d a maximum p r e s s u r e in a given problem.
The p r e s s u r e d i a g r a m i n F i g u r e ( 7 ) shows t h a t a c o n s t a n t g r a d i e n t p r e s s u r e was developed in the channel. This pressure gradient is expected f o r l a r g e p o s i t i v e x with a c o n s t a n t film thickness, but Figure (7) indicates t h a t t h i s constant gradient is rapidly achieved. A l s o , u p s t r e a m of t h e i n l e t , t h e p r e s s u r e t e n d s t o a c o n s t a n t v a l u e , - a s i n d i c a t e d by t h e p r e s s u r e c u r v e f o r t h e s p e e d u = 0.0. F o r a f i x e d f l o w r a t e and v a r y i n g speed t h e r e i s only a small range o f u f o r which t h e p r e s s u r e u p s t r e a m i s close t o z e r o . I n F i g u r e ( 7 ) t h e c u r v e f o r u = 2.5 l i e s c l o s e t o t h i s r a n g e a n d t h e c u r v e f o r u = 1.0 i s c l e a r l y w e l l beyond i t .
6
A similar a n a l y s i s t o t h a t p r e s e n t e d h e r e c a n b e d e v e l o p e d t o d e a l w i t h t h e f l o w o u t of t h e b e a r i n g a n d h e n c e t h e c o m p l e t e p r o b l e m of t h e flow through t h e b e a r i n g can b e solved. Further, if the bearing has a step o r constricti o n i n i t , t h e n a s i m p l e m o d i f i c a t i o n of t h e theory p r e s e n t e d h e r e can b e performed and t h e s o l u t i o n o b t a i n e d . S i n c e t h e Boundary Element Method f o r m u l a t i o n i s i n d e p e n d e n t of t h e geometry of t h e b g a r i n g we c a n s o l v e , w i t h a m i x t u r e of a n a l y t i c a l a n d n u m e r i c a l t e c h n i q u e s , a wide v a r i e t y .of l u b r i c a t i o n p r o b l e m s .
KHADER, M. S. ' A g e n e r a l i s e d i n t e g r a l n u m e r i c a l s o l u t i o n method f o r l u b r i c a t i o n p r o b l e m s ' , T r a n s . ASME, J n l . of T r i b . , 106, 1984, 255-259.
INGHAM, D. B . a n d KELMANSON, M. A. 'Boundary i n g e g r a l e q u a t i o n a n a l y s i s of s i n g u l a r p o t e n t i a l and biharmonic problems', Springer-Verlag, B e r l i n / New York, 1984.
BREBBIA, C. A . , TELLES, J . C . F . a n d WROBEL, L. C . 'Boundary e l e m e n t t e c h n i q u e s : Theory a n d a p p l i c a t i o n s i n e n g i n e e r i n g ' , S p r i n g e r - V e r l a g , B e r l i n / N e w York, 1984. TICHY, J . A . a n d CHEN, S-H. ' P l a n e s l i d e r b e a r i n g l o a d due t o f l u i d i n e r t i a e x p e r i m e n t a n d t h e o r y ' , T r a n s . ASME, J n l . of T r i b , 107, J a n . 1985, 32-38. TIPEI, N . ' F l o w c h a r a c t e r i s t i c s a n d p r e s s u r e h e a d b u i l d - u p a t t h e i n l e t of n a r r o w p a s s a g e s ' , T r a n s . ASME, J n l . of Lub. Tech., 100, J a n . 1978, 47-55. TIPEI, N. 'Flow a n d p r e s s u r e h e a d a t t h e i n l e t of n a r r o w p a s s a g e s , w i t h o u t u p s t r e a m f r e e s u r f a c e s ' , T r a n s . ASME, J n l . of I u b . Tech., 104, A p r i l 1982, 196-202.
CONCLUSIONS
I n t h i s p a p e r a m a t h e m a t i c a l model f o r t h e low Reynolds number f l o w i n t o t h e i n l e t of a t h r u s t pad b e a r i n g h a s b e e n d e v e l o p e d u s i n g t h e Boundary Element Method (B.E.Y.). I n o r d e r t o d e a l adeq u a t e l y w i t h t h e boundary c o n d i t i o n s a t smal l d i s t a n c e s from t h e e n t r a n c e t o t h e b e a r i n g a n The asymptotic s o l u t i o n h a s been obtained. B.E.M. h a s t h e g r e a t a d v a n t a g e o v e r f i n i t e d i f f e r e n c e a n d f i n i t e e l e m e n t methods i n t h a t i t r e d u c e s t h e d i m e n s i o n of t h e s o l u t i o n domain This and h e n c e t h e c o m p l e x i t y of t h e problem. i s a c h i e v e d by u s i n g a n i n t e g r a l e q u a t i o n t r a n s f o r m a t i o n which i s geometry i n d e p e n d e n t . The problem r e d u c e s t o s o l v i n g a n i n t e g r a l e q u a t i o n f o r unknowns on t h e b o u n d a r y of t h e domain o n l y a n d t h i s c a n t h e n b e e x p r e s s e d i n t h e f o r m of a s e t of a l g e b r a i c e q u a t i o n s . R e s u l t s a r e p r e s e n t e d which show t h e d e v e l o p ment o f t h e f l u i d f l o w a s i t e n t e r s t h e b e a r i n g a n d t h e y i n d i c a t e t h e r a p i d i t y w i t h which t h e a n a l y t i c a l l y p r e d i c t e d asymptotic flow i s approached i n t h e b e a r i n g channel. Also p r e s e n t e d
PAN, C . H. T. ' C a l c u l a t i o n of p r e s s u r e , shear and flow i n l u b r i c a t i n g flows f o r h i g h s p e e d b e a r i n g s ' , T r a n s . ASME, J n l . of Lub. Tech., 9 6 , J a n . 1974, 168-173. CONSTANTINESCU, V. N . a n d GALETUSE, S. 'On t h e p o s s i b i l i t i e s of i m p r o v i n g t h e a c c u r a c y of t h e e v a l u a t i o n of i n e r t i a f o r c e s , i n laminar and t u r b u l e n t f i l m s ' , T r a n s . ASME, J n l . of Lub. Tech., 9 6 , J a n . 1974, 69-79. BATCHELOR, G . K . 'An i n t r o d u c t i o n t o f l u i d d y n a m i c s ' , Cambridge U n i v e r s i t y P r e s s , Cambridge, 1967.
440
////////////////////////////////////////////////// b
a
d
C
VELOCITY STREAMLINES
I I I I
I
-- . .
I I I
c
I
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c
b
a
, I ,,
I
,, I
e
c
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-.*
d
C
c
.
\
I
e
VELOCITY PROFILES
Figure 5.
Example of p r e s s u r e driven f l o w
3 b
a
d
C
e
VELOCITY STREAMLINES
\
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, \
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VELOCITY PROFILES Figure 6 .
.. .
..
Example of p l a t e driven f l o w
e
441
FLOW RATE = 0.4
Figure 7 .
The variation of pressure on the runner of the bearing
This Page Intentionally Left Blank
443
Paper XIV(iii)
Thermohydrodynamic analysis for laminar lubricating films Harold G. Elrod and David E. Brewe
1. INTRODUCTION
The purpose of this paper is to present a new method for incorporating thermal effects into the calculated performance of laminar lubricating films. There is enormous interest in the inclusion of such effects, as the recent reviews of Khonsari (1.2) attest. The reason for this interest is well founded, since the viscositytemperature dependence of typical lubricants is such that the viscosity can vary many fold across and along a bearing film, with attendant effects on load capacity. The up to date, extensive reviews in (1.2) remove the need here for a survey of prior literature, and reference will be made principally to those works used for comparative purposes. Suffice to say that earlier theoretical contributions on the subject of thermohydrodynamic lubrication divide themselves roughly into two categories. In the first category are those which embody a full transverse (cross-film) treatment of the energy equation using finitedifference or finite-element methods, and in the second category are those which incorporate rather drastic approximations to the transverse phenomena, usually representing the local film temperature distribution by a single value. Both approaches certainly possess merit. But the first approach obtains accuracy at the expense of computational speed, and the second obtains speed at the expense of accuracy. We shall show here that if just temperatures, chosen at "Lobatto points", are used to characterize the transverse temperature distribution in a laminar lubricating film, the effects of that distribution can be surprisingly well predicted. The calculations we have so-far performed have been directed solely towards demonstrating this fact, and only rudimentary numerical methods have been used in the plane of the film. Accordingly, no meaningful computation times can yet be reported. But we believe the technique will prove to be quite suitable for practical calculations. In the present analysis, fluid properties are taken as constant and uniform, except for the viscosity. The flow is presumed to be laminar, with negligible inertia effects. The fluidity (reciprocal viscosity) is represented by a polynomial in terms of position across the film, with coefficients related to the local film temperature distribution. Use of this procedure permits a closed-form expression for the local lineal mass flux, albeit there is some difference between a fluidity profile which would everywhere correspond to the temperature profile and one which is thus approximated. The Lobatto-point temperatures, or mathematical
equivalents, appear in two simultaneous partial differential equations obtained from the basic energy equation by a Galerkin procedure. Implementation of the present approach has involved considerable tedious algebra, which, however, once done, causes no further embarrassment. The procedure should conveniently couple with cavitation algorithms, and preliminary testing indicates that no special handling is required to cope with moderate recirculation at film entry. Moreover, it can accommodate to some extent the temperature streaking from hotoil carryover. We therefore expect to be able to exploit its use in a number of interesting directions. 2. NOHENCLATURE cP
specific heat at constant pressura, J/kg-K
h
film thickness, m
i
Cartesian tensor index
j
Cartesian tensor index
k
thermal conductivity, J/m2-(K/m)
4
m
lineal mass flux, kg/m-s
Pi
Legendre polynomial, ith order (PO = 1; PI = 5 ; P2 = (3C2 - 1)/2
P
pressure, Pa
T
temperature, K
t
time, s
U
x-wise velocity, m/s
V
y-wise velocity, m/s
vi
i-th component of fluid velocity vector, m/s
W
z-wise (cross-gap) velocity, m/s
wi
Lobatto weight function for i-th quadrature position, Ci
X
lateral coordinate in direction of surface motion, m
Y
lateral coordinate transverse to surface motion, m
Z
coordinate perpendicular to gap midsurf ace, m
444
Greek: a
fluidity functions (see Table I)
E
fluidity functions (see Table I)
d
fluidity functions (see Table I)
€
fluidity functions (see Table I)
c
dimensionless coordinate transverse to film, 2z/h
In these equations we shall treat the fluid viscosity as temperature dependent, and treat other fluid properties as constant. To these equations must be added the mass continuity equation for an incompressible fluid. Thus :
9
fluid viscosity, N-s/m2
K
thermal diffusivity, m2/s
E
fluidity functions (see Table I)
E
fluidity (reciprocal viscosity), m/N-s
P
fluid density, kg/m3
T
fluidity functions (see Table I)
@
dissipation function, J/m3-s
9
fluidity functions (see Table I)
+ + v.v=o
L3.071
The coordinate system used with these equations is defined iq Fig. 1. For convenience, a reference surface is taken midway between the film walls. A local coordinate system is substituted for a fixed Cartesian system, with the local x-y plane tangent to this reference surface. The film walls are rigid, but may be moving. The Galerkin-style analysis used here involves the expansion of the temperature in a truncated series of Legendre polynomials. Satisfaction is required of as many moments of the energy equation as there are unknowns in this series. The ensuing partial differential equations for the Legendre components are then solved. In the present treatment, only two unknown components are used. And for these it is feasible to carry out explicit integration, as follows:
3. BASIC EQUATIONS In the absence of gravity, the momentum equation for a Newtonian fluid without dilational viscosity is: avi
Dv Dt
13.081
av 13.011
J
-1 + axa J : t
where repeated subscripts imply summation. And the corresponding energy equation is:
zT dz
zuT dz
+
'J
- zvT dz
aY
-
J
wT
13.021 where :
D-d+v aT j ax Dt - at ~
13.031
j
is the time derivative following the fluid (Lagrangian derivative) and:
dz
13.091 All of the above integrals are taken from the "bottom" to the "top" of the film. The subscripts and -2 are used to denote the upper and lower walls, respectively. The coordinate "z" is measured from the midplane. Continuity has been used to convert transverse velocity constructs to lateral constructs, wherever possible. 4. LOBATTO POINTS; DISTRIBUTION FORHULAS
13.041 is the dissipation function. In lubricating films, lateral diffusion of momentum and heat is usually much less than transverse. Furthermore, inertia and pressureenergy effects are frequently negligible, and the transverse variation of pressure is small. Therefore, we adopt the following equations for laminar lubricating films. 13.051 and :
An expression for the lineal mass flux can be developed directly from Eq. 13.051. (See, for example, Dowson and Hudson (3)). But the convective terms in Eq. 13.061 involve integrals of the velocity-temperature product, and so require detailed knowledge of the respective distributions. Such information for the velocity is already available. For the temperature, we must develop our own expression. Consider the integral JUT dz which relates to the cup-mean energy flow in the x-direction. Let the temperatures on the two walls be known, and the velocity be available where needed. If two sampling positions for the temperature -and only two -- are to be allowed for estimating this integral, where should these positions be chosen? Without knowledge of end-point values,
445
it is best to choose the well-known Gaussian quadrature points. W i t h knowledge of the endpoint values, the optimum locations are the so-called "Lobatto points" (41, N interior points permitting exact numerical integration of a polynomial of degree (2N + 1). The case N = 1 yields Simpson's rule. Here we take N = 2 , but it is evident that the procedure to be used is quite general, and that further refinement is possible with further analytical effort. In terms of the variable C = 2z/h, the Lobatto points providing exact numerical integration over the interval -1 to +1 of polynomials up to the fifth degree are:
+
116 5/6 5/6 116
1
+
[4.06]
In these expressions, the wall temperatures T2 and T-2 are considered as known for purposes of the film calculation. It is then useful to note that:
-
T = 2
T + T 2 -2 2
-
- To
-T3 = T2 -2 T-2 - -T1
I Location, I Weight, I Subscript 116 -l/& -1
T-2 - (T1
L4.081
[4.091
An expression for the fluidity might be developed a number of ways. Here we collate the fluidity with the temperatures at the Lobatto points; that is, at the walls and at two interior points. The Legendre expansion for the fluidity is then developed in a manner completely analogous to that for the temperature. For example,
2 1 -1 -2
Then, for example, by Lobatto numerical quadrature:
and the fluidity distribution is:
4
P ,
C6 dC
=
0.341
[4.111
(exact = 217)
The temperature distribution which passes through the Lobatto-point values is most easily expressed in terms of an expansion in Legendre polynomials. Thus if we write:
c
5 . VELOCITY EXPRESSIONS; UASS FLUX
A double integration of Eq. [3.05] (with F, = l/n) gives the tangential velocity vector. Thus:
3
=
14.011
YkPk(5.)
k=O then the Legendre coefficients are easily evaluated by integration. C5.021
Or:
-
T = -2 k + l k
and : 3 wiTiPk(Ci)
[4.031
i=o
The linear set of equations in r4.031 can be solved for the Ti, providing us with two modes of description of the temperature distribution in the lubriczting film. The Ti give us detail, and the Ti-give us overall properties. In particular, To is the space-mean temperature at the point (x,y) and Ti is the first moment. For N = 2:
B +
=
($
?p
In view of the Legendre expansion for can be alternatively written as:
r5.03
E . [5.02
[5.041
N
4.041 4.051
Now to obtain the lineal mass flux, the velocity expression L5.011 is integrated again across the passage, with the result: + + + h 2-+h 2 + ~ '=(V2+V )---F,A---B(E + 52 ?2) P - 2 2 3 1 2 3 0
446
Simplification gives the following more recognizable expression:
The expressions for the tangential and cross velocities will be essential for evaluating the contributions to convective heat transfer. 6. THE TEMPERATURE EQUATIONS
-
Here the fluidity parameters Ti are the vehicle for the temperature-flow interaction. The parameter represents asymmetry of the fluidity distribution. It is interesting to note that for symmetric temperature (and fluideffect of temperity) distributions, the ature on the mass flow is through the arithmetic average of the fluidities at the Lobatto points. This result is a special case of the following formula applicable for any symmetric cross-film temperature distribution:
5
With the aid of Legendre series for the temperature and fluidity, we are now in a position to evaluate the integrals appearing in the zeroth and first moment of the energy equation; namely, Eqs. [3.081 and [3.09]. Implementation is straightforward, but tedious. All requisite coefficients are given in Table I. Equation [3.081 becomes:
, +I cO
f6.011
pcP
where “CO” terms :
stands for the collection of 3 [6.021 k=O
k=O
N
For such cases, Eq. [5.07] justifies the effective viscosity concept, and shows the relative importance of the temperatures near the walls. Hass continuity applied to the mass flux expression [5.061 leads to a generalized Reynolds equation, the divergence of the mass flux involving spatial derivatives of pressure. Thus :
The temperature TO appearing here is the <-space mean temperature, and not the mixedcup temperature. The integral of the dissipation function is: I
-3
-ah+ v -3 . m = o at P
[5.081
Equation [3.091 becomes:
The first moment of the energy equation involves the cross velocity, w. With a little care, this velocity can be found via mass continuity. We have:
Transforming coordinates from (x,y,z) to (x,y,c) we find:
where
“C1“
stands for the terms:
+
=
(3
G)c
-
$h
. aaC
[5.10]
+ h2Fl C 1 = - h T0W -2 + V * 6 V - 2 N
+
(Here the velocity vector is understood to consist only of (u,v).) Substituting this result into E5.091 and integrating, we get:
+
3
Yk3c
(t, + dBk)
l6.051
k=O
[5.111 Finally, integrating by parts, we obtain:
..-
The moment of the dissipation function is:
441 demonstrated that, with more realistic wall boundary conditions for the temperature, the above-shown load support vanishes. Finally, Figs. 7 to 9 show the velocity, temperature and pressure profiles for a flat slider with 4/1 film-thickness ratio. These calculations were performed to test the sensitivity of the analysis to flow reversal at the entrance. As mentioned earlier, such reversal can cause problems for point-by-point prediction methods. We have yet to make any comparisons with other investigations.
Equations r6.011 and [6.04] are two simultaneous partial-differenLia1 equations in the two variables, To and TI. Where they appear, T2 and T3 can be eliminated via Eqs. 14.081 and [4.09] in favor of these dependent variables. Coupled with the generalized Reynolds Eq. [5.071, they provide our approximate thermohydrodynamic treatment for laminar films. As mentioned previously, our numerical techniques for dealing with these differential equations have so far been extremely simple. The steady-state solutions to be presented in the next section were found by solving the foregoing equations in their transient form, and no study has been made of the possibilities for economy in the fluidity evaluations. Prior to this work, enough success has been obtained by others with analyses which neglect entirely any crossfilm viscosity variations so that it seems unlikely that it will prove necessary to update the terms in Table I at every step of a solution. N
N
8. CONCLUSIONS A Galerkin-type analysis has been made of temperature effects in laminar lubricating films. The procedure capitalizes on the suitability of so-called "Lobatto points" for sampling of the temperature distribution. Preliminary indications are that the use of just two such sampling points enables satisfactory prediction of bearing performance even in the presence of substantial viscosity variation. The procedure described herein yields two partial differential equations, one for the local space-mean temperature, and one for the first transverse moment of the temperature distribution. These temperature equations are coupled to a generalized Reynolds equation. Results have been presented for the steadystate, infinitely-wide flat slider bearing, and comparisons with the earlier, detailed work of Dowson and Hudson are very encouraging. The procedure is quite general, and our intent is now to refine the numerical techniques, and to carry out calculations for more realistic configurations and boundary conditions.
7. RESULTS FOR THE INFINITELY-WIDE SLIDER BEARING
In 1963, Dowson and Hudson (3) performed some finite-difference calculations on the infinitelywide flat slider bearing, including variableproperty effects. They employed 100 increments along the length of the bearing, and a minimum of 20 increments transversely. These investigators were concerned with the relative effects of density and viscosity variations upon load capacity, with the effects of solid-fluid thermal interaction, and with the possibilities for load support from parallel surfaces. Their findings serve as basis for our assumption of a constantdensity fluid, and two special cases of their exploratory calculations will be used here for direct numerical comparisons. The fluid properties used by Dowson and Hudson were as follows (SI units):
9. ACKNOWLEDGMENTS
3
Density: 874.5 Kg/m Thermal diffusivity: 7.306~10-8 m2/s Viscosity: II = 0.13885*exp(-0.045(T-311.11))Pa S p e c i f i c Heat: 2010 J/Kg-C Lubricant entrance temperature: 311.11 K Wall temperatures: Uniformly at 311.11 K Runner velocity: 31.946 m/s Bearing length: 0.18288 m Uinimum gap: 0.00009144 m In the first case for comparison, the film thickness ratio is 2/1. Figures 2 and 3 show the velocity and temperature profiles obtained by us at the entrance, at 0.65 of the length, and at the exit. Figure 4 shows the corresponding pressure profile along the length of the bearing. Included in this last figure is the profile that would result if the entrance value of viscosity persisted throughout the film. The circles were read from the Dowson-Hudson curves, and the agreement is almost within reading accuracy. Figure 5 compares predicted temperature contours. In the second case, the bearing surfaces are parallel. The possibility of load support through a "viscosity wedge" is being explored. Figure 6 compares pressure distributions. Again, excellent agreement is achieved. We note parenthetically that Dowson and Hudson
s
It is a pleasure for HGE to express thanks for sponsorship as a Visiting Professor at the Technical Univ. of Denmark in 1977, during which time Dr. Jorgen Lund suggested thermohydrodynamics as a fruitful area for research. Also, to acknowledge some subsequent in-house support from The Franklin Research Institute, Phila., PA, USA. 10. REFERENCES 1. Khonsari, M.U., "A Review of Thermal Effects in Hydrodynamic Bearings. Part I: Slider and Thrust Bearings," ASLE Preprint Uo. 86-An-2A-3, 41st Annual Ueeting, Toronto, Ont., Uay 12-15, 1986. To be published in the Trans. of the ASLE, 1986. 2. Khonsari, U.U.. "A Review of Thermal Effects in Hydrodynamic Bearings. Part 11: Journal Bearings," To be published in the Trans. of the ASLE, 1986. 3. Dowson, D. and Hudson, J.D., "Thermo-Hydrodynamic Analysis of the Infinite SliderBearing: Part I, The Plane-Inclined Slider-Bearing. Part 11, The ParallelSurface Bearing," in Lubrication and Wear, May 23-25, 1963, (Institute of Uechanical Engineers, London, 1964) 34-51.
448
4. Villadsen, J. and Xichelsen, X . L . , "solution of Differential Equation Xodels by Polynomial Approximation," 1978, (PrenticeHall, New York) p. 127.
TABLK I.
5. Abramowitz. X. and Stegun, I.A., "Handbook of Xathematical Functions," 1964 (U.S. National Bureai of Standards, Applied Uathematics Series No. 55).
- FLUIDITY FUNCTIONS
uFIGURE 1 .-
40
COORDINATE DEFINITIONS FOR SLIDER BEARING.
r
a
20
E @ 10
0
.2
FIGURE2.-
.4 .6 POSITION IN GAP.?
1 .o
.8
X-VELOCITY VERSUS POSITION I N GAP.
340 c
c c c
0
1
2 3
Y
= o
-
= 2(Yo
- Y2/5)/3
2 ( 1 / 3 - T3/7)/5 1
= 2y2/35
330
= W
5
E
320
310
300 0
.4 .6 .8 POSITION IN GAP.? FIGURE 3.- TEMPERATUREVERSUS POSITION IN .2
1.0 GAP.
449
0
I
01 REF.
REF. 3 (DOWSON AND HUDSON) PRESENT ANALYSIS
3 (DOWSON AND HUDSON)
-J
FIGURE 5 . - TEMPERATURE
CONTOURS,
T/T,.
W
CT
n
0 -1
'0
n
.2
REF. 3 (DOWSON AND HUDSON) PRESENT ANALYSIS
I
.4
I
.6
I
.8
POSITION I N BEARING. FIGURE 6.- PRESSURE
VERSUS P O S I T I O N I N BEARING.
I
1.0
450
40
30
0
-10
, .4
.2
.6
POSITION I N GAP,
1 .o
.8
5
FIGURE 7.- X-VELOCITY VERSUS P O S I T I O N I N GAP.
I
15
I
r
12 a n
3
0
.2
.4
.6
.8
POSITION I N BEARING. X / l
FIGURE 9.-
PRESSURE VERSUS P O S I T I O N I N BEARING.
1.0
451
Paper XIV(iv)
The lubrication of elliptical contacts with spin D. Dowson, C.M. Taylor and H. Xu
1.
INTRODUCTION
It is well known that a spinning motion is superimposed upon the rolling and sliding motions in the lubricated conjunctions in a number of machine elements. Perhaps the best known example is the angular contact ball bearing, where spin frequencies of hundreds or thousands of revolutions per minute may occur. Furthermore, experimental studies ( 1 , 2 , 3 ) have indicated that the effective coefficient of friction in lubricated, spinning contacts can readily equal or exceed that encountered in both pure rolling and sliding situations. It is thus evident that spinning motion can contribute to and even dominate the power loss in important, lubricated machine elements such as the angular contact ball bearings in certain aircraft gas turbines. There has been considerable progress in the development of theoretical solutions to the elastohydrodynamic lubrication point contact problem in recent years, yet such solutions have been restricted to pure rolling or rolling together with some sliding in the direction of lubricant entrainment. In the present paper we examine the role of spinning motion upon the lubrication of a notional point contact formed between two ellipsoids lubricated by either isoviscous or piezoviscous fluids. With pure spinning motion the surface velocities of the ellipsoidal solids are zero in the centre of the conjuction, but since they increase steadily with distance from the axis of spin it is necessary to specify with some precision the periphery of the conjunction and the boundary conditions applied thereon. Snidle and Archard ( 4 ) recognized this important distinction between rolling and spinning motions in hydrodynamic lubrication in their theoretical studies of a sphere spinning in a groove. If hydrodynamic pressure is permitted at infinite distances from the spin axis, an infinite load carrying capacity is predicted, even for an isoviscous lubricant. It is, therefore, necessary to consider carefully the realistic extent of the pressure field, the boundary pressures and film rupture and reformation if satisfactory estimates of pressure distribution and load carrying capacity are to be achieved. In the present paper we present the results of a numerical analysis of the hydrodynamic lubrication of two spinning, rigid ellipsoids. Solutions were initially obtained for a half-Sommerfeld cavitation boundary condition
to enable the results to be compared with the earlier findings of Snidle and Archard ( 4 ) , but in due course the more realistic Reynolds' cavitation boundary condition was adopted. Solutions were obtained for both isoviscous and piezoviscous fluids and empirical expressions have been derived to enable the minimum film thickness and the location and magnitude of the maximum pressure to be predicted for given operating conditions. The analysis was conducted in cylindrical coordinates and the Vogelpohl (4) solution was adopted in the solution of the Reynolds equation. 1.1
Notation
a,b,c
semi-principal axes of equivalent ellipsoid. f load capacity. h film thickness. i,j mesh locations with finite difference approximation. P pressure 9 reduced pressure r radial location of maximum pressure r r r r radii of curvature of bodies Ax, Ay* Bx, By at contact point. S geometrical separation between bodies. u,v mean entraining velocities (u = (u + u ) / 2 , v = (v + v ) / 2 B in x anh y Jirections. A x,y,z Cartesian coordinates. A ,B ,C, constants in Reynolds' equation. D' D' 'enhancement factors (Sections 3.1 1' 2 and 3 . 2 ) . equivalent elastic constant. dimensionless materials parameter ( = aE'). dimensionless film thickness ( = h/R ). X 2 dimensionless pressure (= pR /rl uab). transformed cylindrical coorzinates. radii of curvature of equivalent ellipsoid at contact. 2
dimensionless load ( = f/E'R ) dimensionless Cartesian coorzinates dimensionless speed-material parameter.
.
dimensionless speed-load parameter. radius ratio (R /R ) and pressureviscosity coeffYcifnt. dynamic viscosity. dynamic viscosity at atmospheric pressure. angular location of maximum pressure. density
.
452
0
Vogelpohl function. entraining angular velocity
w
(w=
w
+ w )
A
R
B
dimensionless spin parameter ( = r l W / E ' ) .
The dimensionless film thickness can thus be written as,
Subscripts A,B HS
contacting bodies. solution according to half-Sommerfeld boundary condition. maximum. minimum. solution according to Reynolds' boundary condition. coordinate axes.
max 0
R x,y 2.
THEORY
2.1
Geometry
H = Ho
+
[ ix][' -{
-
1
X2-
or, in cylindrical coordinates,
H
=
Ho
1-{
+[ ix]
1
-
It2}
Y2}
'1
'1
(6)
(7)
where ,
If two rigid ellipsoids have common principal z-axes at their point of contact and principal radii of curvature (rAx, r ) and ), the separation between tkgm at a (rBx' 'By location close to the contact point is the same as that between a plane (z=O) and an equivalent ellipsoid having principal radii (Rx, R ) given Y by Y 1
1
R
2.2
=[ [:]
+ ( 91
'1
f and 0 6 R < 1
(8)
The Reynolds equation
In Cartesian coordinates the steady state Reynolds equation for a plane disc spinning beneath an ellipsoid as shown in Figure 1 can be written as,
In lubrication analysis of more conventional configurations the effective region of pressure generation normally occurs within a space in the (x, y) plane of restricted dimensions compared with the principal radii of curvature (Rx, R ) and hence it is usually deemed to be accgptable to adopt a parabolic profile for the surface of the equivalent ellipsoid. The film thickness (h) can then be expressed in terms of the minimum film thickness (ho) with adequate accuracy by the relationship,
"
"
h
=
ho+
L
- + -2R X
2RX
L
X
Y
In the present problem, where the surface velocities increase steadily with distance from the spin axis, it is necessary to retain a more accurate representation of the film thickness. This is achieved for an ellipsoid having semi-principal axes (a,b,c) in the (x,y,z) directions by noting that, h = h + s
(3)
-
where ( 8 ) is the separation between the plane and the ellipsoid at location (x,y). Figure 1
For an ellipsoid,
(4)
and hence ,
A plane disc spinning beneath an ellipsoid
where entraining velocities (u,v) can be expressed in terms of the angular spin velocity (w) as,
453
When $he rii.,’-*ratio (a) is equal to unity, A = R ; R1 = 1 and C1 = 0. 1 If an incompressible, isoviscous lubricant is considered the Reynolds equation reduces to,
2.3
Lubricant Properties
The initial analysis was carried out in terms of an isoviscous lubricant of viscosity (rl ), but the solutions were subsequently extendgd to yield predictions for the asymtotic film thickness for a piezo-viscous fluid. This asymptotic film thickness corresponds to an infinite pressure attained in a lubricant obeying the Barus relationship between viscosity and pressure. It is convenient to transform the elliptical field of computation into a circular area of unit radius by introducing the dimensionless variables,
-
where ( a ) is here the pressure coefficient.
-
viscosity
In order to produce solutions for a piezo-viscous lubricant it is convenient to solve the Reynolds equation in terms of a reduced pressure (9) given by,
y = l
b
H = -h Rx 2 P I -PRX nub
The piezo-viscous solutions for pressure (p) can thus be obtained directly from solutions to the isoviscous (q) problem.
The Reynolds equation can then be written in dimensionless form as;
(12)
-
where
[ i]
2
R = -Y = a, the radius ratio. RX
In cylindrical coordinates (R,e) this becomes,
where, A1 = R2
2
[ a cos 8 + sin
2 B1 = a sin
+ cos2
8
el
2.4
Boundary Conditions
Snidle and Archard (4) demonstrated that in pure spin between a ball and a groove the load carrying capacity theoretically approached an infinite value as the domain of integration was extended indefinitely. They therefore limited the domain of integration to a circle projected from the spinning ball, with a non-zero pressure on the boundary.
In the present study we assumed that the pressure in the lubricant returned to atmospheric (zero) on the circumference of an ellipse projected onto the spinning disc. The transformation of the elliptical field of interest into a circular region, as shown in Figure 2, facilitated the application of this pressure boundary condition. It is convenient in analytical studies of lubrication to adopt the half-Sommerfeld cavitation boundary, in which the pressure returns to atmospheric (zero) on the boundary between the converging and diverging regions and remains at zero throughout the cavitation zone. This assumption was also adopted initially in the present study to enable comparisons to be made with the predictions of Snidle and Archard (4). However, this assumption violates the physical requirement for continuity of flow and hence the following, more realistic, Reynolds’ cavitation boundary condition was also considered.
454
I’
mesh at larger radii. The computer program accommodated variable mesh sizes automatically for different geometries and film thicknesses. The nodal interval varied from 1/8th of the distance from the spin axis to the radius of maximum pressure to l/lOth of the radius of the transformed circle of contact. Integration Domain. When the half-Sommerfeld cavitation boundary condition was adopted it was necessary to solve the Reynolds equation, and to integrate the predicted pressures to obtain load carrying capacity only in one of the convergent film quadrants. Half the full elliptical domain w a s considered when the Reynolds cavitation boundary condition was imposed. Finite Difference Presentation. Since the hydrodynamic pressure distribution exhibits peaks in the regions where the film thickness is small it is convenient to seek solutions to the Reynolds equation in terms of the Vogelpohl variable (4) defined as,
TRANSFORMATION
3/2
@ = PH
The Reynolds equation can be written in terms of ( 4 ) as,
Figure 2
Transformation of the elliptical field of interest into a circular region, showing nodal points used in the finite difference approximation
It was further assumed that the film would reform along the boundary between the diverging and converging film regions. This also violates the continuity requirement, but the determination of a valid reformation boundary is a major numerical task beyond the scope of the current analysis. 2.5
.
Numerical Analysis
The Reynolds equation was written in finite difference form and the resulting equation was solved numerically by means of the Gauss-Seidel iterative procedure. Over-relaxation was adopted to speed up the convergency of the process. With pure spin the variation of pressure around the contact can be rapid, particularly at small radii, and a fine mesh size was necessary to improve the numerical stability and accuracy. Nodal Structure. The use of cylindrical coordinates greatly assisted the introduction of a fine mesh near the axis of spin and a coarser
The numerical values of (4) can readily be restored t o values of dimensionless pressure by means of equation (16).
In the numerical solution of the Reynolds equation it was assumed that all the variables were distributed quadratically in the coordinate directions. Derivatives were thus expressed in terms of three adjacent points, (i,j-1), (i,j) and ( i , j + l ) in the radial (R) direction and (i-l,j), (i,j) and (i+l,j) in the circumferential (€9 direction as shown in Figure 2. The Gauss-Seidel iterative over-relaxation method was used to solve equation (17) written in finite difference form. Numerical Procedure. In the computer program one main loop was employed to solve the Reynolds equation in its approximate fom. Two convergency criteria were adopted, one for the maximum difference on any nodal pressure from the previous cycle and the other for a global difference check. Several thousand nodal points were adopted for each solution and careful trials established that a local criterion of 0.001 and a global one of 0.01 resulted in excellent accuracy whilst minimizing the computer run time.
455
Integration for load carrying capacity was effected numerically once the above criteria had been satisfied. It was assumed that the pressure in each cell surrounding the mesh points was constant and the load carrying capacity was determined by summing the contributions from all the cells. For the rigid-piezoviscous problem it was assumed that the asymptotic film thickness was achieved whenever the maximum pressure approached an infinite value. It is evident from equation (15) that this limiting situation corresponds to a value of reduced pressure (q) of (l/a). An additional computing loop was designed to adjust the film thickness until (q) satisfied this requirement. 2.6
Non-dimensionalized Results
Although the study presented here was concerned with rigid bodies, it formed part of a wider examination incorporating elastohydrodynamic effects. To facilitate comparison of results, non-dimensional groups appropriate to elastohydrodynamic analysis with spin have been adopted. Thus, Dimensionless load parameter (W) =
f -
E'R~~ Dimensionless materials parameter (G) = a E '
Dimensionless spin parameter
(n)
rlOw
= 7
E
Thus the dimensionless products used in the presentation of results,
and
GR
=
~,wa
are independent of the elastic properties of the solids. The former product (termed a speed-load parameter) is in effect a normalized load, whilst the latter is encountered in rigid-piezoviscous lubrication with spin. 3.
RESULTS AND DISCUSSION
Numerical solutions for the determination of the normalized pressure distribution generated when an ellipsoid is located against a spinning disc have been undertaken for a wide range of conditions. Non-dimensional minimum film thickness has been set as follows,
whilst radius ratios of the following values have been considered,
a
=
making 40 cases in all, was considered whilst the ratio (c/Rx) was fixed at a value of 1.0. The effect of the variation of (c/R ) was established in a further series of ?6 solutions in which this ratio of the semi-principal axis of the ellipsoid in the z-direction to the effective radius of curvature at contact in the x-direction was set at,
-
=
0.5, 1.0, 1.5, 2.0, 2.5, 3 . 0 , 3.5, 4.0
RX
with both the Reynolds' continuity and half-Sommerfeld boundary conditions at film rupture. It was established to a good accuracy that (Ho) was directly proportional to (c/Rx). Having determined the pressure distribution the corresponding speed-load parameter (W/n) could be obtained by numerical integration. A regression analysis of the data resulted in the following formulae for prediction of non-dimensional minimum film thickness according to the rupture boundary condition adopted, Reynolds' Boundary Condition
Ho
=
0.06
1.16
1
RX
Half-Sommerfeld Boundary Condition
H
=
0.056
exp -0.249 RX
R a(a-1)
The maximum absolute error incurred in the use of the above formulae to predict non-dimensional minimum film thickness in comparison with the numerically determined value was 4.9% for the Reynolds' boundary condition case and 2.4% for the half-Sommerfeld boundary condition (with respect to the value set in the numerical solution). The corresponding mean errors over the range of conditions examined were 1.2% and -0.3% respectively. The excellence of the fits represented by the equations (18) and (19) are shown in Figures 3 and 4 respectively. With the half-Sommerfeld boundary condition for film rupture the magnitude and location of the maximum pressure was given to a good approximation by the expressions derived by Snidle and Archard (4). The introduction of the Reynolds' boundary condition resulted in an increase in the value of maximum pressure and a modification in its location. The position of maximum pressure moved almost in a direction perpendicular to the y-axis of the ellipsoid (the major axis of the projected ellipse) i.e. parallel to the minor axis direction. A regression analysis of the numerical data led to the following expressions for maximum pressure and its location with the Reynolds' rupture boundary condition,
1.5, 3 , 6, 12, 24.
A full permutation of these two variables,
w
C exp
WRx
Pmax = 0.59
hO
1.04
[s)
3 a
(20)
456 vary with pressure according to the Barus relationship (equation (14)). The five values of radius ratio 'detailed previously were adopted whilst the non-dimensional speed-material parameter (GR) was set as follows,'
---
XlOZ 16
4.h cl 1.s NUlLP1C.l H I ~ c P L c I I data a 3.0 H I m e ~ l c a l data U 6.0 ~ a e r l c 1 1data 01 12.0 N u m c ~ l c a l data 01 24.0 Fit QY regression analvalr
-
IRayraldS QOundlPy condltlon "lth C I R . 1.01
GR
=
8.621~10-~, 1.724~10-',
3.449~10-~,
A full permutatiqn of these variables was
-7
-6
-8
-3
-4
Lo610 No"-dlm.nslon.l
.l"lW.
111.
tn1cun.sa
lHol
Figure 3 Non-dimensional speed-load parameter as a function of non-dimensional minimum film thickness (Reynolds' boundary condition).
examined leading to the following best fit formulae for the non-dimensional limiting minimum thickness. This is the film thickness when an infinite pressure is first attained as load is increased, other things remaining fixed. Further increase in load does not result in any change in film thickness; thus the expressions are independent of load. Reynolds' Boundary Condition (Piezoviscous Lubrication)
Half-Sommerfeld Boundary Condition (Piezoviscous Lubrication) Ho
=
0.5
CE ba+l -I] a
The latter expression is the same as that given by Snidle and Archard (4) with their analytical solution. Here again, because the value of (c/R ) does not influence the intense conditions loeal to the minimum film thickness position, it does not appear in the formulae. -7
-5
-6
-4
rn
-3
Figure 4 Non-dimensional speed-load parameter as a function of non-dimensional minimum film thickness (Half-Sommerfeld boundary condition).
0 = tan-1
((1.3a)
4)
r = (hoRx (a+ 0.8) ) 3
(22)
Here the angular co-ordinate (€I) is measured from the line of the x-axis (the minor axis of the projected ellipse). These formulae were good fits to the numerical data and it will be noted that they do not involve the ratio (c/R ). X This is because this ratio does not influence to any significant extent conditions local to the maximum pressure. A comparison of the numerical results with analytical expressions will be made later. Computations have also been carried out for the case of rigid-piezoviscous lubrication in which, although the solids were taken to be rigid, the lubricant viscosity was permitted to
Figure 5 shows a plot of non-dimensional minimum film thickness (H as a function of the normalised grouping iholving the speed-load parameter (Wh)and radius ratio (a) appearing in the regression fit given as equation (18). A value of (c/R Of unity has been taken for the plot. On a fog-linear basis the rigid-isoviscous solution with Reynolds' boundary condition at rupture as expressed by equation (18) is a straight line of negative slope. Also shown for the various values of radius ratio and speed-material parameter ( G R = 6.446 x lo-') are the curves leading to the non-dimensional limiting minimum film thickness as given by equation (23) with rigid-piezoviscous lubrication. A s can be seen, for each radius ratio, as the loading is increased a situation is reached where the film thickness reduces no more. This behaviour is analogous to that described by Dowson, Dunn and Taylor ( 5 ) for rolling/sliding concentrated contacts lubricated with a piezoviscous fluid. In Figure 6 an example of the pressure distribution occurring with pure spin is presented. This is for the Reynolds' boundary condition at rupture and the pressure has been set to zero gauge along the line of the minor axis of the projected ellipse. The values of the influential variables are detailed and two contour plots are shown. The one on the left covers the entire region of computation, whilst
457 LOG 10
RIGIO-ISOVISCOUS
&
n = 1.5
--
U
+ -+-
I q -
\ /
y
-C--
= 3.0
n :6 . 0 u = 12.0 a = 24.0
"
1.14
E
f
in8
2-6-
2 1.06
Gfl-
6.L66E-7
L
c
E
-5-
RIGIO-PIEZOVISCOUS LIMITING
B
-
i
a
. .
..
u n n
n
-+
I.04
LOO !7
. . . . . ..,
.
-6
.
. ..
,
---
-
1.5
3.0 6.0 12.0 24.0
,
-5
.
-4
,
,
.
-
-3
LO610
15 2 0 2 5 30 3 5 LO 45 5 0 5 5 60 65 70 7 5 80 8 5 90 9 5 1 0 0 1 0 5 110 Non-dmcnsional proup
I
NOn-di.snsiona1
lnd
1i ~n' ) l . l r I
R n-1
Figure 7 Figure 5 Non-dimensional minimum film thickness showing transition from rigid-isoviscous lubrication to the limiting minimum film thickness with rigid-piezoviscous lubrication.
that on the right is for the localised region indicated in the vicinity of the minimum film thickness position. The symmetry of the pressure distribution in the two halves of the contact separated by the minor axis of the projected ellipse is apparent. Two pressure peaks are found at symmetrical locations on either side of the centre of the contact. The pressure gradients in the vicinity of the maxima are very large but the pressure decreases much more gradually as the elliptical boundary is approached. The angular location of the position of maximum pressure (equation ( 2 1 ) ) was found to be in the sector between ( ~ 1 4 )and ( ~ 1 1 2 )radians from the appropriate minor axis. An increase in the radius ratio ( a ) caused this location angle to increase.
It is instructive to consider the influence of the incorporation of Reynolds' boundary condition into the analysis in some depth. The effect upon ( i ) the pressure profile, maximum pressure and its location ( i f ) the load capacity for a fixed film thickness and (iii) film thickness for a fixed load will be addressed. 3.1
nlninum 111. t n i c k n e I I
Pressure
If the half-Sommerfeld boundary condition was employed to define film rupture then the pressure distribution was confined to the two converging portions of the film. The introduction of the Reynolds' continuity boundary condition resulted in the generated pressure creeping over the major axis with the rupture cavitation boundary being located in the diverging portion of the film. In Figure 7 the ratio of the maximum pressures obtained with the Reynolds' and half-Sommerfeld rupture boundary conditions is plotted as a function of the non-dimensional minimum film (H ) for the range of values of radius ratio (ay considered. It is seen that this ratio depends only upon the radius ratio
Ratio of maximum pressures obtained with Reynolds' and half-Sommerfeld boundary conditions as a function of non-dimensional minimum film thickness.
( a ) with the value of minimum film thickness exerting no effect. For radius ratio varying from 1.5 to 2 4 , the enhancement of maximum pressure with the Reynolds' boundary condition varies from 1.106 to 1.166.
Results have shown that the angular location of the position of maximum pressure is independent of film thickness and a function only of radius ratio. In Figure 8 the ratio of this angle to the analytical value determined by Snidle and Archard ( 4 ) has been plotted for both the Reynolds' and half-Sommerfeld boundary conditions. It can be seen that with the half-Sommerfeld condition the agreement with the Snidle and Archard values is excellent.
-A"D"llP
-5
108
2
1.06
IC/R,I
-
location 1.0
:I 0 4 m
-1
..:
1.02 100
:: 0.38
'
0.96
Figure 8 Ratios of angular and radial location of maximum pressure to the Snidle and Archard solution ( 4 ) for both the Reynolds' and half-Sommerfeld boundary conditions. This would be expected, despite the differing pressure conditions adopted at the extremity of the contact region, as this does not affect significantly the pressure generation local to the contact centre. The introduction of the Reynolds' film rupture condition causes the angular location of the maximum pressure posJtion to be moved downstream. The data on
458
Zero pressure along mlnor-axls 8 Reynolds rupture boundary condltlon
1 2 3
k &
a 21
1.0
0.5 I
,
\
ProJectIon In Y dlrect Ion
X
0.0
101 0.1OSE 91 0.98m 81 0.801% 71 0. S881E 61 0.2524E 51 0 . l l l l E 41 0.S04M 31 0.23SE 21 0. 10S6E 11 0.18OoE
-a 5
P-
=
04
03 03
03 03 03 02 02
02 01
0.1069E 04
Splnnlng dlrect Ion# Ant I -c loc ku I se
0
-1.0
-0.5
ao
a5
1.0
-0.1ft4
0.18
Non-dlmenslonal pressure wlth pure spln (a) = 0.5OE-03, (U/N = W R x ) = 0. lOOE 01
Redlus r e t l o
HO =
0.240E 02 0.1036E 04
Left, Pressure d l s t r l b u t Ion u t t h l n comput tng regloni Right1 Locel pressure dlotrlbutlona
Figure 6
Pressure distribution with pure spin (Reynolds' boundary condition).
459 Figure 8 also presents the ratio of the radius of location of the maximum pressure to that determined by Snidle and Archard. Again results for the two rupture boundary conditions considered in the numerical analysis are shown. This latter ratio is also independent of film thickness and good agreement with the Snidle and Archard analytical prediction was observed when the half-Sommerfeld rupture boundary condition was adopted in the numerical analysis. With the introduction of the Reynolds' boundary condition, the radius of location of the position of maximum pressure is decreased, this effect reducing with increase in the radius ratio ( a ) . 3.2
This equation predicts the enhancement to be 13% for a radius ratio ( a ) of 1.5 and 22% for a radius ratio of 24.
3.3
F i l m Thickness
The influence of the incorporation of Reynolds' rupture boundary condition upon non-dimensional film thickness, with the speed-load parameter fixed, may be determined in a similar manner to above from equations (18) and (19). Defining an enhancement,
Load Capacity
The effect of incorporating the Reynolds' rupture boundary condition upon load capacity, other paramters remaining fixed, may be seen from Figures 3 and 4. As would be expected there is an enhancement of load capacity in comparison with the value determined with the half-Sommerfeld boundary condition. The formulae for film thickness with the two rupture conditions (18 and 19) may straightforwardly be rearranged to render expressions with the speed-load parameter as the subject. For the Reynolds' condition this is,
4.93 (a-1)
we find, r
(a+') R (0.249-0.203 da-1)
(27)
This indicates that the enhancement is very sensitive to both the speed-load parameter (W/R) and the radius ratio ( a ) because of the exponential relationship. This situation is clarified in Figures 9 and 10. In Figure 9 ,
a
-
ney : 1.-
2 : 3: 4 :
whilst for the half-Sommerfeld condition, 0.056
5 '
c
m I I
4
0
--k
o
m
---
1 5 3.0 8 0
12.0
- 2 4 0
Defining a factor measuring the enhancement of load capacity,
Figure 9
we obtain,
-1 (25)
Noting that normally,
0.06 >>c HO
RX
equation ( 2 5 ) shows that the enhancement of load capacity is virtually independent of the non-dimensional minimum film thickness and to a good approximation,
D1 = 1.23
[
2 0.16 a+l]
-1
Ratio of mon-dimensional minimum film thickness calculated according to the Reynolds' and half-Sommerfeld boundary conditions as a function of the half-Sommerfeld value.
for the range of radius ratios considered, the ratio of non-dimensional minimum film thickness determined with the Reynolds' condition to that calculated with the half-Sommerfeld condition is plotted as a function of the latter (for c/R = 1 > . This ratio ( ( H o > , / ~ ) s ) , which is equaXl to ( D 2 + l > , increases witR fecreasing (Ho) and increasing radius ratio ( a ) . For condy?ions which would give a non-dimensional minimum film thickness wi_kh the half-Sommerfeld boundary condition of 10 , with the Reynolds' boundary condition this film thickness would be about 7.5 times larger (D =6.5) for a radius ratio of 24 and about 3.7g times larger (D 2 =2.75) for ( a ) equal to 1.5. An alternative way of presenting the
460
enhancement is given in Figure 10 where the same film thickness ratio on the ordinates is now plotted as a function of the speed-load parameter (W / a ) . The dramatic influence of the Reynolds' rupture boundary condition is here clearly evident. The rate of change of the film thickness ratio with speed-load parameter increases with reducing radius ratio (a), and is very substantial at the smaller values of (a) considered here. For a speed-load parameter (W/Q) of 1000 the film thickness is predicted to increase by a factor just less than 8 for a radius ratio of 2 4 . 12
- ----
-t 01
111
- u -
a
PI -8- I -i c
Figure 10
1.5 3 0 6.0
/
12.0 24.0
Ratio of non-dimensional minimum film thickness calculated according to the Reynolds' and half-Sommerfeld boundary conditions as a function of the speed-load parameter.
It is immediately apparent that whilst there is a modest increase in load capacity when the Reynolds' rather than the half-Sommerfeld boundary condition is adopted with film thickness being taken a constant (and other factors remaining unchanged), if the load capacity is fixed the change in film thickness is substantial. The physical interpretation of this does help to clarify the understanding of contacts in which there is spin, as opposed to the better known behaviour of point contacts with rolling and/or sliding. With the latter kinematic conditions, the effect of the introduction of the Reynolds' boundary condition on load capacity with constant film thickness is similar in magnitude, however, the influence on film thickness with constant load is significantly smaller. The essence of the difference between concentrated contacts with rolling and/or sliding and those with spin is that with the former the influencial pressure region is very local to the point of minimum film thickness, whilst with the latter, although the maximum pressure is located very close to the position of closest approach, the more modest pressures well removed from this point are important. This is because the spin velocity increases with increasing distance from the axis of rotation, a situation which is not found with velocities in rolling and sliding contacts. Thus if the minimum film thickness in a point contact with spin changes substantially, then the maximum pressure changes in inverse proportion (equation ( 2 0 ) ) . However, the remaining pressures in the contact will not show
the same degree of change because of the kinematic conditions in the contact. Here it must be recalled that although the minimum film thickness may change substantially, the thickness of the lubricant film away from this point will only change by a smaller amount, and well away from the nominal contact point the percentage change will be minute. Since pressures well away from the maximum pressure position are significant in the total load capacity (because of the spin velocity situation mentioned above) the load capacity will only vary slightly despite the fact that the minimum film thickness may have varied substantially. This may be seen from Figures 3 and 4 . This is why the introduction of the Reynolds' boundary condition to replace the half-Sommerfeld condition can have such a significant effect on film thickness with load kept constant. The situation described above for an isoviscous lubricant does not apply for piezoviscous lubrication, at least in terms of the limiting minimum film thickness when an infinite pressure in the lubricant film is attained. It can be seen from equations ( 2 3 ) and ( 2 4 ) that the effect on limiting minimum film thickness of the introduction of the Reynolds' boundary condition at rupture is modest. From Figure ( 5 ) it may be seen that piezoviscous effects in the limit may increase the rigid-isoviscous minimum film thickness for this situation by a factor of about 3. The boundary condition for rupture only has a small effect because the asymptotic limiting film-thicness shown in Figure 5 is dependent upon the maximum pressure and this is not greatly affected. 4.
COMPARISONS WITH ANALYTICAL PREDICITONS
It is instructive to compare the results of numerical analysis presented in this paper with approximate analytical solutions. Snidle and Archard ( 4 ) in their analysis of a sphere spinning in a groove derived expressions for the maximum pressure and its location, together with a formula for load capacity which can be cast in terms of the non-dimensional parameters adopted by the present authors. The Snidle and Archard analysis employed the following simplifications, parabolic film profile ... half-Sommerfeld rupture boundary condition circular area of contact to determine load capacity. The latter simplification was necessary since an integration of the pressure field extending to infinity would render an infinite load capacity. The adoption of the circular area of integration for load capacity (with radius equal to the radius of the spinning sphere) implies a non-zero pressure boundary condition. The Snidle and Archard expressions for maximum pressure and its location were,
(28)
461
These formulae may be compared' with those obtained by the present authors ( (201, (21), (22) ) for an ellipsoid with the Reynolds' rupture boundary condition. The relationship between load capcaity and minimum film thickness from Snidle and Archard can be expressed in non-dimensional form as,
[ (a-1) Ln
w
4a R - (a+l)
(
1 ) -
ULn
a ] (31)
2H0
Figure 11 shows that the agreement between the prediction of non-dimensional maximum pressure after Snidle and Archard and that according to a numerical solution with the half-Sommerfeld boundary condition is excellent. I
x1d
.P
n u .. . .
Figure 11
...
I
.
. . .. ....
' 0 .
,
. , . . , . 15. . . . . , . . . 20. . , 1
Figure 12 Non-dimensional speed-load parameter as a function of non-dimensional minimum film thickness - Snidle and Archard ( 4 ) and authors' solution for half-Sommerfeld boundary condition. according to the Snidle and Archard analytical solution and the numerical solution obtained by the authors with the half-Sommerfeld boundary condition. For a given value of non-dimensional minimum film thickness, the Snidle and Archard prediction underestimates the authors' non-dimensional speed-load parameter at radius ratio values of 2 4 and 12, is in excellent agreement at (a=6) and marginally overestimates the parameter at the two smaller radius ratios adopted. The authors data is here for a value of (c/Rx) of unity. For a given value of non-dimensional speed-load parameter, the error in prediction of normalised film thickness between the two solutions would be very substantial for the higher radius ratios because of the nature of the (w1a-H ) relationship.
' ' 7
25
Non-dimensional maximum pressure as a function of radius ratio for various solutions.
Indeed across the whole range of radius ratio A similar agreement is found in relation to the location of this maximum pressure. This demonstrates that the concentrated nature of the pressure distribution in the vicinity of nominal contact is essentially unaffected by the boundary conditions on pressure at the extremity of the region of interest. Also shown on Figure 11 is the non-dimensional maximum pressure according to the present numerical analysis with the adoption of the Reynolds' rupture boundary condition. The enhancement compared with the predictions using the half-Sommerfeld boundary condition is evident.
( a ) the agreement is within 0.5%.
A plot of non-dimensional speed-load parameter (W/n) against non-dimensional minimum film thickness (H,) for the various values of radius ratio (a> is shown in Figure 12,
The explanation of the results presented in Figure 12 lies in two conflicting effects. On the one hand the non-zero pressure boundary condition adopted by Snidle and Archard in their prediction of load capacity will result in an overestimation of load capacity for the circular projected area. On the other hand the use of the circular projected area will lead to an underestimate of load capacity compared with an elliptical region of interest. The latter underestimate will get worse as the radius ratio increases. (It will be noted that since the authors' data of Figure 12 is for (c/&=l>, the minor axis of the projected ellipse representing the area of pressure generation is equal to I&. This equates to the radius of the spinning sphere as used by Snidle and Archard). Thus for a large radius ratio the Snidle and Archard prediction of load for a given film thickness will be an underprediction compared with the authors' value. Similarly at low radius ratios the reverse will be true. 5.
AN EXAMPLE
Consider an ellipsoid with the following semi-principal axes,
462
a = 12.7 mm b = 50.8 mm c = 12.7 mm This gives R =12.7mm, R =203.2mm and R /R =16. Take the ellfpsoid to bg located againxt f disc having a rotational frequency of 25 Hz and lubricated by a fluid of dynamic viscosity 0.2 Pas. The following table, Table 1, shows the prediciton of (a) minimum film thickness for a fixed load of 2N, (b) load capacity for a fixed minimum film thickness of loW6, according to the analytical solution of Snidle and Archard and the numerical solutions of the authors for the two rupture boundary conditions considered, according to the regression formulae developed. (a) Minimum film thickness for a Load of 2N Snidle and Archard analytical solution
0.30 pm
(b) Load Capacity for a Minimum film thickness of 1 pm
(iii) The solutions undertaken have shown the adoption of the Reynolds' boundary condition at rupture to have the following influences vis-a-vis the half-Sommerfeld condition. With other factors remaining constant, (a)
the maximum pressure is increased and its location moved 'downsteam' almost in a direction parallel to the minor a x i a of the projected ellipse.
(b)
there is a modest increase in load capacity for a given minimum film thickness.
(c)
there is a substantial reduction in minimum film thickness for a given load capacity.
1.66N
Numerical Solut ion (halfSommerfeld boundary condition) Numerica1 solution (Reynolds boundary condition) TABLE 1
6.
to predict non-dimensional minimum film thickness in terms of the influential normalised parameters, for both film rupture conditions, with rigid-isoviscous lubrication. In addition, for rigid-piezoviscous lubrication, expressions to predict the non-dimensional limiting minimum film thickness have been presented. For isoviscous lubrication, the formulae developed by Snidle and Archard to predict the maximum pressure and its location for the half-Sommerfeld boundary condition have been found to be entirely consistent with the present numerical solutions using the same film rupture condition. New expressions for these parameters have been presented for the case with the Reynolds' boundary condition.
CONCLUSIONS
(i) The lubrication of an ellipsoid located ageinst a spinning disc has been considered. Thls study extends that of Snidle and Archard(4) who analysed the case of a sphere spinning in a groove. The latter solution was entirely analytical and in order to achieve this a parabolic film shape was taken and the half-Sommerfeld rupture boundary condition adopted. In order to avoid the possibility of an infinite load capacity, Snidle and Archard chose a circular region of interest with a non-zero pressure boundary condition. The present numerical analysis has adopted an elliptical pressure domain with zero gauge pressure on the boundary. Both the half-Sommerfeld and Reynolds' rupture boundary conditions have been considered. (ii) From the numerical solutions regression analyses have been undertaken to give formulae to predict non-dimensional minimum film thickness in terms of the influential normalised parameters, for both film rupture conditions, with rigid-isoviscous lubrication. In addition, for rigid-piezoviscous lubrication, expressions
(iv) Comparison of the non-dimensional speed-load versus non-dimensional minimum film thickness characteristic obtained by the authors using the half-Sommerfeld boundary condition with the analytical prediction of Snidle and Archard has been undertaken. One influential factor in the comparison is the difference of the areas of pressure generation which leads to an underestimation of load capacity (for a given film thickness) by the Snidle and Archard formula for the present situation. This is particularly true at high values of the radius ratio. The other factor relates to the non-zero pressure boundary condition adopted by Snidle and Archard. (v) In utilizing the data presented in this paper to analyse the behaviour of non-conformal contacts with pure spin it will be necessary to represent the contact by the classical equivalent of the ellipsoid/plane. There are difficulties in achieving this effectively. The nature of the problem with pure spin is that the extent of the region of interest is of importance in predicting the load capacity - film thickness characteristic. Significant differences, particularly in predicting the minimum film thickness for a given load capacity, may result from the adoption of alternative domains of pressure generation. The same observation is pertinent to the situation of the elastohydrodynamic lubrication of point contacts with pure spin which the authors are presently addressing and details of which will be published in due course.
463
APPENDIX 1 Reference s Parker, R.J., Zaretsky, E.V. and Anderson, W.J. 'Spinning friction coefficients with three lubricants'. J.Lubn.Tech.,Trans. ASME (Series F ) , Vol. 9 0 , Jan. 1 9 6 8 , p. 330.
Dietrich, N.W., Parker, R.J., Zaretsky, E.V. and Anderson, W.J., 'Contact conformity effects on spinning torque and friction'. J.Lubn.Tech., Trans. ASME, April, 1 9 6 9 , Paper No. 68 - Lub. 10, p. 308.
Allen, C.W., Townsend, D.P. and Zaretsky, E.V., 'Elastohydrodynamic lubrication of a spinning ball in a non-conforming groove'. J.Lubn. Tech. Trans. ASME, Vo1.92, Jan. 1 9 7 0 , p. 89. Snidle, R.W. and Archard, J.F., 'Theory of hydrodynamic lubrication for a spinning sphere'. Proc. I.Mech.E., Vol. 184. Part 1 , No. 44. 1969-70, p. 8 3 9 . Dowson, D., Dunn, J.F., and Taylor, C.M., 'The piezo-viscous fluid, rigid solid regime of lubrication'. Proc. I.Mech.E., Vol. 1 9 7 c , March 1983, p. 43.
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SESSION XV BEARING DYNAMICS (2) Chairman: Mr. P.G. Morton PAPER XV(i)
Oil film rupture under dynamic load? Reynolds' statement and modern experience
PAPER XV(ii)
The influence of cavitation on the non-linearity of velocity coefficients in a hydrodynamic journal bearing
PAPER XV(iii) Effects of cavity fluctuation on dynamic coefficients of journal bearings PAPER XV(iv) Investigation of static and dynamic characteristics of tilting pad bearing
This Page Intentionally Left Blank
467
Paper XV(i)
Oil film rupture under dynamic load? Reynolds' statement and modern experience Otto R. Lang The problem o f boundary c o n d i t i o n s . f i l m r u p t u r e , c a v i t a t i o n o r C i i m e x t e n d i s b r o a d l y under d i s c u s s i o n f o r s t e a d i l y loaded b e a r i n g s wi.thout s i g n i f i c a n t e f f e c t t o b e a r i n g c h a r a c t e r i s t i c s . Much more i n f l u e n c e d a r e d y n a m i c a l l y loaded b e a r i n g s by d i f f e r e n t t h e o r e t i c a l boundary c o n d i t i o n s . By e x p e r i mental i n v e s t i g a t i o n s o f b e a r i n s under dynamic l o a d - measuring j o u r n a l c e n t r e p a t h and p r e s s u r e development - w i t h s p e c i a l l o a d p a t t e r n s r e v o l v i n g sudden change a c t i v e f i l m y i e l d , t h a t under n o r mal p r e s s u r e o i l f e e d t h e r e i s no r u p t u r e , p r e s s u r e d i s t u r b a n c e o r r e d u c t i o n o f l o a d c a p a c i t i y .
1.0
INTRODUCTION.
The problem o f f i l m r u p u r e o r c a v i t a t i o n i s a s u b j e c t o f d i s c u s s i o n s i n c e Osborne Reynolds i n 1866 d i s c u s s e d t h e e x i s t e n c e o f p r e s s u r e s below ambient by d i v e r g i n g wedges i n o i l f i l m s . Anal y s i n g T o w e r ' s r e s u l t s he found, t h a t t h e p r e s sure ended v e r y s h o r t b e f o r e t h e end o f t h e h a l f - b e a r i n g under t h e e x c e n t r i c i t y o f 0.5 -Fig. 1. H i s numerical s o l u t i o n rendered a t higher e x c e n t r i c i t i e s negative pressures a t the end. He concluded, t h a t a t h i g h e r e x c e n t r i c i t i e s t h e c o n t i n u i t y o f t h e p r e s s u r e would be m a i n t a i n e d by t h e a t m o s p h e r i c p r e s s u r e .
Fig.
1.
was t h e o r e t i c a l l y j u s t i f i e d i n t h e e a r l y 1930 i n d e p e n d e n t l y f r o m each o t h e r by S w i f t and S t i e b e r . The f i r s t r e l a t e d t h i s f r o m minimum p o t e n t i a l energy theoreme, t h e second f r o m f l o w continuity. Up t o today t h e boundary c o n d i t i o n s f o r p r e s s u r e end a r e s t i l l under d i s c u s s i o n , n o t a t l e a s t b y t h e v e r y s i m p l i f i e d pure a n a l y t i c a l s o l u t i o n s f r o m Sommerfeld and Gumbel w i t h t h e boundary c o n d i t i o n s p ( 9 = p(lp+Zrr) resp. p(f 27) = 0.
...
: T o w e r ' s p r e s s u r e measurement.
Moreover Reynolds a l s o t r e a t e d t h e l u b r i c a t i o n o f a r o t a t i n g c y l i n d e r near a p l a n e - F i g . 2- t o e x p l a i n f l o w c o n t i n u i t y o r boundary c o n d i t i o n s . For a s y m m e t r i c a l l y p l a n e - F i g . 2a- h i s s o l u t i o n gave a p r e s s u r e d i s t r i b u t i o n w i t h z e r o l o a d c a p a c i t y due t o t h e m i r r o w e d n e g a t i v e p r e s s u r e s . The p r e c o n d i t i o n t h e r e f o r e i s - as he s t a t e d -, t h a t t h e s u r r o u n d i n g l u b r i c a n t i s under such a p r e s s u r e , t h a t t h e p r e s s u r e m i n i mum i s t o m a i n t a i n . For an a s s y m m e t r i c a l l y p l a n e - F i g . 2b- he found a s o l u t i o n w h i c h gave z e r o p r e s s u r e a t t h e end of t h e p l a n e . I n t h e l a s t s t e p he showed, t h a t f o r a s y m m e t r i c a l l y p l a n e under l i m i t e d o i l s u p p l y a p o i n t c e x i s t s a t t h e end o f t h e p o s i t i v e p r e s s u r e d i s t r i b u t i o n where o i l can n o t f i l l anymore t h e f i l m because o f c o n t i n u i t y . So we can say, t h a t Osborne Reynolds has formul a t e d t h e c a v i t a t i o n boundary c o n d i t i o n p h y s i c a l l y q u i e t c l e a r l y . The m a t h e m a t i c a l f o r m u l a tion p dp/dcP = 0
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F i g . 2. : P r e s s u r e development between r o t a t i n g c y - l i n d e r near a p l a i n b y Osborne Renolds
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Much work has been done i n former times t o exp l a i n sub-atmospheric p r e s s u r e s l o p e , a i r e n t rainment etc., s u p p o r t e d b y v i s u a l s t u d i e s on g l a s b e a r i n g s a t v e r y low l o a d n a t u r a l l y . For s t e a d i l y loaded b e a r i n g s f i l m r u p t u r e o r c a v i t a t i o n has o n l y a l i t t l e e f f e c t on b e a r i n g p e r formance c o n d i t i o n . More complex a r e t h e boundary c o n d i t i o n s i n d y n a m i c a l l y loaded b e a r i n g s . The e x i s t i n g s o l u tions under instationary conditions d i f f e r
468
m o s t l y by t h e boundary c o n d i t i o n s . T h i s i s e v i d e n t by t h e f a c t o f two d i f f e r e n t p r e s s u r e developments due t o wedge and squeeze. E v i d e n t l y t h e n o b i l i t y Hethod f o r d y n a m i c a l l y loaded b e a r i n g s i s no l o n g e r based on t h e simp l i f i e d Ocvirk solution for short bearing l e n g t h , b u t on n u m e r i c a l s o l u t i o n s f o r f i n i t e l e n g t h / 1 t o 3/ such as F i n i t e Element methods. U s i n g t h e s e methods, t h e problems o f f i l m ext e n t and r u p t u r e became o f new i n t e r e s t . In /4/ t h i s p r o b l e m i s t r e a t e d i n a comprehensive s t u d y and t h e e f f e c t o f t h e r u p t u r e d , o n l y p a r t i a l l y - f i l l e d f i l m w i t h changing p o s i t i o n i s demonstrated f o r a c o n n e c t i n g r o d b e a r i n g F i g . 3. For i n s t a n c e , t h e r e w o u l d b e a sudden change o f p r e s s u r e development i n such a r u p t u r e d a r e a between 300 and 420° c r a n k a n g l e .
w i t h n e g a t i v e squeeze. Due t o t h e e x c e s s i v e l y h i g h n e g a t i v e squeeze p r e s s u r e s i n t h e loaded area, t h e r e s u l t a n t p r e s s u r e from Hahn (a) i s s i g n i f i c a n t l y s h o r t e r and lower, w h i c h must g i v e an e x t r a o r d i n a r i l y lower l o a d c a p a c i t y .
-
a
? .)
i
4
DO
ID0
110
100
41
Angular Dhpl#cam#nt
F i g . 4 : T h e o r e t i c a l p r e s s u r e development a c c o r d i n g t o Hahn (a) o r o r Hol land/Lang (b) posit i v e squeeze
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F i g . 3 : F i l m e x t e n t i n a c o n n e c t i n g rod bear i n g /4/ I n t h e FRG t h i s p r o b l e m was d i s c u s s e d i n t h e m i d - s e v e n t i e s w i t h t h e background o f two d i f f e r e n t s o l u t i o n s by Hahn /5/ and H o l l a n d / L a n g /6,7/. The d i f f e r e n c e s between t h e two a r e s i g n i f i c a n t f o r b o t h f i l m e x t e n t and r u p t u r e . T h i s p r o b l e m r e l a t e s i n a m a t h e m a t i c a l sense the conditions o f f i l m extension, superposition o f wedge and squeeze a c t i o n and p a r t i a l a c t i o n o f n e g a t i v e p r e s s u r e components o f wedge and/or squeeze. Hahn d i d account f o r n e g a t i v e p r e s s u r e s f r o m wedge/squeeze so l o n g as t h e r e s u l t a n t pressure i s p o s i t i v e . As shown i n F i g . 4 . because o f h i g h n e g a t i v e p r e s s u r e s PIJ t h i s s o l u t i o n i n d i c a t e s a s t e e p and e a r l y end o f t h e resultant pressure PI^ In contrast, H o l l a n d / L a n g i s based upon t h e assumptions t h a t t h e p r e s s u r e developments o f wedge and squeeze a r e independent o f one a n o t h e r , due t o t h e i r s p e c i a l v e l o c i t y p r o f i l e s and c o n t i n u i t y c o n d i t i o n s , and t h a t t h e p a r t i a l p r e s s u r e s a r e a l s o v a l i d f o r combined wedge and squeeze. The su(squeeze) and Pw (wedge) p e r p o s i t i o n o f P, y i e l d s a steady pressure curve o f the r e s u l t a n t PI( w i t h a h i g h e r p r e s s u r e and a s i g n i f i c a n t l y l o n g e r p r e s s u r e a r e a compared t o Hahn.
.
W h i l e F i g . 4 r e l a t e s t o b o t h p o s i t i v e wedge and p o s i t i v e squeeze (as t h e j o u r n a l approaches t h e b e a r i n g ) , F i g . 5 shows t h e even g r e a t e r d i s c r e pancy i n t h e case o f p o s i t i v e wedge combined
-
n ? . . ) I
i
wedge pressure squeeze pressure
p - ~ p---
resultant pressure p r , . I
rio
2m
ID0
Angular Di8placamant
Fig.5:
Analogous F i g . 4
-
n e g a t i v e squeeze
Comparing t h e two s o l u t i o n s by use o f p o l a r j o u r n a l d i s p l a c e m e n t p a t h s under t h e same l o a d p a t t e r n , t h e r e s u l t f r o m Hahn shows h i g h e r ecc e n t r i c i t i e s under c o n d i t i o n s where t h e d i s p l a cement g e t s s m a l l e r . One example i s g i v e n i n F i g . 6.
469
S E m
Fig.
8 : P o l a r l o a d diagram f o r t e s t F L l f r o m
/8/ Fig. 6 : Journal c e n t r e p a t h o f Diesel main bearing, according t o theory o f Holland/Lang (A) o r Hahn (B) 2.0
DISPLACEHENT AEASUREAENTS BY H E I S E L .
A t K a r l s r u h e U n i v e r s i t y , where t h e Hahn s o l u t i o n was founded, i n 1977 H e i s e l /8/ completed experiments on d y n a m i c a l l y loaded b e a r i n g s i n which he measured t h e d i s p l a c e m e n t and p r e s s u r e d i s t r i b u t i o n s t o c l e a r up t h i s problem. For d e t a i l s o f t e s t i n g and measurement methods see /8/. The l o a d p a t t e r n was chosen i n t h i s way, t o o b t a i n sudden p r e s s u r e changes o f s e v e r a l orders i n ruptured, p o s s i b l y o n l y p a r t l y - f i l l e d areas. T h i s was a c h i e v e d by means o f a sudden change f r o m p o s i t i v e t o n e g a t i v e wedge, as shown i n F i g . 7 . U s i n g t h i s method, i t was hoped t o s o l v e t h e p r o b l e m i n q u e s t i o n . B rO
*-P
Li.0
D4
*=p
Y..y
F i g . 9 : Hydrodynamic v e l o c i t i e s t o F i g . 8 F i g . 10 shows t h e p o l a r d i a g r a m o f measured and calculated (by Holland/Lang) displacement. T h e r e i s no s i g n i f i c a n t d i f f e r e n c e between t h e two, e s p e c i a l l y i n t h e a r e a 8-C-D-A.
F i g . 7 : Sudden p r e s s u r e development i n r u p t u r e d areas-by change o f e f f e c t i v e hydrodynamic v e l o c i t y G, = 0 H e i s e l e x p e r i m e n t e d w i t h many l o a d p a t t e r n s , b u t we s h a l l r e s t r i c t o u r s e l v e s h e r e t o j u s t two r e p r e s e n t a t i v e examples. A t Karlsruhe i t was d e c i d e d t o compare t h e s o l u t i o n s u s i n g d i f f e r e n t , b u t v e r y s i m i l a r loads. For comparison w i t h Hol land/Lang, t h e l o a d p a t t e r n shown i n F i g . 8 was chosen. Between B-C-D-A r a p i d p r e s s u r e changes o f s e v e r a l o r d e r s were undergone by " r u p t u r e d " a r e a s . T h i s becomes more o b v i o u s i n F i g . 9. where t h e a n g u l a r v e l o c i t y W r of t h e measured f i l m i s g i v e n . The l i n e c o r r e s p o n ,ding t o h a l f the s h a f t v e l o c i t y crosses the w h i c h means a c u r v e a t t h e p o i n t s B-C-D-A. change i n t h e e f f e c t i v e wedge v e l o c i t y .
VERtAGERUNG
Fig.10 : J o u r n a l c e n t r e p a t h for F i g . 8 and 9; comparison between measurement and Holland/Lang Comparison w i t h t h e Hahn s o l u t i o n was made w i t h t h e l o a d p a t t e r n shown i n Fig.11. Fig.12 shows t h e v e l o c i t y p r o f i l e w i t h j u s t two changes o f wedge d i r e c t i o n a t B and A. The p o l a r d i s p l a c e ment - F i g . l j c a l c u l a t e d by t h e Hahn method d i f f e r s s i g n i f i c a n t l y from measurement i n t h e a r e a between B and A. The t h e o r e t i c a l eccent r i c i t y i s h i g h e r t h a n measured due t o t h e e f f e c t s a l r e a d y mentioned.
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470 n o m i c a l . The assumption o f u n d i s t u r b e d and f i xed p r e s s u r e development o f independant wedge and squeeze a c t i o n a l l o w s g e o m e t r i c a l a d d i t i o n o f t h e two r e s p e c t i v e l o a d c a p a c i t i e s . This a s s u m p t i o n g i v e s smoothness o f p r e s s u r e c u r v e s and s t e a d i n e s s o f c i r c u m f e r e n t i a l f l o w , w h i c h means t h a t c o n t i n u i t y c o n d i t i o n s a r e f u l l y satisfied. The b e t t e r t h e o r e t i c a l r e s u l t s o f H o l l a n d / L a n g were a l s o p r o v e n i n p r a c t i c e , b o t h i n t h e case o f t h e i n v e s t i g a t i o n o f worn a r e a s o f combustion e n g i n e b e a r i n g s and i n t h e case where s t a r t i n g wear measured b y r a d i o n u c l i d e s was i n good c o r r e l a t i o n w i t h t h e m i n i m u m c a l c u l a t e d f i l m t h i c k n e s s and t h e s u r f a c e roughness. The H o l l a n d / L a n g method g i v e s good, a c c u r a t e r e s u l t s so l o n g as t h e r u n n i n g c o n d i t i o n s ( l o a d p a t t e r n , o i l p r o p e r t i e s , t e m p e r a t u r e ) a r e chosen c l o s e t o t h o s e used i n r e a l i t y . '
Vwsurh KO I7
Fig.11:
Polar
KRAFT -
l o a d diagram f o r
t e s t Ka17 f r o m
3.0
PRESSURE HEASUREHENTS BY HEISEL.
H e i s e l a l s o p e r f o r m e d p r e s s u r e measurements u s i n g t h e l o a d s m e n t i o n e d above. Fig.14 shows t h e measured p r e s s u r e p r o f i l e s i n t h e m i d d l e o f the bearing over the load c y c l e . The c o r r e sponding t h e o r e t i c a l p r e s s u r e p r o f i l e b y t h e H o l l a n d / L a n g method i s t o be seen i n Fig.15. Regarding the d i f f e r e n t pressure scales, t h e r e i s a v e r y good agreement i n shape, o r i e n t a t i o n and f i l m e x t e n s i o n between measurement and c a l c u l a t ion.
Fig.12:
Hydrodynamic v e l o c i t i e s t o F i g . 1 1
Fig.14: Measured p r e s s u r e d i s t r i b u t i o n i n t h e middle p l a i n o f bearing - t e s t F L l
Varsuch KO 17
VERLAGERUNG
Fig.13 : J o u r n a l c e n t r e p a t h f o r F i g . 1 1 and 12; comparison between measurement and Hahn From t h i s a n a l y s i s - w h i c h i s g r e a t l y e l a b o r a t e d upon i n /8/ - i t seems t o be c l e a r t h a t t h e n a t u r e o f t h e hydrodynamic f i l m i s c l o s e r t o Holland/Lang t h a n t o Hahn, o r e l s e t h e r e a r e no n e g a t i v e p a r t i a l p r e s s u r e s a c t i n g and no r u p t u r e o c c u r s under dynamic load. I t should be mentioned t h a t t h e b e a r i n g was p r e s s u r e - f e d w i t h o i l f r o m t h e top, as i s n o r m a l l y t h e case i n dynamically-loaded bearings. The consequenc e o f K a r l s r u h e U n i v e r s i t y ' s i n v e s t i g a t i o n s was t o m o d i f y t h e Hahn method by n e g l e c t i n g a l l negative p a r t i a l pressures. The H o l l a n d / L a n g method was found t o be i n e x c e l l e n t agreement w i t h measurement, and f u r t h e r m o r e i s more eco-
- v
-
Clramfmranm
Fig.15: Calculated pressure d i s t r i b u t i o n Holland/Lang; analogous t o F i g . 14
by
The comparison w i t h t h e Hahn method i s g i v e n i n F i g . 1 6 (measured) and Fig.17 ( c a l c u l a t e d ) . It i s evident t h a t the pressure amplitudes from measurement a r e lower and t h e f i l m e x t e n s i o n s b r o a d e r when compared w i t h t h e Hahn r e s u l t s , due t o t h e assumption o f r u p t u r e and n e g a t i v e p a r t i a l pressures.
471 boundary c o n d i t i o n s . T h i s e n t a i l s assumptions o f t h e o i l f i l m r u p t u r e c o n d i t i o n s as w e l l as t h e e x t e n d o f t h e l u b r i c a t e d f i l m . Most o f t h e s o l u t i o n s w i t h s p e c i a l boundary c o n d i t i o n s a r e l o o k i n g q u i t e unbalanced , because t h e y a r e r e s t r i c t e d t o s m a l l o r i n f i n i t e l y l o n g bear i n g s . The d i f f e r e n c e i n t h e b e a r i n g c h a r a c t e r i s t i c s i s much more a f f e c t e d by f i n i t e b e a r i n g l e n g t h t h a n by boundary c o n d i t i o n s .
Fig.16: Measured p r e s s u r e d i s t r i b u t i o n i n t h e m i d d l e p l a i n o f b e a r i n g - t e s t Ka17
Besides t h e d i f f i c u l t i e s t o e v a l u a t e n u m e r i c a l s o l u t i o n s f o r f i n i t e l e n g t h i n t h e pre-computer-time, t h e ifnpulses f o r a l l t h e s e a b s t r a c t t h e o r e t i c a l work a r e r e s u l t i n g f r o m s i m i l a r abstracted experiments. All the t e s t s w i t h transparent g l a s s o r perspex bearings a r e suff e r i n g under a s t r o n g l y r e s t r i c t e d l o a d c a p a c i t y due t o t h e n a t u r o f t h o s e b e a r i n g s . So a l l t h e s e r e s u l t s a r e b r i n g i n g up more c o n f u s i o n , b u t no r e a l impetus t o p r a c t i c a l b e a r i n g h y d r o dynamic. B e a r i n g under low l o a d have besides o f i n s t a b i l i t y problems such a h i g h s e c u r i t y , t h a t t h e r e i s no n e c e s s i t y f o r f u r t h e r comprehensive studies. For h i g h l y l o a d e d b e a r i n g s I would l i k e t o c i t e t h e c o n c l u s i o n s o f a l a t e work by Mokhtar and Ameen /g/:
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Fig.17: Calculated pressure Hahn; analogous t o F i g . I6
4.0
distribution
by
F R I C T I O N UNDER DYNARIC LOAD.
H a r t i n /4/ discussed s o l u t i o n s f o r f r i c t i o n losses a f f e c t e d by f i l m rupture. Under s t a t i c load the c o n d i t i o n s o f o i l extension should be clear. Under p r e s s u r e l e s s o i l feed, t h e d i v e r g i n g gap w i l l n o t be f i l l e d d i r e c t l y , but by s u c t i o n from t h e o i l c o l l a r o n t h e s h a f t a t b o t h b e a r i n g ends. Under p r e s s u r e feed, w i t h o i 1 e n t e r i n g s h o r t l y a f t e r t h e p r e s s u r e has reduced t o z e r o , t h e d i v e r g i n g gap i s completel y f i l l e d a c c o r d i n g t o t h e assumption o f a -bearing. D y n a m i c a l l y loaded b e a r i n g s a r e a l ways p r e s s u r e fed, w i t h one e x c e p t i o n - t h e s m a l l end c o n n e c t i n g r o d b e a r i n g . According t o s t a t e o f t h e a r t t h e o r y , t h e o i l f e e d i s chosen i n an o p t i m a l way, i . e . a t t h e a r e a where l o a d and p r e s s u r e a r e low and o f s h o r t d u r a t i o n . T h i s c h o i c e can be made even more c l e a r u s i c g calculated pressure d i s t r i b u t i o n s . t o f i n d t h e optimum o i l e n t r y p o s i t i o n a t an a r e a w i t h o u t r e s u l t a n t p r e s s u r e o r w i t h o n l y low p r e s s u r e s of short duration. For c o n n e c t i n g r o d b e a r i n g s t h i s i s a c h i e v e d w i t h t h e common arrangement o f pressure feed v i a a d r i l l i n g o f the crank-pin a t a b o u t 45 b e f o r e TDC. I n p r e s s u r e f e d , dyn a m i c a l l y loaded b e a r i n g s w i t h optimum o i l e n t rance, one may b e s u r e t h a t t h e r e i s no rupture, but a t o t a l l y f i l l e d f i l m .
5.0
OTHER WORK ON F I L A RUPTURE UNDER STATIC
LOAD. There a r e innumerable p u b l i c a t i o n s on s o l v i n g Reynolds e q u a t i o n under v a r i o u s and a p p r o p r i a t e
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"High l o a d p a r a m e t e r s r e s u l t i n w i d e r f u l l I m extent with l e s s c a v i t a t e d zones. P r o p e r l y p o s i t i o n e d 1 inlet port w i t h i n t h e d i v e r g i n g gap up t o t h e maximum 1 m t h ickness o r even near t h e l o a d v e c t o r g i v e s optimum I m b e h a v i o u r .I' From v i e w o f p r a c t i c e t h i s i s a s i m p l e t e c h n i c a l r u l e , w h i c h s h o u l d be much more c o n s i d e r e d i n b e a r i n g d e s i g n . My r e s u l t f o r d;mamically loaded b e a r i n g s i s t h e adequate r u l e .
6.0
CONSEQUENCES.
I n d y n a m i c a l l y l o a d e d b e a r i n g s w i t h optimum pressure feed according t o s t a t e o f the a r t t h e o r y , t h e r e i s no r u p t u r e and t h e gap i s t o tally filled. For t h e p h y s i c a l c o n d i t i o n s o f p r e s s u r e development, t h i s means t h a t t h e p r e s s u r e e x t e n s i o n has t o s a t i s f y t h e c o n d i t i o n o f c o n t i n u i t y o f c i r c u m f e r e n t i a l o i l flow, which i s o n l y r o u g h l y d e s c r i b e d b y pa0 and dp/d =O s i n c e t h i s i s o n l y a secondary e f f e c t o f c o n t i nuity. W i t h i n t h e loaded f i l m t h e v e l o c i t y p r o f i l e s d e v e l o p a c c o r d i n g t o hydrodynamic t h e o r y i n s p e c i a l forms f o r wedge and squeeze. Each v e l o c i t y p r o f i l e has t o f u l f i l l c o n t i n u i t y c o n d i t i o n s independently. Outside of the pressure extension t h e f i l m i s completely f i l l e d , p a r t l y by o i l coming o u t o f t h e p r e s s u r e zone and p a r t l y by o i l feed. The r e l a t i v e l y low f e e d pressure - n e g l i g i b l e t o the load - i s s u f f i c i e n t t o f i l l the f i l m although i t i s r o t a t i n g w i t h the shaft. But f e e d v e l o c i t y p r o f i l e s a r e n e g l i g i b l e i n comparison w i t h t h e l i n e a r v e l o c i t y gradient from the s h a f t r o t a t i o n . Therefore, f r i c t i o n i n the pressureless diverging gap r e s u l t s f r o m t h e l i n e a r v e l o c i t y g r a d i e n t . The e x p e r i m e n t a l r e s u l t s f rom Hei s e l c o n f i r m n o t o n l y t h a t no r u p t u r e o c c u r s , but a l s o t h e independent p r e s s u r e development under wedge and squeeze a c t i o n and t h e i r u n d i s t u r b e d superp o s i t i o n . T h i s i s no c o n t r a d i c t i o n t o mathemat i c a l r u l e s f o r v a r i o u s p h y s i c a l problems i n
412
t h e f i e l d s o f v i b r a t i o n s and thermodynamics. The s u p e r p o s i t i o n o f f i x e d p r e s s u r e development by wedge and squeeze c o u l d b e f u r t h e r s i m p l i f i e d by i n t e g r a t e d l o a d c a p a c i t i e s and geomet r i c a l a d d i t i o n o f two v e c t o r s . This exactly i s the Holland/Lang s o l u t i o n f o r dynamically loaded b e a r i n g s , w h i c h i s w i d e l y used i n t h e FRG because o f i t s good r e s u l t s i n p r a c t i c a l a p p l i c a t i o n s c o n c e r n i n g t h e o n s e t o f wear, wear p a t t e r n s and f a t i g u e .
REFERENCES: /1/
/2/
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/4/
/5/
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/7/
/8/
Booker, J.F., Goenka, P . K . and van Leeuwen.H.J.: 'Dynamic A n a l y s i s o f Rocking J o u r n a l Bearings with multiple Offset Segments'; Trans. ASME, J . Lub. Techn. Oct. 1982/104 G0enka.P.K.: ' A n a l y t i c a l c u r v e f i t s f o r s o l u t i o n paramet e r s o f d y n a m i c a l l y loaded b e a r i n g s ' ; ASME Paper 83-Lub-33, p r e s e n t e d a t t h e ASME/ASLE L u b r i c a t i o n Conference, Hartford, Conn., Oct. 1983 Goenka, P .K .: ' D y n a m i c a l l y loaded j o u r n a l b e a r i n g s ; F i n i t e Element method a n a l y s i s ' ; ASME Paper 83Lub-32, p r e s e n t e d a t t h e ASME/ASLE Confer e n c e H a r t f o r d , Conn., Oct. 1983 M a r t i n , F.A.: 'Friction i n internal combustion engine bearings'; Inst. Mech. Engr. Conference Combustion Engines - R e d u c t i o n o f F r i c t i o n and Wear, London March I985 Hahn, H.W.: 'Das zylindrische Gleitlager endlicher B r e i t e unter z e i t l i c h v e r a n d e r l i c h e r Belas t u n g ' ; D i s s , U n i v e r s i t a t K a r l s r u h e 1957 H o l l a n d , J.: ' B e i t r a g zur Erfassung der Schmierverhaltn i s s e i n Verbrennungskraftmaschinen'; VDIF o r s c h u n g s h e f t 475/1959 Lang, O.R.: 'Gleitlager'; Konstruktionsbucher Band 31/1978; S p r i n g e r - V e r l a g B e r l i n - H e i d e l b e r g New York H e i s e l , U.: 'Messungen am i n s t a t i o n a r b e l a s t e t e n R a d i algleitlager E i n B e i t r a g zum Problem d e r S p a i t f u l l u n g ' ; Diss. U n i v e r s i t a t Karlsruhe
-
/9/
1977 Mokhtar, M.O.A. and Ameen. M.E.: 'An E x p e r i m e n t a l Study o f C a v i t a t i o n in H y d r o d y n a m i c a l l y L u b r i c a t e d B e a r i n g s ' : Stud i e s o f Engine B e a r i n g s and L u b r i c a t i o n (1983). S-539. N r . 830305
473
Paper XV(ii)
The influence of cavitation on the non-linearity of velocity coefficients in a hydrodynamic journal bearing R.W. Jakeman
p r e s e n t s t h e r e s u l t s o f a t h e o r e t i c a l s t u d y of t h e o i l f i l m forces, a r i s i n g from combined hydrodynamic s q u e e z e and wedge a c t i o n s , i n a d y n a m i c a l l y l o a d e d b e a r i n g . In particular, i t shows how t h e n o n - l i n e a r i t y of t h e f o r c e - j o u r n a l v e l o c i t y r e l a t i o n s h i p i s d e p e n d e n t upon cavitation. S i m p l e e q u a t i o n s f o r t h e t o t a l o i l f i l m force components, a t any g i v e n e c c e n t r i c i t y These e q u a t i o n s i n t r o d u c e f i v e v e l o c i t y r a t i o , are f i t t e d t o t h e p r e d i c t e d f o r c e - v e l o c i t y data. c o e f f i c i e n t s , which t a k e a c c o u n t o f t h e n o n - l i n e a r b e h a v i o u r . A p p l i c a t i o n o f t h e s e e q u a t i o n s t o a f a s t j o u r n a l o r b i t a n a l y s i s , i n c l u d i n g comparison w i t h e x p e r i m e n t a l r e s u l t s , is d e s c r i b e d i n reference (5). T h i s paper
INTRODUCTION
1.
Diametral c l e a r a n c e
cd
The work described in this paper was i n s t i g a t e d p r i m a r i l y t o p r o d u c e a means of p r e d i c t i n g t h e o i l f i l m force components, i n a d y n a m i c a l l y loaded j o u r n a l b e a r i n g , f o r u s e i n a f a s t j o u r n a l o r b i t a n a l y s i s method. This lead to a theoretical study of the r e l a t i o n s h i p between o i l f i l m f o r c e s and journal lateral velocities. A s t h e work p r o g r e s s e d t h e s i g n i f i c a n c e o f ravitation, i n relation t o the non-linearity o f t h e above r e l a t i o n s h i p , became a p p a r e n t . This a s p e c t of t h e work was t o t a l l y d e p e n d e n t on t h e c a v i t a t i o n model u s e d , which t o o k account o f flow c o n t i n u i t y throughout t h e c a v i t a t i o n zone. The l i t e r a t u r e c o n t a i n s much e x p e r i m e n t a l e v i d e n c e of t h e c o m p l e x i t y of r e a l c a v i t a t i o n phenomena, which i n d i c a t e s t h e s u b s t a n t i a l d e g r e e of a p p r o x i m a t i o n l i k e l y i n any theoretical model. However , the c a v i t a t i o n model employed i n t h i s work is b e l i e v e d t o r e p r e s e n t t h e c u r r e n t s t a t e of t h e a r t , for p r a c t i c a l a n a l y s i s purposes. It undoubtedly o f f e r s a c o n s i d e r a b l e improvement on t h e s i m p l e r c a v i t a t i o n models t h a t have been w i d e l y u s e d , p a r t i c u l a r l y t h o s e i n which t h e c a v i t a t i o n b o u n d a r i e s were f i x e d ( e . g . t h e
n film). The most s i g n i f i c a n t a p p r o x i m a t i o n i n t h e t h e o r e t i c a l a n a l y s i s is t h e r i g i d b e a r i n g assumption. Recent work by La Bouff and Booker (1) h a s i n d i c a t e d t h a t t h e c o m p u t a t i o n time associated with modelling bearing e l a s t i c i t y i n a j o u r n a l o r b i t a n a l y s i s is e x c e s s i v e . S i n c e t h e i n i t i a l o b j e c t i v e was t o a fast journal orbit analysis, develop consideration of bearing elasticity was i n c o m p a t i b l e w i t h t h i s aim.
Journal diameter
D
t a n g e n t i a l components of o i l f i l m force
Radial, Frv Ft h,
*
,hminVaximum, minimum
f i l m thickness
j,i
Circumferential, position reference
N
Angular v e l o c i t y of j o u r n a l a b o u t i t s axis. rev/s
PC
Cavitation pressure
PS
O i l supply pressure
Pspec
S p e c i f i c bearing p r e s s u r e W/(LD)
qv
Element g a d v a p o u r volume flow r a t e
i
Radial v e l o c i t y of j o u r n a l
U
Journal surface velocity
Vn
Normal v e l o c i t y o f r e l a t i v e t o element
W
T o t a l b e a r i n g load
A a , Ac
Axial, c i rcumf e r e n t i a l dimensions
&
Eccentricity ratio
UI
A t t i t u d e angle
0
Effective
dynamic
axial
journal
viscosity
element
surface
element
of
oil
film 1.1
Notation
b
Axial width o f each b e a r i n g "land"
Brr,
etc Velocity c o e f f i c i e n t s [ 5 1 , [61
i
-
see e q u a t i o n s
.
eo
Angular velocity about bearing a x i s Equivalent
(= - wi/ 2 ) *
of
angular
journal
axis
velocity
474 W
A n g u l a r v e l o c i t y of j o u r n a l a b o u t i t s , axis.rad/s *
2.3
See f i g u r e 1.
I n i t i a l a p p l i c a t i o n of t h e n u m e r i c a l a n a l y s i s method ( 2 ) t o j o u r n a l l a t e r a l v e l o c i t i e s , t y p i c a l o f t h o s e encountered i n a dynamically l o a d e d b e a r i n g , i n d i c a t e d some a n o m o l i e s i n the computed o i l f i l m f o r c e components. W h i l s t t h e a p p a r e n t errors were small, t h e a b o v e a n o m o l i e s were f o u n d t o b e e n t i r e l y associated with cavitation. It was t h e r e f o r e considered important t h a t they should be i n v e s t i g a t e d , a n d e l i m i n a t e d . F u l l d e t a i l s of t h i s development w i l l b e r e p o r t e d s e p a r a t e l y , a n d t h e following n o t e s are a b r i e f o u t l i n e o f the essential features:
D i m e n s i o n l e s s p a r a m e t e r s a r e i n d i c a t e d by a " b a r " a b o v e them, a n d are d e f i n e d i n t h e t e x t .
Development of t h e A n a l y s i s f o r Large Lateral Velocities of t h e J o u r n a l
( a ) The squeeze film term Vn.Aa.Ac was d e l e t e d from t h e c o n t i n u i t y e q u a t i o n f o r cavitating elements. The hypothesis u n d e r l y i n g t h i s c h a n g e was t h a t i n a c a v i t a t i n g e l e m e n t , t h e o i l d i s p l a c e d by the normal velocity of the journal s u r f a c e Vn, w i l l r e s u l t mainly i n a n a x i a l v e l o c i t y of t h e b o u n d a r i e s o f t h e o i l 'streams. The s q u e e z e f i l m term d o e s n o t , t h e r e f o r e , r e s u l t i n any o i l flow across t h e e l e m e n t b o u n d a r y , a n d t h u s d i s a p p e a r s from t h e c o n t i n u i t y e q u a t i o n for such a n element.
Effective angular velocity Eccentricity ratio
L
6,=6 - 2 2
(stationary bearing case)
=&
Cd
Flg 1 Polar Oil Film Force
- Journal Velocity System
2.
BACKGROUND TO THE THEORETICAL ANALYSIS
2.1
Introduction
The n u m e r i c a l a n a l y s i s method u s e d f o r t h i s work i s b a s e d on t h a t described i n r e f e r e n c e ( 2 ) . F u l l d e t a i l s of t h e a s s u m p t i o n s made a r e given in reference (2), these include : incompressible, isoviscous lubricant of n e g l i g i b l e i n e r t i a , r i g i d c i r c u l a r j o u r n a l and bearing, etc. T h i s a n a l y s i s method h a s b e e n successfully applied to steadily loaded b e a r i n g s , a n d t o small j o u r n a l d i s p l a c e m e n t and v e l o c i t y p e r t u r b a t i o n s r e q u i r e d f o r t h e computation of linearised stiffness and damping c o e f f i c i e n t s . 2.2
Previous Application t o Journal Orbit Analysis
The a b o v e method h a s a l s o b e e n a p p l i e d t o journal o r b i t a n a l y s i s taking account o f o i l film h i s t o r y and journal mass (3). A noteworthy f e a t u r e o f t h e o i l f i l m h i s t o r y model is t h a t o i l f i l m e l e m e n t s s u b j e c t t o c a v i t a t i o n are n o t r e q u i r e d t o s a t i s f y flow continuity. In these circumstances the downstream o i l flow from a c a v i t a t i n g e l e m e n t is c a l c u l a t e d i n a c c o r d a n c e w i t h i t s d e g r e e o f filling. T h i s is d e t e r m i n e d by c o n t i n u o u s l y monitoring t h e n e t t o i l flow t o t h e element o v e r s u c c e s s i v e time s t e p s , a s t h e o r b i t is marched o u t .
( b ) Tbe o r i g i n a l c a v i t a t i o n model f a i l e d t o s a t i s f y continuity i n c a v i t a t i n g elements when circumferential flow reversal relative to hmin occurred; i.e. 0 > W/2.Elimination of t h i s problem simply required recognition t h a t , i n t h e a b o v e c i r c u m s t a n c e s , q v ( j + l ,i ) r e f e r r e d t o t h e upstream element boundary and q v ( j , i ) t o t h e downstream boundry. For cavitating elements subject to flow r e v e r s a l i t was t h e r e f o r e n e c e s s a r y t o compute q v ( j , i ) i n order to satisfy continuity.
was ( c ) The journal surface velocity u c a l c u l a t e d on t h e b a s i s o f t h e e q u i v ? l e n t angular velocity, i.e. u = b - 2 0 ) D / 2 instead t h e o r i g i n a l u = w D/2. In a d d i t i o n , 0, was deleted from the computation of V,: therefore Vn became a f u n c t i o n of R o n l y . The a b o v e measures effectively segregated the h y d r o d y n a m i c s q u e e z e a n d wedge a c t i o n s i n the analysis. This segregation is unnecessary i n f u l l f i l m e l e m e n t s , b u t is advantageous with c a v i t a t i n g elements. The reason for this is that when e l i m i n a t i n g t h e s q u e e z e f i l m t e r m from t h e continuity equation for cavitating e l e m e n t s , a s i n d i c a t e d i n item ( a ) , i t was f o u n d tha! only that part of Vn.Aa.Ac d u e t o R s h o u l d b e e l i m i n a t e d .
09
2.4
P r e v i o u s R e l a t e d Work
No p r e v i o u s work is known t o e x i s t , w h i c h is r e a l l y comparable t o t h a t d e s c r i b e d i n t h i s paper. The a n a l y s i s by B a n n i s t e r ( 4 ) t o o k a c c o u n t of n o n - l i n e a r i t y e f f e c t s i n a 1 2 0 0 partial arc bearing, subject to static m i s a l i g n m e n t , by i n c l u d i n g t h e s e c o n d o r d e r terms of T a y l o r ' s series. T h i s i n t r o d u c e d 2 0 additional second derivative coefficients. Good correlation between predicted and
475 measured o r b i t s was r e p o r t e d , b u t t h e work covered o n l y o u t of b a l a n c e e x c i t a t i o n and small o r b i t s i n r e l a t i o n t o t h e c l e a r a n c e circle. D u r i n g t h e c o u r s e of t h e a u t h o r s development work, the above non-linear c o e f f i c i e n t a p p r o a c h was i n v e s t i g a t e d . This i n c l u d e d t h e u s e of b o t h C a r t e z i a n a n d P o l a r c o d r d i n a t e systems and f u r t h e r expansion t o include third derivative coefficients. Satisfactory oil film force prediction, t h r o u g h o u t t h e range of j o u r n a l d i s p l a c e m e n t and velocity conditions encountered in r e f e r e n c e (51, was n o t a t t a i n e d . .It s h o u l d b e noted t h a t t h e o i l f i l m f o r c e e q u a t i o n s presented in this paper are virtually unrestricted with respect to journal displacement and v e l o c i t y amplitudes.
3.
O I L FILM RESPONSE LATERAL VELOCITIES
3.1
Introduction
TO
LARGE
JOURNAL
In t h i s s e c t i o n , t h e r e s u l t s of a t h e o r e t i c a l s t u d y o f t h e r e l a t i o n s h i p between o i l f i l m f o r c e s and large l a t e r a l v e l o c i t i e s w i l l b e presented and d i s c u s s e d . A p o l a r s y s t e m was used for velocity directions, which f a c i l i t a t e d s e g r e g a t i o n o f t h e hydrodynamic s q u e e z e a n d wedge a c t i o n s . In o r d e r to a c c o u n t f o r t h e t o t a l wedge a c t i o n , an effective angular velocity ( 6 , ) was u s e d , which c o m b i n e s t h e a n g u l a r v e l o c i t y o f t h e j o u r n a l a b o u t i t s own a x i s (W) w i t h t h e a n g u l a r v e l o c i t y of t h e j o u r n a l a x i s a b o u t t h e bearing a x i s ( 8 ) . For t h e stationary- b$aring case we may therefore write: 8,=8-W/2. R e f e r e n c e t o l a r g e l a t e r a l v e l o c i t i e s means v e l o c i t i e s of t h e o r d e r of t h o s e a n t i c i p a t e d i n a first o r d e r o r b i t t r a v e r s i n g a l a r g e y o p o r t i o n of the clearance c i r c l e i.e. e o s 0 , fi2 o c d / 2 . I n o r d e r t o e n a b l e c o m p a r i s o n s t o b e made o f t h e j o u r n a l o r b i t s p r e d i c t e d by t h i s work w i t h experimental d a t a , t h e bearing d e t a i l s used i n t h i s s t u d y corresponded t o test c o n d i t i o n s u s e d by P a r k i n s ( 6 ) : S h a f t Diameter = 63.5 mm. Bearing Length 2 x 9 . 3 mm. l a n d s Diametral C l e a r a n c e 0.0836 mm O i l Groove = 5.08 mm x 360° J o u r n a l S p e e d = 1 1 8 0 rpm O i l Supply Pressure 0.0517 MPa ( g a u g e ) -0.175 MPa ( g a u g e ) Cavitation Pressure E f f e c t i v e V i s c o s i t y = 0.0186 Pa.s. R a d i a l O i l Film F o r c e ( F r )
3.2
The r e l a t i o n s h i p b e t w e e n F r a n d R a t E = 0.7 is shown i n F i g u r e 2 a , from w h i c h t h e f o l l o w i n g c h a r a c t e r i s t i c s may b e n o t e d :
6 0 / ~= 0 t h e r e i s -a marked c h a n g e i n s l o p e a s t h e s i g n of R c h a n g e s , t h a t f o r p o s i t i v e R b e i n g r e l a t i v e l y s t e e p and terfectly linear whilst t h a t f o r negative R is f a i r l y f l a t and c l e a r l y non-linear. The r e a s o ? f o r t h i s b e h a v i o u r is t h a t p o s i t i v e R generates high squeeze f i l m p r e s s u r e s i n t h e h m i n r e g i o n a n d n; cavitation. Conversely negative r e s u l t s i n low s q u e e z e f i l m p r e s s u r e s i n t h e hmax r e g i o n , a n d r e a d i l y g e n e r a t e s c a v i t a t i o n i n t h e hmin region.
(a) A t
( b ) Where Go/ w # 0 hydrodynamic wedge action occurs, which results in cavitation i n the h . region both a t p o s i t i v e and negativemh: T h i s r e s u l t s i? a s m o o t h e r t r a n s i t i o n of t h e , F r R c u r v e from n e g a t i v e t o p o s i t i v e R , w i t h - a d e g r e e of n o n - l i n e a r i t y a t p o s i t i v e R. Note t h a t t h e c u r v e s a r e v a l i d . f o r b o t h p o s i t i v e a n d n e g a t i v e v a l u e s of Bo/w.
-
( c ) Cavitation due t o wedge a c t i o n is s u p p r e s s e d a t h i g h e r p o s i t i v e R , thu: l e a d i n g to. c o n v e r g e n c e w i t h t h e Fr-R c u r v e f o r ' 8 0 / w = 0 , a n d l i n e a r i t y beyond t h e convergence point. T a n g e n t i a l O i l Film F o r c e ( F t )
3.3
.
c o r r e s p o n d i n g r e l a t i o n s h i p between F t The a n d go a t E = 0.7 is shown i n F i g u r e 2 b , a n d h e r e t h e f o l l o w i n g o b s e r v a t i o n s may b e made:
(a)
curves are given f o r p o s i t i v e only. For negative 8, the d a t a is i d e n t i c a l e x c e p t t h a t t h e s i g n of Ft is r e v e r s e d .
?'he
go
( b ) A s n o t e d i n 3.2 ( c ) , p o s i t i v e k t e n d s t o s u p p r e s s t h e c a v i t a t i o n i n d u c e d by wedge action: T h i s y i e l d s l i n e a r i t y of Ft w i t h 8, Lo t h e p o i n t a t which t h e positive R fails to s u p p r e s s wedge cavitation. The m a g n i t u d e o f g o , a b o v e which t h e F t 8, r e s p o n s e becomes n o n - l i n e a r , !epends on t h e magnitude o f t h e P o s i t i v e R.
-
3.4
General O b s e r v a t i o n s on t h e O i l Film Force J o u r n a l Velocity R e s u l t s
-
( a ) Fr is p r i m a r i l y a f u n c t i o n of A , t h e secondary influence of 8, being a result of cavitation induced by hydrodynamic. wedge a c t i o n . The p a r t of Fr d u e t o 8, t h u s becomes z e r o when R i s h i g h enough t o s u p p r e s s t h e a b o v e cavitation. This explains the p r o g r e s p i v e c o n v e r g e n c e of t h e f a m i l y o f Fr R c u r v e s f o c €lo # 0 , w i t h t h f s t r a i g h t l i n e f o r 8, 0 at positive R i n F i g u r e 2a.
-
(b). F t is primarily a function of 6 Hydrodynamic s q u e e z e action (i.e. f j does n o t i n itself, r e s u l t is a f i n i t e F t , t h e r e f o r e a l l curves pass through t h e origi? i n Figure 2b. The influence of R on Ft' i n d i c a t e d by the f a m i l y of curves i n Figure 2b, results purely from the i n t e r a c t i o n of s q u e e z e a c t i o n w i t h wedge cavitation.
( c ) The
p o s i t i v e f i l m p r e s s u r e r e g i o n and c a v i t a t i o n zone a s s o c i a t e d w i t h squeeze a c t i o n are c i r c u m f e r e n t i a l l y s y m m e t r i c a l w i t h r e s p e c t t o t h e l o c a t i o n s of h m i n a n d h,,.
( d ) The p o s i t i v e f i l m p r e s s u r e r e g i o n and c a v i t a t i o n z o n e a s s o c i a t e d w i t h wedge acti o n are c i rcum f e re n t i a11y qssymmetrical with respect to the l o c a t i o n s o f h m i n a n d hmax.
476
lEo0r
lOOOr
I
I
-4
-3
-2
I
-1
I
I
I
I
J
0
1
2
3
4
Radial velocity Fig. 2a
Radial Force Data at
E
k
mm/s
0
100
200
300
400
Effective angular velocity 6o rads/s Fig. 2b Tangentlal Force Data at
E
= 0,7
= 0,7
A s a r e s u l t of ( c ) and ( d ) , s q u e e z e action is capable of virtually e l i m i n a t i n g t h e pas-itive f i l m p r e s s u r e r e g i o n (by n e g a t i v e R)., o r t h e c a v i t a t i o n d u e t o wedge zone (by p o s i t i v e R ) , action. This results in the trend t o y a r d s c o n v e r g e n c e of t h e f a m i l y of F r R - c u r v e s i n F i g u r e 2a a s t h e m a g n i t u d e of R i n c r e a s e s i n b o t h t h e p o s i t i v e and negative sense. Conversely, in no c i r c u m s t a n c e s d o e s wedge a c t i o n h a v e a dominant i n f l u e n c e i n r e l a t i o n to s q u e e z e action. The f a m i l y of F t 8, curves i n F i g u r e 2b d o e s n o t t h e r e f o r e i n d i c a t e and t e n d a n c y t o c o n v e r g e a s s o c i a t e d w i t h i n c r e a s i n g €lo.
-
-
The i n t e r a c t i o n o f s q u e e z e and wedge with cavitation, zction, associated the principle of invalidates s u p e r p o s i t i o n w i t h r e s p e c t t o t h e oil f i l m forces r e s u > t i n g frpm s i m u l t a n e o u s Where t h e a p p l i c a t i o n of R and 8,. c o n d i t i o n s are s u c h t h a t a l l c a v i t a t i o n is supressed, the principle of s u p e r p o s i t i o n is r e d u n d a n t s i n c e Fr is a l i n e a r f u n c t i o n ?f R o n l y and F t i s a l i n e a r f u n c t i o n of B0 o n l y . The o i l f i l m b e h a v i o u r u n d e r l y i n g t h e i n t e r a c t i o n of s q u e e z e and wedge a c t i o n s is i l l u s t r a t e d by F i g u r e 3 , which shows the family of circumf$rential film pressure profiles f o r 8,/w= 1 . 0 , at E=0.7. I n F i g u r e s 2a and 2b, t h e p o i n t s to these profiles are corresponding identified. It may be n o t e d t h a t f o r p o i n t ( A ) , k h a s a t t a i n e d a l e v e l where i t h a s almost e l i m i n a t e d wedge i n d u c e d cavitatio;. A t p o i n t (A). i n f i g u r e 2a, for B0/w 1.0 has the Fr-R curve t h e r e f p r e v i r t u a l l y converged w i t h t h a t f o r Bo/w =O. Conversely t h e p r o f i l e
-0.2L
FY 3
f o r p o i n t (El shows how t h e p o s i t i v e f i l m p r e s s u r e d u e t o wedge a c t i o n h a s b e t n s u b s t a n t i a l l y r e d u c e d by t h e n e g a t i v e R, and t h e c a v i t a t i o n zone e x t e n d e d .
3.5
of Very Influence J o u r n a l Ve 1oc i t i e s
Large
Lateral
C o n s i d e r a t i o n was a l s o g i v e n t o t h e e f f e c t o f l a t e r a l j o u r n a l v e l o c i t i e s a p p r o a c h i n g two o r d e r s o f magnetude g r e a t e r t h a n t h o s e c o v e r e d i n t h e foregoing results. With regard t o practical applications, this may appear This additional analysis somewhat academic. was n e v e r t h e l e s s found t o b e of v a l u e i n
477 enhancing a n u n d e r s t a n d i n g o f t h e o i l f i l m response t o large l a t e r a l j o u r n a l v e l o c i t i e s , p a r t i c u l a r l y w i t h regard of t h e s i g n i f i c a n c e of cavitation.
maintaining a constant o i l supply pressure Ps and c a v i t a t i o n p r e s s u r e Pc, e f f e c t i v e l y r e s u l t s i n a tendency towards o i l s t a r v a t i o n and t h u s a n e x t e n t of c a v i t a t i o n e x c e e d i n g 72% i n F i g u r e 5.
8or 70
Data refers to lefthand scale
-----
i,/o
-
t
I
= +_ 1.0
e'*Data refers to I ""-
=O
-..-
= 2 2.0 B,/w
-a-
-.-
.............
= ? 2.0 =O = r1.0
Data refers lo righthand sw le
'O1 0
\
10
M i a 1 velocity
Flg.4
F,
20
30
R = +3,6 m / s
-x-
40
0
10
30
20
&/w
Fig. 5
R e s u l t s computed f o r a n k r a n g e of -25 t o +50 m m / s are shown i n t i g u r e 4 , t h i s c o m p r i s i n g t h e g r a d i e n t dFr/dR and t h e c o r r e s p p n d i n g A t negative R the e x t e n t of c a v i t a t i o n . dFr/dk c u r v e s show t h e c o n v e r g e n c e referred t o i n s e c t i o n 3.4 ( e ) . The c o r r e s p o n d i n g p o r t i o n s of t h e c a v i t a t i o n c u r v e s c o n f i r m t h a t t h e c o n v e r g e n c e is a s s o c i a t e d w i t h a t e n d e n c y towards "saturation" ?f the extent of cavitation. A t p o s i t i v e R, t h e i n i t i a l l i n e a r response (constant dFr/dk) is seen to coincide with zero or virtually zero cavitation. Above R = 27.5 m m / s c a v i t a t i o n s t a r t s t o o c c u r i n t h e hax r e g i o n , and r e s u l t s i n a s l i g h t d r o p ( 1 . 2 % ) i n dFr/dk. The r e a s o n f o r t h e above e f f e c t b e i n g v e r y small is t h a t t h e c h a n g e i n oil f i l m force i n t h e hmax r e g i o n , a r i s i n g from t h e o n s e t of c a v i t a t i o n , is low i n r e l a t i o n t o t h e change i n o i l f i l m force i n t h e h m i n r e g i o n , d u e , t o t h e s q u e e z e a c t i o n associated w i t h p o s i t i v e R. F i g u r e 5 p r e s e n t s F t r e s u l t s f o r t h e h0/w r a n g e of 0 to 90, together with the It is corresponding e x t e n t of c a v i t a t i o n . e v i d e n t t h a t t h e F t c u r v e s remain d i s t i n c t l y non-linear throughout t h i s very l a r g e v e l o c i t y a range. The c a v i t a t i o n c u r v e s e x h i b i t similar b e h a v i o u r , and t h e p e r s i s t a n c e o f non-linearity i n t h e F t c u r v e s is c l e a r l y associated with t h e f a i l u r e of t h e extent of cavitation t o reach a "saturation" level.
It is i m p o r t a n t t o n o t e t h a t t h e a b s o l u t e maximum e x t e n t of c a v i t a t i o n associated w i t h F o r wedge a c t i o n , s q u e e z e a c t i o n is 505. however, t h e e x t e n t o f c a v i t a t i o n may a p p r o a c h 100% u n d e r o i l s t a r v a t i o n c o n d i t i o n s . The application of very large 6,/w ,whilst
50
= 123.57
60
70
80
90
rads/si
Ft and Cavitation Data at Very Large
B,/w
R mm/s
R Amplltude (c=0.7)
40 (W
mm/s
D imens i on less effective angu I ar velocity
!
- R Gradient and Cavltatlon at Very Larg.
-.- R = O ............. i = - 3.6
Data refers to righthand scale
Amplitude ( ~ = 0 . 7 )
The above f a c t o r s are best e x p l a i n e d by consideration of the requirements for hydrodynamic s i m i l a r i t y . The d i m e n s i o n l e s s l o a d c a p a c i t y p a r a m e t e r h a s been commonly used for steadily loaded hydrodynamic journal bearings :
r11
T h i s is c o n s t a n t f o r a g i v e n b/D r a t i o and e c c e n t r i c i t y r a t i o E , and is t h e i n v e r s e of t h e well known Sommerfeld No. A t any i n s t a n t a dynamically loaded bearing, the in appropriate dimensionless load capacity p a r a m e t e r s associated w i t h wedge and s q u e e z e a c t i o n may s i m i l a r l y b e e x p r e s s e d r e s p e c t i v e l y as :
-W,= "("J; ,pol
Sg=P
C
r21
6
However, hydrodynamic s i m i l a r i t y i n b o t h t h e s t e a d i l y and d y n a m i c a l l y l o a d e d s i t u a t i o n s is a l s o d e p e n d e n t upon t h e geometric s i m i l a r i t y of t h e c a v i t a t i o n z o n e boundary r e l a t i v e t o t h e b e a r i n g s u r f a c e boundary. The c a v i t a t i o n z o n e boundary i s d e p e n d e n t on t h e o i l f i l m boundary p r e s s u r e s Ps and Pc. In order t o f u l f i l t h e above r e q u i r e m e n t f o r geometric s i m i l a r i t y with respect t o cavitation, the f o l l o w i n g d i m e n s i o n l e s s p a r a m e t e r s must a l s o be h e l d c o n s t a n t : F o r wedge a c t i o n :
133
478 For squeeze action: 141
S i n c e P,, Pc a n d w e r e held constant for tests c o v e r e d i n F i g y r e 5 , t h e a p p a r e n t o i l s t a r v a t i o n a t h i g h B0/w i s d u e t o t h e corresponding r e d u c t i o n in P,, and Pcw. Had P,, a n d Pcw b e e n m a i n t a i n e d c o n s t a n t , t h e n t h e Ft c u r v e s i n F i g u r e 5 w o u l d h a v e been l i n e a r . the
4.
DEVELOPMENT EQUATIONS
4.1
OF
OIL
FILM
u s e d a s t h i s r e l a t e s t o t h e c o n d i t i o n k = 0. The degree of approximation involved in l i n e a r i s i n g t h e c o e f f i c i e n t s Brt and Brrt is i.ndicat$d by t h e c u r v e s f o r Fr and $Fr/dR a t R = 0, w h i c h are p l o t t e d a g a i n s t B0/w f o r E = 0.7 i n F i g u r e 6. It m?y b e noted that Fr 1s zero a n d dFr/dR is c o n s t a n t u p t o B0/w = 0.2, due to t h e a b s e n s e ?f c a v i t a t i o n i n d u c e d by wedge a c t i o n ?t low €lo. S i n c e t h e f a m i l y of c u r v e s f o r Bo/w+O i n F i g y r e 2a are c l e a r l y a s y m p t o t i c t o t h a t Cor Bo/w=O f o r b o t h p o s i t i v e a n d n e g a t i v e R , t h e v a l u e o f Fr p r e d i c t e d by e q u a t i o n .[5] is s u b j e c t t o t h e c o n d i t i o n Fc Brr R .
+
FORCE 700
Tnt r o d u c t i o n
600
In o r d e r to facilitate t h e operation o f a fast j o u r n a l o r b i t a n a l y s i s programme, it was n e c e s s a r y t o d e v e l o p o i l f i l m force e q u a t i o n s w h i c h would g i v e a s a t i s f a c t o r y a p p r o x i m a t i o n t o computed d a t a o f t h e t y p e g i v e n i n F i g u r e s 2a a n d 2b. T h i s d a t a is f o r E =0.7. At r e d u c e d e c c e n t r i c i t y r a t i o t h e form o f t h e force velocity curves is essentially similar, b u t r e d u c e d c a v i t a t i o n d u e t o wedge a c t i o n r e s u l t s i n t h e f a m i l i e s of c u r v e s b e c o m i n g more l i n e a r a n d c l o s e r t o g e t h e r . The reverse trend occurs with increased e c c e n t r i c i t y ratio.
500
400
I1
300 5
-
The form o f t h e f o r c e - v e l o c i t y c u r v e s is c l e a r l y c o m p l e x , a n d a n e x t e n s i v e s e a r c h was made f o r e q u a t i o n forms t h a t w o u l d a c c u r a t e l y f i t t h i s data. No s o l u t i o n was f o u n d w h i c h would y i e l d s a t i s f a c t o r y r e s u l t s o v e r a wide range o f e c c e n t r i c i t y ratios. In a t t e m p t i n g t o f i n d a n a c c u r a t e f i t , t h e r e was a n i n e v i t a b l e t r e n d towards complex e q u a t i o n s w i t h e x c e s s i v e numbers o f c o e f f i c i e n t s . The c o m p l e x c u r v e f i t a p p r o a c h was t h e r e f o r e a b a n d o n e d i n f a v o u r o f t h e much s i m p l e r partially linearised solution. This solution is d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n , and h a s r e s u l t e d i n s a t i s f a c t o r y fast j o u r n a l o r b i t p r e d i c t i o n s i n tests c a r r i e d o u t t o d a t e ( 5 ) . 4.2
P a r t i a l l y Linearised Equations
E x a m i n a t i o n of F i g u r e . 2a i n d i c a t e s a n e e d t o u s e d i f f e r e n t Fr R l i n e a r i s e d +opes for t h e p o s i t i v e a n d n e g a t i v e ranges o f R . It is t o t h e c o n s e q u e n t u s e o f two s l o p e s t h a t t h e term " p a r t i a l l y l i n e a r i s e d " refers. F i g u r e 2? also shows a p r o g r e g s i v e i n c r e a s e i n Fr R s l o p e f o r n e g a t i v e R , a n d r e d u c t i o n i n sloEe for positive R, a s t h e m a g n i t u d e o f 8, increases. I n a d d i t i o n , t h e m a g n i t u d e o f Fr a t R = 0 i n seen to increase progressiv7ly with increase in the magnitude of. Bo. Assuming t h a t t h e a b o v e i n f l u e n c e s o f Bo a r e approximately linear, then the following e q u a t i o n may b e w r i t t e n f o r F,:
0
.uz
' U
200 100
D
I
0
I
I
1
6,/0
2 (w = 123.57 r a d d s )
Dimensionless effective angular velocity
Fig.6
F, and dF,/dR at R = O
A s i m i l a r l i n e a r i s a t i o n may b e a p p l i e d t o t h e
Ft d a t a shown i n F i g u r e 2b. Since the s t r a i g h t l i n e f i t t e d t o a l l t h e c u r v e s may c l e a r l y p a s s t h r o u g h F t = 0 a t Bo = 0, t h e n n o B t r term i s r e q u i r e d , i . e . we may write :
-
-
[51 A s i n d i c a t e d above, d i f f e r e n t v a l u e s o f Brr a n d B r r t a r e u s e d f o r R > 0 a n d R < 0. The Brrt coefficient effectively represents the p r e v i o u s l y d e s c r i b e d i n t e r a c t i o n of s q u e e z e a n d wedge a c t i o n s . A s i n g l e v a l u e o f B r t is
The c u r v e s f o r n e g a t i v e 8, a r e i d e n t i c a l t o t h o s e shown i n F i g u r e 2 b , e x c e p t t h a t t h e s i g n of F t is r e v e r s e d , t h e r e f o r e o n l y a s i n g l e value of Btt is required. Btrt also represents squeeze wedge i n t e r a c t l o n i n a similar m a n n e r t o B r r t , a n d t h e u s e . of d i f f e r e n t v a l u e s of t h i s c o e f f i c i e n t f o r R-0 and R < O a g a i n g i v e s a b e t t e r f i t t o t h e computed d a t a .
-
It is i m p o r t a n t t o n o t e t h a t t h e l i n e a r i s e d displacement and velocity coefficients, commonly u s e d i n l a t e r a l v i b r a t i o n a n a l y s i s , o n l y f a c i l i t a t e t h e e s t i m a t i o n of c h a n g e of o i l f i l m force c o m p o n e n t s from a n e q u i l b r i u m condition. In c o n t r a s t with t h i s , t h e o i l f i l m f o r c e c o m p o n e n t s g i v e n by e q u a t i o n s [51 and [ 6 ] are t h e t o t a l v a l u e s . The e s t i m a t i o n of t h e o i l f i l m force c o m p o n e n t s a t a n y l o c a t i o n of t h e j o u r n a l w i t h i n t h e b e a r i n g
479
c l e a r a n c e s p a c e , r e q u i r e s t h e c o m p u t a t i o n of the velocity c o e f f i c i e n t s B r t , etc over t h e range of possible eccentricity ratios. S u i t a b l e i n t e r p o l a t i o n is t h e n used f o r t h e eccentricity ratio corresponding to the specified location. The e r r o r s associated w i t h t h e l i n e a r i s a t i o n r e q u i r e d t o p r o d u c e e q u a t i o n s [51 and C61 w i l l be minimised by computing. t h e velocity 8 , pertubations c o e f f i c i e n t s w i t h k and to the maximum velocities corresponding a n t i c i p a t e d f o r t h e case u n d e r c o r i s i d e r a t i o n . J o u r n a l o r b i t t e s t s u s i n g e q u a t i o n s [ 5 1 and [6], have i n d i c a t e d t h a t t h e p r e d i c t e d o r b i t s are n o t unduly s e n s i t i v e t o t h i s r e q u i r e m e n t ,
4.3
D imen si on l e s s Ve 1o c i t y Co e f f i c i e n t s
The above e x p r e s s i o n s are s u b j e c t t o t h e u s u a l b e a r i n g geometric s i m i l a r i t y r e q u i r e m e n t , i.e. t h e y are v a l i d f o r a g i v e n b/d r a t i o . In a d d i t i o n , a s i n d i c a t e d i n s e c t i o n 3.5, t h e s e e x p r e s s i o n s are a l s o s u b j e c t t o t h e geometric s i m i l a r i t y requirements w i t h r e s p e c t t o the c a v i t a t i o n zone boundary.
Dimensionless-
coefficient+ccentricity
ratio
data, corresponding t o t h e c o n d i t i o n s g i v e n i n
a r e p r e s e n t e d i n F i g u r e s 7 and 8. It may E 0.4, c a v i t a t i o n d i s a p p e a r s , h$nce t h e d u e t o s q u e e z e act@ c u r v e s f o r R-0 and 2 o n v e r g e n c e of t h e Brr R q O s e e n i n F i g u r e 7. Cavitation arising from wedge a c t i o n d i s a p p e a r s a l i t t l e below E 0.6, thus resulting i n B r t &coming and F i g u r e 8 shows how t h e B r r t .-z e r o . Btrt c o e f f i c i e n t s s i m i l a r l y d i s a p p e a r below E = 0.6 s i n c e t h e y r e l a t e t o t h e about i n t e r a c t i o n of s q u e e z e a c t i o n upon wedge cavitation. 1 ooc 75c -b/D= 0,1465 50C Psw = 0,0390 FCw=-0.1321 pss= 0,3361 25C Fcs =-1.1377 3.1,
1oc
75 2 C
The r e s u l t o f t h i s i s :
-r
A noteable exception to the above a p p r o x i m a t i o n , a r i s i n g from f a i l u r e t o s a t i s f y t h e c a v i t a t i o n zone s i m i l a r i t y r-equirements, i s t h e Brr coefficient for R 7 O . This c o e f f i c i e n t i s a f u n c t i o n of t h e b/D r a t i o only, d u e t o t h e absepce o f c a v i t a t i o n associated w i t h p o s i t i v e R. Some. c a v i t a t i o n i n t h e hmax r e g i o n h a s .been shown t o o c c u r a t v e r y l a r g e p o s i t i v e R, b u t t h e e f f e c t on dFr/dR, and heyce on E r r , was shown t o be negligible.
be n o t e d t h a t below a b o u t
For g e n e r a l i s a t i o n of v e l o c i t y c o e f f i c i e n t data, the following non-d i m e n s i o n a l e x p r e s s i o n s may b e u s e d :
Coefficient:
corresponding values pertaining to any d i m e n s i o n l e s s v e l o c i t y c o e f f i c i e n t data used.
-
5c
.-Q,
.-V
Valid for given:
't 0
25
V
> .-
Br
u
0
x>
10
-i 7,5 c
0
'v, 5,c The above v a l i d i t y l i m i t a t i o n s a p p e a r t o make the generalisation of these velocity
coefficients t o t a l l y impracticable. However, errors r e s u l t i n g from f a i l u r e t o s a t i s f y t h e c a v i t a t i o n zone s i m i l a r i t y r e q u i r e m e n t s , are comparable to the errors arising from mismatching of t h e v e l o c i t y p e r t u b a t i o n s used the dimensional velocity to derive with the maximum velocity coefficients, components o c c u r i n g i n t h e j o u r n a l o r b i t u n d e r consideration.
. . The R, 2,
-
,
-0
2.5
t
0
z
180
0,75
.'
-
used to derive psw, Pew, Pss, Pcs and used in the for and dimensionless expressions s h o u l d c o r r e s p o n d to t h e above maximum of velocity components. Insensitivity p r e d i c t e d o r b i t s t o t h e m a t c h i n g of R, go p e r t u b a t i o n s , n o t e d i n s e c t i o n 4.2, s h o u l d thereforesimilarl.apply t h e m a t c h i n g of psw 8 PCW! pss, Pcs with the
Gt,
5
.-E
vases
Kt
0 Fig. 7 Separate Squeeze and Wedge Action
Velocity Coefficients
480
1 ooc
750 0,1465 0,0390 -FCw=-0,1321 P,, = 0,3361 Fc,=-1,1377 b/D=
50C 25C
'i 1oc '0 -- 75
. c
P,,=
* Note:
~ ~ >o] ~ tcoeffi [ i ci en t values are negative
'3
.2
The r e s u l t s g i v e n i n t h i s p a p e r are f o r a n aligned 360° circumferential g ro o v e bearing. For t h i s t y p e o f b e a r i n g , the velocity coefficients in the o i l f i l m force equations are f u n c t i o n s of E only. The equations are also applicable to n o n - c i r c u m f e r e n t i a l l y s y m m e t r i c a l b e a r i n g s , by t h e d e r i v a t i o n o f v e l o c i t y c o e f f i c i e n t s as f u n c t i o n s o f b o t h E and a t t i t u d e a n g l e W. 7.
ACKNOWLEDGE ENT
The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o t h e Committee o f L l o y d ' s R e g i s t e r o f S h i p p i n g f o r p e r m i s s i o n t o p u b l i s h t h i s paper.
c
.-
V
0 25 al > m c .-0
g
lo
.-?i
7.5
Ref e re nc e s
(1) LA BOUFF, G.A. and BOOKER J.F. "Dynamically Loaded J o u r n a l Bearings : A F i n i t e Element T re a t me n t f o r R i g i d and Elastic Surfaces". A.S. M.E./A.S.L. E. Conf. Oct. 1984. A.S.M.E. Paper 84 TRTB 11.
I
-
-
-0
5,o
c
0
R.W. IIA Numerical Analysis Method b a s e d on Flow C o n t i n u i t y f o r Hydrodynamic Journal B e a ri n g s " T r i b o l o g y I n t e r n a t i o n a l . Vol. 17. No. 6 Dec. 1984.
( 2 ) JAKE",
z
.
2.5
1 .O 0.75
( 3 ) JAKEMAN, R.W. "Journal Orbit Analysis t a k i n g a c c o u n t o f O i l Film H i s t o r y and J o u r n a l k s s " Proc. o f Conf. : Numerical Methods i n Laminar and T u r b u l e n t Flow. Swansea J u l y 1 9 8 5 pp 199-210.
0.50 0.25
(4) BANNISTER,
0.10
Fig. 8 Interactive Squeeze and Wedge Action Velocity Coefficients 6.
CONCLUSIONS
T h i s p a p e r h a s p r e s e n t e d d a t a on t h e o i l f i l m forces associated with large lateral v e l o c i t i e s o f t h e j o u r n a l i n a hydrodynamic j o u r n a l bearing. The t h e o r e t i c a l c a v i t a t i o n model u s e d h a s e n a b l e d t h e r o l e o f c a v ' i t a t i o n in relation to non-linearity, and in p a r t i c u l a r t o t h e i n t e r a c t i o n o f hydrodynamic s q u e e z e and wedge a c t i o n s , t o b e c l e a r l y shown. E q u a t i o n s have been i n t r o d u c e d f o r t h e o i l film f o r c e components, ba se d on partial l i n e a r i s a t i o n o f t h e computed f o r c e - v e l o c i t y data. By u s i n g a P o l a r s y s t e m , it was p o s s i b l e t o s e g r e g a t e hydrodynamic wedge and squeeze action. The r o t a t i o n a l v e l o c i t y o f t h e j o u r n a l a b o u t its a x i s was combined w i t h t h e angular v e l o c i t y o f t h e j o u r n a l axis about t h e bearing axis,- t o give an equivalent a n g u l a r v e l o c i t y 8,. The o i l f i l m f o r c e components g i v e n by t h e above e q u a t i o n s are t h e r e f o r e t o t a l values, r a t h e r than changes from some equilibrium condition. These equations are s u i t a b l e f o r fast j o u r n a l o r b i t a n a l y s i s , t h i s a p p l i c a t i o n b e i n g c o v e r e d by r e f e r e n c e (5).
R.H. "A Theoretical and Experimental I n v e s t i g a t i o n illustrating the influence of N o n -L i n e a ri t y and Misalignment on t h e E i g h t O i l Fi l m Fo r c e Coefficients". Proc. 1. Mech. 6. Conf.: Vibration in Rotatinmg bchinery. Cambridge, 1976. Paper C219/76.
R.W. and PARKTNS D.W. ( 5 ) JAKEMAN, " T h e o r e t i c a l and E x p e ri me n t a l O r b i t s o f a Dynamically Loaded Hydrodynamic J o u r n a l B e a ri n g " . To be published: 13th Leeds-Lyon Symposium on T ri b o l o g y . Leeds. S e p t . 1986. ( 6 ) PARKINS, D.W. "Theoretical and E x p e ri me n t a l D e t e r m i n a t i o n o f t h e Dynamic C h a r a c t e r i s t i c s o f a Hydrodynamic J o u r n a l A.S.M.E. Journal of B e a ri n g " L u b r i c a t i o n Technology. Vol. 101. A p r i l 139. 1979. PP. 129
.
-
481
Paper XV(iii)
Effects of cavity fluctuation on dynamic coefficients of journal bearings Ken lkeuchi and Haruo Mori
The dynamic characteristics of a cylindrical journal bearing with an axial oil supply groove is numerically analyzed by taking the film rupture and the reformation boundaries into consideration. The results, based on the assumptions of frozen cavity, constant cavity pressure, constant gas volume in the cavity and adiabatic change of the gas, are compared to each other. It is found that the oil film coefficients depend on the movement of the cavity boundary and the fluctuation of the cavity pressure. Adiabatic change is the most appropriate assumption if the gas is conserved in the cavity. However, the dynamic properties of a heavily loaded bearing can reasonably be edtimated by constant pressure assumption. On the other hand, those of a heavily loaded bearing by constant volume assumption. 1
INTRODUCTION
The authers (1) presented the equivalent flow model for an analysis of journal bearings with cavitation. The modified Reynolds equation is applied to the oil film region and the cavitation region in common. This model is based on the extreme difference of the fluidity between the oil and the gas in the cavity, and considers the oil mass conservation throughout the bearing clearance. The result agrees well with those by JakobssonFloberg (2) and Elrod (3, 4 ) for the static performance, and it is expected to agree with the one by Olsson ( 5 ) for the dynamic performance. In a conventional linear analysis of the dynamic characteristics of a journal bearing, the cavity is assumed to be fixed to the static state. However, the variation of the cavity pressure and the movement of the boundary affect the dynamic oil film coefficients, even if the displacement of the shaft and/or its velocity are infinitely small. The purpose of this paper is to indicate the effects of the variation of the cavity pressure and the movement of the cavity boundary on the dynamic oil film coefficients and the stability threshold in a plain journal bearing with an axial oil supply groove. 1.1
Not ation
C
Radial clearance
i' j D
g h
a' PS
r
Time
w a
Bearing load Cavitation coefficient
p
Oil viscosity
w
Angular velocity of shaft
Dimensionless parameters C Damping coefficient (=c wc/(Sw)) ij ij K . . Stiffness coefficient (=k..c/(Sw)) 1J 2 2 P Pressure (=pc /(par ) ) 3 2 S Sommerfeld number (=2pNr L/(wc ) ) So
Bearing number (=PNr2 /(p c2 ) ) (=WC2 /(pwr
W
Bearing load
y
Local oil content
E
Eccentricity ratio
K
Ratio of specific heats ( = 1 . 4 )
2
3a L))
BASIC EQUATIONS
The oil is assumed to be isoviscous and incompressible. The two phase fluid in the cavity is replaced by single phase equivalent fluid, and the Reynolds equation for the equivalent fluid is
Damping coefficient Shaft diameter (=2r) Gravitational acceleration qilm thickness
k.. Stiffness coefficient 1J L Bearing length P
t
Gauge pressure Ambient pressure Oil supply pressure Shaft radius
where - denotes the equivalent fluid. Since the density and the viscousity of the gas are negligibly smaller than those of the oil, those of the equivalent fluid may be expressed as
P P
= YP =
YP
where y is local oil content. Substituting Eq. (2) and Eq. (3) into Eq. (l), the following modified Reynolds equation is derived.
482
Since the viscosity and the density of the gas is much smaller than those of the oil, only the gas flows by the pressure gradient, consequently the oil content is reduced at low pressure position. Accordingly, the oil content may be given as p>p
(5)
(oil film): y = 1
(6)
p
If a is taken to be large enough, the calculated cavity pressure is slightly lower than p In the oil film, the squeeze effecteterm is
.
The models of the cavity fluctuation are listed in Table 1 . Recently, Ono et al. ( b ) analyzed the dynamic properties of a journal bearing with a circumferential oil supply groove for the equivalent flow model. Since they employed perturbation method in the calculation of the dynamic coefficients, their result corresponds to FC in this paper.
1 1; 1 FC
(7)
Table 1
Constant Pressure Moving Boundary Constant Gas Volume Moving Boundary Adiabatic Change Moving Boundary
AD While in the cavity, since only the gas is squeezed and yh does not vary, this term is expressed as
Constant Pressure Fixed Boundary
Models of cavity fluctuation
4 RESULTS AND DISCUSSION P is set to be zero at the refered static state. Tge dynamic coefficients are calculated by finite difference method and Newton-Raphson's scheme. , shaft displacem nt from the refered If 6 ~ the position, is less than 10 , the stiffness coefficients are independent of 6 ~ and , if4the displacement velocity t / w is less than 10 , the damping coefficients are independ nt of t / w . -5 in the calcuSo 6~ and t / w are taken to be 10 lation of the dynamic coefficients.
-z
Fig. 1 shows the schematic view of the journal bearine. In the calculation. the parameters are set lje; L/D=I, 8 =15", L /L=0.636, Ps=O.l, apwr / c =50 Fig. 2 shows the film pressure under static condition at ~=0.5. Such a result is used as a refered value in the calculation of the dynamic coefficients.
50
-
W
1
3 THE MODELS OF THE CAVITY FLUCTUATION In a conventional linear analysis, the cavity is approximated to be 'frozen' under dynamic load, which means that the pressure is constant in the cavity and the boundary of the cavitation region is fixed. (FC) In a heavily loaded bearing, or if the cavity is ventilated, the cavity pressure may be approximated to be constant, while the boundary moves under dynamic load. (CP) When sufficient oil is supplied, the gas in the cavity is isolated from the ambient by the meniscuses of the oil formed at the ends of the bearing. If the cavity is stable under dynamic load, the gas is conserved in the cavity. In a lightly loaded bearing, the gas may be assumed to be incompressible, while the pressure varies. (CV) In this case, the following equation must be satisfied to define p
Fig. 1
Bearing configuration
.
where the subscript o indicates the refered static state. In general cases, it seems to be most appropriate to assume that the gas changes adiabatically. (AD) In this case, the following equation must be satisfied to define p
.
8.0 Fig. 2
7
1
hmin
2r
Film pressure at steady state (~=0.5)
483 4 . 1 Effect of movement of cavitation boundary
In this section, the frozen cavity model (FC) and the constant pressure model (CP) are compared in order to indicate the effect of the movement of the cavity boundary. Fig. 3 (a) and (b) illustrates the ratio of the pressure variation to the vertical shaft displacement velocity for FC and CP respectively. The ratio is slightly negative in the left upstream region due to the negative squeeze film effect, while the ratio is positive in the middle region due to the positive squeeze film effect, and the pressure is constant in the cavity. Fig. 3 (c) shows the difference between FC and CP. The left two peaks are located just inside the film reformation boundaries, and a larger plateau is located just inside the film rupture boundary.
Fig. 4 compares the stiffness coefficients of the oil film for FC and CP. Considerable differences are observed for K and K Fig. 5 compares thex8ampingxgoefficients of the oil film. FC gives higher damping coefficients at high eccentricity ratio, and there are discrepancies at low eccentricity ratio too. For CP, C is larger than C at low eccentricity ratio?’on the other hand:xthey agree for FC. Fig. 6 shows the stability thresholds for a horizontal rigid shaft. CP predicts higher critical speed than FC at low and high eccentricity ratio.
.
100
50
0 -25
Fig. 4
1
t hmin
(c) Fig. 3
FC-CP
Ratio of pressure variation to vertical shaft displacement velocity (waP/ae ) Y
Stiffness coefficients for CP and FC
484
4 . 2 Effect of pressure variation in the cavity
6
Fig. 7 (a) and (b) illustrates the ratio of pressure variation to vertical shaft displacement velocity for constant pressure model (CP) and constant volume model (CV) respectively. (b) shows that the ratio is positive in the cavity because the cavity is compressed by the surrounding oil film. Fig. 7 (c) shows the difference between (a) and (b). The discrepancy between CP and CV is observed just inside the film reformation boundary, just inside the film rupture boundary and in the cavitation region. Fig. 8 compares the stiffness coefficients and smallfor CP and CV. CV predicts larger K than CP. YX er K X fig. 9 compares the damping coefficients for CP and CV. CV estimates C to be close to zero at low eccentricity ratioYX
4
g 3” 2
0.2
0 Fig. 6
0.4
&
0.6
0.8
I.o
Stability thresholds for CP and FC
40
Fig. 8
Kxx
Stiffness coefficients for CP and CV
100
80
hmin
60 :=
u
40
20
hmin
(c) Fig. 7
cv-CP
Ratio o t pressure variation to vertical shaft displacement velocity (waP/ai ) Y
Fig. 9 Damping coefficients for CP and CV
485
Fig. 10 shows the stability thresholds for CP,CV and AD. CP predicts higher critical speed than CV, and the threshold for AD is in between them. However, AD can be approximated by CP if So is larger than 100 (a heavily loaded bearing), on the other hand, AD can be approximated by CV if So is less than 1 (a lightly loaded bearing); whe e the bearing number So is defined as So= $ 2 PNr /(pat 1. 5 CONCLUSIONS
(1)
The movement of the cavity boundary affects the pressure variation just inside the oil film boundary, and the dynamic coefficients of the oil film. A higher critical speed is predicted by taking the movement of the cavity boundary into account. (2) If the gas in the cavity is conserved under dynamic load, the dynamic coefficients are affected by the variation of the cavity pressure. Constant cavity volume assumption leads to lower critical speed than constant cavity pressure assumption. (3) If the gas is conserved in the cavity, adiabatic change seems to be the most appropriate assumption. However, it can be approximated by constant pressure assumption for a heavily loaded bearing, on the other hand, it can be approximated by constant gas volume assumption for a lightly loaded bearing. References IKEUCHI, K. and MORI, H. 'An analysis of the lubricating films in journal bnarings -Effects of oil supply condition on the static performance-', J. JSLE International Ed. 1983, NO. 4 , 57-60. JAKOBSSON, B. and FLOBERG, L. 'The finite journal bearing, Considering vaporization', Trans. Chalmers Univ. Technol. 1957, No. 190. ELROD, H.G. and ADAMS, M.L. 'A computer program for cavitation and starvation problems', Proc. 1st Leeds-Lyon Symp. on Trib. 1974, I1 (ii), 37-41.
Y
0
0.2
0.4
&
0.6
0.8
Fig. 10 Stability thresholds for CP, CV and AD
(4)
ELROD. H.G. 'A cavitation algorithm', Trans. ASME, 1981, J. Lub. Technol., 103-3, 350-354. (5) OLSSON, K. 'Cavitation in dynamically loaded bearings', Trans. Chalmers Univ. Technol. 1965, No. 308. (6) ONO, K., MICHIMURA, S. and TAMURA, A . 'Analysis of dynamic characteristics for cylindrical journal bearing based on average flow theory', Trans. JSME, Ser. C, 1985, 51-471, 3026-3033, in Japanese.
This Page Intentionally Left Blank
481
Paper XV(iv)
Investigation of static and dynamic characteristics of tilting pad bearing Tingting Huang, Yinglong Wang and Shizhu Wen
I n t h i s paper, t h e t h e o r e t i c a l and experimental i n v e s t i g a t i o n s on t h e s t a t i c and dynamic c h a r a c t e r i s t i c s of t i l t i n g
pad bearing are d e s c r i b e d . The experiments f o r i d e n t i f y i n g t h e dynamic p r o p e r t i e s
of journal bearing have been performed by using t h e mechanical PRBS e x c i t e r . Good agreement between the measured and t h e
c a l c u l a t e d value of t h e s t a t i c and dynamic p r o p e r t i e s h a s been shown. E i t h e r
c a l c u l a t i o n o r experiment j u s t i f i e s t h a t t h e dynamic c o e f f i c i e n t s of t i l t i n g pad b e a r i n g vary v e r s u s the e x c i t i n g f r e q u e n c y . I t i s demonstrated i n p r a c t i c e t h a t t h e
system under t e s t i n g can be e x c i t e d
v a l i d l y w i t h i n a wide frequency range by means of mechanical PRBS e x c i t e r .
1 INTRODUCTION PRBS (pseudo random binary sequence) s i g n a l i s
and only t h e responses o f t h o s e f r e q u e n c i e s can
widely used i n i d e n t i f y i n g t h e unknown parameters
be o b t a i n e d ( 3 ) .
of a system because of i t s good s t a t i s t i c proper-
( 3 ) Under background of low s i g n a l n o i s e
t i e s , and t h e p o s s i b i l i t y o f reducing t h e n o i s e
r a t i o , by means of t h i s method, t h e i d e n t i f i c a -
to the least level.
t i o n of t h e parameters can be much more p r e c i s e .
I n r e c e n t y e a r s , PRBS has been a p p l i e d i n
The method f o r i d e n t i f y i n g t h e dynamic
estimating t h e c o e f f i c i e n t s of b e a r i n g (1,2) a l s o .
c h a r a c t e r i s t i c s of b e a r i n g i n terms of pseudo
In 1984, The Tribology Laboratory of Tsinghua
random code i s as f o l l o w s :
University i d e n t i f i e d t h e dynamic performance of
PRBS e x c i t i n g f o r c e s a r e a p p l i e d t o t h e
e l l i p t i c a l and t i l t i n g pad b e a r i n g by means o f
t r i a l system, t h e f o r c e s and t h e responses a r e
pseudo random code. I n t h i s paper, t h e t h e o r e t i -
measured simultaneously. Through power spectrum
c a l and experimental r e s u l t s of t h e s t a t i c and
analysis, the
dynamic p r o p e r t i e s of t i l t i n g pad p e a r i n g are
can be o b t a i n e d , and by computer processing o f
introduced.
t h e s e f u n c t i o n s c h a r a c t e r i s t i c s of t h e bearing
The advantages f o r e s t i m a t i n g t h e dynamic
t r a n s f e r f u n c t i o n s of t h e system
can be determined.
c o e f f i c i e n t s of t i l t i n g pad b e a r i n g by PRBS a r e a s follows:
(1) W i t h only a small e x c i t i n g amplitude,
1.1
Notation
much more energy can be generated, s o t h a t t h e normal o p e r a t i o n o f t h e system w i l l not be i n f l u -
magnitude of small d i s t u r b a n c e
enced, and obviously t h i s i s very s u i t a b l e f o r
r a d i a l clearance
on-line measurement.
damping c o e f f i c i e n t s
(2)
The dynamic c o e f f i c i e n t s of t i l t i n g
diameter of b e a r i n g
pad bearing are varying with t h e v a r i a t i o n of
incremental component of f i l m f o r c e
t h e e x c i t i n g frequency. I n terms of wide band
gain of t r a n s f e r function
e x c i t i n g s i g n a l such as t h a t of t h e PRBS, t h e
i n e r t i a moment of pad about p i v o t
t r a n s f e r f u n c t i o n of a system can be obtained
stiffness coefficients
a t once. However, t r a d i t i o n a l l y , t h e e x c i t i n g
width of b e a r i n g
s i g n a l i s imposed a t one o r two f r e q u e n c i e s ,
mass o f t r i a l b e a r i n g block
488 AM
moment of t h e incremental p r e s s u r e
AP
increment of p r e s s u r e
Px,P
exciting force
Y
bearing a r e
Gb+
CXxk
+
C XY
9 +
k + C 9 YY
Kxxx
+ K
+
K XY
y = p (1)
x + K y = p YX YY Y
R
r a d i u s of bearing
XYY
r e l a t i v e displacement between j o u r n a l
I n terms of Laplace t r a n s f o r m a t i o n , t h e s e equa-
and t r i a l b e a r i n g
t i o n s can be w r i t t e n as:
a b s o l u t e displacement of t r i a l b e a r i n g
Xb "b x,y
coordinates
W
s t a t i c load
6
o s c i l l a t o r y angle
A&
increment of o s c i l l a t o r y a n g l e angular frequency of small d i s t u r b a n c e
W
angular frequency of j o u r n a l
$
phase of t r a n s f e r f u n c t i o n
n
dynamic v i s c o s i t y
E
eccentricity r a t i o
e
e s t i m a t i v e parameter
(2) Let
H =
+K Y Y Y Y
SC
We have
Superscripts
-
-
..
dimensionless v a l u e s
(3)
time d e r i v a t i v e
The t r a n s f o r m a t i o n block-diagram i s shown i n
second degree time d e r i v a t i v e
Fig.2. Through Fourier transformation of equations (1) and r e p l a c e t h e corresponding terms with
2.
RELATIONSHIP WITHIN THE TRANSFER FUNCTIONS
t r a n s f e r f u n c t i o n s , and i f t h e e x c i t i n g f o r c e
AND THE IDENTIFICATION OF DYNAMIC CHARACTERIS-
a p p l i e d f i r s t i n X d i r e c t i o n , then i n Y we have
T I C S OF A ROTOR-BEARING SYSTEM
t h e following e q u a t i o n s :
Fig. 1 i s t h e sketch of a t r i a l bearing system. o,o'
a r e t h e c e n t e r s of t h e r o t o r and t r i a l bear-
i n g , xb,yb a r e t h e a b s o l u t e displacements o f t h e t r i a l b e a r i n g , x,y are t h e r e l a t i v e d i s p l a c e ments between t h e r o t o r and t h e t r i a l b e a r i n g , px,py a r e t h e e x c i t i n g f o r c e s on t h e t r i a l bearing i n X and Y d i r e c t i o n s r e s p e c t i v e l y , and m
i s t h e mass of t h e t r i a l b e a r i n g block.
ty
t r i a l bearing
Here G(jw)=G(Cos$+jSin$)
By e q u a l i z i n g t h e r e a l and imaginary p a r t s on both s i d e s of t h e s e e q u a t i o n s , t h e following s t a t i c load
m a t r i x e q u a t i o n s can be o b t a i n e d , i . e .
e1=w-1z1 A
e2=w-1z2 A
Fig. 1 The sketch of t h e t r i a l b e a r i n g system i n which The d i f f e r e n t i a l e q u a t i o n s of t h e t r i a l
(4)
489
s e n t t h e accelerate responses o f t h e t r i a l beari n g , and t h e second s u b s c r i p t such a s x of G ux represents exciting force i n X, Y or y of G UY’
direction respectively. By s u b s t i t u t i n g t h e g a i n s and phases obtained experimentally f o r t h e t r a n s f e r f u n c t i o n s a t each frequency i n t o equation (41, t h e estimated parameters 81 and 0 2 can be i d e n t i f i e d .
3
THEORETICAL CALCULATION FOR THE STATIC AND DYNAMIC CHARACTERISTICS OF TILTING PAD
BEARING
Fig. 2
Transformation block-diagram
By s o l v i n g two dimensional Reynolds e q u a t i o n , two
el=ccxx
cxy
e,=cc
c
dimensional
energy e q u a t i o n ,
tempera-
Kxx
Kxy IT
t u r e - v i s c o s i t y e q u a t i o n , thermal d i s t o r t i o n
K
K IT YY
taneously and i t e r a t i v e l y , t h e s t a t i c p r o p e r t i e s
Sin$
Reynolds e q u a t i o n , Coleman method ( 4 ) i s adopted,
equation and e l a s t i c deformation equation simul-
-
YY
YX
-wG
xx
YX
Sin$ xx
-wG
wGxxCos $ xx
w =
-wG
Sing
-wG
XY
XY
-
wG
Cos$
wG XY
YX
YX
YX
wG
G
YY
yx
between temperature and flow of t i l t i n g pad bear-
YX
i n g , E t t l e ’ s groove mixing t h e o r y ( 5 ) h a s been a p p l i e d . However, some improvements h a s been made on them.
Cos$ YY
Through combination of t h e Reynolds equa-
-
Cosb YX
YX
t i o n and t h e o s c i l l a t i n g equation o f t h e pad, t h e s o l u t i o n of t h e dynamic c h a r a c t e r i s t i c s of t h e t i l t i n g pad b e a r i n g can be determined. The
Sin$
G
GxxSinOxx
and f o r t h e establishment of t h e r e l a t i o n s h i p
Cos$
Sin$ YY YY
XY
GxxCos$xx
of t h e b e a r i n g can be determined. I n s o l v i n g
YX
process w i l l be described i n t h e following. Cos$
G
G Cos$ YY YY
XY
XY
Sin$
G XY
I-mG
G
YY
XY
Sin$ YY
A t f i r s t , t h e r o t o r i s e x c i t e d with
a
small harmonic d i s t u r b a n c e h o r i z o n t a l l y , i . e .
-
a Sinw t
Ax
1
CosQ
ux
Assuming t h a t t h e incremental components of t h e f i l m f o r c e a r e varying synchronously i n
Z = 1
UY
UY
UY
Sin$ uy
-mG
-
-mGnCos$vx -mG
z* =
I-mG
-
frequency , t h e n
Cos$
-mG
-mG
vx
’
Sin$= According t o t h e d e f i n i t i o n of t h e f i l m Cos$
dynamic c h a r a c t e r i s t i c c o f f i c i e n t s ,
v y v y
vy
Sin$
vy-
Here G s r e p r e s e n t g a i n s and $ s r e p r e s e n t phases,
a l l i n f u n c t i o n of f r e q u e n c i e s . The first subs c r i p t such a s x of G
XY
or y of $
YX’
represents
t h e r e l a t i v e displacement r e s p o n s e s ; u , v r e p r e -
aKxx
=[aKYx
aw
c
awt Cz:,
Sinwlt [Coswlt]
490 Thus, t h e f o u r dynamic c h a r a c t e r i s t i c
coef-
4
EXPERIMENTAL APPARATUS AND PROCEDURE
f i c i e n t s are (1) AFxx a , Kxx =
The e x p e r i m e n t a l b e a r i n g i s shown i n F i g . 3 ,
(2) AFxx
-
cxx =
aw.’
and t h e s k e t c h o f t h e t e s t r i g i s g i v e n i n F i g . 4 .
Where, t h e f i r s t s u b s c r i p t such as x o f K y of C
or xx represents t h e direction of t h e force
YX
and t h e second s u b s c r i p t r e p r e s e n t s t h e d i r e c t i o n of displacement o r v e l o c i t y . S i m i l a r l y , when a small harmonic d i s t u r b a n c e
i s imposed v e r t i c a l l y , t h e o t h e r f o u r c o e f f i c i e n t s can be determined. I n t h i s way, t h e problem o f t h e c a l c u l a t i o n
for t h e e i g h t c o e f f i c i e n t s can be reduced t o t h a t o f f i n d i n g t h e a p p r o p r i a t e i n c r e m e n t a l components o f t h e f i l m f o r c e such a s AF
xx
( i1
’
AFyx
( i, ( i = 1 , 2 )
Fig. 3
The e x p e r i m e n t a l b e a r i n g
etc. The i n c r e m e n t a l components of t h e f i l m force exciting force Y
c a n be d e t e r m i n e d a s follows. While t h e r o t o r i s e x c i t e d , t h e p r e s s u r e p
and t h e o s c i l l a t o r y a n g l e 6 of t h e pad w i l l b e varying. Therefore, i n c a l c u l a t i n g t h e incremental p a r t s o f t h e p r e s s u r e Ap, t h e a p p r o p r i a t e i n c r e ments of t h e o s c i l l a t o r y a n g l e A6 must be c a l c u l a t e d , i . e , t h e Reynolds e q u a t i o n and t h e e q u a t i o n
exciting force
of t h e pad motion must b e s o l v e d s i m u l t a n e o u s l y .
x
1
W
s t a t i c load
From Reynolds e q u a t i o n , t h e d i m e n s i o n l e s s i n c r e m e n t s o f p r e s s u r e are:
From t h e e q u a t i o n o f t h e pad m o t i o n , t h e
I-motor 2-transmission coupling 4-support t r i a l bearing 6-- s t r a i n gage displacement sensor accelerometer 9velometer
35-78-
i n c r e m e n t s of o s c i l l a t o r y a n g l e are: Fig. 4
Sketch o f t h e t e s t r i g
The r o t o r w i t h a f l o a t i n g t r i a l b e a r i n g on
Where AM 1s t h e moment of t h e i n c r e m e n t a l p r e s s u r e , and
3,
t h e moment of t h e pad i n e r t i a , botli
and
c a l l y . Upward s t a t i c l o a d i s a p p l i e d t o t h e t r i a l b e a r i n g by p r e s s u r i z e d a i r t h r o u g h a f l e x i b l e
o f them are d i m e n s i o n l e s s . I t e r a t i o n of
i t s c e n t r e i s s u p p o r t e d by two b e a r i n g s symmetri-
h a s been c a r r i e d
waved p i p e . Dynamic e x c i t i n g f o r c e s g e n e r a t e d by
on u n t i l t h e mean r e l a t i v e error o f Ap(i) i s l e s s
t h e mechanical PRBS e x c i t e r are a p p l i e d t o t h e
t h e n 0.001.
t r i a l b e a r i n g through s t e e l c a b l e s h o r i z o n t a l l y
By i n t e g r a t i n g Ap(i) for e a c h p a d ,
t h e h o r i z o n t a l and v e r t i c a l components c a n be
and v i r t i c a l l y . The c a b l e s must be p r e t e n s i o n e d .
d e t e r m i n e d , and t h e sum o f t h e forces on a l l t h e
Dynamic forces a r e measured by s t r a i n g a g e s , and
p a d s becomes t h e i n c r e m e n t a l c o n p o n e n t s of t h e
r e l a t i v e d i s p l a c e m e n t s between t h e t r i a l b e a r i n g
f i l m force.
and t h e r o t o r are measured by f o u r eddy c u r r e n t proximity s e n s o r s mounted i n X and Y d i r e c t i o n s
491 a t both ends of t h e bearing. Two accelerometers
lowest shoe, i n which t h e temperature r i s e i s
are used t o measure t h e a b s o l u t e a c c e l e r a t i o n of
most s e v e r e , but t h e two shoes l o c a t e d a t o p were
t h e t r i a l bearing.
unloaded.
On t h e mechanical -PRBS e x c i t e r , r e p e a t e d t e e t h of PRBS t y p e are arranged around t h e c i r c l e
Fig. 6 shows good c o r r e l a t i o n between l o c u s of t h e
measured and c a l c u l a t e d a x i s
of a metal d i s k . When t h e d i s k r o t a t e s uniformly, t h e PRBS e x c i t i n g f o r c e can be o b t a i n e d by means
(atm)
2
of orthogonal c u t t i n g . Through t h e v a r i a t i o n of
3
4
5
t h e width and depth of c u t t i n g , t h e magnitude of t h e e x c i t i n g f o r c e can be a d j u s t e d . S t a t i c experiment i s performed under c e r t a i n r o t a t i n g speed with d i f f e r e n t l o a d s , and t h e s t a t i c equilibrium curve of t h e j o u r n a l i s p l o t -
(OC
pressure d i s t r i b u t i o n
1
ted by a x-y p l o t t e r . Dynamic experiment is performed under each equilibrium p o s i t i o n of t h e j o u r n a l . A t f i r s t , PRBS e x c i t i n g f o r c e i n X d i r e c t i o n i s a p p l i e d t o t h e t r i a l b e a r i n g , and t h e f o r c e and responses
Temperature d i s t r i b u t i o n
of displacement and a c c e l e r a t i o n both i n X and Y d i r e c t i o n are measured simultaneously. Then
P e r i p h e r i a l p r e s s u r e and temperature
Fig. 5
f o r c e i s a p p l i e d i n Y d i r e c t i o n and a l l t h e
d i s t r i b u t i o n on t h e f i v e shoes of
f o r c e and responses measured s i m i l a r l y a s above
t h e bearing
mentioned. A l l t h e measured s i g n a l s are r e g i s t e r ed by a 14 channel magnetic r e c o r d e r and subseque n t l y t r a n s m i t t e d t o a CF-500 s i g n a l processor.
t i l t i n g pad bearing. On account of t h e p i v o t
Through FFT(Fast F o u r i e r Transformation) and
d e v i a t i o n and t h e mass of pads an a t t i t u d e angle
power spectrum a n a l y s i s , t h e e i g h t t r a n s f e r
of t h e t i l t i n g pad b e a r i n g h a s occurred.
f u n c t i o n s r e l a t i n g t h e e x c i t i n g f o r c e s and t h e
I n t i m e r e g i o n , t h e e x c i t i n g f o r c e and
responses of t h e b e a r i n g can be determined. The
i t ’ s response of displacement, both i n Y d i r e c -
constant component of t h e recorded s i g n a l s w a s
t i o n , a r e given i n Fig. 7. It i s obviously t h a t
f i l t e r e d o u t , and i n o r d e r t o avoid a l i a s t h o s e
t h e e x c i t i n g f o r c e curve i s similar t o t h a t
frequencies above 200 Hz were a l s o f i l t e r e d o u t .
of t h e PRBS s i g n a l and t h e response follows it
F i n a l l y , by s u b s t i t u t i n g t h e corresponding gains and phases of t h e t r a n s f e r f u n c t i o n s a t each frequency i n t o matrix’s equation ( 4 ) , t h e dynamic c h a r a c t e r i s t i c c o e f f i c i e n t s of t h e t r i a l
RESULTS
A
program
for c l a c u l a t i n g t h e
0
co
0
bearing can be determined.
5
I
co 0
s t a t i c and
dynamic c h a r a c t e r i s t i c s of t i l t i n g pad bearing under varying temperature f i e l d h a s been developed. Two t i l t i n g pad b e a r i n g s with a diameter of 100 mm and f i v e pads were t e s t e d and c a l c u l a t e d .
---- c a l c u l a t e d
locus measured l o c u s
Fig.5 shows p e r i p h e r i a l p r e s s u r e and t e m perature d i s t r i b u t i o n on t h e f i v e shoes of t h e
Fig. 6
The measured and c a l c u l a t e d a x i s
bearing a t t h e i r middle s e c t i o n s . Obviously
l o c u s o f t h e t i l t i n g pad bearing
t h e major p a r t o f t h e l o a d i s supported by t h e
for C/R=.00135
492
CH-A+B
-
A:
1
B:
2V 1 0 0 H z
100
I
0
I
0/64
FREQ:
.
+180° $
:
T R A N S F E R FUNC
2 0 0 Hz A:.5V
B:.5V
-L
0°'
-180O"
Y'
time
2 0 . .
-
200
w (Hz)
Fig. 9
The e x c i t i n g f o r c e and i t ' s
Fig. 7
The t r a n s f e r f u n c t i o n between e x c i t i n g f o r c e and corresponding
response of displacement
displacement very well. Fig.8 shows t h e a u t o c o r r e l a t i o n funct i o n of PRBS e x c i t i n g f o r c e . A t t h e p o i n t o f mul-
abscissa i s a l i n e a r scale of t h e a n g u l a r I n t h e f i g u r e , up t o t h e
t i p l e p e r i o d of pseudo random code t h e r e appears
frequency w ( H z ) .
an impulse sequence, which i s c o n s i s t e n t with
frequency of 120 H z t h e coherence f u n c t i o n i s
t h e c h a r a c t e r i s t i c s of PRBS.
normally g r e a t e r t h a n 0 . 8 . This means t h a t
The Bode diagram of t r a n s f e r f u n c t i o n betwe-
t h e t r a n s f e r f u n c t i o n i n Fig.9 i s r e l i a b l e . In
en e x c i t i n g f o r c e and corresponding displacement
frequency over 130 Hz, t h e s i g n a l h a s been a t -
on an average of 64 times i s shown i n Fig.9.
t e n u a t e d t o t h e magnitude o r d e r of n o i s e and
In
t h e low frequency r e g i o n , t h e g a i n approaches
can be omitted s i n c e it i s beyond t h e scope of
t o a c o n s t a n t w i t h a phase a n g l e approximates
t h e desired region.
t o Oo; t h e r e o c c u r s a peak of g a i n between 40-80
Hz, meanwhile t h e phase a n g l e has an
a l t e r n a t i o n ; and i n t h e high frequency r e g i o n ,
It should be noted t h a t i n both t h e t r a n s f e r f u n c t i o n and i t ' s coherence f u n c t i o n , t h e r e appear some p i n d i s t o r t i o n s a t c e r t a i n f r e -
t h e g a i n is reducing and t h e phase i s l a g g i n g g r a d u a l l y . Fig.10 shows t h e coherence f u n c t i o n
.5V
200
S
64
B:
.5V
200Hz
S
64/64
1
CH-A:
0.75
AUTO C O R R E L A T I O N
2
Y (w) 0.5
0.25
0 0
time
M
L
I td
200
100 w
Fig.10
J
A:
(Hz)
The coherence f u n c t i o n between e x c i t i n g f o r c e and correspondi n g displacement
Fig. 8
Auto c o r r e l a t i o n f u n c t i o n of PRBS e x c i t i n g f o r c e
quencies. They a r e induced from e x t e r n a l n o i s e and are r e s p e c t i v e l y corresponding t o :
which i s corresponding t o t h e above mentioned
( 1 ) Rotating frequency of f a n motor 47.5 H z ;
t r a n s f e r f u n c t i o n . The o r d i n a t e i s a l i n e a r
( 2 ) Rotating frequency of r o t o r 75 Hz;
scale of t h e coherence f u n c t i o n y L ( w ) , and t h e
( 3 ) P e r t u r b a t i o n of main power 50 Hz and
493 t h e s t a t i c and dynamic p r o p e r t i e s f o r t i l t i n g
i t ' s double-frequency 1 0 0 Hz. Fig. 11,12 show t h e t h e o r e t i c a l c a r v e s and
pad b e a r i n g . ( 2 ) Since t h e s t i f f n e s s and damping coeff-
experimental d a t a p o i n t s f o r t h e dimensionless s t i f f n e s s and damping c o e f f i c i e n t s v e r s u s dimen-
i c i e n t s of t i l t i n g pad b e a r i n g a r e varying
s i o n l e s s l o a d c a r r y i n g c a p a c i t y o f C/R=.00135
with t h e v a r i a t i o n o f e x c i t i n g frequency, t h e r e -
bearing. I n comparison with t h e i s o t h e r m a l
f o r e , while t h e c r i t i c a l speed of i n s t a b i l i t y
c a l c u l a t e d curves, t h e c a l c u l a t e d c u r v e s with
o r damping of a system i s being t r e a t e d when
v a r i a b l e temperature are i n b e t t e r approach t o
t h e r o t o r i s supported on t i l t i n g pad b e a r i n g ,
t h e experimental r e s u l t s .
t h e i n f l u e n c e of frequency v a r i a t i o n must be considered. ( 3 ) It i s r a t h e r s u c c e s s f u l t o i d e n t i f y t h e dynamic c h a r a c t e r i s t i c c o e f f i c i e n t s of
3.0
2.0
*
b e a r i n g by means of mechanical PRBS e x c i t e r .
Kxx -
:yy -t cxx O F
The main advantages can be l i s t e d as simple t r i a l data
assembly, higher e x c i t i n g energy, h i g h e r r a t i o of s i g n a l t o n o i s e and s h o r t e r experimental
YY
t i m e . I n a d d i t i o n , system under t e s t i n g within
1.0
a wide frequency range can be e x c i t e d v a l i d l y by means of t h i s device and i t s a p p l i c a t i o n
0.5
i n r e a l - t i m e monitoring i s p r o s p e c t i v e .
0.2
7
0.3
ACKNOWLEDGEMENT
The a u t h o r s would l i k e t o thank Associate F'rofessor L i j i Chen o f T s b g h u a University f o r h i s h e l p f u l s u g g e s t i o n s during t h e course of
-
v a r i a b l e temperature c a l c u l a t e d curves
----
isothermal c a l c u l a t e d curves
t h i s work.
REFERENCES DOGAN, I. U., BURDESS, J. S. and HEWIT, J. R.
Fig. 11 The dimensionless d i r e c t s t i f f -
' I d e n t i f i c a t i o n of j o u r n a l bearing c o e f f i -
n e s s and damping c o e f f i c i e n t s
c i e n t s u s i n g a pseudo random binary
versus load carrying capacity
seqence'
, I. Mech. E. 1980,4,277-
BURROWS,C.R.
f o r C/R=.00135
281.
and SAHINKAYA, M.N.'Frequency-
domain e s t i m a t i o n of l i n e a r i z e d o i l - f i l m The t h e o r e t i c a l and experimental c u r v e s of dimensionless s t i f f n e s s and damping c o e f f i c i e n t s v e r s u s t h e r a t i o o f frequency wl/w
f o r C/R=.00215
bearing a r e p l o t t e d i n Fig. 13. It should b e poted t h a t e i t h e r i n measured o r c a l c u l a t e d
c o e f f i c i e n t s ' , J. o f Lub, Tech. ,1982,104,210-215.
d&*a'4.4 wwdim&*g#4&*t jM3t
1975, 20, N0.3,191-197.
COLEMAN, R.
'
The numerical
s o l u t i o n of
l i n e a r e l l i p t i c equations' ,Trans. Am. SOC.
r e s u l t s , t h e dynamic c o e f f i c i e n t s are varying
Mech. Engrs. 1968, Ser. F, 90, 773- 776.
with t h e v a r i a t i o n o f t h e e x c i t i n g frequency.
ETTLES, C.M.McC.
'The a n a l y s i s and perform-
ance of pivoted pad j o u r n a l b e a r i n g conside r i n g thermal and e l a s t i c e f f e c t s ' , J . of 6
CONCLUSIONS
(1) There i s good agreement between t h e t h e o r e t i c a l and experimental r e s u l t s o f both
Lub. Tech.
, 1980,102,
183- 192.
494
c
0.4
A K - XY
0
0.3
'
Kyx
+ cxy o c
trial data
YX
0.2
0.1
0.0
0.1
0
0.2
0.3
LDwn
----
v a r i a b l e temperature calculated curves isothermal calculated curves,
Fig.12 The d i m e n s i o n l e s s c r o s s s t i f f n e s s and damping c o e f f i c i e n t s v e r s u s l o a d c a r r y i n g c a p a c i t y f o r C/R = .00135
R,E
0.511 -
1.0
-
_-- -- - -. YY
c J-
A-
9
-- -
0.2
0.1
Kxx
t
I
0.25
-
----
0.50
1
1
0.75
1.00
1
1.25
calculated curves measured c u r v e s
I
1.51 w1 /w
Fig.13 The d i m e n s i o n l e s s s t i f f n e s s and damping coefficients versus t h e r a t i o of frequency w / w 1
f o r C/R=.00215
SESSION XVI OIL FILM INSTABILITY Chairman: Professor B.O. Jacobson
PAPER XVl(i)
Instability of oil film in high-speed non-contact seal
PAPER XVl(ii)
Instability of the oil-air boundary in radial-groove bearings
PAPER XVl(iii) An experimental study of oil-air interface instability in a grooved rectangular pad thrust bearing
This Page Intentionally Left Blank
497
Paper XVI(i)
Instability of oil film in high-speed non-contact seal M. Tanaka and Y. Hori
This paper studies the malfunction of smooth bore, floating ring oil film seal for the shaft sealing of high-speed turbocompressor. The seal prevents compressed gas from leaking out by means of sealing fluid of oil supplied in the seal clearance at a pressure a little higher than the gas pressure. When the operating speed is increased, however, the gas is found to leak through the seal clearance into the atmosphere. The hydrodynamic lubrication theory is applied to analyse the problem. In experiments the oil flow in the seal clearance is visualised and the behaviour of the gas-oil interface is observed. The theory agrees well with measurements both quantitatively and qualitatively. 1
INTRODUCTION
z
C Fluid film seals are widely used for the shaft sealing of high-speed turbocompressors where the discharge pressures are as high as some tens or hundreds MPa and also the operating shaft speeds sometimes exceed 50 m/s. Fig. 1 shows schematic of the fluid film seal. The seal assembly consists of two separate rings, that is, the inner one at the gas side and the outer one at the atmosphere side. The radial clearances between each of the two rings and the journal are usually as small as that of journal bearing. Oil is fed to the space between the two rings at the pressure a little higher than that of the compressed gas. Therefore, due to the pressure differential, very small part of the oil flows into the compressor housing through the annular space under the gas side ring and has to be drained off. Most part of oil flows to the atmosphere under the outer ring and is returned to oil tank for reuse. The seal rings are prevented from rotating by an anti-rotation pin but are intended to move radially to keep concentric with the journal whose centre shifts its position with the variation of operating condition, for example, the shaft speed. With the increase in shaft speed, the oil leakage on the gas side starts decreasing and the gas is sometimes found to leak out to the oil side of the seal and eventually to the atmosphere. This report investigates this malfunction of the fluid film seal theoretically and experimentally. Attention is paid to the behaviour of the gas-oil interface in the clearance of the inner seal ring. The interface is observed to shift its position in the circumference, depending on the operating conditions and the seal dimensions. The predicted shape of the gasinvaded region is compared with measurements. 1.1
h ho p r u
w
Notation film thickness recess of seal at oil end pressure in the oil film radial coordinate circumferential velocity component of oil film axial velocity component of oil film
L N pg
PO
Pr
AP R
Rj Za E
n 0 81 P w
axial coordinate mean radial clearance of seal seal length shaft speed, rpm gas pressure oil supply pressure corrected pressure at seal end pressure differential, Po - p g seal radius journal radius axial coordinate of gas-oil interface eccentricity ratio of journal oil viscosity circumferential coordinate angle from minimum film thickness oil density angular velocity of journal
Oil, Po
Atmosphere I
.-
Drain Fig. 1
To Tenk
Schematic of Fluid Film Seal
498 2 TEST RIG AND PRELIMINARY EXPERIMENT Fig. 2 shows schematic of the test rig used in the experiments. A straight bore seal(6) of transparent plastics is supported on a horizontal journal(7) of steel driven by an electric motor. The nominal diameter 2R and the length L of the test seal are 150mm and 21mm respectively. The mean radial clearance C is 0.201mm. Oil film pressure is measured at equally-spaced six points in the midplane of the seal. The seal can be fixed at an arbitrarily eccentric position with respect to the journal centre. The eccentricity ratio can be calculated from measurements by noncontact gap sensors(4) mounted on the seal holder ( 5 ) . Cas(air) is supplied from the port(1) and oil is fed from the port(8). Po, oil pressure, is set to be a little higher than Pg, air supply pressure. The -gas-oil interface was observed to move into the seal clearance when the incresing shaft speed exceeded a certain limit. The air-invaded region kept extending axilally and circumferentially with the shaft speed and eventually the air burst out when the interface reached the oil end of the seal.
Fig. 2
Schematic of Test Rig
3 THEORY The following laminar Reynolds equation is assumed to be applicable to the steady-state oil film in the seal clearance.
The velocity components of oil film, u and w, are given as follows:
(3)
i a w =2n
az
Fig. 3 Gas-Oil Interface
(r-R) (r-R+h)
A tangent of the interface curve on the lubricating plane is assumed to coincide with the resultant velocity vector of u and w at the point of contact(Fig. 3).
To Pressure Gauge
ItI
Surface Both equations(1) and (4) are solved simultaneously by means of numerical methods. The pressure on the gas side of the interface is assumed to be P and that at the oil end of the g seal is assumed to be Pr given as follows:
=
3 mm
Journal Surface Fig. &
Oil supply pressure Po is adjusted to be constant by reference to the pressure measured at the recessed surface of the seal 3mm distant from the journal surface(Fig. 4). With journal rotation, the circumferential Couette flow of oil is assumed to take place in the annular space at
ho
Pressure Tap Position for Pg
the oil end of the seal and Equation(5) is derived with the centrifugal effect being considered.
499
4 RESULTS AND DISCUSSIONS 5 shows t h e p r e d i c t e d and measured p r o f i l e s t h e g a s - o i l i n t e r f a c e on t h e s e a l s u r f a c e f o r c a s e of E = 0.5 and AP = 0.005 MPa. The gasi n t e r f a c e g e t s i n t o t h e seal c l e a r a n c e from a l i t t l e beyond t h e minimum f i l m t h i c k n e s s p o s i t i o n and i t s a x i a l d e p t h Za i n c r e a s e s i n t h e d i r e c t i o n of she.ft r o t a t i o n . After c i i l m i n a t i n y , t h e i n t e r face retreats t o t h e g a s end o f t h e s e a l g r a d u a l l y and Z, d e c r e a s e s t o n a u g h t e v e n t u a l l y . Both t h e d e p t h and t h e a n g u l a r e x t e n t o f t h e invaded r e g i o n i n c r e a s e w i t h s h a f t s p e e d and i t i s noted t h a t t h e r e g i o n extends i t s e l f i n t o t h e c o n v e r g i n g f i l m r e g i o n ( 8 , > 180). Measurements are i n good agreement w i t h t h e t h e o r e t i c a l p r e d i c t i o n s c a l c u l a t e d w i t h t h e v a l u e o f w/u a t t h e seal s u r f a c e i n E q u a t i o n ( 5 ) . F i g . 6 shows t h e similar p r o f i l e s f o r t h e c a s e o f E = 0.3 and AP = 0.010 Mpa. The invaded region reduces with t h e decrease i n E o r t h e i n c r e a s e i n AP. P r e d i c t i o n s a r e i n good a g r e e ment w i t h measurements w i t h t h e e x c e p t i o n o f t h e c a s e f o r N = 3000 rpm which w i l l be d i s c u s s e d later. F i g . 7 shows t h e v a r i a t i o n o f t h e maximum value o f t h e dimensionless a x i a l depth of gas i n v a s i o n w i t h s h a f t s p e e d f o r t h e c a s e o f AP = 0.005 MPa. The l a r g e r t h e e c c e n t r i c i t y r a t i o , t h e l o w e r t h e s h a f t speed a t which t h e g a s i n v a s i o n starts. However b o t h t h e t h e o r y and t h e measurements show t h a t t h e i n t e r f a c e r e a c h e s t h e o i l end o f t h e seal a t t h e same s h a f t speed (Nb), r e g a r d l e s s o f E , though t h e i n v a s i o n s t a r t s a t d i f f e r e n t speeds. Predictions are in good agreement w i t h measurements. The c a l c u l a t i o n o f t h e i n t e r f a c e p r o f i l e t u r n e d t o be u n s t a b l e w i t h t h e i n c r e a s e i n s h a f t speed, t h a t i s , with t h e d e c r e a s e i n Pr. T h e r e f o r e , t h e t h e o r e t i c a l NbS were assumed t o be t h e s h a f t s p e e d s which made Pr i n E q u a t i o n ( 5 ) e q u a l t o Pg. Fig. of the oil
(b)
Idegl
9, (C) Fig. 5 (E =
Interface Profile 0.5, AP = 0.005 ma)
--exp'
0
l
i
I
J 360 1M
7- exp.
l $ N
1M
0 9, Fig. 6 (E
tC)
353 Idegl
Interface Profile = 0.3, AP = 0.010 MPa)
Fig. 7
Maximum Depth o f I n v a s i o n w i t h N (AP = 0.005 MPa)
500
Fig. 8 shows the similar data for the case of AP = 0.015 MPa. Compared with the case shown in Fig. 7, the gas invasion proceeds more gradually with N and reaches the oil end at a speed higher than that for the case in Fig. 7, due to higher pressure differential. However the measured Nb is lower than predicted. Presumably this is due to the reduce of the clearance caused mainly by the inward dilation of the plastic seal which has a low thermal conductivity. This presumtion is supported by the fact that the measured pressure profile fluctuates with 0 more violently than predicted at N = 3000 rpm while predictions agree well with measurements at N = 1000 rpm (Fig. 9 ) . Frictional heat generated in the oil film is known to increase with the second power of shaft speed. Therefore the real clearance at N = 3000 rpm can be much smaller than that at N = 1000 rpm. 5
BC=O. I
CONCLUDING REMARKS
The malfunction of smooth bore, floating ring oil film seal was investigated theoretically and experimentally. The mechanism of gas leakage to the oil side in the test rig can be explained by the fact that the positive pressure differential AP decreases with shaft speed due to the centrifugal effect of Couette oil flow at the oil end of the seal. Gas invasion into the seal clearance is enhanced by the undesirable eccentricity of journal which lowers the bottom pressure in the oil film. The causes of decreasing the oil pressure in actual machines have not been identified yet but further investigation on some suspects are under way. The end face of actual floating seal ring forms a face seal which contacts the mating surface of the stationary seal housing. As the machine speed increases the axial load on the ring due to pressure unbalance increases, which in turn the Coulomb friction at the end face increases too. Presumably the inability to keep floating rings concentric with journal results from the increasing friction which may cause the ring lock up.
6 ACKNOWLEDGEMENT The authors gratefully acknowledge the assistance of Mitsui Engineering & Shipbuilding Company, Ltd and Eagle Industry Company, Ltd in supporting part of this research.
Fig. 8 Maximum Depth of Invasion with PI (AP = 0.015 MPa)
0.35
n (P
n
auo n
exQ
- cal.
N =1000rpm
0.30
tJt I t
N=3003rpm
i0 Fig. 9 (E
Pressure Profile at Midplane = 0.3, AP = 0.015 MPa)
501
Paper XVl(ii)
Instability of the oil-air boundary in radial-groove bearings A. Leeuwestein
ABSTRACT Radial-groove bearings seal themselves against loss of oil. At h g h slider speed however streaks of oil are continuously separated from the full oil film. The mechanism of this fenomenon is explained and it is shown how this fenomenon can be calculated. For this purpose the film gap height has been taken in the form of a constant plus a sinus. I
slider speed is raised enough, streaks of oil will continuously be separated from the full oil film (see figure 2). These streaks of oil will be attached to the smooth slider surface. According to ref.(l) the full oil film breaks down when
INTRODUCTION
In ref.(l) Bootsma and Tielemans have described the breakdown of the full oil film they observed in their experiments with radial-groove bearings. The boundary that separates the region where the oil fills the film gap height completely and the region where the oil does not, will be the free boundary for the Reynolds equation. If the slider speed of a radial-groove bearing is raised, the form of the free boundary will change strongly (see figure I). If the
In the present paper an explanation for the change of the form of the free boundary and the loss of oil from the full oil film via the smooth slider surface is given.
.. .. .. u Figure 1
u
u
4
+
In the shaded area air pushes oside the oiL on the grooved bearing surface. On the smooth bearing surfoce the o i L is stiLL up
to the dotted Line.
The speed u of the smooth beoring surface increases from Left to right. Figure 2 At high d i d o r speed u streaks of oil are separated from the f u L L fiLm. Shaded area is air.
" Ca
1.1
Notation
P a
h,@I /
see figure 3.
U
slider speed in
P
viscosity of the oil in E !L
T
surface tension of the oil in
R, r,
m
Y
J
m2
N x.
K
a,bJ
capillarity number . N pressure in -. in' groove angle defined in fig. 3. defined in figure 3. disturbance of the free boundary. Jacobian mapping from 11 to extra oil stream across the free boundary. the highest real part of the eigenvalues are defined in eq.(6)
502 2
THE FREE BOUNDARY
On the free boundary we have a Dirichlet boundary condition and a boundary condition for the derivative of the pressure as well. Take for the moment the pressure equal to zero on the free boundary and take as boundary condition for the derivative of the pressure the condition 'no oil flow across the free boundary'
Consider the Reynolds equation in Q (fig.3a). h' + 0.5;h) = 0 . = . div( - -grud(p) 12P For the calculations we need a film gap height h(x,y) which is sufficiently smooth ,for example a constant plus a sinus (fig.3b).
(ff)
h' it.( - -grad@)
I2P
+ 0.5uh) Ir,
= 0.
r In On On
n
: t h e Reynolds e q u a t i o n
r,,: p e r i o d i c b . c . r : nofLow across r.
.
is t h e f r e e boundary.
X
Then the solution p depends linearly on the slider speed u and the free boundary is independent of u, in contrast with the experiments in reL(1). Coyne & Elrod ref42) have used the Navier-Stokes equations to calculate outlet boundary conditions for the one dimensional Reynolds equation. They have calculated the thickness h,, (see fig. 3b) of the oil layer that leaves the full oil film. This oil layer is caused by the slider speed across the free boundary. Now the boundary condition for the derivative of the pressure becomes (.;
h3 - -grOd(p) 12P
+ 0.5uh) Ir1 = ( n . u ) . h o with (n.;)
t 0 . (1)
ho IS . a monotonously increasing function of The function h
the capillarity number
p.(i.i)
T '
3
STABILITY O F THE FREE BOUNDARY
An extended discussion of the stability of the free boundary for the one-dimensional Reynolds problem can be found in ref.(3) and ref.(4). 17P The cause of instability is the positive sign of -. This derivative Sn will be positive on the part of the free boundary where n.; t 0 is. see eq.(l). We will now show why a positive derivative causes instability. For that purpose we will dicuss the effect of a disturbance of the free boundary. Let s be a parameter along the free boundary and let y ( s ) be a disturbance of the free boundary:
r/(s) r,o)+ y ( s ) . i ( s ) = rnw +
If the capillarity number equals zero, the thickness of the oil layer h, is also zero. For high value of the capillarity number the thickness h, will become equal to half the film gap height h and the following equations hold.
Let p be the solution of the Reynolds equation with boundary I-,. On rnthe Dirichlet b.c. becomes (surhce tension included):
ho $0)
We can calculate the derivative of p towards the normal, we use this derivative to approximate the value of p. on I-/:
h0 h
= 0 and -(
+ GO) = 0.5
The free boundary is a solution of eq.(l). Now if the slider speed u increases the influence of grad(p) in eq.(l) will also increase. This is due to eq.(2) and the form of the free boundary will become more parallel to grad(p). The existence of the free boundary as a solution of the discretized equations will be discussed in one of the following sections.
So the Dirichlet b.c. on
-
r, is changed by
2P d2g(s) 6p(s) = - -.y(s) - T.an
as2
503 If the derivative of p and y ( s ) are positive then the first term of the right hand side in eq.(3) will lower the Dirichlet b.c. and extra oil will flow towards the free boundary. Of course the second term will have a stabilizing effect on the free boundary. In contrast with eq.(2) ( i . G ) . ( h - 2h,,)
is an increasing function of
p.(ri.Ii) T ‘
an
h ( s , J’) = o
p.(i.G)
T ’
So with an increasing capillarity number the free boundary will become unstable. One difference with the stability of the free boundary for the one-dimensional problem is the slider speed tangent to the free boundary and due to this the disturbance of the free boundary will give an extra oil stream equivalent to a oil stream over the undisturbed boundary of
--
dY
- O.~(U.S ).(A - 2h,,).7.
(4)
as
This extra oil stream will have a stabilizing effect on the free boundary. This will be discussed further on. 4
5 EXISTENCE OF THE SOLUTION FOR THE DISCRETIZED EQUATIONS
Let a be the groove angle and let the film gap height h be of the form:
With eq.(l) the derivative aP
is also an increasing function of
for the pressure near the boundary ) the amount of work for the calculations is drastically reduced.
-
b. cos[2L(x
I
-
L )] tan(a)
For b = 0 the solution of the free boundary problem is known: p(x,y) is constant in R. T, is parallel to the x-axis. We now want to use the implicit function theorem to prove the existence of the solution for the discretized equations with b close to zero. It is necessary that the mapping J is a continuous function of the discretized free boundary. By Fourier analysis one can easily show that for b = O the linear mapping J is nonsingular and the implicit function theorem can be applied. As long as J is nonsingular we can apply a continuation method for b 2 0 and use the implicit function theorem to calculate the free boundary. The singularity of J is of course closely related to the stability of th free boundary.
CALCULATION O F THE FREE BOUNDARY
Suppose we have an approximation of the free boundary. If we use the Dirichlet b.c. on the free boundary, we can calculate the oil stream over that boundary. We can use that oil stream to improve the initial approximation. To do that we split up the free boundary into line pieces of equal length L. Now we can discretize R in fig.2 with rhombs of side length L and split up the rhombs into two triangles (linear-conformal element) to get a discretization of the Reynolds equation. This procedure will give a very regular M matrix and the solution of the linear equations will fulfil the maximum-minimum principle. For each disturbance y(s) of the free boundary we can calculate the extra oil stream over the boundary (extra Reynolds problem) and calculate the linearly Jacobian mapping J, for simplicity we take only the two most important terms: J : y(s)
--t
n.(- -grad(bp)) h3 1%
- O . S ( u . s ) . ( h- 2h0).-. a7
as
(5)
Now we can apply Newton’s method to calculate the free boundary. The calculation of J costs a lot of effort, but because the Reynolds equation is elliptic, we can use a disturbance of the Dirichlet b.c. on the undisturbed boundary (see eq.(3)) instead of a disturbance of the free boundary. Now we only need one matrix inversion of the discretized Reynolds equation to calculate J. We have used a central difference scheme to calculate the mapping J. If we use enough elements on the free boundary ( some hundreds of elements ) the second term on the right-hand side in eq.(3) will make the matrix J diagonal dominant. T o further reduce the amount of work for the calculation of the mapping J, we make use of the regularity of the discretization of R. Except for the equations for the b.c. on r this regularity gives a block Toeplitz matrix for the Reynolds problem. With a stable version of cyclic reduction ( we only have to know the solution
6
CALCULATION OF THE STABILITY
Suppose that the free boundary has been calculated. Let y be again a disturbance of the free boundary. For the stability of the free boundary we have to calculate the eigenvalues 1of the equation: (7)
Now J does have the dimension of flow integrated over the film gap height, so if we divide J by the film gap height we obtain the speed of the disturbance y in One shouldn‘t divide by the full film gap height but divide%(;. the part not filled by the oil. If we take b = 0 in eq.(6) then we have to divide J by the constant film gap height. Consider again eq.(5) for J. The first term on the right-hand side is the oil stream caused by the curvature times the surface tension ( b = 0 and p is constant ) and gives rise to a symmetric matrix A with real negative eigenvalues. The second term on the righthand side gives rise to an a-symmetric matrix B (central difference scheme ). It is easy to prove that the real part of the eigenvalues of A + B will lie in between the smallest and highest eigenvalue of A. So at b=O the free boundary will be stable. Suppose that the free boundary is unstable for some b larger than zero. If a solution for the free boundary still exists we cannot use a continuation method towards b ( starting with b=O ) because the mapping J will be singular for some intermediate value of b. Now we use a global method to minimize h’
r/ [i.( - -grad@) I2P
+ 0.5;.(h
- 2ho))l2.K
We can check the obtained solution by using Newton’s method which should give quadratic convergence for eq.(l).
504 7
ARESULT
-=
(6.;)
We kept the groove angle a , the ratio a/l (see eq.(6)) and the c:ipiIlarity number fixed: tan(a) = 0.5 a/l=0.011 -= 5.0. T b (see eq.(6)), lar er than 0.06 the free We calculated that for
'J
5
4
boundary is unstable, whle at least up to 7 = 0.2 a solution for the free boundary exists. Let
K
1 be greatest real part of the eigenvalues in set. of eq.(7).
For each point on the free boundary we can define the capillarity number:
'(ii.U)
T .
b = 0.06 we calculated a maximum capillarity number on For 7 the free boundary of 0.15 . h = 0. I we calculated a maximum capillarity number on For 7 the free boundary of 0.275 and a maximum real part of the eigenvalues oC
0.0091.
In figure 4 an eigenvector corresponding to this K has been plotted. The disturbance of the free boundary in fig.4 is transported along the free boundary by the slider speed (complex eigenvalue). One can see that the disturbance is damped on that part of the free boundary where (n.;) is negative. One would expect that the oscillations are transported with half the slider speed, however the function h, (fig.3) in eq.(5) reduces this to 1/3 of the slider speed. In eq.(5) we took h. independent of the disturbance y. This disturbance will change the capillarity number on the free boundary and so it will also change the thickness ho of the oil layer on the smooth slider surface. This change of thickness could be visible as streaks of oil. If the capillarity number on the free boundary increases, then the thickness of the oil layer on the smooth slider surface will also increase and the speed of the transport of the disturbance along the free boundary will be reduced relatively to the slider speed. This reduction in speed will make the streaks of oil on the smooth slider surface visible, see ref.(]).
FIGURE 4 UNSTASLE DISTURBANCE OF TnE FREE BOUNDARY.
I X
8 References
( I ) J.Bootsma and L.P.M. Tielemans; Conditions of leakage free operation of herringbone grooved journal bearings, A.S.M.E. J.Jub.Tech.99f.April 1977 pp.215-223. (2) J.C.Coyne and H.G.Elrod jr; Conditions for the rupture of a lubrication film, A.S.M.E. J.Lub.Tech.92f.July 1970 pp.45 I . (3) G.Dalmaz; thesis "Le film mince visquex dans les contacts Herziens en regimes hydrodynamique et elastohydrodynamique", I'lnstitut National des Sciences Appliquees de Lyon et I'Universite Claude Benard. 1979, p.220. (4) K.J.Ruschak;"A Three-Dimensional Linear Stability Analysis for Two-Dimensional Free Boundary Flows by the Finite-Element Method", Computers and Fluids, Vol.lI,No 4,pp.391-401,1983.
505
Paper XVl(iii)
An experimental study of oil-air interface instability in a grooved rectangular pad thrust bearing D.J. Hargreaves and C.M. Taylor
As part of a continuing study of the effects of film reformation in plain journal bearings, an experimental programme with grooved, rectangular pad, slider thrust bearings has been undertaken. For such a pad with lubricant supplied under pressure to a central groove an oil-air interface may form between the pad and the opposing runner. Under specific operating conditions instability of this interface was observed and the study reported was undertaken to elucidate the mechanism of this instability. The importance of the non-dimensional grouping called the capillarity number in characterizing the instability is identified, and the ingress of air into the lubricant film inmediately prior to interface collapse I s shown to occur. It is postulated that this is the basis of the mechanism of Instability. 1
INTRODUCTION
Since the late 1970's the occurrence of film reformation in plain journal bearings has been a topic of research interest for workers in the Institute of Tribology at the University of Leeds. Possibly no freely available design procedures for liquid lubricated journal bearings incorporate an explicit consideration of the effects of such film reformation. Some design routines ( I ) do attempt to quantify the consequences of the reformation both in thermal and flow rate terms, however, the reliabllity of such predictive techniques is still uncertain particularly at critical boundaries representing design limitations. More recently reported studies of numerical analysis techniques developed to predict both the rupture and the reformation boundaries and obtain experimental validation have been directed towards steadily loaded plain journal bearings ( 2 , 3 , 4 ) . Consideration of the influence of cavitation regions in dynamically loaded bearings (an in particular the film reformation boundary) upon lubricant flow rate and other influential operating parameters is still inadequate. In .early work the authors undertook both theoretical and experimental studies relating to the much simpler rectangular pad thrust bearing configuration. Such a geometry offered the possibility of developing numerical analysis techniques to predict the location of the oil-air interface formed which would be directly applicable to the journal bearing situation, since the formulation of the boundary on a continuity basis is in essence the same as for the reformation boundary in a plain journal bearing. In addition it was reasoned that the use of a simpler arrangement for experimental studies would give valuable experience, particularly in relation to the theoretical/experimental correlation of flow rate, before moving onto the more complex journal bearing arrangement. Details of the theoretical and experimental aspects of this work have been published ( 5 , 6 , 7 ) .
During experimental work on the rectangular thrust pad geometry an instability of the oil-air interface which was formed was encountered. Such an instability has been reported by other researchers (e.g. 8) in relation to cavitation in bearings, however, little physical or quantitative basis for its occurrence has been presented. The instability restricted the range of parameter variations over which experimental work could be carried out. It was felt important therefore to characterise the effect in the hope that it could be avoided in future studies. 2
THE EXPERIMENTAL APPARATUS AND THE OIL-AIR INTERFACE INSTABILITY
A description of the experimental apparatus is available in the published literature (5,9). In essence it comprised a 1.0 m diameter steel disc mounted on a vertical shaft and supported in rolling element bearings. A series of glass/steel rectangular thrust pads e.g. Figure 1 , each with a supply hole or groove could be pressed against the stationary or rotating disc with lubricant supplied under pressure. Full details of the pad geometries may be found in reference ( 5 ) . Figure 1 shows a glass pad with a rectangular supply groove mounted in two outriggers supplied with adjustable legs. The legs consisted of grub screws tipped with polyethylene and held in position by lock buts. By this means (5,9) the film profile between the test pads and runner could be set with good accuracy. This in turn facilitated the correlation between theoretical predictions and experimental measurements of important parameters, particularly lubricant flow rate. Photographs in plan view through a glass pad with a supply hole are shown in Figure 2. Here the runner beneath the pad is moving from right to left. The leading edge of the pad is at the right extremity of the field of view whilst the supply hole at the pad centre may be identified at the left of the photographs. A stable oil-air interface (with oil on the left and air on the right) is evident in Figure 2(a).
506
Figure 1
A photograph of a t y p i c a l g l a s s t h r u s t pad showing t h e a d j u s t a b l e polymer b e a r i n g pad l e g s l o c a t e d on the outriggers
T h i s h a s been formed where t h e P o i s e u i l l e o r p r e s s u r e f l o w f o r c e d u p s t r e a m from t h e s u p p l y h o l e is j u s t b a l a n c e d by t h e C o u e t t e or v e l o c i t y f l o w due t o v i s c o u s d r a g a s s o c i a t e d w i t h t h e runner. T h e r e is no l u b r i c a n t s u p p l i e d h e r e t o t h e l e a d i n g e d g e o f t h e pad. The p h y s i c a l n a t u r e o f t h e o i l - a i r i n t e r f a c e of F i g u r e 2 ( a ) may be compared w i t h t h e f i l m r e f o r m a t i o n boundary formed i n a p l a i n j o u r n a l b e a r i n g s e e n in F i g u r e 3 ( r e f e r e n c e 4 ) . Here l u b r i c a n t is b e i n g s u p p l i e d u n d e r p r e s s u r e t h r o u g h a h o l e a t t h e p o s i t i o n o f maximum f i l m t h i c k n e s s ( z e r o d e g r e e s a s s e e n on t h e s c a l e in t h e photograph). The s h a f t is r o t a t i n g from r i g h t t o l e f t a s viewed in p l a n and t h e r e g i o n o f c a v i t a t i o n c a n be c l e a r l y s e e n t h r o u g h t h e g l a s s bush in t h e d i v e r g e n t c l e a r a n c e s p a c e . The f i l m r e f o r m a t i o n boundary i s formed u p s t r e a m o f t h e s u p p l y g r o o v e in a manner similar t o t h a t d e s c r i b e d f o r t h e r e c t a n g u l a r g l a s s t h r u s t pad. The v i s u a l s i m i l a r i t y between t h e two i n t e r f a c e s may be s e e n and i n d e e d t h e i r a n a l y t i c a l d e s c r i p t i o n is b a s e d on t h e same c o n t i n u i t y agreement ( 3 , 6 ) . The p r e s e n c e o f a r e c i r c u l a t i n g flow f o r the journal bearing i n t h i s case d o e s n o t d e t r a c t from t h e In f a c t c o m p a r a b i l i t y o f t h e two s i t u a t i o n s . s t u d i e s with a l u b r i c a n t supplied t o t h e leading edge o f t h e r e c t a n g u l a r t h r u s t pads shown in F i g u r e 2 , t h e r e b y s i m u l a t i n g more c o m p l e t e l y t h e r e f o r m a t i o n p r o c e s s in a p l a i n j o u r n a l b e a r i n g , have been u n d e r t a k e n ( 9 ) . The p h y s i c a l a s p e c t s o f t h e i n t e r f a c e s formed in t h e p r e s e n t work, t h e i r c o n s i d e r a t i o n in a n a l y s i s and i n d e e d t h e i n s t a b i l i t y d e s c r i b e d below were u n a f f e c t e d by t h e p r e s e n c e o f t h e l e a d i n g e d g e pad flow. Under c e r t a i n c o n d i t i o n s t h e s t a b l e o i l - a i r i n t e r f a c e s e e n in F i g u r e 2 ( a ) c o u l d become u n s t a b l e and c o l l a p s e . A p h o t o g r a p h of one i n s t a n t d u r i n g s u c h a n i n s t a b i l i t y is shown i n F i g ur e 2(b). P r e l i m i n a r y e x p e r i m e n t a l work revealed t h a t ,
Figure 2
Photograph of a s t a b l e o i l - a i r i n t e r f a c e ( u p p e r ) and t h e i n s t a b i l i t y ( l o w e r ) viewed in p l a n t h r o u g h a g l a s s t h r u s t pad. Runner r o t a t i o n from r i g h t t o l e f t .
( i ) Vigorous rubbing o f t h e s u r f a c e o f t h e s t e e l d i s c j u s t u p s t r e a m of a g l a s s b e a r i n g pad caused a s t a b l e i n t e r f a c e t o ' d a n c e ' back and f o r t h . T h i s i n d i c a t e d t h a t t h e s u r f a c e condition of t h e runner e n t e r i n g t h e b e a r i n g may be i n f l u e n t i a l in t h e instability. ( i i ) Complete d i s r u p t i o n o f t h e i n t e r f a c e o c c u r r e d r e g a r d l e s s o f t h e b e a r i n g pad geometry o r l u b r i c a n t f i l m p r o f i l e .
( i i i ) The c o l l a p s e o f t h e i n t e r f a c e was a l w a y s i n i t i a t e d on t h e nose o f t h e f r e e boundary. ( i v ) The s u r f a c e s p e e d o f t h e d i s c a t which t h e i n s t a b i l i t y w a s i n i t i a t e d was d e p e n d e n t on t h e pad geometry and l u b r i c a n t f i l m thickness. (v)
F o r a g i v e n b e a r i n g pad w i t h a f i x e d f i l m p r o f i l e t h e i n t e r f a c e c o l l a p s e was a t t h e same s p e e d i r r e s p e c t i v e o f s u p p l y p r e s s u r e , although t h e l o c a t i o n of t h e i n t e r f a c e varied with t h e supply pressure.
( v i ) F o l l o w i n g t h e i n s t a b i l i t y , a r e d u c t i o n in r u n n e r speed a l l o w e d t h e i n t e r f a c e t o reform t o a s t a b l e c o n d i t i o n .
507 The e x p e r i m e n t a l s t u d y now t o be d e s c r i b e d was u n d e r t a k e n t o e l u c i d a t e f u r t h e r q u a n t i t a t i v e and p h y s i c a l a s p e c t s of t h e i n s t a b i l i t y which had been e n c o u n t e r e d . I t s o c c u r r e n c e l i m i t e d t h e range of i n f l u e n t i a l v a r i a b l e s € o r which v a l i d a t i o n o f t h e t h e o r e t i c a l model c o u l d be undertaken. A s such t h e i n s t a b i l i t y was a n u n f o r t u n a t e f a c t which had n o t been f o r e s e e n i n t h e d e s i g n of t h e a p p a r a t u s and a n improved u n d e r s t a n d i n g o f i t would be d e s i r a b l e and p o s s i b l y of a s s i s t a n c e t o many r e s e a r c h e r s .
l u b r i c a n t employed was a S h e l l o i l d e s i g n a t e d The v i s c o u s c h a r a c t e r i s t i c s were d e t e r m i n e d u s i n g a series of suspended l e v e l U-tube v i s c o i n e t e r s i n t h e r m o s t a t i c b a t h s . The :I1 had a k i n e m a t i c v i s c o s i t y of 310 c S t a t 20 C ( c o r r e s p o n d i n g a p p r o x i m a t e l y t o a n IS0 VG 2 g r a d e ) and i t s d e n s i t y a t 21.6 OC was 882 kg/m
HVI 160.
3.
An o r d e r of magnitude a n a l y s i s of t h e f l u i d f i l m f o r c e s p o s s i b l y of s i g n i f i c a n c e a t t h e o i l - a i r i n t e r f a c e and hence i m p o r t a n t i n t h e c o l l a p s e mechanism ( i n e r t i a , g r a v i t y , v i s c o u s and s u r f a c e t e n s i o n f o r c e s ) c l e a r l y i n d i c a t e d t h e dominance o f v i s c o u s and s u r f a c e t e n s i o n f o r c e s (9). This suggested t h a t t h e non-dimensional g r o u p i n g known as t h e c a p i l l a r i t y number, C a p i l l a r i t y number
=
'Tl"
(where T i s t h e s u r f a c e t e n s i o n o f o i l t o a i r ) r e p r e s e n t i n g a measure of t h e r a t i o of v i s c o u s t o s u r f a c e t e n s i o n f o r c e s , might be a n i m p o r t a n t parameter. I n f a c t i t is w i d e l y known t h a t t h e c a p i l l a r i t y number i s o f s i g n i f i c a n c e i n t h e mechanics of l i q u i d - g a s i n t e r f a c e s (e.g. 10, 11, 1 2 ) .
Figure 3
3.
Film r e f o r m a t i o n i n a p l a i n j o u r n a l bearing (4). Shaft r o t a t i o n r i g h t t o l e f t a s viewed.
EXPERIMENTAL MEASUREMENTS
The test a p p a r a t u s w a s modified by t h e a d d i t i o n a t t h e l u b r i c a n t s u p p l y i n l e t of a l e n g t h of 10 mm d i a m e t e r c o p p e r t u b e formed i n t o a c o i l of some 150 mm d i a m e t e r and immersed i n a b a t h of w a t e r . The water t e m p e r a t u r e c o u l d be a d j u s t e d t o e € f e c t a change i n l u b r i c a n t v i s c o s i t y . A l l t h e r e s u l t s r e p o r t s were c a r r i e d o u t w i t h one g l a s s pad (number 2 of r e f e r e n c e ( 6 ) ) which had a l e n g t h ( i n t h e s u r f a c e motion d i r e c t i o n ) o f 100 mm, a w i d t h of 45 mm and a c e n t r a l s q u a r e s u p p l y groove o f 9.5 mm s i d e . The d e p t h of t h e g r o o v e was 1.5 mm. The l u b r i c a n t s u p p l y p r e s s u r e ( p ) was measured by means of p r e s s u r e t a p p i n g l o c a t e d j u s t upstream o f t h e s u p p l y groove and c o n n e c t e d t o a mercury U-tube manometer. The r o t a t i o n o f t h e d i s c a c t i n g as t h e t e s t b e a r i n g r u n n e r was recorded w i t h a m e c h a n i c a l c o u n t e r and by t i m i n g a g i v e n number o f r e v o l u t i o n s t h e s u r f a c e speed of t h e d i s c a t t h e b e a r i n g l o c a t i o n (U) c o u l d b e d e t e r m i n e d . The t e m p e r a t u r e of t h e l u b r i c a n t s u p p l i e d was measured by a nickel-chromium/ nickel-aluminium thermocouple l o c a t e d i n t h e s u p p l y p i p e j u s t u p s t r e a m o f t h e s u p p l y groove. An e l e c t r o n i c thermometer r e c o r d e d t h e s u p p l y t e m p e r a t u r e hence e n a b l i n g t h e l u b r i c a n t The v i s c o s i t y a t i n l e t ( n ) t o be determined.
The s u r f a c e t e n s i o n of t h g t e s t o i l w i t h a i r w a s measured u s i n g a Du Nouy t e n s i o m e t e r t o be 0.03 N/m a t 19.5 C. E x p l i c i t d a t a on t h e v a r i a t i o n of t h e s u r f a c e t e n s i o n w i t h t e m p e r a t u r e f o r t h e test l u b r i c a n t c o u l d n o t be o b t a i n e d . For water o v e r t h e t e m p e r a t u r e r a n g e 20-70 OC ( t h e v a r i a t i o n c o n s i d e r e d i n e x p e r i m e n t s u n d e r t a k e n by t h e a u t h o r s ) , t h e s u r f a c e t e n s i o n w i t h a i r d r o p s from 0.073 t o 0.064 N / m ( 1 3 ) , a f a l l o f a b o u t 12%. For t h e purpose of t h e c a l c u l a t i o n undertaken h e r e t h e s u r f a c e t e n s i o n of t h e t e s t l u b r i c a n t was t a k e n t o be c o n s t a n t a t 0.03 N/m. A series o f e i g h t t e s t s were c a r r i e d o u t . P a r a l l e l f i l m p r o f i l e s were employed w i t h t h e f i l m t h i c k n e s s v a r y i n g from 200-450 p ( 5 ) . The s u p p l y p r e s s u r e v a r i a t i o n was from 2.0 111.5 kPa and l u b r i c a n t t e m p e r a t u r e ranged from 67.9 OC ( c o r r e s p o n d i n g t o a dynamic 20.6 v i s c o s i t y range 0.26 - 0.023 Pas. The r e s u l t s a r e r e c o r d e d i n T a b l e 1. The s u r f a c e v e l o c i t y of r u n n e r a t t h e b e a r i n g l o c a t i o n a t which i n s t a b i l i t y of t h e o i l - a i r i n t e r f a c e was o b s e r v e d is l i s t e d and c a l l e d t h e c o l l a p s e v e l o c i t y (U). As w e l l as d e t a i l i n g t h e important o p e r a t i o n a l parameters i n Table 1, t h e d i s t a n c e of t h e nose of t h e i n t e r f a c e from t h e l e a d i n g edge o f t h e g l a s s test pad j u s t b e f o r e t h e i n s t a b i l i t y o c c u r r e d ( 6 ) i s given. I n a d d i t i o n a v e r a g e v a l u e s of c o l l a p s e v e l o c i t y , l u b r i c a n t t e m p e r a t u r e and c a p i l l a r i t y number are r e c o r d e d from t h e t e s t s i n which t h e t e m p e r a t u r e v a r i a t i o n was i n s i g n i f i c a n t .
-
-
Figure 4 p r e s e n t s d a t a r e l a t i n g t o test s e r i e s 1 , 4 and 7. F o r e a c h series t h e i m p o r t a n t test p a r a m e t e r s are shown and a p l o t o f t h e c o l l a p s e v e l o c i t y (U) as a f u n c t i o n of t h e supply p r e s s u r e ( p ) presented. It is clear t h a t f o r a given operating f i l m thickness the s u r f a c e v e l o c i t y a t which c o l l a p s e o c c u r r e d is e s s e n t i a l l y independent of t h e s u p p l y p r e s s u r e . Further, although t h e average lubricant t e m p e r a t u r e (and hence dynamic v i s c o s i t y ) v a r i e d s l i g h t l y f o r t h e tests, t h e c o l l a p s e
508 v e l o c i t y decreased markedly with i n c r e a s i n g f i l m thickness. This r e s u l t s i n a f a l l i n g v a l u e of t h e c a p i l l a r i t y number (qU/T) with i n c r e a s i n g f i l m t h i c k n e s s although i t h a s a f i x e d value f o r a given f i l m thickness. T h i s might be expected s i n c e an a l t e r n a t i v e non-dimensional grouping involving t h e product of t h e c a p i l l a r i t y number and n o m a l i s e d group i n c o r p o r a t i n g t h e f i l m t h i c k n e s s could be a more s i g n i f i c a n t parameter i n characterizing the interface.
i n v o l v i n g cine-photography of the cross-film mentscus of t h e i n t e r f a c e j u s t before and during c o l l a p s e . T h i s study is reported i n t h e next section.
FILM THICKNESS 250 vrn VARYING SUPPLY PRESSURE
O l ' 0
'
"
0.05
0.1
0.15
"
0.2
'
' 025
'
-
LUBRICANT OYNAMIC VISCOSITY IPa sl
0.22 0.20 0.18
"
0.18 M)
m
M)
so iw
20
Lo 60 80 loo
o 10 20
30
LO so
SUPPLY PRESSURE lhPil
Figure 4
P l o t of c o l l a p s e v e l o c i t y a g a i n s t supply p r e s s u r e f o r t e s t s 1, 4 and 7.
Too much s i g n i f i c a n c e must not be a t t a c h e d t o t h e near constancy of t h e c a p i l l a r i t y number f o r a given test series. T h i s i n essence only shows t h a t c o l l a p s e v e l o c i t y is independent of supply p r e s s u r e s i n c e l u b r i c a n t dynamic v i s c o s i t y and s u r f a c e t e n s i o n were i n v a r i a n t f o r a given test s e r i e s . However, f u r t h e r confirmation of t h e p o t e n t i a l importance of t h e c a p i l l a r i t y number is demonstrated i n t h e graph shown i n Figure 5. Here t h e i n t e r f a c e c o l l a p s e v e l o c i t y has been p l o t t e d as a f u n c t i o n of l u b r i c a n t dyunamic v i s c o s i t y , v a r i a t i o n of t h e l a t t e r being achieved through supply temperature v a r i a t i o n s . The r e s u l t s of Figure 5 a r e f o r a f i l m t h i c k n e s s of 250 pm and taken from t e s t s e r i e s 4, 5 and 6. A c o n s t a n t c a p i l l a r i t y number (nU/T) f o r t h i s given f i l m t h i c k n e s s would imply t h a t t h e curve (U) a g a i n s t (q) would be n e a r l y a hyperabola, s i n c e t h e s u r f a c e t e n s i o n (TI would only be expected t o v a r y s l i g h t l y with temperature and hence (nu) would be fixed. T h i s is borne out i n F i g u r e 5 where t h e c o l l a p s e v e l o c i t y is seen t o i n c r e a s e markedly as t h e l u b r i c a n t dynamic v i s c o s i t y is decreased. It did not prove p o s s i b l e t o undertake tests with l i q u i d s having a s i g n i f i c a n t d i f f e r e n c e i n s u r f a c e t e n s i o n t o a i r compared with S h e l l o i l HVI 160. This would be a n important f u r t h e r s t e p i n comfirming t h e s i g n i f i c a n c e o f , and f u r t h e r q u a n t i f y i n g , t h e c a p i l l a r i t y number over a range of test v a r i a b l e s . I t was decided t h a t a more e f f e c t i v e r o u t e t o gaining an understanding of t h e i n s t a b i l i t y mechanism within t h e time period a v a i l a b l e would be t o undertake a programme
Figure 5
P l o t of c o l l a p s e v e l o c i t y a g a i n s t l u b r i c a n t dynamic v i s c o s i t y f o r a film-thickness of 250 ( t e s t s 4, 5 , 6)
C I N E PHOTOGRAPHY
Cine f i l m s were used t o record t h e sequence of e v e n t s l e a d i n g t o and d u r i n g t h e i n t e r f a c e c o l l a p s e . Figure 6 shows a schematic diagram of t h e arrangement used t o record t h e v a r i a t i o n i n cross-film meniscus shape. A 16 mm Bolex HR 16 camera f i t t e d with a 50 mm S w i t a r l e n s and 50 mm e x t e n s i o n tubes and mounted on a t r i p o d was employed. The exposure time was (1/125 8 ) a t ( f l l ) w i t h a f i l m speed of 64 frames/s. The HP5 f i l m was developed i n 1-6 Teknol a t 29 OC a t a speed of 25 mm/s. D i f f i c u l t i e s were encountered i n t h e photography of t h e meniscus shape. It was important t o Cocus t h e camera on t h e nose of t h e i n t e r f a c e otherwise t h e t r u e shape would not be viewed. The a x i s of t h e camera w a s h o r i z o n t a l and normal t o t h e d i r e c t i o n of runner motion. A m i r r o r was used t o r e f l e c t t h e i l l u m i n a t i n g l i g h t p a s t t h e meniscus and i n t o t h e camera l e n s a s shown i n Figure 6. This method produced r e f l e c t e d l i g h t i n a l l d i r e c t i o n s whereas i d e a l l y a p a r a l l e l source would be b e t t e r . The p o l i s h e d metal s u r f a c e of t h e r o t a t i n g d i s c a l s o r e f l e c t e d l i g h t from the m i r r o r i n t o t h e camera l e n s . Therefore, not o n l y did t h e camera record t h e meniscus shape but a l s o i t s r e f l e c t e d image. This was v e r i f i e d by t h e i n s e r t i o n of a f e e l e r gauge i n t o the bearing gap and the observation t h a t two such gauges were observed through t h e camera. The knowledge of the film-thickness s e t and t h e camera m a g n i f i c a t i o n allowed t h e p o s i t i o n i n g of t h e upper and lower bearing s u r f a c e s on p r i n t s taken of t h e f i l m t o be made. The e x i s t e n c e of t h e r e f l e c t e d meniscus w a s v e r i f i e d again and d e s p i t e considerable e f f o r t e l i m i n a t i o n of t h e r e f l e c t i o n could not be achieved.
509 t h e moving s u r f a c e w h i l s t f i r s t converging (moving from r i g h t t o l e f t on t h e s t i l l ) b e g i n s t o d i v e r g e . T h i s s u g g e s t e d t h a t t h e meniscus had ' s e p a r a t e d ' from t h e moving s u r f a c e and a n a i r b u b b l e had been drawn i n t o t h e f i l m . The d e f i n i t i v e shape o f t h e i n t e r f a c e c l o s e t o t h e moving s u r f a c e w a s n o t c l e a r l y e v i d e n t because o f unwanted r e l f e c t e d l i g h t .
m
Glass bearing pod
Mirror
Figure 7
I Figure 6
Motion
A s c h e m a t i c diagram in p l a n of t h e a r r a n g e m e n t used t o f i l m t h e c r o s s - f i l m meniscus shape
S t i l l frames of t h e f i l m t a k e n t h e r e f o r e have t h e a p p e a r a n c e shown in F i g u r e 7. A l l t h e photography r e p o r t e d h e r e w a s f o r t h e g l a s s pad d e t a i l e d e a r l i e r (pad number 2 ) w i t h a p a r a l l e l f i l m t h i c k n e s s o f 800 pm. In F i g u r e 7 t h e moving and s t a t i o n a r y s u r f a c e s are i d e n t i f i e d and t h e c r o s s - f i l m meniscus between t h e l u b r i c a n t and a i r is shown by t h e w h i t e band. The r e f l e c t i o n o f t h e meniscus s h a p e below t h e l i n e marking t h e moving s u r f a c e is e v i d e n t . The series o f photographs shown in F i g u r e 8 d e m o n s t r a t e t h e change in shape of t h e o i l - a i r meniscus immediately p r i o r t o c o l l a p s e . Frame 1 was t a k e n a t a s u r f a c e speed of 0.12 m / s which was known t o be c l o s e t o t h e i n s t a b i l i t y c o n d i t i o n . The e l o n g a t i o n of t h e meniscus r e l a t i v e t o t h e s t a b l e s i t u a t i o n a t a lower s u r f a c e v e l o c i t y shown in F i g u r e 7 is a l r e a d y a p p a r e n t . The r e m a i n i n g frames (2-5) were t a k e n a t i n t e r v a l s of ( 1 5 1 6 4 ) second, however, t h e precise v e l o c i t y a s s o c i a t e d w i t h e a c h s i t u a t i o n could n o t be determined. The i n c r e a s e in s u r f a c e speed w a s s l i g h t and c o l l a p s e o c c u r r e d precipitously. The sequence of frames in F i g u r e 8 shows t h a t a s l i g h t i n c r e a s e in s u r f a c e s p e e d n e a r t o the v e l o c i t y of c o l l a p s e h a s a marked e f f e c t on t h e meniscus shape. The meniscus e l o n g a t e s c o n s i d e r a b l y and indeed i t is a p p a r e n t from frame 6 t h a t t h e g a p between t h e i n t e r f a c e and
A s t i l l frame w i t h o v e r l a i d i d e n t i f i c a t i o n of t h e b e a r i n g s u r f a c e s and c r o s s - f i l m meniscus
The development o f a n a i r bubble in t h e l u b r i c a n t f i l m is i l l u s t r a t e d i n F i g u r e 9 where t h e sequence w a s t a k e n in p r e c i s e l y t h e same manner as f o r t h e p r e v i o u s f i g u r e , e x c e p t t h a t t h e s u r f a c e v e l o c i t y was reduced immediately t h e i n t e r f a c e c o l l a p s e was observed t h e r e b y a l l o w i n g t h e i n t e r f a c e t o reform. After the formation of a n a i r b u b b l e it w a s n e c e s s a r y t o rotate t h e camera through a s m a l l a n g l e a b o u t i t s v e r t i c a l In a x i s in o r d e r t o k e e p t h e b u b b l e i n view. d o i n g so, r e f e r e n c e p o i n t s were l o s t so t h a t t h e l o c a t i o n of t h e bubble w i t h i n t h e l u b r i c a n t f i l m was unknown. It d i d n o t prove p o s s i b l e t o o b t a i n a d e f i n i t i v e sequence of e v e n t s embracing e f f e c t s p r i o r t o c o l l a p s e , t h e entrainment of a n a i r b u b b l e and movement of t h a t a i r bubble. The c l a r i t y of F i g u r e s 8 and 9 is b e t t e r in reference (9). 5.
DISCUSSION AND CONCLUSIONS
A s p a r t o f a wider s t u d y o f t h e i n f l u e n c e o f f i l m reformation i n p l a i n j o u r n a l bearings, an e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n o f a n o i l - a i r i n t e r f a c e i n a grooved, r e c t a n g u l a r pad, t h r u s t b e a r i n g h a s been undertaken. t h e range of parametric v a r i a t i o n s p o s s i b l e i n t h e e m p i r i c a l work proved t o be l i m i t e d by t h e o c c u r r e n c e of a n i n s t a b i l i t y of t h e o i l - a i r i n t e r f a c e . An i n v e s t i g a t i o n of t h i s i n s t a b i l i t y h a s been r e p o r t e d i n t h i s paper. The s t u d y h a s comprised two t h r e a d s , t h e f i r s t o f which i n v o l v e d measurements of t h e s u r f a c e v e l o c i t y a t which c o l l a p s e o c c u r r e d f o r a r a n g e o f o p e r a t i n g c o n d i t i o n s , w i t h in a l l cases a p a r a l l e l f i l m p r o f i l e in t h e t e s t b e a r i n g .
(i)
The c o l l a p s e v e l o c i t y w a s independent of t h e l u b r i c a n t supply pressure with o t h e r p a r a m e t e r s k e p t c o n s t a n t . Thus a l t h o u g h t h e l o c a t i o n o f t h e o i l - a i r i n t e r f a c e in t h e b e a r i n g c o u l d v a r y , t h e s u r f a c e speed a t which t h e i n s t a b i l i t y commenced was e s s e n t i a l l y fixed.
510 (ii)
For thicker films the value of the collapse velocity reduced.
(ill)
For conditions when the lubricant dynamic viscosity was reduced by heating the supply o i l , the collapse velocity was shown to increase substantially. This i s consistent with the influence of viscous forces expressed through the grouping (QU) and order-of-magnitude analysis has demonstrated that the other important influential film force at the interface would be that due to surface tension.
(1+3)
Figure 8
Thus the capillarity number (qU/T) i s seen to be important in characterizing the meniscus and its collapse. However, no direct empirical work in which the surface tension of the lubricant has been varied substantially has been undertaken. The second aspect of the study undertaken has been related to cine photography of the cross-film meniscus immediately prior to and during the instability process.
( 4 -+6)
A sequence of frames illustrating the oil-air cross-film meniscus during interface collapse: the surface velocity at frame 1 was 0 . 1 2 m/s with the velocity increasing slightly from frames 2 through 5 which were at ( 1 5 / 6 4 ) second intervals.
511
(1 Figure 9
+
(4
3)
-f
6)
The development of a n a i r bubble i n t h e f i l m : frame 1 corresponded t o a s u r f a c e v e l o c i t y of 0.12 m / s and t h e s u b s e q u e n t frames were e a c h a t (35/64) second
(iv)
The e l o n g a t i o n o f t h e meniscus as t h e c o l l a p s e v e l o c i t y approached h a s been c l e a r l y d e m o n s t r a t e d . The i n s t a b i l i t y occurred f o r a very s l i g h t i n c r e a s e i n b e a r i n g s u r f a c e speed. During t h i s r a p i d p r o c e s s t h e l e n g t h of t h e meniscus i n t h e d i r e c t i o n o f motion i n c r e a s e d s u b s t a n t i a l l y and e v e n t u a l l y ' s e p a r a t i o n ' o f t h e meniscus from t h e moving s u r f a c e was e v i d e n c e d . T h i s w a s i n t e r p r e t e d a s b e i n g due t o t h e i n g r e s s of a i r under t h e meniscus i n t o t h e f i l m .
(v)
The e n t r y o f an a i r bubble i n t o t h e f u l l l u b r i c a n t f i l m h a s been p h o t o g r a p h e d , a l t h o u g h because of t h e r e q u i r e m e n t t o rotate t h e camera t o a c h i e v e t h i s t h e l o c a t i o n of t h e bubble w i t h r e s p e c t t o t h e r e f e r e n c e frame of t h e b e a r i n g s u r f a c e s was l o s t . The f i n a l f a t e o € t h e bubble and i t s i n f l u e n c e i n t h e t o t a l i n s t a b i l i t y o c c u r r i n g c o u l d n o t be observed.
It i s postulated t h a t the o i l - a i r i n s t a b i l i t y which h a s been o b s e r v e d a r o s e from t h e i n v a s i o n of a n a i r b u b b l e i n t o t h e f i l m under t h e meniscus. The v e r y dynamics of t h e
i n t e r f a c e d u r i n g t h i s p r o c e s s may l e a d t o t h e t o t a l i n s t a b i l i t y o r i t is possible t h a t the air which h a s e n t e r e d t h e f i l m may m i g r a t e t o a r e g i o n of lower p r e s s u r e , expand and d i s r u p t t h e i n t e r f a c e . F u r t h e r e x p e r i m e n t a l work and t h e o r e t i c a l s t u d i e s which may f u r t h e r e l u c i d a t e t h e p r o c e s s e s i n v o l v e d are a p p a r e n t . E x p e r i m e n t a l l y t h e v a r i a t i o n of l u b r i c a n t s u r f a c e t e n s i o n n e e d s t o be i n v e s t i g a t e d and t h e p o s s i b l e i n f l u e n c e of p h y s i c a l / c h e m i c a l a s p e c t s of t h e lubricant/surface i n t e r a c t i o n explored (14,15,16). A n a l y t i c a l l y b o t h t h e i n g r e s s of a i r i n t o t h e f i l m (in t h e form of a i r l u b r i c a t i o n ) and t h e i n s t a b i l i t y o f t h e meniscus form ( b a s e d on a p p r o a c h e s s u c h a s t h c s e i d e n t i f i e d i n 11, 12) would be v a l u a b l e . 6
ACKNOWLEDGEMENTS
One of t h e a u t h o r s (DJH) was i n r e c e i p t of a U n i v e r s i t y of Leeds s c h o l a r s h i p d u r i n g t h e p e r i o d t h a t t h e r e s e a r c h d e s c r i b e d i n t h i s paper w a s c a r r i e d o u t . The work was s u p p o r t e d under a n SERC R e s e a r c h g r a n t GR/A 66185. The c o n t r i b u t i o n of M r Stephen B u r r i d g e , a member o f t h e t e c h n i c a l s t a l f i n t h e Department of Mechanicacl E n g i n e e r i n g a t t h e U n i v e r s i t y o f L e e d s , t o t h e p h o t o g r a p h i c a s p e c t s of t h e s t u d y r e p o r t e d is g r a t e f u l l y acknowledged.
512 and LABY, T.H., ' T a b l e s of ( 1 3 ) KAYE, G.W.C. p h y s i c a l and c h e m i c a l c o n s t a n t s , Longman, 1962.
APPENDIX REFERENCES
------
( 1 4 ) ROZEANU, L and SNARSKY, L. 'The u n u s u a l b e h a v i o u r of a l u b r i c a n t boundary l a y e r ' , WEAR, 43, 1977, pp 117-126.
' C a l c u l a t i o n methods f o r s t e a d i l y l o a d e d a x i a l g r o o v e hydrodynamic j o u r n a l b e a r i n g s ' , E n g i n e e r i n g S c i e n c e s Data U n i t I t e m No 84031, 1984.
and TAYLOR, DOWSON, D., MIRANDA, A.A.S. C.M., 'Implementation of an a l g o r i t h m e n a b l i n g t h e d e t e r m i n a t i o n of f i l m r u p t u r e and r e f o r m a t i o n b o u n d a r i e s i n a l i q u i d f i l m bearing'. P r o c e e d i n g s of t h e 1 0 t h Leeds-Lyon Symposium on T r i b o l o g y Developments i n Numerical and E x p e r i m e n t a l Methods Applied t o T r i b o l o g y , R u t t e r w o r t h s , 1984, pp 265-270. DOWSON, D., MIRANDA, A.A.S. and TAYLOR, C.M., 'The p r e d i c t i o n of l i q u i d f i l m j o u r n a l b e a r i n g performance w i t h a c o n s i d e r a t i o n of f i l m r e f o r m a t i o n , P a r t 1 t h e o r e t i c a l r e s u l t s ' , Proc. I.Mech.E., Part C, Vol. 199, C2, 1985, pp 95-102.
-
DOWSON, D., MIRANDA, A.A.S. and TAYLOR, C.M., 'The p r e d i c t i o n of l i q u i d f i l m j o u r n a l b e a r i n g performance w i t h a c o n s i d e r a t i o n of f i l m r e f o r m a t i o n , P a r t I1 - e x p e r i m e n t a l r e s u l t s ' , Proc. I.Mech.E., p a r t C, Vol. 199, C2, 1985, pp 103-111. HARGREAVES, D . J . and TAYLOR, C.M., 'The d e t e r m i n a t i o n of f l o w rate from a s t a t i c grooved r e c t a n g u l a r t h r u s t b e a r i n g u s i n g a stream f u n c t i o n a n a l y s i s ' , J n l . Mech. Eng. Scs., 14, 1, 1982, pp 51-53. 'An HARGREAVES, D . J . and TAYLOR, C.M., e x p e r i m e n t a l and t h e o r e t i c a l s t u d y of l u b r i c a n t f l o w rates i n s t a t i c , g r o o v e d , r e c t a n g u l a r t h r u s t b e a r i n g s ' , J n l . Mech. Eng. SOC., 24, 1 , 1982, pp 21-29. 'An HARGREAVES, D.J. and TAYLOR, C.M., e x p e r i m e n t a l and t h e o r e t i c a l s t u d y o f l u b r i c a n t f i l m e x t e n t and f l o w r a t e i n grooved r e c t a n g u l a r pad t h r u s t b e a r i n g s ' , P r o c . I.Mech.E., P a r t C, 198C, 1 6 , 1984, pp 225-233. 'Film NEWTON, M.J. and HOWARTH, R.B., recession i n hydrostatic t h r u s t bearings', P r o c . I.Mech.E., 57, 1973, pp 725-731. 'An e x p e r i m e n t a l and HARGREAVES, D . J . , t h e o r e t i c a l s t u d y of f i l m f o r m a t i o n and l u b r i c a n t f l o w rates i n grooved r e c t a n g u l a r t h r u s t b e a r i n g s ' , Ph.D. Thesis, Dept. Mech. Eng., Univ. of Leeds, 1981. ' C a v i t a t i o n of a v i s c o u s ( 1 0 ) TAYLOR, G . I . , f l u i d i n narrow p a s s a g e s ' , J n l . F l u i d Mechs, 16, 4, 1963, pp 595-619. 'Conditions ( 1 1 ) COYNE, J.C. and ELROD, H.G., f o r t h e r u p t u r e of a l u b r i c a t i n g f i l m . P a r t I: t h e o r e t i c a l model', J. Lub. Tech., T r a n s . ASME, 90, J u l y , 1970, pp 451-456. 'Cavitation i n lubrication. ( 1 2 ) SAVAGE, M.D., P a r t 1. On boundary c o n d i t i o n s and cavity-fluid interfaces', J n l . Fluid Mechs., 8 0 , 4 , 1977, pp 743-755.
( 1 5 ) ROZEANU, L. and SNARSKY, L., ' E f f e c t o f s o l i d s u r f a c e l u b r i c a n t i n t e r a c t i o n on t h e load capacity of s l i d i n g bearings', Jnl. Lub. Tech., T r a n s . ASME, 100, 1978, pp 167-175. ( 1 6 ) ROSEANU, L. and SNARSKY, L., 'Second o r d e r t h e r m a l e f f e c t s in l u b r i c a t i o n ' , P r o c e e d i n g s o f 5 t h Leeds-Lyon Symposium on Tribology Thermal E f f e c t s in L u b r i c a t i o n , MEP, 1980, pp 95-100
-
513
rest lo.
Film Thickness (vm)
Supply Pressure (p) (kPa)
6
(OC)
Viscosity (11) (Pas)
(mm)
Collapse Velocity (U) (m/sec)
Temp.
:::;
Tem
0.227 0.231 0.236
1.51 1.49 1.50
U = 0.231 m / s
41 37 35 33
0.226 0.226 0.227 0.235
1.72 1.69 1.71 1.74
U = 0.229 m / s 'I(eemg = 22.9 OC
0.147 0.141 0.128 0.121
40 39 38 38
0.259 0.272 0.273 0.276
1.27 1.28 1.17 1.11
0.053 0.031 0.023
41 42 42
0.364 0.540 0.810
0.64 0.56 0.62
22.2 23.1 23.3 23.8
0.236 0.222 0.219 0.212
40 36 35 33
0.241 0.250 0.243 0.253
2
200 200 200
91.8 102.5 111.5
21.6 21.9 22.2
0.245 0.241 0.236
36 34 33
0.232 0.234 0.236
3
250 250 250
43.2 52.3 58.8
24.7 25.1 25.3
0.199 0.194 0.191
-
250 250 250 250
21.6 42.8 58.6 71.5
22.7 22.9 23.0 23.1
0.228 0.225 0.223 0.222
250 250 250 250
34.7 33.4 32.4 31.0
29.5 30.2 31.5 32.3
250 250 250
13.3 4.5 2.0
49.3 63.4 67.9
5
6
~
7
8
-
350 350 350 350
16.9 24.6 31.1 37.5
20.6 20.9 21.0 21.4
0.262 0.257 0.255 0.249
38 35 34 32
0.182 0.185 0.190 0.189
450 450 450 450
5.7 8.0 11.5 14.4
22.7 23.3 23.3 23.3
0.228 0.219 0.219 0.219
40 38 36 34
0.183 0.196 0.195 0.195
Table 1
[%I
em
.~
6
-
U = 0.247m/s
57.8 91.7 99.8 108.2
4
Averages
1.90 1.84
200 200 200 200
1
[F]
1.59 1.59
= 25.0
OC
OC
~
U = 0.187 m/s Tem
= 21.0
OC
[F]
1.61 1.57 1.39
23.1
U
the distance from the leading edge of the bearing pad t o the nose of the i n t e r f a c e .
Experimental data €or the study of the o i l - a i r i n t e r f a c e collapse
= 0.192
m/s
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SESSION XVII GAS BEARINGS Chairman: Professor H.G. Elrod
PAPER XVll(i)
The performance of an out-of-balance rotor supported in self acting gas bearings
PAPER XVll(ii)
Comparison of theoretical characteristics of t w o types of externally pressurized, gas lubrication, compliant surface thrust bearings
PAPER XVll(iii) An experimental investigation of the steady-state performance of a compliant surface aerostatic thrust bearing PAPER XVll(iv) The effect of finite width in foil bearings: theory and experiment
This Page Intentionally Left Blank
517
Paper XVII(i)
The performance of an out-of-balance rotor supported in self acting gas bearings H. Marsh
An a n a l y s i s i s presented f o r t h e behaviour of an out-of-balance r o t o r ' i n a p l a i n s e l f a c t i n g gas journal bearing. The theory is used t o p r e d i c t t h e synchronous o r b i t a l motion and good agreement with t h e experimental r e s u l t s i s obtained. The e f f e c t of r o t o r out-of-balance on h a l f speed whirl i s examined. It is shown t h a t t h e h a l f speed w h i r l i s suppressed when t h e synchronous o r b i t a l motion exceeds a c r i t i c a l value, t h e e c c e n t r i c i t y a t t h e onset of i n s t a b i l i t y with a well balanced rotor.
1
INTRODUCTION
The o p e r a t i o n of high speed r o t o r s supported i n p l a i n s e l f a c t i n g gas j o u r n a l bearings i s o f t e n l i m i t e d by t h e onset of self e x c i t e d i n s t a b i l i t y , u s u a l l y h a l f speed w h i r l i n e i t h e r t h e t r a n s l a t i o n a l o r c o n i c a l mode, r e f . 1. This i n s t a b i l i t y is well understood and f o r c e r t a i n bearing geometries, t h e design engineer i s a b l e t o p r e d i c t t h e speed a t which it w i l l occur and ensure t h a t t h i s i s n o t i n t h e normal range of o p e r a t i o n . I f t h e rotor-bearing system is l i k e l y t o encounter h a l f speed w h i r l below t h e normal o p e r a t i n g speed, then v a r i o u s s t e p s can be taken t o avoid t h e i n s t a b i l i t y , such as lobed o r grooved members, f l e x i b l y mounted bearings o r t h e i n t r o d u c t i o n of a small amount of rotor-outof-balance. For many y e a r s it has been known t h a t a small amount of out-of-balance can improve t h e s t a b i l i t y of a r o t o r supported i n p l a i n s e l f a c t i n g gas j o u r n a l bearings. R e l a t i v e l y f e w experiments have been c a r r i e d out t o study t h i s phenomenon. The r e s e a r c h reported h e r e was undertaken t o discover how a r o t o r responded t o out-of-balance and t h e e f f e c t which t h i s had on t h e onset of h a l f speed w h i r l . I f t h e r o t o r response can be p r e d i c t e d and a c r i t e r i o n e s t a b l i s h e d f o r suppression of h a l f speed w h i r l , then it should be p o s s i b l e t o a s s e s s whether t h e i n t r o d u c t i o n of r o t o r out-of-balance might improve t h e performance of a bearing system. 2
NOMENCLATURE
C
r a d i a l clearance
g
gravitational constant
IT
transverse i n e r t i a
I
p o l a r i n e r t i a of r o t o r
L
bearing l e n g t h
m
r o t o r mass
P
'ni
1 m =mt6m
M
load
pa
ambient p r e s s u r e
r
r a d i u s f o r out-of-balance mass
r
0
r a d i u s of synchronous o r b i t
R
bearing r a d i u s
x
BG
z
BR
B
n Pa R$ ml c w
6
o f f s e t of c e n t r e of g r a v i t y
6m
out-of-balance mass
E
eccentricity r a t i o
e
a t t i t u d e angle
w
r o t o r angular v e l o c i t y
wL
load v e c t o r angular v e l o c i t y
3
RESPONSE TO A ROTATING LOAD
In t h e a n a l y s i s of bearing behaviour, t h e response t o a constant u n i - d i r e c t i o n a l load can be shown diagrammatically as i n f i g s . l a and l b . The bearing c e n t r e B i s f i x e d , while t h e r o t o r c e n t r e R t a k e s up a p o s i t i o n such t h a t t h e f l u i d f i l m f o r c e balances t h e applied load. The load v e c t o r i n f i g s . l a and lb behind t h e l i n e of c e n t r e s , BR, by t h e angle 8, t h e a t t i t u d e angle. Fig. 2a shows a bearing system where t h e r o t o r i s s u b j e c t t o a constant r o t a t i n g load. The bearing is f i x e d , t h e r o t o r speed i s w and
518 t h e load v e c t o r r o t a t e s a t a speed WL i n t h e same d i r e c t i o n as t h e r o t o r . The r o t o r c e n t r e , R , follows a c i r c u l a r o r b i t around t h e bearing c e n t r e B a t a speed WL. Now consider t h e system a s seen by an observer r o t a t i n g with t h e l i n e of c e n t r e s . The motion as seen by t h i s observer i s obtained by adding an angular motion -WL t o t h e system shown i n f i g . 2a t o o b t a i n t h a t of f i g . 2b where t h e r e i s : (a) (b) (c)
(d)
a r o t o r spinning a (w-WL) a bearing r o t a t i n g a t (-wL) a load v e c t o r and l i n e of c e n t r e s which a r e now f i x e d i n d i r e c t i o n r e l a t i v e t o t h e observer, and an e c c e n t r i c i t y r a t i o E and an a t t i t u d e angle 8 .
I f t h e e f f e c t s of l u b r i c a n t i n e r t i a can be neglected, then t h e performance of a j o u r n a l bearing is determined by t h e entrainment v e l o c i t y , t h e sum of t h e s u r f a c e v e l o c i t i e s of t h e r o t o r and t h e bearing. Applying t h i s b a s i c p r i n c i p l e , t h e system shown i n f i g . 2b h a s e x a c t l y t h e same behaviour as t h a t i n f i g . 2c where :
( a ) the rotor spins a t ( W - ~ W L ) , ( b ) t h e bearing is f i x e d , ( c ) t h e load v e c t o r and l i n e of c e n t r e s a r e f i x e d , and ( d ) t h e e c c e n t r i c i t y and a t t i t u d e angle a r e t h e same as i n f i g s . 2a and 2b. The process shown i n f i g s . 2a, 2b and 2c provides
a g e n e r a l method f o r r e l a t i n g t h e response of a bearing system with a c o n s t a n t r o t a t i n g load t o t h e response t o a u n i - d i r e c t i o n a l load. F o r a bearing system with a r o t o r speed w and a constant load r o t a t i n g a t WL, t h e e c c e n t r i c i t y r a t i o and a t t i t u d e angle a r e e x a c t l y t h e same as i n a bearing system with a r o t o r speed (w-2wL) and a u n i - d i r e c t i o n a l load of t h e same magnitude. There a r e f i v e s p e c i a l cases which must be considered:
1.
WL
0
This i s t h e bearing with a u n i - d i r e c t i o n a l load, so t h a t f i g s . 2a and 2c a r e i d e n t i c a l . 2.
WL < ( d 2 )
If WL < ( w / 2 ) , then ( w - 2 ~ ~ i)s p o s i t i v e s o t h a t i n moving from f i g . 2a t o 2c, t h e d i r e c t i o n of r o t o r motion remains unchanged and is a n t i clockwise. Fig. 2c corresponds t o f i g . l a and t h e load vector l a g s behind t h e l i n e of c e n t r e s , so t h a t t h e r e l a t i v e p o s i t i o n of t h e l i n e of c e n t r e s and load v e c t o r i s c o r r e c t i n f i g s . 2a, 2b and 2c.
3.
W L = w/2
When WL = w/2, then (w-2w~) i s zero. In f i g . 2c t h e r o t o r speed is zero so t h a t t h e r e is no mechanism by which a pressure d i s t r i b u t i o n can be generated t o c a r r y t h e applied load. When t h e bearing system i s s u b j e c t t o a constant load r o t a t i n g a t one h a l f of t h e r o t o r speed, t h e r e can be no pressure d i s t r i b u t i o n around t h e r o t o r , t h e r e is no load c a r r y i n g c a p a c i t y and t h e beari n g f a i l s . For t h i s case t h e e c c e n t r i c i t y r a t i o can be regarded as u n i t y and t h e a t t i t u d e angle
is zero. 4.
WL
w/2
When WL > w / 2 , then ( w - 2 w L ) i s a negative q u a n t i t y s o t h a t i n f i g . 2c t h e r o t o r i s spinning clockwise, corresponding t o t h e s t e a d i l y loaded bearing of f i g . l b . However, comparison of f i g s . 2c and l b shows t h a t t h e p o s i t i o n of t h e load v e c t o r i n f i g . 2c i s now i n e r r o r . The c o r r e c t s e t of diagrams f o r wL > w / 2 i s shown i n f i g . 3 and f i g . 3c then corresponds with f i g . l b . When t h e constant load r o t a t e s a t a speed g r e a t e r than ( w / 2 ) , then t h e load vector l e a d s t h e l i n e of c e n t r e s BR by an angle 8 which is e x a c t l y t h e same a s f o r a bearing with a r o t o r speed ( w - 2 ~ ~and ) a s t e a d y load of t h e same magnitude. 5.
WL
= w
When wL w , then ( w - 2 w L ) = -w and i n f i g . 3c, t h e r o t o r speed i s w i n t h e clockwise d i r e c t i o n . The e c c e n t r i c i t y r a t i o and a t t i t u d e angle for t h e synchronous r o t a t i n g load a r e t h e r e f o r e t h e same a s f o r a bearing s u b j e c t t o t h e same unid i r e c t i o n a l load. The load vector lags behind t h e l i n e of c e n t r e s i n f i g . l b , while f o r t h e synchronous r o t a t i n g load, t h e load vector l e a d s t h e l i n e of c e n t r e s , f i g . 3a. I t should be noted t h a t t h i s a n a l y s i s i s not r e s t r i c t e d t o gas l u b r i c a t e d b e a r i n g s ; it i s a p p l i c a b l e t o j o u r n a l bearings with any l u b r i c a n t where t h e e f f e c t s of l u b r i c a n t i n e r t i a may be neglected.
4
ROTOR OUT-OF-BALANCE
In a system with a p e r f e c t l y balanced r o t o r of mass m , t h e c e n t r e of g r a v i t y l i e s on t h e r o t o r geometrical a x i s . If a small mass 6m i s added a t a r a d i u s r i n t h e plane containing t h e c e n t r e of g r a v i t y , then t h e c e n t r e of g r a v i t y is displaced t o a p o i n t G where t h e o f f s e t from t h e r o t o r a x i s , RG, i s given by r. 6m r 6m R G = 6 =
~'
m
+
--
6m
.__.-
ml
(1)
This o f f s e t of t h e c e n t r e of g r a v i t y i s then a fixed quantity. Now consider a bearing which supports an out-of-balance r o t o r which is spinning a t a speed w . I n f i g . 4 , B r e p r e s e n t s t h e fixed bearing c e n t r e , R is t h e r o t o r geometric a x i s and G i s t h e c e n t r e of g r a v i t y . The e f f e c t of t h e out-of-balance i s t o impose a synchronous r o t a t i n g load which causes t h e r o t o r c e n t r e R t o t r a c e out a synchronous c i r c u l a r o r b i t around t h e bearing c e n t r e . The load v e c t o r , a c t i n g along BG, l e a d s t h e l i n e of c e n t r e s BR by t h e angle 8 . The response t o t h e out-ofbalance load can be determined by applying t h e a n a l y s i s shown diagrammatically i n f i g s . 3a, 3b and 3c. The f l u i d f i l m f o r c e i n t h e bearing can be determined from t h e published d a t a on t h e performance of gas l u b r i c a t e d j o u r n a l bearings and t h i s must balance t h e out-of-balance load, film force
=
fi(L/D, H ,
=
(m
+
E)
am). BG. w2
(2)
519
where x
=
d=
BG and
m
+
H x = B GH2
or
6m
. z
(7)
The angle between t h e out-of-balance load and t h e l i n e of c e n t r e s i s a l s o taken f r o m t h e published d a t a , 8 = f2(L/D,
H,
(4)
E)
Applying t h e Cosine r u l e t o t h e t r i a n g l e BRG,
In a d d i t i o n , t h e r e i s a geometric c o n d i t i o n which is obtained by applying t h e Cosine r u l e t o t h e t r t a n g l e BRG, RG2 = BR2
+
BG2
-
=
z
o r s e t t i n g BR 62
22
+
e
(5)
A LONG BEARING SOLUTION
A s o l u t i o n o f t h e governing equations can be obtained f o r a system o p e r a t i n g with a s m a l l e c c e n t r i c i t y r a t i o w i t h bearings where t h e end e f f e c t s can be n e g l e c t e d . Ausman ( 2 ) has shown t h a t t h e low e c c e n t r i c i t y performance o f such a bearing i s given by 'TI
Pa RL
f i l m stiffness =
H &+H2
(6)
1
H
-
z2 t x2
2z.x. c o s
1 t H2 (1 1
x2 - 2zx cos
and Tan 8 =
=
2BR.BG.Cos 8
Equations 3 , 4 and 5 a r e solved i t e r a t i v e l y using a computer program which c o n t a i n s d a t a on t h e performance of gas l u b r i c a t e d j o u r n a l b e a r i n g s , t h e f u n c t i o n s f l ( L / D , H , E ) and f 2 ( L / D , H , E ) . The process converges r a p i d l y , but under-relaxation i s necessary a t high eccentricity ratios. Fig. 5 shows how t h e t r i a n g l e BRG v a r i e s with r o t o r speed f o r a s m a l l out-of-balance. A t low speeds, f i g . 5a, t h e r o t o r c e n t r e R l i e s c l o s e t o t h e bearing c e n t r e B s o t h a t t h e r o t o r tends t o s p i n about i t s geometric a x i s through R. A s t h e speed i n c r e a s e s , t h e a t t i t u d e a n g l e decreases and t h e r a d i u s o f t h e synchronous o r b i t i n c r e a s e s , f i g s . 5b and 5c. A t high speeds, f i g . 5d, t h e c e n t r e of g r a v i t y , G , moves towards B and t h e r o t o r t h e n t e n d s t o s p i n about t h e p r i n c i p a l a x i s through t h e c e n t r e of g r a v i t y . The process of moving from f i g . 5a t o 5d is u s u a l l y c a l l e d i n v e r s i o n . Since t h e e c c e n t r i c i t y r a t i o cannot exceed u n i t y , it is c l e a r from f i g . 5d t h a t i n v e r s i o n cannot occur when t h e o f f s e t of t h e c e n t r e of g r a v i t y exceeds t h e bearing r a d i a l c l e a r a n c e and t h i s is discussed i n more d e t a i l i n t h e next s e c t i o n . With a l a r g e out-of-balance, 6>c, i n v e r s i o n cannot occur and t h e t r i a n g l e BRG v a r i e s a s shown i n f i g . 6. 5*
62
+
-
e
El2
H2
For any speed o f r o t a t i o n , it i s p o s s i b l e t o determine t h e geometry o f t h e t r i a n g l e BRG by c a l c u l a t i n g z, x and 8. A t low speeds, B is v e r y l a r g e and t h e s o l u t i o n i s given by 8=90°, z=O and xz6, corresponding t o f i g . 5a. A s t h e speed is i n c r e a s e d , a c o n d i t i o n is reached where 1 f3=1+-
H2 l e a d i n g t o t h e s o l u t i o n z=6H=6 Cot 8 , corresponding t o f i g . 5b. With a f u r t h e r i n c r e a s e i n speed, B = l and f o r t h i s condition z
=
6
G
~
F I
= 2Sin e
corresponding t o f i g . 5c. F i n a l l y , a t very high speeds, H + m and B 9 0 , g i v i n g 8 + O and z + 6 , f i g . 5d. A t t h i s high speed condition, t h e f i l m f o r c e , which i s equal t o t h e out-ofbalance f o r c e , i s given by
f i l m f o r c e = n pa
RL
E)
but it should be noted t h a t t h i s is only a p p l i c a b l e t o a system which can p a s s through i n v e r s i o n t o reach t h e c o n f i g u r a t i o n o f f i g . 5d. This closed s o l u t i o n i s extremely u s e f u l f o r providing a first e s t i m a t e f o r t h e response t o out-of-balance loading. For example, t h e geometrical c o n f i g u r a t i o n of f i g . 5c i s only reached when B = 1 and t h e r a d i u s o f t h e r o t o r synchronous o r b i t i s t h e n 6 G H 2 . If t h i s r a d i u s is t o be less t h a n c , t h e b e a r i n g c l e a r a n c e , t h e n t h i s implies t h a t t h e o f f s e t o f t h e c e n t r e of g r a v i t y must be less t h a n For t h e apparatus described i n t h e c / fi2. next s e c t i o n , t h e c o n d i t i o n B = l i s reached a t 36,660 rev/min with Hz9.29, so t h a t t h e r a d i u s of t h e synchronous o r b i t , BR, is 9.346. This l i n e a r i s e d a n a l y s i s s u g g e s t s t h a t t h e system w i l l only pass through i n v e r s i o n i f 6<0.107c. I n p r a c t i c e , t h e non-linear behaviour of t h e b e a r i n g w i l l permit a higher value of 6 , but t h e simple a n a l y s i s provides a u s e f u l guide.
6 uu R2 where H
6
Pa c2
Equating t h e out-of-balance i n the fluid f i l m ,
f o r c e with t h e f o r c e
EXPERIMENTS WITH AN OUT OF BALANCE ROTOR
For t h e experiments, t h e a p p a r a t u s took t h e form of a r i g i d r o t o r supported i n a s i n g l e r i g i d l y mounted bearing. This simple symmetrical arrangement overcomes many of t h e d i f f i c u l t i e s a s s o c i a t e d with a two b e a r i n g system where it i s necessary t o have two i d e n t i c a l b e a r i n g s . This i s a system which i s w e l l - s u i t e d t o fundamental r e s e a r c h on r o t o r - b e a r i n g dynamics.
520 I f t h e e f f e c t s of r o t o r out-of-balance can be e s t a b l i s h e d f o r t h i s simple system, t h e n t h e c r i t e r i a f o r w h i r l s u p r e s s i o n might then be extended t o more complex systems. The b a s i c parameters f o r t h e a p p a r a t u s are: mass of r o t o r
4.116 kg
m
2
transverse i n e r t i a
IT
0.0363 kg m
polar i n e r t i a
I
0.0015 kg mL
load
M
46.85 N
bearing l e n g t h
L
152 mm
bearing r a d i u s
R
25.4 mm
r a d i a l clearance
C
17 pm
viscosity (air)
u
1.8llO-5 kg/ms
ambient p r e s s u r e
pa
1 bar
P
When o p e r a t i n g with no out-of-balance, t h e w e l l balanced r o t o r , t h e system e x h i b i t e d c o n i c a l h a l f speed w h i r l a t a l l speeds g r e a t e r t h a n 1780 rev/min. The r o t o r out-of-balance w a s obtained by a t t a c h i n g small p i n s t o t h e r o t o r . For t h e f i r s t experiment, a mass o f 20 gm was added a t a r a d i u s of 26.7 mm i n each o f two p l a n e s s i t u a t e d 1 2 5 mm from t h e c e n t r e of t h e r o t o r . With t h e two masses s e t ' i n - p h a s e ' , t h e c e n t r e of g r a v i t y is o f f s e t from t h e r o t o r geometrical a x i s by 222 pm, a g r o s s out-of-balance. F i g . I shows t h e r a d i u s of t h e synchronous o r b i t a l motion a s a f u n c t i o n of speed. The o r b i t w a s p r e d i c t e d u s i n g t h e computer program t o s o l v e f o r t h e t r i a n g l e BRG f o r a range of speeds. There i s good agreement between t h e model and t h e experiments and t h e d i f f e r e n c e may be caused by r o t o r o u t of round or t h e e f f e c t of t h e g r a v i t a t i o n a l s t e a d y load which i s neglected i n the analysis.
I
THE EFFECT OF OUT OF BALANCE ON STABILITY
With a w e l l balanced r o t o r , t h e bearing system e x h i b i t s c o n i c a l h a l f speed w h i r l a t a l l speeds higher than 1780 rev/min. However, t h e experiments with t h e out-of-balance r o t o r , f i g . 7, show t h a t it i s p o s s i b l e t o suppress t h e h a l f speed w h i r l , l e a v i n g only t h e synchronous motion caused by t h e out-of-balance f o r c e . This effect has been s t u d i e d i n more d e t a i l by conducting a s e r i e s of experiments with varying amounts of out-of-balance. Fig. 8 shows t h e e f f e c t of out-of-balance on h a l f speed w h i r l . There is a c r i t i c a l value f o r t h e out-of-balance mass such t h a t f o r higher v a l u e s , t h e onset of h a l f speed w h i r l i s t o t a l l y suppressed. For an out-of-balance l e s s than t h i s c r i t i c a l value, t h e system e x h i b i t s an onset o f h a l f speed w h i r l and a c e s s a t i o n of h a l f speed w h i r l . The onset of h a l f speed w h i r l , where t h e small synchronous o r b i t a l motion g i v e s way t o a l a r g e amplitude h a l f speed w h i r l , remains c l o s e t o 1780 rev/min, t h e value f o r t h e w e l l balanced r o t o r . The upper boundary f o r t h e r e g i o n of h a l f speed w h i r l , where t h e l a r g e amplitude h a l f speed w h i r l g i v e s way t o a s m a l l e r amplitude of synchronous w h i r l , i s dependent on t h e out-of-
balance mass, t h e speed i n c r e a s i n g a s t h e o u t of-balance i s reduced.
8 DISCUSSION The experimental r e s u l t s shown i n f i g . 8 confirm t h a t with an out-of-balance r o t o r , it i s p o s s i b l e t o p a s s through t h e region of h a l f speed w h i r l and with s u f f i c i e n t out-of-balance, t h e i n s t a b i l i t y can be t o t a l l y suppressed. For t h e w e l l balanced r o t o r , t h e onset of c o n i c a l h a l f speed w h i r l occurs a t 1780 rev/min when t h e e c c e n t r i c i t y r a t i o i s 0.12, s o t h a t t h e r o t o r c e n t r e is a t a d i s t a n c e of 2.0 pm from t h e b e a r i n g c e n t r e . With t h e out-of-balance r o t o r , t h e r e i s a synchronous o r b i t a l motion and i n t h e absence of h a l f speed w h i r l , it i s assumed t h a t t h i s o r b i t a l motion t a k e s place about t h e p o s i t i o n which would be taken up by t h e w e l l balanced r o t o r . When t h e out-of-balance is j u s t s u f f i c i e n t t o suppress t h e o n s e t of h a l f speed w h i r l , 44 gm, t h e system e x h i b i t s a synchronous o r b i t a l motion a t a l l speeds. The computer program has been used t o p r e d i c t t h e amplitude of t h i s motion and a t a speed of 1780 rev/min t h e r a d i u s of t h e synchronous o r b i t i s estimated t o be 1 . 9 pm. The r a d i u s i s 2.0 pm a t a speed s l i g h t l y h i g h e r than 1780 rev/min. This suggests t h a t t h e o n s e t of h a l f speed w h i r l i s suppressed when t h e amplitude of t h e synchronous o r b i t a l motion a t t h e normal w h i r l onset speed i s g r e a t e r than t h e steady s t a t e e c c e n t r i c i t y . An a l t e r n a t i v e i n t e r p r e t a t i o n i s t h a t t h e onset o f t h e h a l f speed w h i r l i s suppressed when t h e out-of-balance f o r c e a t t h e w h i r l onset speed exceeds t h e steady load on t h e bearing. When t h e out-of-balance mass i s l e s s than t h e c r i t i c a l v a l u e , t h e r e is an onset of h a l f speed w h i r l and a c e s s a t i o n . The w h i r l onset occurs a t a speed c l o s e t o 1700 rev/min., t h e w h i r l onset speed with a w e l l balanced r o t o r . If t h e out-of-balance i s less t h a n t h e c r i t i c a l v a l u e , t h e n t h e synchronous o r b i t a l motion a t 1180 rev/min has an amplitude l e s s t h a n t h e e c c e n t r i c i t y and t h e out-of-balance f o r c e i s l e s s than t h e steady g r a v i t a t i o n a l load on t h e b e a r i n g . A t t h e w h i r l c e s s a t i o n , with a s l i g h t i n c r e a s e i n speed, t h e l a r g e amplitude h a l f speed w h i r l g i v e s way t o a s m a l l e r amplitude of synchronous w h i r l . By u s i n g t h e computer program, it i s p o s s i b l e t o p r e d i c t t h e magnitude of t h e synchronous o r b i t a l motion and t h e f o r c e involved f o r any speed. Fig. 9 shows 1 contours o f constant e c c e n t r i c i t y r a t i o a s a f u n c t i o n of speed and out-of-balance mass. The s u r p r i s i n g f e a t u r e of f i g . 9 is t h a t t h e experimental r e s u l t s f o r t h e c e s s a t i o n of h a l f speed w h i r l a l l l i e along a l i n e corresponding t o an e c c e n t r i c i t y r a t i o o f 0.12 t o 0.13, corresponding t o a synchronous o r b i t a l motion with a r a d i u s o f 2.0 pm t o 2.2 pm. This is c o n s i s t e n t w i t h t h e t o t a l suppression of t h e h a l f speed w h i r l when t h e synchronous o r b i t a t 1780 rev/min i s g r e a t e r than 2.0 pm. The a l t e r n a t i v e i n t e r p r e t a t i o n i n terms of t h e out-of-balance f o r c e has been t e s t e d f o r t h e w h i r l c e s s a t i o n boundary. It has been found t h a t t h e out-of-balance f o r c e a t t h e w h i r l c e s s a t i o n i n c r e a s e s r a p i d l y as t h e out-ofbalance mass i s reduced.
521 The experimental r e s u l t s suggest t h a t t h e h a l f speed w h i r l is suppressed when t h e amplitude of t h e synchronous o r b i t a l motion exceeds a c r i t i c a l value. This c r i t i c a l value appears t o be equal t o t h e e c c e n t r i c i t y a t t h e onset o f h a l f speed w h i r l with a w e l l balanced r o t o r .
9
CONCLUSIONS
To understand t h e behaviour o f an out-ofbalance r o t o r , it i s first necessary t o analyse t h e response of a j o u r n a l bearing t o a c o n s t a n t r o t a t i n g l o a d . A g e n e r a l method of a n a l y s i s has been presented which i s a p p l i c a b l e t o a j o u r n a l b e a r i n g o p e r a t i n g with any l u b r i c a n t where t h e e f f e c t of l u b r i c a n t i n e r t i a can be neglected. I t has been shown t h a t f o r a c o n s t a n t load r o t a t i n g a t t h e r o t o r speed, t h e load v e c t o r l e a d s t h e l i n e of c e n t r e s . A model has been developed which can be used t o p r e d i c t t h e performance of an out-ofbalance r o t o r i n a s e l f a c t i n g gas j o u r n a l bearing. With a small out-of-balance, t h e r o t o r w i l l first tend t o r o t a t e about t h e geometrical a x i s , but a s t h e speed i s i n c r e a s e d , i n v e r s i o n w i l l occur and a t high speed, t h e r o t o r w i l l tend t o r o t a t e about a p r i n c i p a l a x i s through t h e c e n t r e of g r a v i t y . A t high speed, t h e out-of-balance f o r c e t e n d s towards a constant value of n pa RL ( 6 / c ) . The c o n d i t i o n for i n v e r s i o n t o occur has been examined u s i n g a l i n e a r i s e d long b e a r i n g s o l u t i o n . With a l a r g e out-of-balance, i n v e r s i o n does n o t occur and t h e out-of-balance f o r c e continues t o i n c r e a s e a s t h e speed i n c r e a s e s . The i n f l u e n c e of out-of-balance on t h e onset o f h a l f speed w h i r l has been examined and it has been shown t h a t t h e r e i s a c r i t i c a l out-ofbalance above which t h e h a l f speed w h i r l i s t o t a l l y suppressed. For an out-of-balance less t h a n t h i s c r i t i c a l v a l u e , t h e r e i s a n o n s e t and a c e s s a t i o n f o r h a l f speed w h i r l . The experimental r e s u l t s have been analysed and it has been found t h a t i f t h e e c c e n t r i c i t y i s C E ~ a t t h e onset of h a l f speed w h i r l with a w e l l balanced r o t o r , t h e n t h e h a l f speed w h i r l i s suppressed when t h e r a d i u s of t h e synchronous o r b i t a l motion exceeds C E ~ . Further experiments a r e r e q u i r e d t o determine whether t h i s i s a g e n e r a l conclusion or whether it a p p l i e s only t o t h i s apparatus.
f LOAD
(b)
Fig. 1
S t e a d i l y loaded bearing
Fixed bearing r o t a t i n g load
load
r o t a t i n g bearing f i x e d load
f i x e d bearing f i x e d load
References load
(1) MARSH, H: 'The s t a b i l i t y of aerodynamic g a s b e a r i n g s ' , Mech. Eng. S c i . No. 2, I.Mech.E., 1965. ( 2 ) AUSMAN, J.S., 'The f l u i d dynamic t h e o r y of gas l u b r i c a t e d b e a r i n g s ' , Trans. A.S.M.E. , Vol. 79, p. 1218, 1957.
Fig. 2
Rotating l o a d , w
L
2
522 l o w speed B I,
n
fixed bearing r o t a t i n g load
load
1
r o t a t i n g bearing fixed load load
@ 2UL-w
fixed bearing f i x e d load
load
R
h i g h speed
Fig. 3
R o t a t i n g l o a d , wL > 2
Fig. 5
I n v e r s i o n w i t h a small out-of-balance
L
low speed
B
/--
/
c
'i ,.
B
B
Fig. 4
T r i a n g l e BRG
G
(d) h i g h speed
Fig. 6
Gross o u t - o f - b a l a n c e ,
no i n v e r s i o n
523
r a d i u s of o r b i t , pm
5
4
/ 1 -
-
00 0
3
\
-
0
0
Experiment
- Theory 4000
speed rev/min
2000
4000
3000
6000 6000-
SPEED, Rev./min.
2000
Fig. 7
O r b i t r a d i u s v s . speed
1000
0.lc
-
-\ -
0
0.02
out-of-balance,
-
synchronous w h i r l
;ooo
O
L
I 0.04
I
0.02
out-of-balance, 0
Fig. 8
kg.
Experiment
E f f e c t of out-of-balance
on b e a r i n g s t a b i l i t y
kg.
experiment
0
Fig. 9
0.04
c o n t o u r s of c o n s t a n t o r b i t
C e s s a t i o n of h a l f speed whirl
This Page Intentionally Left Blank
525
Paper XVll(ii)
Comparison of theoretical characteristics of t w o types of externally pressurized, gas lubricated, compliant surface thrust bearings K. Hayashi and K. Hirasata
In this paper, two types of the externally pressurized, gas lubricated and circular thrust bearings with the flexible surface are proposed for the purpose of developing the high performance bearings. Their static and dynamic characteristics have been theoretically derived and the effects of the flexibilities of the bearing surfaces on them have been made clear. From the calculated results, it was discussed which type of the bearings was better in the practical applications, and one of these two types of the compliant surface bearings have been recommended as the high performance bearing. 1.
INTRODUCTION
static condition hd
In recent years, the requirements to the bearings have become severe more and more with the movements of the super-precisions and the high performances of machines. In order to overcome such severe requests imposed on the bearings, many types of bearings have been proposed and the investigations to develop the high performance bearings have been vigorously carried out. As one of them, there is an invention in which one constructs some parts of the bearing surface with the compliant material and intends to improve the bearing performances with the aids of the flexibility of the bearing surface, and the several types of the compliant surface bearings have been proposed. The present paper also belongs such direction of the investigations. Two types of the externally pressurized, gas lubricated and circular thrust bearings in which some parts of the bearing surfaces are constructed with the thin and flexible metal plates are offered and their static and dynamic characteristics, such as the load-carrying capacity, the stiffness, the coefficient of damping and the gas consumption, will be theoretically derived and the effects of the flexibilities of the bearing surfaces on them will be made clear. Then, these two types of the compliant surface bearings will be compared with each other and the possibilities as the high performance bearings will be discussed.
1.1 Notation
A B Cij c CD D E e G H Hd Hdi h ho
: constant, 7pa-ro2.~a/(1440p2) : : : : :
: : : : : :
: :
dimensionless coefficient of damping constant of integration, (i=1,2,3; j=O,1,2) coefficient of damping flow coefficient Pa.ro4/{12(1-v2)Eto3.h~} Young’s modulus amplitude of oscillation dimensionless stiffness normalized bearing gap, h/ho hd/ho hdi/ho, (i=O,1,2) bearing gap bearing gap without surface deformation in
: variation of bear ng gap by elastic defor-
mation of bearing surface hdi : static or dynamic component of surface deformation, (i=O 1,2) K : flexibility number, 6A1’4(1-v2)Pa.r03/ (Eto3) k : stiffness ms : mass flow rate of gas in static condition P : normalized pressure, €’/Pa Pb : normalized back pressure, Pb/Pa Pi : static or dynamic component of normalized pressure, Pi/Pa, (i=O,1,2) PS : normalized supply pressure, Ps/Pa p : pressure Pa : ambient pressure Pb : back pressure PS : supply pressure Pi : static or dynamic component of pressure, (i=O,l,2) Q : normalized flow rate of gas, ms / (CD * ho 4J?;T) R : normalized coordinate, r/rO Ri : ri/rO, (i=1,2,3) : coordinate r ro : radius of bearing rl : radius of feeding hole r2 : radius of chamber r3 : radius of chamber T : absolute temperature of gas t : time to : thickness of thin metal plate W : load-carrying capacity I‘ : feeding parameter, -To*Rl-lnR1 ro : 12pcD-rgfi/(Pa*ho2) E : e/ho K : specific heat ratio of gas X : squeeze number, 12pwro2/(Pa.ho2) J ! : viscosity v : Poisson’s ratio Pa : dencity of gas at ambient condition T : normalized time, w-t w : angular frequency of oscillation R : gas constant
m/
Subscripts : value in the region I I1 : value in the region I1 111 : value in the region 111
I
526 2. 2.1
THEORY
of oscillations, so the gas film pressure and
the deformation of the bearing surface may be expressed with
Derivation of governing equations
The schematic drawings of two types of the compliant surface bearings under considerations are shown in Fig.1. Both of them are the externallypressurized bearings with a central feeding hole. The bearing named Type 1 constructs its surface with the flexible metal plate of the thickness to putting on the concave of radius r2, and the bearing Type 2 does so with such a metal plate putting on the deep groove of the inner radius r2 and the outer radius r3 and furthermore the pressurized gas is fed into this groove and operates behind the metal plate as the back pressure. For these bearings, the equations governing the bearing performances will be derived below. ro
P(R,T) = Po(R)+Pl(R) Hd(R,T)
.E*sinT+P2(R) - E - C O S T
(4) (5)
= HdO(R)+Hdl(R).E.SinT+Hdp(R).E.COST
In the region I, P0=Ps and P1=P2=0 since P=Ps. Substituting eqs. ( 4 ) and (5) into eqs. (1) and (2), and neglecting the higher than the second order terms of E, one obtains the following expressions from the Reynolds equation. In the region 11,
I
Ph
1
I
and in the region 111,
Fig.1
Schematic views of two types o f compliant surface bearings
Bearing Type 1: First, the bearing Type 1 is treated. For the convenience of the theoretical analysis, the bearing region is divided into three parts as the region I (Ri>RrO), the region I1 (R2LRS1) and the region 111 (lzR>R2). The normalized Reynolds equation for a laminar, viscous, compressible and isothermal gas flow is i aaR ( R H 3 PaP _._ ) aR
= - A a(PH)
ar
For the case in which the small amplitude of oscillations is imposed around the static equilibrium position, the dimensionless bearing gap is expressed by 1
+
E-SinT + Hd(R,T) in the region I1
H(R,T) =
(2)
1 + €.sin7 in the region I11 Next, the deformation of the bearing surface will be treated, For the fluctuations of the deformation of the thin metal plate caused by those of the gas film pressure, the effect of the inertia force from the vibration of the metal plate should be considered, but this effect may be neglected for the case in which the frequency of its vibration is not so high. Then, for such a case, the following relation comes into existsnce between the gas film pressure and the elastic deformation of the bearing surface in the regions I and 11.
I, a [R-{--a.-(R1 a -.-) R aR aR R aR
aHd aR
I]
=
1 -(P-Pb)
D
(3)
From the aforementioned equations, the pressure distribution of the gas film and the bearing clearance can be determined. The first order purterbation may be valid for the small amplitude
Meanwhile, applying eqs.(4) one obtains
and (5) to eq.(3),
(i=0,1,2)
(12)
where Pbo=Pb and Pbl=Pb2=0. Furthermore, the conditions Po=Ps and P,=P2=0 come into existance in the region I. Therefore, in the region, the expression
can be derived by solving eq.(12) with the consideration of the symmetry of the deformation of the bearing surface, where C10=1, Cll=C12=0. By solving these equations derived above with some boundary conditions, the pressure distributions of the gas film (PO, P1 and P2) and the geometries of the bearing surface (Hdo, Hdl and Hd2) are obtained. After then, the loadcarrying capacity, the stiffness and the coefficient of damping can be calculated from
-*
1
B =
= -2/oP2*R.dR nro Pa respectively. Bearing Type 2: Next, the bearing Type 2 is treated. As same as for the bearing Type 1, the bearing region is divided into three parts as the region I (RpR>Rl), the region I1 (R3>R> For the gas R2) and the region I11 (l2R2R3). flow in the bearing gap, the normalized Reynolds equation (1) comes into existance as same as in the bearing Type 1. The bearing gap is given by
527
H(R,T)
=
t
1 + e*SinT + Hd(R,.r)
1
+
(17) in the region I1 E-sinT in the regions I and I11
Meanwhile, the relation (3) exists for the elastic deformation of the bearing surface in the region 11. Therefore, the pressure distribution of the gas film and the bearing gap are determined from eqs.(l),(3) and (17). Assuming the pressure distribution and the bearing gap in the forms of eqs.(4) and (5) respectively as same as in the case of the bearing Type 1, and disposing of them in the same manner as in the aforementioned case, one obtains eqs.(9),(10) and (11) in the regions I and 111 from the normalized Reynolds equation (1). Similarly, eqs. (6) ,(7) and (8) can be derived in the region 11. Meanwhile, eq.(12) is derived for the deformation of the bearing surface in the region 11. In this equation, Pbo= Pb and pbl=Pb2=0. Therefore, for the bearing Type 2, the pressure distributions of the gas film (Po,P1 and P2) and the geometries of the bearing surface (Hdo, Hdl and Hd2) can be determined by solving the aforementioned equations with some boundary conditions. After then, the characteristics of the bearing are calculated by applying them to eqs. (14),(15) and (16).
With the consideration of the continuity of the bearing gap at r=r2 (hII I r=r2=hIII1 r=r2), the above expression may be rewritten into F l R = R 2
=
1-
aR
R=R2
Moreover, the condition
pIII R=R2
(23) pIII I R=R2 is given from the continuity of the pressure of the gas film at r=r2. Meanwhile, the condition =
comes into existance since the pressure of the gas film becomes ambient at the periphery of the bearing, r=ro. In the above, the boundary conditions for the gas flow and the gas film pressure have been treated. So, the boundary conditions for the deformation of the bearing surface, i.e. that of the metal plate, will be considered in the following. Since the deformation of the thin metal plate is axisynnnetric and the shearing force acting on the metal plate vanishes at the center, the conditions
2.2 Boundary conditions The boundary conditions for the aforementioned governing equations will be derived from the conditions of the continuities of the gas flow, the pressure and the bearing gap in the following. Bearing Type 1: With the assumption of the inherent comp-ensation, the mass rate of flow from the feeding hole into the bearing gap, ml, is given by
can be derived. Meanwhile, since the deformation of the thin metal plate, its gradient, the shearing force and the bending moment acting on the metal plate should be continuous at r=r1, the following conditions come into existance.
where
O(PS,P) =
1
P for a = - 2 PS
2 K/(K-I) (x)
Moreover, the conditions HdIIIR=R2 = Meanwhile, assuming that the temperature of gas falls once because of the adiabatic expansion when tha gas passes through the minimum cross section but it recovers to that in the feeding hole, T, just after flowing into the bearing gap, and then the gas flows in the bearing gap as the isothermal, viscous and laminar flow, one obtains the following expression for the mass flow of gas from the inherent compensator to the outer part of the bearing, mg.
Therefore, from the condition of continuity of gas flow (ml=m2), the condition
is derived. Next, from the continuity of the mass flows of gas in the regions I1 and 111 at r= rg, one obtains
are given from the condition how to fix the thin metal plate. These conditions mentioned above are the boundary conditions for the bearing Type 1. In the practical use, these conditions are reformed to those with respect to Pi and Hdi (i= 0,1,2) by applying eqs.(4) and (5) to them as eqs. (1)-(3) have been rewritten to eqs.(6)-(13). Bearing Type 2: For the bearing Type 2, the boundary conditions concerning with the continuities of the gas film pressure and the gas flow are essentially the same as those for the bearing Type 1. From the conditions of the continuity of the gas flow at r=r1, and from those of the gas flows and the gas film pressures at r=r2 and r=r3, the following conditions are derived corresponding to eqs.(21)-(23) in the aforementioned case.
528
(33)
Moreover, the condition (24) should be satisfied at the periphery of the bearing, r=ro. Meantime, for the deformation of the bearing surface, the conditions
are given from the condition how to fix the thin metal plate. These are the boundary conditions for the bearing Type 2 and they are reformed to the conditions with respect to Pi and Hdi (i=O,l, 2 ) for the practical use as mentioned in the case of the bearing Type 1. 2.3
Solution of zeroth order of
E
The solution of the zeroth order of E ( E O ) derived from eqs.(6) and (9) as follows: For the bearing Type 1:
can be
(37)
and for the bearing Type 2 :
where
calculations are essentially same for both types of the bearings. So, those for the bearing Type 1 will be explained below as the representative. The finite difference method is used to solve the aforementioned governing differential equations. First of all, the values of the parameters giving the dimensions of the bearing and the operating conditions are suitably chosen. Then, the static components of the gas film pressure and the deformation of the bearing surface (PO and Hdo) are calculated from eqs.(37), (38) and ( 1 2 ) . In this step, the deformation of the bearing surface are properly assumed (Hd0=0 is used as its starting value), and the pressure PO corresponding to this value of HdO are CalCulated from eqs.(37) and (38). By applying thus obtained pressure to eq.(12), the deformation of the bearing surface may be estimated. After then, the gas film pressure corresponding to the newly-corrected values of the deformation are recalculated from eqs.(37) and ( 3 8 ) . This procedure of the numerical calculations is repeated till the values of the static components of the pressure distribution and the deformation of the bearing surface converge their final ones, respectively. Next, by using the static components of the gas film pressure and the deformation of the bearing surface obtained above to eqs.(7),(8), (10),(11) and (12), their dynamic components, PI, P2, Hdl and Hd2 can be calculated. In this step of the calculations, it is known from the governing equations and the boundary conditions that the treatments of the pressures in the forms of products, Po.P1 and Po'P2, instead of P I and P2 themselves may be more convenient. The procedure of the numerical calculations is as follows. From applying the suitably-assumed values of the deformations, Hdl and Hd2 (Hdl=Hd2 =O is used as their starting values) to eqs.(7), (8), (10) and (ll), the dynamic components of the gas film pressure, PI and Pz (Po'P1 and Po'P2 in practice), are calculated. After then, the deformations, Hdl and Hdg, corresponding to thus obtained pressures are estimated. The dynamic components of the gas film pressure for the newly-corrected values of the deformations are recalculated. This procedure should be repeated till the values of Pi, P2, Hdl and Hd2 converge to their final ones, respectively. With theusage of the pressure distributions determined above, the load-carrying capacity, the stiffness and the coefficient of damping are calculated from eqs.(l4), (15) and (16), respectively. Meantime, the mass flow rate of the gas in the static condition can be obtained from eq. (18) as follows:
The characteristics of the bearing Type 2 can be also calculated with the same procedure of the numerical calculations described above. Therefore, if the static component of the deformation of the bearing surface, HdO, are previously known, the static component of the gas film pressure, Po, may be determined from eqs.(37) and (38) or from eqs.(39)-(41). Meanwhile, HdO may be determined from eq. (12). 3.
NUMERICAL CALCULATIONS
3.1 Procedure of calculations The method and the procedure of the numerical
3.2
Calculated results and discussions
First, the calculated results for the bearing Type 1 will be shown. The condition Pb=l has been adopted for all of them. It is shown in Fig.2 how the flexibility of the bearing surface affects on the static pressure distribution. With the increment of the flexibility of the bearing surface (namely the increment of the value of the flexibility number K), the variation of the pressure in the
529 compliant r e g i o n becomes less and less. F i g . 3 t e l l s t h e e f f e c t s of t h e f l e x i b i l i t y of t h e b e a r i n g s u r f a c e on t h e l o a d - c a r r y i n g capaci t y . The l o a d - c a r r y i n g c a p a c i t y of t h e b e a r i n g Type 1 i s remarkably improved w i t h t h e increment of t h e f l e x i b i l i t y of t h e b e a r i n g s u r f a c e i n comp a r i s o n w i t h t h a t of t h e c o n v e n t i o n a l r i g-i d s u r f a c e b e a r i n g , and t h i s tendency becomes more not a b l e a s t h e b e a r i n g gap i s smaller, i . e . a s t h e v a l u e of t h e f e e d i n g p a r a m e t e r r i s l a r g e r .
How t h e s t i f f n e s s v a r i e s under t h e i n f l u e n c e of t h e f l e x i b i l i t y of t h e b e a r i n g s u r f a c e i s shown i n F i g s . 4 and 5. For t h e c o n d i t i o n of n o t W
K = O(rigid)>
:ru +I .A
w
v)
10-'c 2.0 = 2,0 = 0.01
c-\
Pb R1
m
a t
Ps =
4,o
Ri
0,Ol
r
0,9 0,3
= R2 = =
a,
Ps =
4
c
0 .rl v)
F:
g
-2-01
1 10 squeeze number A Fig.5 Effects of flexibility number on stiffness-squeeze number curves (Type 1) lo-'
a .?-I
::
x10-*
-
1,s
-
N
x b
1.0
a,
E
0
-4
.d
b
a
m
a
v
Fig.6
2 0,5
I
'
' '
l L * * l l
'
'
'
=
a 2 1 t h 1
0,l '
'
"*LLJ
1 10 lo2 feeding parameter r Effects of flexibility number on coefficient of damping-feeding parameter curves (Type 1)
lo-'
0
1 10 lo2 feeding parameter r Effects of flexibility number on load carrying capacity-feeding parameter curves (Type 1)
10-l
Fig.3
c 5 1 v)
In a,
c
ru ru .rl U v) v)
10-
v)
a, r(
c
0 .r(
i/
K =
'--igid)\\
U(I
c
.r(
Q
Fig.4
' '
2,o
Rz
0,9
x
= =
\
0,Ol 0,l
\
v)
g lo-;o-~
,
Ps = Ri =
'
'
1
' '
\
'
'""I
10
' ' '
lo2
feeding parameter r Effects of flexibility number on stiffness-feeding parameter curves (Type 1)
.rl
a
Fig.7
lo-'
1
10
squeeze number A Effects of flexibility number on coefficient of damping-squeeze number curves (Type 1)
530 so large squeeze number ( A < l ) , the flexibility of the bearing surface raises notably the stiffness, but this effects turn over forthe condition of the large value of X (The value of X is usually much less than unity in the practical case.). Figs.6 and 7 are the calculated results showing the effects of the compliant surface on the coefficient of damping of the gas film. The value of the coefficient of damping goes down as the bearing surface becomes easy to deform and it may become negative at last. Next, the characteristics of the bearing Type 2 will be told below. The static pressure distribution of this type of the compliant surface bearing is shown in Fig.8. The effects of the flexibility of the bearing surface and the back pressure on the loadcarrying capacity are shown in Figs.9 and 10, respectively. It is known from these results that, for the case with the sufficiently-high back pressure, the load-carrying capacity great Y rises with the increment of the flexibility of the bearing surface, but it becomes inferior to that of the conventional rigid surface bearing n the case of no back pressure (Pb=l.O). The influences of the flexibility of the bearing surface on the stiffness are shown in
n
Ps =
a
Pb =
I
Ri
rn
.a.-
=
R2 = R3 =
r
"
Fig.8
=
Figs.11 and 12. For the case in which the back pressure added behind the thin metal plate is high enough, the stiffness of this type ofbearing is higher than that of the conventionalrigid surface bearing, and the difference between them becomes more pronounced as the flexibility of the bear.ing surface increases and/or the bearing gap decreases. For the condition of the large squeeze number, however, the stiffness is scarcely improved with the flexibility of the bearing surface. Figs.13 and 14 show how the coefficient of damping is affected by the compliance of the bearing surface. The effects of the compliant surface on the cdefficient of damping in the bearing Type 2 are not so marked as those in the bearing Type 1, and the deviations of the coefficient of damping for the bearing Type 2 from that for the rigid surface bearing are small. For the case inwhich the back pressure is high enough (Pb=Ps) and the squeeze number is rather small as shown in these figures, the coefficient of damping of this type of the compliant surface bearing may be progressed a little as compared with that of the rigid surface bearing. Meanwhile, it may be gathered from the above mentioned results that the characteristics
2.0 0.01 0.2 R 3 = 0.9 K = 50
2,o 2.0 0.01 0.2 0,9
Ps =
Ri
= R2 =
0.5
1.0
r/ro Effects of flexibility number on pressure distribution (Type 2)
feeding parameter r Effects of back pressure on load carrying capacity-feeding parameter curves (Type 2 )
Fig.10
Pb =
2.0 2.0
E
R1 =
0,Ol
h
R3'=
0,9
Ps =
pb/
i,o
c
a,
'i
Ps = Pb =
2.0 2,0
0.01 0,2 = 0,9
R1 = R2 = R3
I
N
c 10-2
a,
E
K = 0 (rigid)
.rl
Fig.9
1 10 lo2 feeding parameter r Effects of flexibility number on load carrying capacity-feeding parameter curves (Type 2)
1 10 lo2 feeding parameter r Effects of flexibility number on stiffness-feeding parameter cueves (Type 2)
lo-'
Fig.11
531
of the hearing Type 2 are strongly influenced by the existance and the magnitude of the back pressure. So, it is shown in Figs.15, 16 and 17 how they vary with the magnitude of the back pressure. The load-carrying capacity, the stiffness and the coefficient of damping for this type of the compliant surface bearing in the case of no back pressure are inferior to those for the conventional rigid surface bearing. But the formers rise with the increment of the back pressure and This tendency become superior to the latters. becomes more remarkable as the flexibility of the bearing surface increases and/or the back pressure becomes higher. For the coefficient of damping, however, there may be the optimum condition. The coefficient of damping may go down when the bearing surface becomes too easy to deform and/or the back pressure is too high. The effects of the back pressure on the gas consumption in the static condition is shown in Fig.18. The gas consumption of the bearing Type 2 in the case of no back pressure (Pb=l.O) is more than that of the rigid surface bearing, but it becomes less and less with the increment of the back pressure and/or the flexibility of the bearing surface. On the other hand, the gas consumption of the bearing Type 1 increases with the increment of the flexibility of the bearing surface. At last, an example of the bearing gap configuration in the static condition is given in
Fig.19. In the case with the sufficiently-high back-pressure, the minimum bearing gap becomes smaller in comparison with the nominal one, hg, depending on the compliance of the bearing surSo, it may be necessary to pay one's face. attention to this in the practical applications.
1F
c,
1
,K = 0
lo-1= Ps = Pb = Ri = Rz =
2,o 2.0
0,Ol 0.2
lo-' -d
Fig.14
1 10 squeeze number Effects of flexibility number on coefficient of damping-squeeze number curves (Type 2 )
U
a,
c
4-1 4-1 .rl
c, ln v)
10- 1
$ 1 ln
t
a,
1u
K = O
(rigid)
0,2
R3 =
0.9
r
c m
Fig.12
R2 =
(rigid)
0.3
=
Effects of flexibility number on stiffness-squeeze number curves (Type 2 )
ln
m
a, E
4-1
CH 0 U M
Ps =
ln *rl
Pb =
m
Ri
c
a
a,E
=
d c d
Rz =
0
R3 =
ca 5
*rl
4-1
x
mo c
=
.ri
\
2,o 2.0 0.01 0.2 0.9 0,l
K = 0 (rigid)
.I+
i
' i L L i l '
1
R2 = R3 =
3,O 0.01
50Y
0.2 0.9
0,
0
m
E
'
=
.I+
a,
0'
R1
a,
c
10-
Fig.13
m m m
2
;
a
0, 3
c,
Ps =
'
l
l
i
l
l
i
(
10
'
a
,.,,'J
lo2
feeding parameter r Effects of flexibility number on coefficient of damping-feeding parameter curves (Type 2 )
:0 , l 0' Fig.16
Y
K = 0 (rigid)
I
I
1
I
I
I
0,5
1,0
1.5
2,O
2,s
3,O
pb Effects of flexibility number on stiffness-back pressure curves (Type 2 )
532
4.
10
e
In the above, the characteristics of two types of the externally-pressurized, gas-lubricated and compliant surface thrust bearings have been theoretically derived and the effects of the flexibility of the bearing surface on them have been discussed. From the aforementioned results, the following conclusions have been obtained. In the bearing Type 1, the characteristics of the bearing are remarkably influenced by the flexibility of the bearing surface. With the increment of the flexibility of the bearing surface, the load-carrying capacity and the stiffness are strongly improved as compared with those of the conventional rigid surface bearing, but the coefficient of damping reversely goes down and may become negative. Meantime, the characteristics of the bearing Type 2 are greatly changed by not only the flexibility of the bearing surface but also the existance of the back pressure. In the case of no back pressure, the bearing performances of this type of bearing are inferior to those of the rigid surface bearing on the whole. But they are improved more and more with the increment of the back pressure, and its load-carrying capacity and stiffness in the case of the sufficiently-high back pressure (Pb=Ps) are much superior to those of the rigid surface bearing. The more the bearing surface becomes easy to deform, the more this tendency is pronounced. On the other hand, the variations of the coefficient of damping are not so marked. Judging from the aforementioned results, the bearing Type 1 may not be recommended from the point of stability in the practical applications since the marked decline of the coefficient of damping happens at the same time even though the improvements of the load-carrying capacity and the stiffness are more remarkable than those of the bearing Type 2. S o , the bearing Type 2 is superior in the practical use as the high performance bearing.
---
m .rl m a Q E
d c d
c-a 9
Ps =
3.0
R1 =
0.01 0.2 0.9
R2 =
0
R3 =
.rl VI
x
m o
ca,
r
E
0,l 0,3
= =
1,s 2,O 2,5 3,O pb Effects of flexibility number o n coefficient of damping-back pressure curves (Type 2)
Fig.17
1,0
0.5
0
-a
xl~-
91-
ffl
-3(rigid) (rig id),lh1,o o,
e,
d
Ri
= R2 = R3 = =
w
27 N
r
.rl
0,Ol 0.2 0.9 1,o
-
-
-
Pb =
/1
3.0
r(
2
CONCLUSIONS
6'
10"
'
'
'
"""
1
'
'
'
""*' 10
'
'
'
""' 10
flexibility number K Gas consumption (Type 2)
Fig.18
1,5
0
r: \
r:
I 0
R2 = R3 =
r
=
0.2 0.9 1,0
n 100
0.5
1.0
r/rg Fig.19 Bearing gap configuration (Type 2 )
References
(1) ROWE, W.B. and KILMISTER, G.T.F. 'A theoretical and experimental investigation of a self-compensating externally pressurised thrust bearing', Proc. 6th International Gas Bearing Symposium, 1974, Paper D1, 1-17 (Southampton, U.K.) (2) LOWE, I.R.G. 'Some Experimental Results From Compliant Air Lubricated Thrust Bearings', Trans. ASME, J. Lub. Tech. 1974, 96, 547-553 (3) BLONDEEL, E., SNOEYS, R. and DEVRIEZE, L. 'Aereostatic Bearings with Variable Gap Configuration', Proc. 7th International Gas Bearing Symposium, 1976, Paper E2, 21-38 (Cambridge, U.K.) ( 4 ) GRAY, S., GESHMAT. H. and BHUSHAN, B. 'Technology progress on compliant foil air bearing systems for comercial applications', Proc. 8th International Gas Bearing Symposium, 1981, 1-18 (Leicester, U.K.) (5) HAYASHI, K. and HIRASATA, K. 'Investigation on the Back-Pressured, Externally-Pressurized, Gas-Lubricated, Circular Thrust Bearing with the Flexible Surface', Proc. JSLE International Tribology Conference, 1985, 743-748 (Tokyo, JAPAN)
533
Paper XVll(iii)
An experimental investigation of the steady-state performance of a compliant surface aerostatic thrust bearing D.A. Boffey and G.M. Alder
Performance tests have been made on a n i n d u s t r i a l a e r o s t a t i c t h r u s t b e a r i n g whose s u r f a c e h a s a concave f l e x i b l e steel diaphragm. E x p e r i m e n t a l r e s u l t s are p r e s e n t e d f o r l o a d , f i l m s t i f f n e s s and f l o w r a t e v e r s u s a i r - g a p f o r v a r i o u s s u p p l y p r e s s u r e s and a l s o f i l m p r e s s u r e and diaphragm d e f l e c t i o n . The r e s u l t s show t h a t t h e c o m p l i a n t b e a r i n g h a s a s u p e r i o r performance i n terms o f l o a d c a p a c i t y and f i l m s t i f f n e s s compared w i t h a f l a t r i g i d b e a r i n g o f t h e same s i z e . The r e s u l t s a r e i n g e n e r a l agreement w i t h o t h e r p u b l i s h e d work.
1
INTRODUCTION
Gas b e a r i n g s c a n be e i t h e r c o m p l i a n t by n a t u r e , such a s w i t h f o i l b e a r i n g s o r f l o p p y disc/magnetic head s i t u a t i o n s , o r d e s i g n e d t o be d e l i b e r a t e l y s o i n o r d e r t o overcome uneveness of s u r f a c e o r enhance performance. I n t h e a r e a o f b e a r i n g d e s i g n where compliance h a s been used t o enhance a e r o s t a t i c t h r u s t b e a r i n g performance a d a t a b a s e s e a r c h revealed o n l y a l i m i t e d amount o f s u p p o r t i n g r e s e a r c h . The work o f Lowe ( 1 ) was concerned with e x p e r i m e n t s on a c i r c u l a r b e a r i n g i n which one o f t h e metal s u r f a c e s was f i t t e d w i t h a l a y e r o f r u b b e r . He showed t h a t , i n g e n e r a l , t h e c o m p l i a n t b e a r i n g had a g r e a t e r l o a d capacity than t h a t o f t h e equivalent r i g i d s u r f a c e b e a r i n g . Blondeel e t a1 ( 2 ) investigated variable air-gap geometries i n circular bearings, including the case of a compliant s t e e l membrane between t h e s u p p l y and the f i l m . They came t o t h e c o n c l u s i o n t h a t v a r i a b l e gap g e o m e t r i e s l e a d t o much h i g h e r l o a d c a p a c i t i e s when compared t o a p a r a l l e l r i g i d c o n f i g u r a t i o n . A more r i g o r o u s a n a l y s i s o f t h e s t e e l membrane c a s e i n ( 2 ) i s p r e s e n t e d by Hayashi ( 3 ) , a l t h o u g h i n t h e a c t u a l model and experiments t h e f l e x i b l e s t e e l p l a t e (membrane) is placed o p p o s i t e t h e c e n t r a l f e e d h o l e . H i s work c o n f i r m s t h e c o n c l u s i o n s i n ( 1 ) and ( 2 ) and he a l s o r e p o r t s a n enhanced f i l m s t i f f n e s s , though a t t h e expense o f mass f l o w r a t e . T h i s p a p e r i s concerned w i t h performance measurements made on a commercially a v a i l a b l e compliant s u r f a c e a e r o s t a t i c t h r u s t b e a r i n g designed f o r h i g h l o a d c a p a c i t y and s t i f f n e s s The a u t h o r s became i n t e r e s t e d i n a g e n e r a l experimental and t h e o r e t i c a l s t u d y o f t h e performance o f t h e b e a r i n g b e c a u s e a manufacturer o f c o o r d i n a t e measuring machines i n t h e United Kingdom i s i n t e r e s t e d i n e x p l o i t i n g t h e i r advantages.
2
TEST BEARING
Fig. 1 shows s c h e m a t i c views o f t h e test b e a r i n g a s s u p p l i e d , which h a s a d i a m e t e r o f 60 nun and
a c e n t r a l o r i f i c e o f d i a m e t e r 0.5 mm i n t h e diaphragm. A i r i s s u p p l i e d t o t h e chamber behind t h e diaphragm, s o t h a t i t i s t h e p r e s s u r e d i f f e r e n c e between t h e s u p p l y and f i l m which c a u s e s t h e diaphragm t o d e f l e c t from i t s i n i t i a l p o s i t i o n when i n s e r v i c e . The f l e x i b e s t e e l diaphragm h a s a c e n t r a l t h i c k n e s s o f 2.29 nun and i s i n i t i a l l y concave. The method o f manufacture i s t o g r i n d t h e o u t e r s u r f a c e o f t h e diaphragm when i t i s s u b j e c t t o a s u i t a b l e back p r e s s u r e . T h i s means t h a t t h e diaphragm thickness v a r i e s very s l i g h t l y with radius. A s u r f a c e p r o f i l e o f t h e diaphragm a s manufactured o b t a i n e d on a S o c i e t y Genevoise U n i v e r s a l Measuring Apparatus i s provided i n F i g . 2 , which shows t h a t t h e d e p t h o f t h e diaphragm a t t h e o r i f i c e i s a p p r o x i m a t e l y 22Jlm.
3
APPARATUS
A s c h e m a t i c o f t h e b e a r i n g t e s t r i g i s shown i n F i g . 3. The s p i n d l e c a r r y i n g t h e l o a d pan and w e i g h t s i s s u p p o r t e d i n two a e r o s t a t i c j o u r n a l b e a r i n g s i n o r d e r t o minimise f r i c t i o n . A b a l l b e a r i n g i s l o c a t e d c e n t r a l l y between t h e s p i n d l e and t e s t b e a r i n g i n o r d e r t o p e r m i t self a l i g n m e n t o f t h e t e s t b e a r i n g and b a s e p l a t e when t h e test b e a r i n g i s v e n t e d . F o r a i r - g a p measurments two Wayune Kerr c a p a c i t a n c e p r o b e s a r e mounted on e i t h e r s i d e o f t h e b a l l b e a r i n g , perpendicular t o t h e plane o f t h e f i g u r e r a t h e r than i n t h e plane o f t h e f i g u r e as i l l u s t r a t e d . The b a s e p l a t e houses a p r e s s u r e t r a n s d u c e r and a t h i r d d i s p l a c e m e n t probe and c a n be t r a v e r s e d p e r p e n d i c u l a r t o t h e p l a n e - o f t h e f i g u r e r e l a t i v e t o t h e test b e a r i n g . A b a r r e l micrometer p l a c e d i n t h e d i r e c t i o n o f t h e b a s e p l a t e motion a c c u r a t e l y l o c a t e s t h e r a d i a l p o s i t i o n o f t h e p r e s s u r e t r a n s d u c e r and d i s p l a c e m e n t probe. F i l t e r e d and d r i e d a i r i s s u p p l i e d t o t h e test b e a r i n g t h r o u g h a p r e c i s i o n p r e s s u r e r e g u l a t o r which m a i n t a i n s a c o n s t a n t downstream p r e s s u r e . The a i r f l o w t o t h e test b e a r i n g i s measured by p a s s i n g i t t h r o u g h a Gapmeter.
534 4
EXPERIMENTAL PROCEDURE
The e x p e r i m e n t a l programme comvrised f o u r Darts: F o r s u p p l y p r e s s u r e s o f 3 , 4 , 5 and 6 b a r gauge measurements o f a i r - g a p and f l o w r a t e were made f o r a series o f l o a d s . Although t h e b a l l b e a r i n g a l l o w e d a l i g n m e n t between t h e t e s t b e a r i n g and b a s e p l a t e when t h e test b e a r i n g was v e n t e d some m i s a l i g n m e n t was e x p e r i e n c e d on " l i f t - o f f " . T h i s was overcome by t h e u s e o f small b a l a n c e w e i g h t s t o make t h e two d i s p l a c e m e n t p r o b e s g i v e t h e same r e a d i n g . F o r a similar r a n g e o f s u p p l y p r e s s u r e s and l o a d s t h e d e f l e c t i o n o f t h e diaphragm was measured a t a d i s t a n c e o f 5 mn from t h e c e n t r e o f t h e test b e a r i n g . I t was found t o b e n e c e s s a r y t o d i s p l a c e t h e t r a n s d u c e r from t h e c e n t r e o f t h e b e a r i n g b e c a u s e dynamic e f f e c t s o f t h e a i r from t h e c e n t r a l o r i f i c e seemed t o d i s t u r b t h e d i s p l a c e m e n t measurements. The p r e s s u r e i n t h e f i l m was measured o v e r a r a d i u s from 5 t o 25 mm a t i n t e r v a l s o f 5 mm f o r s u p p l y p r e s s u r e s o f 3 , 4 , 5 and 6 b a r gauge a t l o a d s o f 125, 164, The p r e s s u r e measurements 213 and 312 N. were n o t made a t l e s s t h a n t e n o r i f i c e d i a m e t e r s from t h e c e n t r e b e c a u s e o f dynamic e f f e c t s i n t h e v i c i n i t y o f t h e orifice. I n order t o provide a b a s i s f o r comparison, tests s i m i l a r t o t h o s e d e s c r i b e d i n t h e f i r s t p a r t were conducted on a f l a t r i g i d b e a r i n g h a v i n g t h e same d i a m e t e r and o r i f i c e s i z e as t h e c o m p l i a n t bearing.
5 RESULTS AND DISCUSSION 5.1 Compliant b e a r i n g Curves o f l o a d v e r s u s a i r - g a p f o r 4 s u p p l y p r e s s u r e s are p r e s e n t e d i n F i g . 4. A t a s u p p l y p r e s s u r e o f 3 b a r gauge t o u c h down o c c u r r e d i n t h e r a n g e 550-600 N and a t 4 , 5 and 6 b a r gauge i n t h e r a n g e 600-650 N. A computer r o u t i n e was employed t o p r o v i d e l e a s t s q u a r e s f i t s t o t h e experimental d a t a u s in g polynomials o f t h i r d o r d e r . T h i s r o u t i n e a l s o a l l o w s one t o d i f f e r e n t i a t e t h e f i t t e d e q u a t i o n s and t h i s provided t h e curv e s o f f i l m s t i f f n e s s v e r s u s airgap i n F i g . 5. However, t h e a c c u r a c y o f t h i s method i s n o t known. These r e s u l t s show t h a t t h e r e i s no s i g n i f i c a n t d i f f e r e n c e i n l o a d c a p a c i t y and f i l m s t i f f n e s s f o r s u p p l y p r e s s u r e s o f 4 , 5 and 6 b a r gauge, which s u g g e s t s t h a t diaphragm t h i c k n e s s c o u l d b e a n i m p o r t a n t p a r a m e t e r and may need t o b e o p t i m i z e d f o r a g i v e n s u p p l y p r e s s u r e and a i r - g a p . The v o l u m e t r i c f l o w r a t e c u r v e s o f F i g . 6 show a continual increase i n flowrate with increase i n air- gap and s u p p l y p r e s s u r e . It seems p r o b a b l e t h a t t h e f i l m area around t h e o r i f i c e p e r i m e t e r c o n t r o l s t h e rate o f f l o w t h r o u g h t h e b e a r i n g . T h e r e i s no e v i d e n c e o f c h o k i n g a c r o s s t h e r e s t r i c t o r , a s i n ( 4 ) f o r example, which i s c o n s i s t e n t w i t h t h e f i l m p r e s s u r e measurements reported later.
The d e f l e c t i o n o f t h e diaphragm from i t s i n i t i a l p o s i t i o n i s plotted a g a i n s t air-gap f o r s u p p l y p r e s s u r e s o f 3 , 4 , 5 and 6 b a r gauge i n F i g . 7, and shows t h a t diaphragm d e f l e c t i o n i s comparable w i t h a i r - g a p . Deflection increases w i t h i n c r e a s e i n a i r - g a p and s u p p l y p r e s s u r e , which i s c o n s i s t e n t w i t h t h e d e s c r i p t i o n g i v e n i n s e c t i o n 2. High l o a d s (and hence h i g h f i l m p r e s s u r e ) c o r r e s p o n d t o low a i r - g a p s . A s high l o a d s ( a n d hence low p r e s s u r e d i f f e r e n c e s a c r o s s t h e diaphragm) c o r r e s p o n d t o low diaphragm d e f l e c t i o n s , i t f o l l o w s t h a t low d e f l e c t i o n c o r r e s p o n d t o l o w * a i r - g a p s . Using t h e r e s u l t s o f F i g . 3, which shows a n i n i t i a l c o n c a v i t y o f a b o u t 22,Um, t h e d e f l e c t i o n s o f F i g . 7 i n d i c a t e t h a t t h e diaphragm becomes convex a t low l o a d s . F i g s . 8 and 9 show t h e measured s t a t i c f i l m pressure a g a i n s t bearing radius f o r supply p r e s s u r e s o f 3 , 4 , 5 and 6 b a r gauge w i t h f i x e d l o a d s o f 125 and 312 N r e s p e c t i v e l y . These r e s u l t s are c o n s i s t e n t w i t h e x p e c t a t i o n s from t h e p o i n t s o f view o f s u p p l y p r e s s u r e , l o a d and r a d i u s . It i s i n t e n d e d t h a t t h e s e r e s u l t s w i l l provide f u t u r e checks wi t h theory. I f t h e p r e s s u r e c u r v e s are e x t r a p o l a t e d towards t h e c e n t r e of t h e bearing i t i s apparent t h a t the r a t i o o f t h e absolute film pressure t o supply pressure across the r e s t r i c t o r is greater than the c r i t i c a l r a t i o , so the bearing r e s t r i c t o r is unchoked. During t h e s t e a d y - s t a t e e x p e r i m e n t s t h e r e was no e v i d e n c e o f any pneumatic hammer instability. 5.2 Comparison o f c o m p l i a n t and f l a t r i g i d bearings I n t h e compliant b e a r i n g t h e d e f l e c t i o n o f t h e diaphragm a f f e c t s t h e l o a d c a p a c i t y , f i l m s t i f f n e s s and f l o w r a t e f o r two r e a s o n s :
As t h e b e a r i n g i s i n h e r e n t l y compensated i t alters t h e a r e a c o n t r o l l i n g t h e flow t o t h e
(1)
film.
(2)
The p r e s s u r e w i t h i n t h e f i l m i s dependent upon t h e t o t a l a i r - g a p .
By a s u i t a b l e d e s i g n o f diaphragm t h e l o a d c a p a c i t y c a n be maximised and a n optimum a i r gap f o r maximum f i l m s t i f f n e s s e f f e c t i v e l y e l i m i n a t e d . These a d v a n t a g e s are demonstrated i n t h e comparison o f l o a d c a p a c i t y and f i l m s t i f f n e s s made between t h e c o m p l i a n t b e a r i n g and t h e f l a t r i g i d b e a r i n g o f s i m i l a r geometry i n F i g s . 10 and 1 1 , which are f o r a s u p p l y p r e s s u r e o f 5 b a r gauge. However, F i g . 12 shows t h a t t h e a d v a n t a g e o f l o a d c a p a c i t y and s t i f f n e s s i s a t t h e e x p e n s e o f f l o w r a t e . These f i n d i n g s are i n g e n e r a l a c c o r d w i t h t h e work r e p o r t e d i n ( 2 ) and ( 3 ) .
6 1.
CONCLUSIONS Experimental curves o f l o a d , f i l m s t i f f n e s s and f l o w r a t e v e r s u s a i r - g a p have been o b t a i n e d f o r a n i n d u s t r i a l a e r o s t a t i c t h r u s t b e a r i n g with a compliant steel s u r f a c e .
535 2.
3.
The r e s u l t s i n d i c a t e t h a t t h e compliant surface can be used t o i n f l u e n c e t h e load and s t i f f n e s s c h a r a c t e r i s t i c s and enhance them compared with a f l a t r i g i d design of t h e same size. The experiments have only given a n i n s i g h t i n t o what i s a complicated problem. What i s required a r e comprehensive t h e o r e t i c a l steadys t a t e and s t a b i l i t y s t u d i e s o f a l l of t h e parameters involved i n o r d e r t o optimize a design f o r given s e t o f o p e r a t i n g conditions.
References LOWE, I.R.G. 'Some experimental r e s u l t s from compliant a i r l u b r i c a t e d t h r u s t b e a r i n g s ' , Trans. ASME S e r i e s F, Val. 96, 1974, pp 547-560BLONDEEL, E., SNOEYS, R and DEVRIEZE, L., ' E x t e r n a l l y pressurized b e a r i n g s with v a r i a b l e gap geometries', Proc. 7 t h I n t e r n a t i o n a l Gas Bearing Symposium, Cambridge, UK, J u l y 13-15 1976, paper E2.
' I n v e s t i g a t i o n on HAYASHI, K. externally pressurized, gas-lubricated, c i r c u l a r t h r u s t bearing with f l e x i b l e s u r f a c e ' , Proc. 8 t h I n t e r n a t i o n a l Gas Bearing Symposium, L e i c e s t e r , UK, 8-10 April 1981, paper 1 . BOFFEY, D.A., WADDELL, M , and DEARDEN, J.K. ' A t h e o r e t i c a l and experimental s t u d y i n t o t h e s t e a d y - s t a t e performance characteristics of i n d u s t r i a l a i r l u b r i c a t e d t h r u s t b e a r i n g s ' , Tribology I n t e r n a t i o n a l , Vol. 18 No. 4, 1985, pp 229233
536
fi I
Fig. 1
Schematic of compliant test bearing
Fig. 3
,Load
Ill I
1
Schematic of bearing test rig
3bar 0 4
5-
+
Z N.
x 5
b4-
06
X
B0 3-I
21-
1 Radius mm
Fig. 2
Compliant bearing surface profile
0
I
I
10
I
I
20 Air-gap
Fig. 4
I
I
30
I
I
40
pm
Load versus air-gap for various supply pressures
537
40 40 -
4
E
-
a
-
C
.O 30c,
30 -
E
\
U
b, Y-
U
-
5- 2 0 -
€200)
0
Y,
L
Ul
b,
C Y-
l= Q
-
0
+ .-
2
-
b,
0 10-
10-
I
1 0
I
1 10
I
I
I
20
I
1
30
Air-gap prn
Fig. 5
Film stiffness versus air-gap for various supply pressures
/ Fig. 6
I
0
10
1
I
20 Air-gap
40 Fig. 7
I
I
30 pm
I 40
Diaphragm deflection versus air-gap for various supply pressures
t\
04 x5
06
Flow rate (ntp) versus air-gap for various supply pressures
0
5
Fig. 8
Film pressure versus radius for various supply pressures Load = 125 N
15 20 10 Radius mm
25
538
40t
.
3 3 bar 04 x 5 +
(5,
L
06
0
n
-2 CJ
L
3 ul ul
CJ
L
Q
E iil 0
10
20 Air-gap
Fig.11 1
1
0
1
1
1
1
10
5
1
1
1
20
15
25
30
prn
40
Film stiffness versus air-gap. Comparison of compliant and rigid surface bearings. Supply pressure = 5 bar
Radius mm Fig. 9 Film pressure versus radius for various supply pressures Load = 312N
.-c
5-
E
\ U
-
-. U
a
4-
c,
t
z N 1 -
0 c
v
-
CJ c,
0 L
3-
3 0 iz
x
u
Q
-
2 21-
t I
0
1
I
10
I
I
I
20 Air-gap
I
I
30
pm
Fig. 10 Load versus air-gap. Comparison of compliant and rigid surface bearings. Supply pressure = 5 bar
20 30 Air-gap prn
0
10
Fig. 12
Flowrate (ntp) versus air-gap. Comparison of compliant and rigid surface bearings. Supply pressure = 5 bar.
40
I
40
539
Paper XVll(iv)
The effect of finite width in foil bearings: theory and experiment J.G. Fijnvandraat
This paper deals with the effect of the finite width of a foilbearing. The film is described by the 2-D Reynolds equation, the foil by a I-D membrane equation. These equations are then put in nondimensional form. First an analytical approximation is given in the case of a large width and a large wrap angle. Then an approximative I-D Reynolds equation is derived, which accounts for side leakage. This equation is solved numerically. Finally the 2-D Reynolds equation is solved numerically and comparisons are made. Experiments have been done also and the calculations are compared with them.
I INTRODUCTION Very much work is done on the infinite wide foil bearing. Eshel and Elrod [I] analysed the case with uncoupled inlet and outlet region. They also treated the influence of bending stiffness (21. Eshel [3]. also together with Wildman [4], analysed the transient response on disturbances. In [S] Wildman summarized the influence of several parameters on the bearing characteristics. The edge effects also have been studied. Eshel and Elrod [6] have made a theoretical investigation of the undulations that will appear at the edges. These undulations have been measured by Licht [7]. The purpose of this paper is to show the influence of side leakage on the film thickness for a symmetric foil bearing of finite width. This side leakage is accounted for in much the same way as is done in [8] for journal bearings. 2 A FOIL BEARING OF FINITE WIDTH
The equation for an incompressible film between foil and cylinder in the 2-dimensional case is
in which p is the difference with ambient pressure. At the sides of the bearing the pressure equals the ambient, so p = O . y = f Yzb.
(2)
Fcr x approaching infinity the pressure tends to ambient, so (3)
p + o . x + +oo.
2.2 Stress eauation When the tensile stiffness of the foil is very high. the form of the foil may be approximated with a developable surface (curved in one direction only). Because of symmetry this means that the foil is bent in the x-direction only, so h depends on x only. The equilibrium condition then becomes
in which h
F(.r) =
P(X,.V)dY. - %h
The curvature of the foil I/p can be approximated by (cf.[l])
For x going to infinity the foil becomes straigth. This condition leads to
Fig. 1
Picture of Foil Bearing
A foil bearing consists of a foil, wrapped around a rotating cylinder (fig. I). The relevant parameters are. the following: a : wrap angle : angular velocity w : radius of cylinder r b : width of foil N : foil force q : viscosity We further define the following constants
T
=
N/h
,
C
=
6qwr.
What we are interested in are: h : film thickness p : pressure in the film 2.1 Reynolds eauation
h=
(x T Xar)’
2r
+ O(I) , x
+
fM.
By defining w ( x ) = - h ( x ) equation (4) together with the Zr boundary condition can be written as
P(x) = Td2a
*
,
(5)
d.r
3 EQUATIONS IN NONDIMENSIONAL FORM
*
When we define the following nondimensional quantities E =
x = - rEr
Y = rEP
B = rch
540 constant. So in fact this approximation can be thought ol as a and Petrov-Galerkin method2"-2with basis functions 1 , using n = 1. weight functions
(%)
(T)
4.3 Numerical solution equations ( I ) and (5). together with the boundary conditions (2),(3) and (6). can be rewritten to
-
P(X)= I
d2H
d2W
dX2
dX2
--=-.
The combined equations (14) and (8) are solved by the finite element method. The boundary conditions at infinity are replaced by conditions on X = +L. L then must be chosen large ensugh. The interval [-L,L] is divided into N equal elements. Now P and H can be approximated by N+ I
N+ I
with boundary conditions in which the di are linear basis functions. Multiplying (14) by t$i for i = I , ... ,N - I and integrating by parts result in
P-+O.X++oC, P=O. Y = l % B ,
=-+
% A , X -+ k 00 .
dX
4 APPROXIMATION O F FINITE WIDTH EFFECT
(15)
4.1 Large width and large wraD angle When the wrap angle is large (ABI), in the largest part of the wrap region the film thickness vanes only slightly, so (cf. (8)) a regionexists between inlet and outlet with nearly constant pressure (P(X)rl). Then in (7) the first term can be neglected, so (also using that H is independent of Y)
Dividing by
R shows that J??
2
P(X. Y) = .f(m (1 - (7) 1. Because yields
P-1 f(X)
=
PO
= PN =
0.
Doing the same with (8) for i = 0, ... ,N yields
depends on X only, so (using (9))
aYz
2Y
The essential boundary conditions are
3/2. Substituting this in (12) and solving
H(X) =
Defining a Reynolds-stress element with nodal degrees of freedom P and H and using 1 point integration rule, the element contribution to the system of discrete equations is
(13) 24H;(X+
%A)
H, is of order unity. And when the finite width does not influence the inlet region H, = 0.643 (cf.[l]). DiKerentiating (13) twice yields L
J
which =,).( % ((.), + (.)2) and A = 2 L / N and with u, = ( P , ,H, , P2,H2)rbeing the element d.0.f. vector. For elements l and N we get an extra (boundary) term -%A in the second and fourth component respectively. The stress equation is linear, the Reynolds equation is not. The combined non-linear equations are solved by a Newton in
SO
5 12 2 W,,,= 3Hp(-) .
B2 So, if 849144 then H"g1. Then, because of (8). also the as-
sumption of constant pressure is justified. We define a characteristic width as the width for which H halves over the wrap angle. Then from (13) the following expression for this width can be derived:
Raphson method. The element "stiffness" matrix comes
Bchm = H c 6 . When B28B:,+,, the influence of side leakage is negligible in the wrap region. 4.2 ADDroximation for Dressure PIx.Y) Because of the foregoing a good approximation for P seems to be P(X,Y) = P ( x ) . -(I3 - (2Y T2 )). 2 Substituting this in (7) and integrating with respect to Y yields d ( H - - I) - -~H dX dX
F
12 ~2
3-p = - . d H
dX
This equation shows that for B2%12the inlet and outlet behaviour is not influenced by the finite width of the foil. As a matter of fact the parabolical approximation for P can be regarded as the first 2 symmetrical terms of a Taylor series expansion, whereas the Reynolds equation is weighted with a
For elements 1 and N the Reynolds boundary conditions have to be taken into account.
541
K 1
-60.0
-40.0
-20.0
20.0
0.0
60.0
40.0
In fig. 2 the influence of the width is shown for a large wrap angle (A = 100). The calculations are done for B = m, B = Bchu,, B = G , B = f i and B = G . For B > f i the inlet region remains unchanged.
-Pm A
1
I= I
w2- w l +A
in which k runs from 1 to K. The element matrix remains of similar form as in 4.3, only the matrix elements K l , , K13,KI1and K,, now become diagonal submatrices with similar diagonal terms. In the same way K12, KI4.KI2and K , become columns and K2, , K23, &, and become rows. 5.3 Comparison between approximation and 2-D solution
5 SOLUTION O F 2-D REYNOLDS EQUATION
For comparing the approximated solution with the real solution the fu!l 2-D Reynolds equation is solved also numerically. For approximation in the Y-direction a series expansion is used.
For the same value for A as in fig. 2 also calculations with full 2-D Reynolds are performed. In fig. 3 the difference between these solutions and the approximation is shown.
5 . I Series expansion in Y-direction The pressure P can be expanded with respect to Y in several ways and also different sets of weighting functions for the Reynolds equation can be used. The choice as indicated in 4.2 seems a natural one. But with this set of basis and weighting functions, and also with other sets of polynomial functions, the resulting matrix appears to be very ill-conditioned for larger sets (N>5). For this reason Fourier series are used, so P is approximated by
Lv,
in which
81 0 0
dl
By this choice boundary condition (10) is fullfilled automatically. Because the basis functions are orthogonal, the substitution of (17) in (7) result in K uncoupled (with respect to the Pi's) equations: d dX
3dPk dX
-(H-)-(
(2k- 1)n
)
2
H 3Pk
p k = - d2 . W
d H ,(18)
(2k - I)n dX
5.2 Numerical solution The coupled equations (18) and (19) can be solved in the same way as in section 4.3. Pk and H a r e approximated by NI I
N II
j=O
j=O
Fig. 3
1
lb
I
I
u
I I , ,
12
I
t
i
B Approximotion versus 2-D solution
I
1
Ha(X) - H2D(X) , in which Ha means the 2(Hn(X) + H2D(X)Y)) approximated solution. As was to be expected the approximation has a finite error for B + 0 , whereas the error goes to zero when B --* M. For B > 10 the error is smaller than 10 %. The maximum error is reached normally in the outlet region. In the wrap region the error is much smaller.
is max
*
(19)
dX2
k= I
1 , 1 1 1 1 1
The measure along the vertical axis for the difference in solutions =
The substitution of P in (8) results in a coupling: 2
,
lbo
Defining the nodal degrees of freedom as PI. ... , PK,H , the element r.h.s. now becomes
6 EXPERIMENTS
Experiments are performed on a rotating cylinder, with a foil wrapped around it. The drum speed was varied by a factor of 4. The airfilm is measured with a Photonic Sensor, which is mounted on a device that rotates concentrically with the axis of the drum. The results for 1, 2 and 4 times a reference speed are shown in fig. 4. The scaling of h as well as x is affected by the speed. For the sake of comparison all measurements are scaled with the factors belonging to the reference speed.
542
1 K-
- 4Wr.f
I
........
2Wrmt
-measurement
I
- - - calculation
-
Lo-
I 1: 1: ’.
-
n
I I
._.._
Je-
.......
I
v
........... ........
‘--- - - - - _ _ _
I
................... ..........
- - - - - - - _ - _ - - - _ -- -_
2-
-‘I
9
dl
-70.0
Fig. 4
-35.0
0.0
35.0
70.0
The results evidently show that the airfilm is thicker for higher speeds. Also the influence of the side leakage is clearly seen. This influence is relatively larger for higher speed. The characteristic undulations in the exit region for this kind of bearing can be seen very clearly. 6.1 Comuarison between calculation and exwriment In fig. 5 the comparison between calculations and experiments is shown.
-measurement - - - - calculation
Y
I
-
to
3.v
I
Y
0
I
0.0
1
1
-35.0
35.0
0.0
x Fig. 5a
I
0.0
I
35.0
1
70.0
The agreement is fairly good, especially at lower speeds. For higher speeds the measured film is thicker than the calculated one. This is expected to be the result of cupping of the foil in transversal direction. Because of the cupping a better sealing exists. That results in a seemingly larger width of the foil: the comparison is very good for calculations with B is 1.5 times the real B. The nondimensional A and B are: A w B 4, 20.3 125. 17.3 We/ 18.1 99.3 13.7 2 0,tr 10.9 16.1 78.8 4 wrtr In the table also the characteristic width is listed, which is larger than the real width, so in the calculations the film thickness diminishes more than a factor 2 over the wrap angle. In the measurements this is only true for the highest speed. 7 CONCLUSIONS
n
8
I
-35.0
X ( wrmi ) Fig. !jC Comparison for 4wrmf
X ( ormi ) Influence of o : measurement
K
-70.0
(
wrei
I
70.0
)
The coupled Reynolds and stress equations can be solved simultaneously via a Newton Raphson procedure. The equations are discretised, using a Fourier series expansion as well as the finite element method. For a large width the 2-dimensional Reynolds equation is approximated very well by a I-dimensional one which accounts for side leakage. For large wrap angles an analytical expression is derived for the variation of the film thickness in the region between inlet.and outlet. The measurements that are performed, are in fairly good agreement with the theoretical results. In the future attention will be paid to the effect of cupping of the foil.
Comparison for wref
-measurement - - - - calculation
-
Y
-
n
-j’s
v
I
a 01 I
d.0
-70.0
-35.0
Fig. gb
X ( Qref ) Comparison for 2wrmi
35.0
70.0
References ESHEL. A. and ELROD, H.G.jr. ‘The Theory of the Infinitely Wide, Perfectly Flexible, Self-Acting Foil Bearing’, Trans A.S.M.E., J. Basic Eng., 1965, pp. 831-836. ESHEL, A. and ELROD. H.G.jr. ’Stiffness Effects on the Infinitely Wide Foil Bearing’, Trans A.S.M.E., J. Lubr. Techn., 1967, pp. 92-97. ESHEL. A. ’The Propagation of Disturbances in the Infinitelv Wide Foil Bearing‘. Trans A.S.M.E.. J. Lubr. TecGn., 1969, pp. 120-125.- ’ ESHEL. A. and WILDMAN. M. ’Dvnamic Behaviour of a Foil in the Presence of a Lubriiating Film’, Trans A.S.M.E., J. Appl. Mech., 1968, pp. 242-247. WILDMAN, M. ’Foil Bearings’, Trans A.S.M.E., J. Lub. Tech., 1969, pp.37-44. ESHEL, A. and ELROD, H.G.jr. ’Finite Width Effects on the Self Acting Foil Bearing’, Report 6, Lubrication Research Laboratory, Columbia University, New York. LICHT, L. ’An Experimental Study of Elastohydrodynamic Lubrication of Foil Bearings’, Trans A.S.M.E., J. Lubr. Techn., 1968, pp. 199-220. SHELLY, P. and ETTLES. C. ’A Tractable Solution for Medium Length Journal Bearing’, Wear, 1970, pp. 221-228.
SESSION XVIII SEALS Chairman: Professor D. Berthe
PAPER XVlll(i)
The influence of back-up rings on the hydrodynamic behaviour of hydraulic cylinder seals
PAPER XVlll(ii)
Study on fundamental characteristics of rotating lip-type oil seals
PAPER XVlll(iii)
Influence of pressure difference and axial velocity on a spiralgroove bearing for a moving piston
PAPER XVlll(iv) Elastohydrodynamic lubrication of an oil pumping ring seal
This Page Intentionally Left Blank
545
Paper XVIII(i)
The influence of back-up rings on the hydrodynamic behaviour of hydraulic cylinder seals Hans L. Johannesson and Elisabet Kassfeldt
The main part of a hydraulic cylinder seal of compact type is the soft seal element. This is the part that usually is taken into consideration when calculating hydrodynamic properties in the seal contact like oil film thickness, leakage flow and friction forces. In this work a symmetric piston seal with back-up rings is analysed. Calculations of oil film thickness, leakage flow and friction forces are carried out using the inverse hydrodynamic theory. Measured pressure distributions, for either the whole seal including the back-up rings, for the soft seal element or for one back-up ring at a time are used as input data. The purpose of this investigation is to determine how the oil film thickness, the leakage flow and the friction forces in the three contact zones are influenced by the back-up rings and the main seal element being coupled in series. 1
INTRODUCTION
The main part of a hydraulic cylinder seal of compact type is the soft seal element. This is usually the only part taken into consideration when calculating hydrodynamic properties in the, seal contact like oil film thickness, leakage flow and friction forces. Johannesson deals in the works ( 5 ) , ( 6 ) and ( 8 ) with this problem using the inverse hydrodynamic theory presented by Block ( 1 ) in 1963. The same method is used in works presented by Fazekas (2), Hirano and Kaneta ( 3 ) and Olsson ( 9 ) . The inverse hydrodynamic theory requires a known pressure distribution in the seal contact as input data. This pressure distribution is inserted in the Reynolds' equation, and then the oil film thickness, the leakage flow and t.he friction force can be calculated. Assumed or measured pressure distributions have been used in the works by Fazekas, Hirano and Kaneta and Olsson. Methods to calculate the pressure distribution in the contact zone are presented by Johannesson ( 4 ) and Johannesson and Kassfeldt (7). In ( 4 ) a semi-empirical method for the calculation of the pressure distribution in an O-ring seal contact for arbitrary sealed pressures is presented, and in (7) an approximate analytical method, for calculation of the pressure distribution in an arbitrary elastomeric seal contact, is suggested. In both these papers measurements verifying the calculated pressure distributions are also presented. When pressure distribution measurements are carried out for seals of compact type it is found that the pressure over the hard back-up rings is much higher than over the softer main seal element. Over the back-up rings there also exist very high and sharp pressure peaks. These results show that the back-up rings must be of importance for the hydrodynamic behaviour of the seal.
In this work a symmetric piston seal with backup rings is analysed. Calculations of oil film thickness, leakage flow and friction forces, using the inverse hydrodynamic theory, are carried out. Measured pressure distributions, for either the whole seal including the back-up rings, for just the soft seal element or for one back-up ring at a time are used as input data. The hydrodynamic pressure build up is neglected compared to the static pressure in all contact zones. Further, pumping-ring effects and pressure build up between the rings do not occur.. 1 . 1 potatiw
a
Diameter (m)
F
Friction force (N)
C
Maximum pressure gradient (N/m 3
h
Oil film thickness (m)
I
Contact point number ( 1 )
J
Integral value
1
Length of clearance (m) Pressure (N/m2)
P
Q
(1)
Leakage flow (m3 1
t
Length coordinate
U
Sliding velocity
X
Length coordinate (m)
Y
Length coordinate (m)
z
Length coordinate (m)
r
"Seal f a c t o r "
(N)
546 6
Parallel clearance gap (m) 2
Pynamic viscosity (Ns/m
Po
P
=
)
Itid ices :
-
(dimensionless pressure)
pi
uo
=
lP u 2
(dimensionless sliding velocity)
6 Pi
0
Pimensionless quantity
1
High pressure side
h O
2
Low pressure side
M
"Motor case"
h
dimensionless film t.hickness)
= -
6 X
dimensionless length coordinate)
xo = -
1
max Maximum . I
P
"Pump case"
r
Resulting
t
Total
Further - the notations and " are used. No such notation refers to the seal clearance. ' refers to the parallel clearance at the high pressure side or the high pressure chamber of the hydraulic cylinder. " refers to the parallel clearance at the low pressure side or the low pressure chamber of the hydraulic cylinder. 2
(dimensionless parallel clearance length - high pressure side) (dimensionless parallel clearance length - low pressure side) The dimensionless leakage flow and friction force can be written:
THEORY
. . 2.1 a o r t descriution of the inverse hvdrodvnamic theorv The assumptions made in references 5) and ( 8 ) are also made here, i.e.: ( 1 ) The oil in the seal contacts is Newtonian.
(2) The flow through all clearances is laminar, isoviscous and two-dimensional. (3) The formation of oil films does not effect static pressure distributions i the seal contacts. ( 4 ) At least one pressure maximum always exists somewhere in the seal contact. (5) Starvation does not occur, i.e. the seal contacts are well lubricated.
F
2
=-
...(2.2)
* F
ndp;l,
Positive xo-direction is in the direction towards the low pressure side of the seal. In this direction the resulting dimensionless leakage flow in the "Pump case" can be written : QPOr =
...
r~~
The total dimensionless friction force acting on the cylinder tube against the motion of the tube in this case is
For a piston seal in a cylinder with recipro-
cating motion two different cases have to be analysed separately. These two cases are denoted the "Pump case" and the "Motor case'' respectively. In the "Pump case" the sliding velocity is directed towards the low pressure side of the seal and in the "Motor case" it is directed towards the high pressure side of the seal. In reference (6) all possible working conditions for piston seals and piston rod seals in hydraulic cylinders have been investigated. It is shown that each one of the possible cases is either a "Pump case" or a "Motor case". Here a case with a piston seal will be treated. It is assumed that the cylinder tube is moving with a reciprocating motion, i.e. both the "Pump case" and the "Motor case" must be considered. According to reference (5) dimensionless quantities (index 0 ) are defined as follows:
1 1 1
1
U
- rpo(i' - - + I 3
A u
dxO -L
5 )
..( 2.4)
hn u
In the "Motor case' the resulting dimensionless leakage flow in positive x -direction, and the total dimensionless fricti8n force counteracting the motion of the cylinder tube become *MOi = 'MO
...(2.5)
547
In this work, a measured pressure distribution of a piston seal is used as input data. In addition to the soft main seal element in the middle, the two hard back-up rings surrounding the main seal element are considered. The pressure distribution can be seen in figure 2 . 1 below.
1
'MO
FM@t
1' 0
dxO
t 1" t j - ) t
2
0
0 ho 1
+
HUO 1 ' t
1"
0
0
dxO --
+ 0
)
. . (2.6)
ho
The demands on the pressure distribution stated in reference ( 6 ) are: ( 1 ) There must be pressure maxima in the
contact zone.
The quantities Tpo and TMO appearing in the equations above are the so called "seal factors" defined in reference (5). Here it is shown that
rpo = 4
-?I
( 2 ) The hydrodynamic pressure, built up at the
...(2.7)
and
...(2.8) is to be interpreted as the maximum positive pressure derivative in the contact pressure distribution. G is to be interpreted as the modulus ofM?he maximum negative pressure derivative in the contact pressure distribution.
The oil film thicknesses in the different seal contacts are calculated in the same manner as in reference ( 5 1 , 1.e. at each point in a seal contact one of the following equations must be solved :
beginning and the end of the seal contacts, shall coinside with the static pressure distribution. ( 3 ) The pressure derivative must have extreme values on both sides of the back-up rings and the main seal element. ( 4 ) The pressure derivative shall be zero on both sides of the back-up rings and the main seal element. The contact length for one part or the whole seal is defined as the distance between two such end points where the derivative is zero. These demands are met by the used pressure distribution in figure 2 . 1 . This is the measured static pressure distribution over the main seal element and the two back-up rings of a piston seal. The pressure at each point in the contact has been measured with a method presented by Johannesson in the references ( 4 ) and (5). The aim of this investigation is to find out how the back-up rings influence the hydrodynamic behaviour of the seal. Therefore the whole pressure distribution in figure 2 . 1 or different parts of it are used as input data to the computer program in four different calculations. The different cases are: A The whole pressure distribution p(x).
B The pressure distribution in the main seal element contact p(y). C
The pressure distribution in the back-up ring contact at the high pressure side P(t).
D
The pressure distribution in the back-up ring contact at the low pressure side P(Z). and z
The appropriate conditions for selection of the correct roots are given in reference ( 5 ) .
The different length coordinates x , are defined in figure 2 . 1 below.
All numerical calculations have been carried out using a computer program mainly developed in the work presented in reference ( 5 ) .
In all four cases the sealed pressure is 14 MPa, and the oil viscosity p is 0.075 Ns/m . Each case is run with three different sliding velocities 0 - 0.01 mJs, 0.05 m/s and 0.25 m/s. The geometrical data are as follows:
2 . 2 The D-ure
.
.
.
distribution - input data
can be seen in reference ( 5 1 , (8), and in the previous chapter, the whole method for calculating the leakage flow, t.he friction force and the o i l film thickness relies on the existence and availability of GPO and GMO. The total pressure derivative distribution is d l s o necessary to know. These quantities can be determined if the pressure distribution in the seal contact region is known.
As
y, t
- Parallel clearance gap 6
= 50 v m . - Parallel clearance length at the high pressure side I' = 0.010 m. - Parallel clearance length at the low pressure side 1" = 0.010 m.
,
In each program run, both the "Pump case" and the "Motor case" are treated as mentioned above.
548
other back-up ring at the low pressure side, which governs the leakage flow in figure 3.2. With the back-up ring alone, the inlet pressure slope of this f'lrst back-up ring is governing the leakage.
20.0
-
I I I
16.0
(2.0
I I I
I 'I I I I
-
In the figures 3.5 and 3.6 the oil film thickness results are shown for the main seal element alone. Here the oil films are much thicker than in the main seal element part of the curves in figure 3.1 and figure 3.2. The reason is that the leakage is now governed by the slopes of the pressure curve in the main seal contact, which are smaller than the maximum slopes in the whole contact. Note that the film thicknesses and shapes are the same as one would get in the main seal contact with drain grooves on the back-up rings.
-
1 8.0
4.0
-
-
*
I
0.0 0.0
1 2.0
Figure 2.1.
3
I
I
I 1 1
4.0
X
I 6.0
I
I
e.0
1
I 10.0
I
I 12.0
I
I 14.0
x
Pressure distributions in the seal contact.
RESULTS
The back-up ring influence on the oil film thickness, the leakage flow and the friction forces in the three contact zones is studied. Note that holding constant pressures between the back-up rings and the main seal element, 10 MPa at the high pressure side and 0 at the low pressure side, corresponds to a real case with either drain grooves on the back-up rings, or a case where the leakage flow allowed through the main seal element contact is always greater than the flow allowed through the back-up ring contacts. Calculated results for each single contact are compared to calculated results for the three parts coupled in series. 3.1 T h e e n c e of the h . c k . c k z ~ e
In the figures 3.1 and 3.2 the calculated oil film thickness results for the whole compact seal contact are shown. Here the soft main seal element always allowes the small leakage flow coming from one of the back-up ring clearances to leak through without influencing it. The magnitude of the leakage flow is totally governed by the pressure slopes of the back-up rings. The oil films are, as can be seen, very thin and almost constant throughout the contact for each sliding velocity. The oil film shapes have;as expected, the minimum film thickness at the outlet (near x = 1 in the "Pump case"'and near x = 0 in tfle 'Motor 0 case"). In the figures 3.3 and 3.4 the calculated oil film thickness for the back-up ring contact at the high pressure side are shown. In the "Pump case" the oil film thickness curves are identical with the left inlet part of the corresponding curves in figure 3.1 as this back-up ring is governing the total leakage flow. A corresponding comparison of the curves in the figures 3.2 and 3.4 concerning the "Motor case" shows that a single back-up ring gives a somewhat thicker oil film. The reason is that the inlet pressure slope of this ring is smaller than the inlet pressure slope of the
The calculated oil film thickness results for the back-up ring at the low pressure side are shown in the figures 3.1 and 3.8. In the "Motor case', the oil film thickness curves are identical with the right inlet part of the corresponding curves for the whole contact, as can be seen by comparing figure 3.2 and figure 3.8. In the 'Pump case" the curves in figure 3.1 are somewhat higher than the inlet part of the curves in figure 3.1. The reason is of course the difference in inlet pressure slope near = 0 and z = 0. In other words - in the Ifflotor case" ehe leakage flow is governed by the back-up ring at the low pressure side both in figure 3.2 and in figure 3.8. In the "Pump case" the leakage flow is governed by the back-up ring at the high pressure side in figure 3 . 1 , and by the back-up ring at the low pressure side in figure 3.1. In some of the treated cases the films are extremely thin, sometimes of the order 0.1 urn. The surface roughness must then be even smaller, and this is not realistic in a practical situation. Nevertheless, in the pure hydrodynamic case the influence of t.he back-up rings is shown by the results. By governing the leakage, these rings can make the seal work in the mixed lubrication regime, thus causing even greater friction losses than those calculated here.
hm 2.0
E-2
PoralI*I
0.0
0.2
Figure 3.1
0.4
eI*aranco
0.6
eap
= 50.L-6
0.e
m
1.0
xo
Dlmensionless oil film thickness in the "Pump case" as function of dimensionless length coordinate for the whole contact.
549
h
m
2.0
1 PwrwlImI clmarancm gwp = 50.E-6
m
1 7 1, ,,;; I
1.2
=
\,
0.;
,
m/:
U .
0.064.
u=
0.01 m / w
J
,
I
0.2
0.05
m/s
U
= 0.01
m/s
-
0.0
0.0 0.0
U
0.6
0.4
0.0
1.0
0.2
0.0
no
0.4
0.6
1.0
0.0
Yo
Figure 3.5 Dimensionless oil film thickness in the "Pump case" as function of dimensionless length coordinate for t.he main seal element.
Figure 3.2 Dimensionless oil film thickness in the 'Motor case* as function of dimensionless length coordinate for t.he whole contact.
l.0
"m 2.0
6.0
1
E-2
ParalI*I
clmaranc.
gap
=
50.E-6
m
5.0
2.0 0.6
-
u =
-
0.0
I
I
0.2
Figure 3 . 3
.o
= 0.01 m/m
U I
0.0
1
0.06 J. I
I
I
I
I
0.0
0.6
0.4
I
0.0
'
1.0
to
Dimensionless oil film thickness in the "Pump case" as function of dimensionless length coordinate for the back-up ring contact at the high pressure side.
1&
0.0
I
I
0.0
0.2
Figure 3 . 4
u = U
1
= 0.01
1
0.4
I
1.
! 0.2
= 0.05
U
1
u
= 0.05
u * 0.01
0.6
I
I
0.0
0.0
I
1.0
1
n/.
0.6
0.4
m/.
I
I
0.25 m/s
U
0.0
1.0
Yo
Figure 3.6 Dimensionless oil film thickness in the "Motor case" as function of dimensionless length coordinate for t.he main seal element.
0.4
0.05 m/s
-
I
0.01 m/s
/
0.0
I: 0.6
=
U
I
0.0
I
0.2
1
1
0..
I
-/.
I
m/. I
0.6
I
I
0.0
Y 1.0
.O
0
Dimensionless oil film thickness in the "Motor case" as function of dimensionless length coordinate for the back-up ring contact at the high pressure side.
Figure 3.7 Dimensionless oil film thickness in the "Pump case" as function of dimensionless length coordinate for the back-up ring contact a t the low pressure side.
550
h
m
2.0
E-2
0.4
Parallel
1
.
L
0.0
I
I
I
0.2
0.0
Fj.gure 3 . 8
~ I m a r a n c bg a p
u =
0.05 In/.
u =
0.01 In/.
I
I
0.4
I 0.6
=
50.E-6
m
I
I
I
0.e
I 1.0
0.0
ZO
Dimensionless oil film t.hickness in the "Motor case" as function of dimensionless length coordinate for the back-up ring contact at the low pressure side.
3 . 2 The influence of the back-up rinqs on the
Figure 3 . 1 0
The curves clearly show that the back--upring at the high pressure side is governing the flow in the "Pump case", and the ring at the low pressure side in the "Motor case", provided that there are no drain grooves on the back-up rings. If there are such grooves, the leakage is governed by the main seal element. This leakage flow rate is much higher than the flow rate in the case without grooves. The difference i s a factor of four in the "Pump case" and a factor of two in t.he "Motor case" at t.he highest sliding velocities treated.
0.2
0.S
u
d e
Leakage flow in the direction of motion through the different contacts in the "Motor case" as function of sliding velocity.
3 . 3 The influence of the back-up rings on the
friction forces
Ieakdqe flow In the figures 3 . 9 and 3 . 1 0 the leakage flow through the different treated clearances as function o f sliding velocity is shown. The flow is always in the direction of motion and is increasing with the sliding velocity.
0. 1
The friction forces as function of sliding velocity are shown in figure 3 . 1 1 for the "Pump case" and in figure 3 . 1 2 for the "Motor case". Here the existence or nonexistence of drain grooves is of minor importance as far as the friction in the back-up ring contacts are concerned. The reason is that the part of the contact area that consists of the drain groove area is very small. The main difference is that the friction in the main seal contact depends on the presence of drain grooves. With grooves, the friction in the main seal contact in the "Pump case" is about 2 0 % of the total friction force at high velocities, and if the back-up rings are without drain grooves the main seal friction is 40 % of the total friction. Corresponding figures in the "Motor case" are 30 % and 40 % respectively.
f?
I
2.8 €I2
2.6
2.0 1 .a 1 .P
0.8 0.4
-0.0
0. I
0.2
0 .f
u
I/.
0.0
Figure 3 . 9
Leakage flow in t.he direction of motion through the different contacts in the "Pump case* as function of sliding velocity.
0. I
0.2
0.3
u
I/.
Figure 3 . 1 1 Friction force counteracting the motion in the different contacts in the "Pump case" as function of sliding velocity.
551
References ( 1 ) Blok, H. Inverse Problems in Hydrodynamic
Lubrication and Design Directives for Lubricated Flexible Surfaces. Symp. Lubrication and Wear, Univ. of Houston, Texas, USA, 1963. (2) Fazekas, G.A. On Reciprocating Torodial Seals. ASME Journal of Engineering for Industry, Aug. 1976. ( 3 ) Hirano, F. and Kaneta, M. Theoretical
Investigation of Friction and Sealing Characteristics of Flexible Seals for Reciprocating Motion, Paper G2. 5th Int. Conf. on Fluid Sealing, 1971, BHRA. 0.0
0.1
0.2
0.3
Figure 3.12 Friction force counteracting the motion in the different contacts in the "Motor case" as function of sliding velocity. It is important to note that the friction forces in the bdck-up ring contacts are dominating. They will dominate even more if the oil films in the back-up ring contacts get so thin that mixed friction occures. 4
CONCLUSIONS
In this work d symmetric piston seal with back-up rings is analysed. Calculations of oil film thickness, leakage flow and friction forces are carried out usiny the inverse hydrodynamic theory. Measured pressure distributions, for either the whole seal including the back-up rings, for the soft seal element o r for one back up ring at a time are used as input data Cdlculated results for each single contact are compared to calculated results when the three parts are coupled in series. Both friction force and leakage flow are influenced by the back-up rings. If these rings have no drain grooves, about 60 % of the total friction force results from these rings, and leakage flow and oil film thickness are totally determined by the pressure distribution in the back-up ring contact zones. With drain grooves, the friction force contribution from the back-up rings is 70 % - 80 %, the leakage flow is determined by the pressure distribution in the main seal element contdct, and the oil film thickness in each part of the total seal contact is determined by its own pressure distribution. 5 ACKNOWLEDGEMENT
The authors would like to express their sincere thanks to Professor Bo Jacobson, head of the Machine Elements Division at Lulea University of Technology, where the computer calculations and the experiments have been carried out. Also thanks to Mr H Wikstrom and to Mr S-I Bergstrom for producing the experimental data, and to the Swedish Board for Technical Development and SKEGA AB for sponsoring the. work.
( 4 ) Johannesson, H. Calculation of the Pressure
Distribution in an O-ring Seal Contact. Paper XI (ii), Proc. 5th Leeds-Lyon Symp. on Tribology, Leeds, England, Sept. 1978. ( 5 ) Johannesson, H. On the Optimization of
Hydraulic Cylinder Seals. Doctoral Thesis 1980-071). Machine Elements Division, University of Lulea, Sweden. ( 6 ) Johannesson, H.L. Optimum Pressure Distri-
butions of Hydraulic Cylinder Seals. Paper C3, Proc. 9th Int. Conf. on Fluid Sealing. BHRA Fluid Engineering, 1981. (7) Johannesson, H.L. and Kassfeldt, E. Calculation of the Pressure Distribution in an Arbitrary Elastomeric Seal Contact. TULEA 1985:02, Lulea University of Technology, Sweden. (8) Johannesson, H.L. Oil Leakage and Friction Forces of Reciprocating O-ring Seals Considering Cavitation. Trans. ASME, JOLT Vol. 105, April 1983. (9) Olsson, E. Friction Forces and Oil Leakage of O-rings on an Axially Moving Shaft. Chalmers University of Technology, Gothenburg, Sweden, 1972.
This Page Intentionally Left Blank
553
Paper XVlll(ii)
Study on fundamental characteristics of rotating lip-type oil seals Masanori Ogata, Takuzo Fujii and Yorikazu Shimotsuma
The puipose of this paper is to investigate the friction and lubrication conditions within the sealing zone of lip-type oil seals. The friction coefficient, oil film breakdown ratio, and dynamic lip motion are simultaneously measured from extreme low-speed of 0.003 m/s ( 1 rpm ) to high-speed of 18 m/s ( 7000 rpm ). Up to the present, it has been accepted that the lubrication conditions within the sealing zone are subject to fluid film lubrication. However, experimental results show that they vary from dry friction in the extreme low-speed region to fluid film lubrication in the high-speed region through boundary and elastohydrodynamic lubrication conditions. 1 INTRODUCTION
It has been explained that the lubrication conditions within the sealing zone are subect mainly to fluid film lubrication. From the results measured for the friction coefficient, Hirano and Ishiwata (1) found that the friction coefficient f was proportiona to the characteristic term (Q*U/P~)~)~ which is equivalent to the bearing modulus. Furthermore, they verified this theoretically applying foil bearing theory by Blok et.al (2). On the other hand, Jagger and Walker (3) defined that the friction coefficient f was roportional to the terms of (rl*U)1/3 and Pa-119 of contact pressure based on the theory for elastohydrodynamic lubrication. In the present paper, not only mesurement of the friction coefficient, but also the oil film breakdown ratio (4) are used as methods for quantities estimation of the lubrication condition to observe the seal surface in datail. The results suggest that the friction and lubrication conditions within the sealing zone of lip-type oil seals cannot probably be explained by one of the lubrication theories alone.
LIP w;E Fig.1
2
APPARATUS
The lip-type oil seals used in this study are shown in Figure 1. They have a metal case embedded and a garter spring. Their Case width is 12.5 mm and outer diameter is 72.0 nun. The bore diameter before installation is 48.2 mm at room temparature, and after installation with a standardized shaft of 50 mm in diameter, the lip is given the interference of 1.8 nun in diameter. The contact band width b generated in contact with the shaft expands to 1.43 mm, and the radial load P, amounts to 326 N. To electrically observe the lubrication within conditions the sealing zone, the seal is given electric conductivity adding carbon black in larger quantities than usual to the NBR. The resistance for unit volume is 528 Q/m. The thermocouples are set to the lip whose position is 1.0 mm just under the lip edge. They are used to measure the body and surface
Sectional view of lip-type oil seal used for test
temeratures. Fur hermore. he strain gauges are'attached to the outside and sealing side of the seal to observe lip motions. A test seal is mounted to the head part of the experimental apparatus shown in Figure 2. The main shaft is driven by a positively infinite variable gear changer from 1 to 7000 rpm. A main shaft of 50.0 mm in diameter is used. The type of sealed lubricating oil is turbine oil of viscosity 56 cSt at 40 OC characteristically. It is filled in the sealing side, and circulated by the oil pump. The temperature of the lubricating oil is controlled to 30 OC at the position of the lip
554
TEMPERATI JRE RECORDER
STRAIN METER
D I G I T A L MEMORY AND SYNCHRONIZED OSCILLOGRAPH
LUbR
I
I
RC C I R C U I T
Fig. 2 Head part of seal lubrication tester type I
surface. The friction torque Tf of the lip at the sealing zone is measured by the load cell connected to the housing. The friction coefficient f of the seal lip at the sealing zone can be expressed as equation (1) employing the radial load P, and the shaft diameter d. f = Tf/ ( Pr.d/2 ) (1) The oil film breakdown ratio is measured by a direct current circuit comprising a resistor, capacitor, shaft, and seal. The charging characteristic of the capacitor differs from cases where the shaft is perfectly contacted o r occasionally contacted with the seal lip. Thereupon, the time elapsed when the voltage of the capacitor reached a certain threshold value are added to t, and ti respectively, the oil film breakdown ratio E is defined by equation 2 E = (t,/tl).100 (2) The oil film breakdown ratio E takes a value of 100 percent when the oil film is perfectly broken the shaft contacts the seal lip as usual, meanwhile, E takes a value of almost zero percent when the shaft and the seal lip are separated by a sufficiently thick oil film. This method is an original development ( 4 ) .
.
3 RESULTS and DISCUSSIONS Simultaneously measured friction and lubrication conditions and dynamic motion of the seal lip within the sealing zone are shown in Figures 3 and 4 . For example, please look at Figure 3 ( S ) .
The figure in the upper row, waveform Waveform @ indicates that an oil film between the shaft and lip was formed and broken. Voltage Vf at zero in order of magnitude indicates a condition where the oil film is broken down, and on the other hand, the voltage Vf at 1 indicates that the oil film is formed. Waveform @ shows the friction torque Tf of the seal lip. Waveforms @ and in the figure in the middle row, show circumferential and radial motions of the seal lip respectively. Waveform @ in the figure in the lower row, shows resultant motion obtained by synchronizing @ with The friction and lubrication conditions within the sealing zone and the temperatures of the seal lip are shown in Figure 5. They are measured when the shaft speed was changed at random from 1 rpm (0.0026 m/s) in the extremely low-speed range to 7000 rpm (18 m/s) in the high-speed range. The friction coefficient f and the oil film breakdown ratio E are obtained from and @ in Figures 3 and 4 employing waveformes equations (1) and (2). And the symbols of S,A,B,C, and D in Figures 3, 4 and 5 indicate the same points of measurement, In the extremely low-speed region from 1 to 5 rpm, the friction coefficient takes a value of 0.08 to 0.07. And the oil film breakdown ratio indicates 80 to 90%, so an effective oil film is scarcely existent. Both values show a contact. Therefore, the possible cause
0, shows shaft rotation.
0
0.
0
555
for this region is dry friction. When point S of shaft speed 1 rpm and point A are investigated at Fig. 3 , oil film breakdown clearly indicates instantaneons breakdown, friction torque waveform @ fluctuates abruptly, and dynamic lip motion waveform @ indicates stick-slip. In the low-speed region from 5 to 50 rpm, the friction coefficient decreases from 0.08 to the minimum value of 0.025. On the other hand, the oil film breakdown ratio decreases from 80 to 40%, though, this does not yet indicate the minimum value. In general, the oil film breakdown ratio also corresponds to the phenomenon of friction coefficient. This difference in phenomenon is explained by waveform @ of the lip motion. The displacement 6 , of data B which indicates the motion in the circumferential direction is smaller than that of data A . This means that the followability of the lip in circumferential direction at data B is inferior to data A. For this reason, although the friction coefficient decreases, the asperities of the shaft and the seal lip easily contact each other, indicating that a high film breakdown ratio is the possible cause. This region is assumed to be the boundary lubrication. In the medium-speed region from 50 to 1000 rpm, the friction coefficient increases to 0.12. The value is a little larger than the 1 rpm value in the extremely low-speed region. As the oil
0
- - - - - --
film breakdown ratio decreases further from 40%, it takes the minimum value of 2% at 1000 rpm. One of the reasons for these phenomena can be explained by the increase of viscous resistance in the oil film which is formed thicker by the speed effect. The other reason is that the lip motion in radial direction at data C is larger than that of B, therefore the tendency stated above is also explained by the effect of inducement of an oil film to the sealing zone caused by the fluctuation of the seal lip. This region is probably subject to the fluid film lubrication. In the high-speed region from 1000 to 7000 rpm, the friction coefficient fluctuates, however, its mean value indicates 0.12 approximately. Then, the oil film breakdown ratio increases from the minimum value of 2% to 10% with fluctuation. This tendency is probably caused by a temperature of 5 " C in the body of the seal lip slightly higher than the temperature of the surface where the temperature of the lubricating oil is controlled to 3CE2 " C . In this region, although the oil film is formed by the speed effect, the viscosity decreases, and the followability of the seal lip to shaft rotation is increased due to heat generation in the seal lip. When these two factors occur simultaneously, the micro asperities on the surfaces of the shaft and the seal lip do not easily contact with each other and also the oil film is broken. Thereby, the waveform @ of lip motion is obviously different up to this point in the elliptical motion
r
Ic.
>
0 n
'm
r
1
> SEPARATION : Vf = 1 V
r
CONTACT : Vf = 0 V
+ . -
I
Tf
r+
(S)
'r+
1 rpm
LIP
Fig. 3 Friction and lubrications within the sealing zone, and motion of a seal lip
556
250 msfdiv
2 sfdiv
P----l
p-----l
W
W
r+
(B) 50 rpm
2.5 msfdiv
p d
(C)
Fig. 4
lC00 rpm
(D) 6000 rpm
Friction and lubrications within the sealing zone, and motion of a seal lip
557
0
50
0
CD W
tlL
40
z IQ
cr W
30
a E W I-
20
cc 1
V
U LL
LL
1 n
c (
SHAFT SPEED Fig. 5
N rpm
Effect of shaft speed on friction coefficient and oil film breakdown ratio
558
tendency that it demostrates. The area of the ellipse indicates the energy consumed by lip motion. It can be easily assumed that energy causes the temperature within the sealing zone to rise high and the viscosity of the lubricating oil to fall. Besides, in the high-speed region, it has been confirmed that the oil film breakdown ratio increases remarkably when the temperature of the lubricating oil is not controlled to a constant value (4). From the above discussion, the oil film thickness within the sealing zone is investigated as follows. The oil film thickness is calculated substituting the operation conditions of the seal for lubrication theories. They are, h~ in case the viscous fluid formed between plain-parallel surfaces is subject to Newton's rule, hg-F by Hirano (1) applying the theory of Blok (2) for foil bearing to a rigid body with surface roughness, and hD by Dowson (5) in consideration for elastic deformation, line contact, and viscosity change by pressure in theory of elastohydrodynamic lubrication. The equation used for the calculations is given in the appendix.
The results are shown in Figure 6 . Hereupon, the operation factor is a characteristic number equivalent to the bearing modulus. The oil film formed between the shaft and the seal lip becomes thicker accordingly as the operation factor increases. By the theoretical equations, the oil film thickness of hg-H and hD take values of 0.67 and 0.7 in gradient, respectively. On the other hand,where the gradient of hN should become 1, it shows fluctuation owing to the variation in the ratio of the shaft speed to the friction coefficient. The film thickness exists in the range from 0.001 pm of hN to 10 pm of hD in order of magnitude. However, from the results measured for the surface roughness of the shaft and the lip shown in Table 1, the oil film thickness between data B of 50 rpm and the data D to almost the maximum speed of 7000 rpm is considered to be reasonable. They are from 0.22 pm in Rrms of the seal lip to 2.04 vm in Rp of the shaft. As mentioned in Figure 5, in these regions the oil film is formed positively by the effect of
10
10 10
10-
A
.
A hNewton
-
0 hBlok-Hirano (1,2)
lo-"
I
10-
LO-'
A
10-
OPERATION FACTOR
rl*U/P1
Fig. 6 Oil film thickness calculated by theories employing experimental conditions speed, and the phenomena whereby the friction coefficient increases by the resistance.of viscosity and the oil film breakdown ratio decreases, are observed. However, in the limited operation factor within the range from to the minimum oil film thickness is measured as 0.05 Um experimentally (4). This value corresponds with the oil film Table 1.
Surface roughness in shaft rotating direction [urn].
Roughness
Shaft
Seal lip
Rmax Rm Ra Rrms
2.04 0.67 1.37 1.28 0.25 0.31
1.89 0.13 1.76 0.91 0.22 0.30
thickness at data S of 1 rpm in the extreme low-speed range calculated by the theory of Dowson et,al. The relation between the friction coefficient, oil film breakdown ratio and oil film thickness, oil film parameter are shown in Figure 7. Here, the oil film parameter indicates the ratio of the oil film thickness to the resultant values of Rrms employing the shaft and the seal lip shown in Table 1. General tendencies are similar to that discussed in Fig. 5. However, if we look at the turning point, the friction coefficient is divided into four regions, and the oil film breakdown ratio is divided into three regions. The difference of the number of regions and turning points indicates a characteristic of the measuring method. Namely, the friction coefficient probably expresses the region A through B where the lubrication condition changes from dry friction to boundary lubrication involving a thin fluid film.
559
10
10
i n
10
10
10
1 0 -2
1 0 -3
10
101
100
h prn
OIL FILM THICKNESS
I 1 0 -3
I
1 1 1 1
1 0 -2
I
I
I l l
I
1 0 -I
I
Ill
100
1
102
I I I
I
I I l l
I
lo1
OIL FILM PARAMETER A
Fig. 7
Relation between friction coefficient, oil film breakdown ratio and oil film thickness, oil film parameter
I I l l
560
On the other hard, the oil film breakdown ratio seisitirely indicates111where microscopic contact between the two surfaces existed in fluid film lubrication. According to Johnson's theory (6), shown as a broken line in Fig. 7, it is said that the oil film breaks down more than 90% under the value 1 of oil film parameter A , and in the value 3 to 4 of A , the oil film does not break to the extent that the two surfaces are almost separated. With regard to this, the theory is partially overlapped with film thickness hD in high region of oil film breakdown ratio E and with hB-D in low-region of E . To look again at Fig. 6 for this, the oil film thickness h~ is fit for data B and b n e a r b y to data C. Therefore, the oil film thickness within the sealing zone is assumed to exist from 0.1 to 1 urn. Thereupon, the elastohydrodynamic lubrication is applicable for the transient region from low speed to medium speed.
BLOK, H. and VAN ROSSUM, J,J. 'The foil bearing - a new departure in hydrodynamic lubrication', Lubic. Engug.,b1953, 9, 316-. JAGGER, E.T. and WALKER, P.T. 'Further studies of the lubrication of synthetic rubber rotary shaft seals', Proc, Instn. Mech. Engrs., 1966-67, 181-Pt.2, 191-204. OGATA, M., KITADA, F., FUJII, T., and SHIMOTSUMA, Y. 'Studies of lip-type oil seal - friction and lubrication conditions within the sealing zone', JSLE. Intern. Edit., 1983, No.4, 135-142. DOWSON, D. and HIGGINSON, G.R. 'Elastohydrodynamic lubrication' , 1966 (Pergamon Press, New York), 187-212 JOHNSON, K.L., GREENWOOD, J.A., and POON, S.Y., 'A simple theory of asperity contact in elastohydrodynamic lubrication', Wear, 1972, 2, 91-108. Appendix
4
CONCLUSIONS
The friction coefficient, oil film breakdown ratio within the sealing zone and the motion of the seal lip are investigated by changing the shaft speed under the constant temperature of the lubricating oil. The results show that the friction and lubrication conditions within the sealing zone changed depending on the shaft speed. This can be explained by, dry friction in extremely low-speed, are boundary lubrication in low-speed. In the transient region from low-speed to medium speed, elastohydrodynamic lubrication is assumed. In medium-speed regions and higher, the results can be explained by fluid film lubrication including partial boundary oil film. Particularly, the contact of micro asperities which break down the fluid film and are considered as the core of seizure are detected in the high-speed region. This suggests the probrem of lubrication when the lip seals are used in further high speed. To determine the oil film thickness width the sealing zone precisely practically and theoretically, however, they are assumed to exist from 0.1 to 1 m approximately. This suggests the necessity for application of the elastohydrodynamic lubrication theory of thin film considering surface roughness or for the starved lubrication theory. 5 ACKNOWLEDGMENT We would like to express our gratitude to Professor Koichi Sugimoto, fh-.Heihachiro Inoue and Mr. Yasushi Atago of Kansai University and Professor Andere Deruyttere, Jacques Paters, Raymond Snoeys, and Hendrick Van Brussel of Katholieke Universiteit Leuven for their kind advices. And also, we would like to thank Mr. Yasuo Shimoji of Koyo Co., Ltd. and Mr. Masanori Nakatani of Honda Co., Ltd. for their good cooperation in the experiments. References
(1)
Hirano, F and Ishiwata, H. 'The lubricating condition of a seal lip', Proc. Instn. Mech. Engrs., 1965-66, 180-Pt.3B. 138-147.
Equations used for calculations of oil film thickness are shown as follows: By Newton's law,
By Dowson et,al. (5) modified from experiment, h, = 2.65.R.GO.54.U0.7.~0.13. Where, A Area influenced by friction F Friction force G Material parameter in EHL h a x Maximum height in surface roughness hi Difference in surface roughness shaft and seal lip Pi Radial load per unit contact band width R Equivalent radius of curvature U Speed parameter in EHL W Load parameter in EHL 11 Viscosity coefficient X Interval of peak to peak in surface roughness
m
2
N m
m N/m m N*s/m2 m
561
Paper XVlll(iii)
Influence of pressure difference and axial velocity on a spiralgroove bearing for a moving piston F. Bremer, E.A. Muijderman and P.L. Holster
This paper describes how one can realise a bearing for pistons that compress or expand a gas, during the complete cycle of movement, via a full film built up by a self-acting bearing and using the same gas. To this end, the piston has a rotational movement added to its translational movement. The bearing occurs through spiral groove patterns. From a theoretical analysis of the gas film carried out with the finite element method (FEM), it appears that “static” instability can occur under certain conditions. Solutions are offered to prevent this instability under certain operational circumstances. 1
INTRODUCTION
In an article by R.J. Vincent et a1 [I], a report is made of a free piston machine, in which the piston has two hydro-dynamic gas bearings (one on each end). The piston is coated to prevent damage during starting and stopping. The piston movement is realised by a linear motor, which has a frequency range of 1 to 60 Hz. The radial play between piston (bearing) and cylinder wall is 13 m and compression pressures up to 10.5 bar can be obtained. The article mentioned here is one of the few articles that deals with hydro-dynamic bearings in free piston machines. It is not clear from the article whether the piston makes a rotational movement as well as a translatory one. Independently of Vincent‘s work, the idea arose in the Philips research laboratory in Eindhoven of a hydro-dynamic bearing for the piston in a free piston machine, by providing the piston with spiral grooves. This idea led to a closer theoretical investigation. All calculations were made with the aid of the finite element method. The programs used work with the finite element package AFEP [2]. In order to be sure that the programs gave the correct answers, a series of comparative calculations was carried out. Results from Floberg 131, Muijderman [4], Bootsma [5] and Hamrock and Fleming [6] were used for comparison. TDC
14 BDC 1
I
I
I
A refrigerator compressor was used as the starting-point for the investigation, see fig. 1. The bearing gap was assumed to be 5 pm and the groove depth was 10 pm. The piston frequency was 50 Hz; bore and stroke were 2 5 mm and 16 nun, respectively.
1.1
Notation
D e h
diameter eccentricity film height groove depth
hO
number of grooves length of the bearing speed
P
pressure
P1
pressure at the low-pressure end
p2 R
pressure at the high-pressure end
S
t T
V U W
W X
Y z tl
1
AP AR
pressure difference radial play relative eccentricity dynamic viscosity
’I
The basic idea of a gas bearing piston “applicable in a refrigerator compressor“. The piston has a herringbone bearing.
volume peripheral velocity axial velocity radial load-carrying capacity co-ordinate in peripheral direction co-ordinate in radial direction co-ordinate in axial direction groove angle, right-hand side groove angle, left-hand side
t
P (I w
V V.
[ml [r
universal gas constant stroke time temperature
CY
2
Fig. 1
[-I
k L n
density attitude angle of load capacity angular velocity gradient divergence
[ml
hl [ml [degrees] [degrees]
“/m2 [ml
1
[-I
[Pa.s] [kg/m3I [degrees] [rad/sl
[l/ml
[-I
562
For an ideal gas (pi' = p/p = RT), equation
A FIRST ANALYSIS
2
(2.1 ) becomes:
In fig. 1, the starting-point for the previously-mentioned compressor is shown. The translation frequency was 50 Hz, the rotational speed of the piston was 6000 rev./min., the maximum compression 13.6 bar and the minimum pressure (suction pressure) 1.26 bar. Air was used as the working medium with a dynamic vis-
V.(-
h3 p Vp) = 12Q
V.(Th
P 1) +
a0
(2.2)
2
(2.3)
at This is a non-linear partial differential equation. Equation (2.2) can be further simplified because it may be assumed that h does not vary with time, ah/at = 0.
-6
cosity of 18.10 Pa.s. If we look at fig. 1, we can see that we have, at the top side of the piston, a different edge pressure for every different piston position. Fig. 2 shows a much simplified p-V diagram of a piston compressor (disturbances as a result of the opening and closing of valves etc. have been omitted).
h3 h v.(- 1 2 n p v p ) = v . ( - 2 p
1) +
h
Equation (2.3) was generalised with the aid of the Galerkin method [TI. This gives:
Partial integration of the left-hand side of equation (2.4) gives:
bl "?
2
+ !0 an
(- h3 P VP
12n
- Th P 1) E
= ! @ h s d n
dan
(2.5)
R
Equation (2.5) can be linearised with the method of Newton. The flow over the edge was neglected, which meant that a class of functions 41 was chosen that became zero at the edges where the pressure is prescribed. Thus, equation (2.5) yields:
Fig. 2 The p-V diagram of a compressor. Because we are dealing with a compressible medium, which also serves as a lubricant for the spiral groove bearings, the Reynold's equation used in the calculations must contain a ap/at term. This term describes the compression and expansion of the lubricant in the film.
Cross multiplication, collecting terms, ignoring second order effects and again writing p for (po + bp0 ) gave: h3
2.1
(?T;; P VP,
Pressure distribution in the film
h3
+
12n Po
VP
-
h
P 1)
L1
The pressure distribution in the film can be described by the Reynold's equation, as valid for a compressible medium. According to the Euler description, this is:
a (-P-) h3 ap + a ax 12n ax y
3
(12q L 2ayq
Here, the following assumptions have been made : - the temperature is constant throughout the film; - inertia terms are neglected; - the bearing surfaces are undeformable and ideally smooth.
(2.7)
Where: p = the unknown pressure; po = the known pressure from the previous step of the iteration process, according to Newton. By using the method reported in [71, equation -(2.7) can be converted to an 'FEM formulation (triangular elements with linear basic functions). This provides a matrix-vector notation of the following shape:
= [CI
To
(2.8)
563
This system can be solved with the implicit method of Crank-Nicholson (trapezium rule) ; this is: ti
The axial piston velocity w can be determined by the
w = a . s . = n. s i n a
(2.13)
+I The main criterion for the correct operation of this type of rotating and translatory piston is:
=
1 2
A t
(ft +
i +1
fti)
(2.9)
-
to Pt . The i + f 1 + 1 advantage of this is that the number of terms in the right-hand side of the equation remains smaller with an error equal to that of the trapezium rule. This gives:
i t.
We can go from
= p
ti
I
THE ATTITUDE ANGLE MUST BE BETWEEN 0 AND 90 DEGREES FOR EVERY PISTON POSITION
I
The attitude angle is the angle between the line, along which the eccentricity e lies and the line, along which the load capacity W lies (see fig. 4 ) .
1
+ - A t f
2
ti++
t.
(2.10)
Substitution in ( 2 . 8 ) gives:
Fig. 4 (2.11)
Radial spiral groove bearing with grooves in the shaft and rotating bush. The attitude angle Q is also given.
The reason for setting this criterion is: attitude angles greater than 90 degrees cause negative stiffness. This means that there is no force counteracting small disturbances around the position of equilibrium, so that the piston's bearing system will not function.
or
(2.12)
Results of calculations
2.2
where :
-
Pt = pressure vector, previous step in the Crank-Nicholson process;
-
Po = pressure vector, previous step in the Newton iteration process; and
5
= pressure vector, unknown pressures.
The calculations were made for a piston of 24 mm length, 2 5 mm bore and 16 mm stroke. The pressure at the upper end was variable and that of the lower end was kept constant at 1 bar during the complete cycle. Fig. 5 shows the course of the pressure at the upper end of the piston and the course of the axial piston velocity during one work cycle.
A computer program was written for the above FEM formulation and calculations were made for different circumstances. To this end, the work cycle was divided into eight equally long time intervals. The edge pressure p2 (see the element distribution, fig. 3 ) can be determined for every piston position from the simplified p-V diagram (fig. 2 ) .
14
12
pI "2
1
x 1 ~ 5 ~ i m
6 4
P-,
2
0
+ u,x Fig. 3
Fig. 5 Element distribution of groove bearing ( 1 5 grooves)
the spiral
The course of the pressure at the upper end of the piston and the axial piston velocity as a function of the piston position.
564
Before it was possible to make realistic statements about the attitude angle as a function of the eccentricity or as a function of rotational speed, two other parameters had to be investigated. These were: (1)
the situation whereby the exhaust valve was constantly open (i.e. no pressure difference across the piston) and the situation whereby the axial piston velocity was set to zero. Results of these latter tests are shown in fig. 7 .
the influence of the number of time steps (the step size) on the accuracy of the calculations.
NOTE: The result is given in fig. 6. It can be seen that an increase in the number of steps (a reduction of the step size) has little influence on the course of the attitude angle as a function of the piston position. This calculation was made for a rotational speed of 6000 rev./min
.
( 2 ) the
influence of a second calculation cycle, which was placed after the first calculation cycle. Fig. 7
NOTE:
Fig. 6 also gives the result of this second calculation cycle. It can be seen that the results of both calculation cycles correspond fairly well. If one continues to expand the number of calculation cycles, then the results for the attitude angles are the same at every moment in time. This calculation was also made for a rotational speed of 6000 rev./min.
Fig. 6 This graph shows the influence of the number of steps on the attitude angle at a certain piston position denoted by ti and (at a certain piston position) between the first calculation cycle ( 8 steps) and the subsequent second calculation cycle ( 8 steps). In both situations, attitude angles of greater than 90 degrees occur. This means that there is an unacceptable stiffness present during a part of the cycle. After confirming the method of calculation, we checked two special situations. These were
The attitude angle as a function of piston position t(s). The dashed line shows the situation with the exhaust valve open and the dot/dash line shows the situation with the piston speed at zero. The relative eccentricity is 0 . 5 .
It can be seen that the attitude angle varies very little when the exhaust valve is kept open. This means that the influence of the axial piston velocity on the pressures in the film is negligible. For the situation where the axial piston velocity is zero (which is a situation that can never be realised), attitude angles greater than 90 degrees occur again. This means that the influence of the pressure in the compression chamber on the operation of the bearing is the main reason why excessive attitude angles occur. Next, calculations were made to show the attitude angle as a function of the piston position t(s) for various speeds and eccentricities, respectively. The results of these calculations are shown in figs. 8 and 9. See fig. 5 for the course of pressure and axial velocity.
Fig. 8 The attitude angle as a function of the piston position t(s) for various rotational speeds. The relative eccentricity is 0.5. Fig. 8 shows that there is an attitude angle of less than 90 degrees for every piston position at speeds above 24000 revs./min. This satisfies the main criterion for the correct operation of the free piston.
565
bush. Next, a pressure difference is placed across-thepiston. It then appears that this pressure difference can cause instability.
The explanation for this is that the pressure build-up in the peripheral direction of the piston (at an eccentricity equal to 0.5) is so large that the negative effects, which arise from a pressure difference across the spiral groove bearing, are compensated.
~
" H e r r i n g - bo n e I' bea r i n g
-
w f O
w = o Ap = 0
I
7777777
w = o
I1
w f O
Ap = 0
Fig. 9 The attitude angle as a function of the piston position t(s) for various eccentricities. Rotational speed is 6000 revs./min. Fig. 9 shows that an attitude angle of greater than 90 degrees occurs in certain piston positions for relative eccentricities greater than 0 . 5 . This means that the main criterion for correct piston operation is not satisfied. Paragraph 3 gives an explanation of the effects that occur here as a result of the pressure difference across a piston and an axial velocity of a piston that is provided with spiral grooves. 3
I11
Fig. 10
The various situations that influence the operation of a free piston.
PHYSICAL EXPLANATION OF RELEVANT PHENOMENA
Ip order to obtain an insight into the pressure build-up in a spiral groove bearing and the effect of a pressure difference across a spiral groove bearing, the factors that are responsible for the pressure build-up in the film were analysed separately. This meant a separation between the rotation and translation of the piston and that of a pressure difference across a piston provided with spiral grooves.
3.1
//////I
Decoupling the factors of influence
3.2
The influence of compressibility on the course of the pressure in a gap
Before giving a physical explanation for the instability that occurs in situations I1 and 111, we would like to say something about the pressure differences across parallel, convergent and divergent gaps. Here, we are studying a compressible medium and an infinitely wide gap. Through the influence of the compressibility of the medium, the course of the pressure across a parallel gap is parabolic in shape, see fig. 11.
There now follows a brief description of the various situations shown in fig. 10. I
This situation shows the normal operation of the spiral groove bearing. The peripheral velocity u is responsible for the pressure build-up in the gap. It can be said that this type of bearing is stable (has positive stiffness) for every value of eccentricity.
I1
This case is realised by pulling a grooved pin, with eccentricity, through an infinitely long cylinder at a speed w. This situation is always unstable, as explained below.
I11 This is the most interesting situation. Here, a piston that is provided with spiral grooves is situated eccentrically in a
Fig. 11
Pressure distributions across parallel, convergent and divergent gaps (infinitely wide gap and a compressible medium). High pressure p2 and low pressure p,.
In the third figure, two
different gaps are indicated.
566
If the gap is divergent, the pressure distribution is less convex in shape than the pressure distribution across a parallel gap. If the gap is convergent, the pressure distribution is more convex in shape than the pressure distribution across a parallel gap. This can be verified with the aid of Reynolds equation, which is, in this case:
(3.1)
dx
It now follows that, with a constant dynamic viscosity and with p and h as function of the place x, the equation for an ideal gas becomes:
From
equation
(3.2), it follows that if h 2 is a constant, then is a constant, which
means that p has a parabolic shape. When h = h(x), the curve is either higher or lower than is the case when h is a constant, depending on the sign of dh/dx. 3.3
Negative stiffness
Now that the pressure distribution of a compressible lubricant is known for various gap configurations, we can investigate how negative load capacity (instability) arises in a piston fitted with spiral grooves (spiral groove bearing), across which there is a pressure difference. Our starting-point is situation I11 of fig. 10 (angular velocity and axial velocity both zero). Fig. 12 shows the investigated situation. On the left-hand end of the bearing, we can see the high pressure (p2 is 13.6 bar). The righthand end is at low pressure (ambient pressure, p1 is 1 bar). Furthermore, the figure contains the x- and z-direction co-ordinates, as well as the directions of velocity, u and w.
3 Y
t
0'
1
I
I
L in m
Fig. 12 The course of the pressure in a spiral groove bearing ( 6 # O), across which there is a pressure difference (compressible medium). No rotation, translation or tilt.
567
The following explanation is valid when the speeds of u and w are equal to zero and when pressure p2 is higher than the ambient pressure p,. Furtherwre, the piston is in the cylinder with a fixed eccentricity (e) and with no tilt. When we look at the course of the pressure by following line a from point 1, through points 2 and 3, to point 4 (in fig. 1 2 ) , we see: a. The gap height increases from point 1 to point 2, which means that it is a divergent gap. Going back to our investigation into different gap configurations, that have a pressure difference across them, we can ccnclude that the pressure from point 1 to point 2 drops more quickly than it would do in a parallel gap. It must be noticed that the flow is mainly through the groove. The flow is directly proportional to the film height h to the power three (the film height above the groove is three times as high as the film height above the ridge). b. From point 2 to point 3 , the gap is parallel, and the pressure drops parabolically. In this case, the end effect is that the pressure across this part of the piston is balanced out by peripheral flow (see the lines of constant pressure in fig. 1 2 ) . C. From point 3 to point 4 , the gap is convergent (above the groove). This means that the pressure across this part of the piston drops more slowly than it would do across a parallel gap. The reduction in pressure here is just as in case a. The reasoning is similar for a trajectory starting at a point 1 80 degrees further along the periphery (going from point 5 , via points 6 and 7 , to point 8 ; see line b in fig. 1 2 ) . Now, however, we have firstly a convergent, then a parallel and, lastly, a divergent gap across the groove, the flat middle part and the groove again, respectively. After this consideration, the following conclusions can be drawn: In area I of fig. 12, the pressures are somewhat lower than in the case of a smooth piston. Consequently, in area 11, the pressures become somewhat higher; the same applies to area 111. In area IV, the pressures are lowered again (see also the lines of constant pressure in fig. 1 2 ) . These higher and lower pressures at the specified areas result in a total load capacity with an attitude angle between 90 and 210 degrees. This latter point denotes that the piston has a negative stiffness and is therefore "statically" unstable. This negative stiffness is valid for every value of eccentricity except zero. A grooved piston that makes an axial movement can be regarded in a similar way to that of a pressure difference across a grooved piston. A grooved piston that makes an axial movement in a bush (cylinder) is therefore "statically" unstable for every value of eccentricity except zero. It should be noted here that the pressures that arise are a result of the
transition from grooved part to smooth part and vice-versa. These "static" instabilities, which arise as a result of pressure difference across a piston or from an axial movement of the piston, can be compensated by rotating the piston. The negative effect resulting from an axial movement of a grooved piston is much less than the negative effect of a pressure difference across a grooved piston. With a pressure difference across a grooved piston, in combination with a translatory movement of the piston, the pressure difference across the grooved piston is the major factor in determining the instability. In our case, the instability resulting from the translatory movement of the piston can be completely compensated by rotating the piston. This is important for the use of a rotating (hydrodynamic bearing) and translatory piston in a compressor. PRACTICAL REALISATION Once more was known about the consequences of a pressure difference and an axial (translatory) movement of a grooved piston, it was possible to look for a practical construction for a piston with a hydrodynamic bearing in a compressor. Obviously, the construction would have to prevent negative stiffness ("static" instability) from occurring. The starting-point for the construction was to attempt to remove the pressure difference across the piston, seeing as this is the main cause for the malfunction of this type of compressor. The influence of the translatory movement of the grooved piston on the "static" stability is many times less. This investigation has resulted in a great number of constructive measures that can prevent "static" instability, see lit. [ E l . Of these measures, one is described here. The rotating piston is subdivided into three sections, see fig. 13.
-
TDC
BDC
Fig. 13 Piston with one smooth and two grooved sections. The left-hand and right-hand sections have a herring-bone groove pattern with a smooth part in the middle. The centre section is completely smooth and acts as a seal between the compression chamber and the surroundings. The piston is made in such a way that there are no pressure differences across the grooved sections. This can be simply realised by making a number of channels in the piston. The result should be that the operation of the spiral grooved sections is the same as that of the "conventional" spiral groove bearings.
One must, however, note that the higher the pressures at both ends of a spiral groove bearing, the higher the attitude angle. Nevertheless, the attitude angles at high compression pressures are still well within the prescribed criterion of 0 to 90 degrees. It is therefore justified to state that the spiral groove sections are stable (have positive stiffness). The smooth section, which serves as a "seal", functions as a radial plain bearing. It is known that this type of bearing always tends to instability for relative values of eccentricity below 0.5. The two adjacent spiral groove sections must have sufficient positive stiffness to compensate the negative effect of the smooth section, thus making the whole construction stable. 5
CONCLUSIONS
- the higher the pressures at the end of the spiral groove bearing, the higher the attitude angle. In other words, the greater the chance of "static" instability. References [ l ] Vincent, R.J., Rifkin, W.D. and Benson, G.M.
"Test results of high efficiency Stirling machine components". ERG, Inc., Oakland 1982. [2] Segal, A. "A Finite Element Package". Dept. of Mathematics, T.H. Delft, 1983. 31 Floberg, L.
"The infinite journal bearing, considering vaporization". Report 2 and 3, Gotheburg 1957.
41 Muijderman, E.A.
"Spiral-groove bearings". Dissertation, T.H. Delft, 1964.
This investigation into the possibilities of applying a piston with a gas bearing in a compressor has led to the following conclusions:
[ 51 Boot sma, J
- a pressure difference across a spiral groove
[6] Hamrock,
-
bearing can cause negative stiffness; an axial movement of a spiral groove bearing can cause negative stiffness; the piston bearing can be made "statically" stable by subdividi'ng the piston into sections with a grooved surface that do not have a pressure difference across them and a section with a smooth surface that does have a pressure difference across it;
.
"Liquid-lubricated spiral-groove bearings". Dissertation, T.H. Delft, 1975. B,J,. and Fleming D.P. "Optimalization of self-acting Herringbone journal bearings for maximum radial load capacity". Lewis Research Center.
[71 Kan,
J. "Numerieke analyse". Mathematics, T.H. Delft, 1983.
[81 Bremer,
Dept. of
F. and Muijderman, E.A. patent, number: 8503031.
Dutch
569
Paper XVlll(iv)
Elastohydrodynamiclubrication of an oil pumping ring seal G.J.J. van Heijningen and C.G.M. Kassels
The p e r i o d i c e l a s t o h y d r o d y n a m i c b e h a v i o u r o f t h e p u m p i n g r i n g h a s b e e n a n a l y s e d . A c o u p l e d s o l u t i o n of t h e Reynolds and t h e e l a s t i c e q u a t i o n s has been o b t a i n e d by u s i n g t h e F i n i t e Element Method. For t h e c a l c u l a t i o n o f t h e c a v i t a t i o n - z o n e we h a v e c h o s e n t h e a p p r o a c h o f t h e v a r i a t i o n a l i n e q u a l i t y (Reynolds boundary c o n d i t i o n s ) . I n t h i s a x i - s y m m e t r i c p r o b l e m t h e c o u p l i n g o f t h e R e y n o l d s and t h e e l a s t i c e q u a t i o n s h a s been r e a l i z e d b y a s p e c i a l l i n e - e l e m e n t p l a c e d a g a i n s t t h e i n n e r b o u n d a r y o f t h e p u m p i n g r i n g . The l i n e a r i z a t i o n h a s been o b t a i n e d b y a p p l y i n g a Newton method; t h e t i m e i n t e g r a t i o n h a s been c a r r i e d out b y u s i n g a Cranck-Nicolson-scheme. I n t h i s way a v e r y f a s t s o l u t i o n c a n b e f o u n d .
1
INTRODUCTION
I n t h e P h i l i p s S t i r l i n g e n g i n e t h e u s e f u l work per s t r o k e can be i n c r e a s e d i f t h e thermodynamic p r o c e s s p r o c e e d s a t a h i g h p r e s s u r e . An i m p o r t a n t problem d u r i n g t h e development o f t h e engine c o n s i s t s i n t h e seal around t h e r e c i p r o c a t i n g p i s t o n r o d (see F i g u r e 1 ) . Not o n l y has t h i s s e a l t o p r e v e n t t h e gas f r o m leaking t o the outside of the cilinder, but also the e n t e r i n g o f o i l i n i t . A p o s s i b l e assembly c o n s i s t s o t a p u m p i n g r i n g a n d a s c r a p e r . The principle of the seal i s i l l u s t r a t e d i n Figure 2. The a x i - s y m m e t r i c p u m p i n g r i n g i n c r e a s e s t h e o i l p r e s s u r e t o t h e l e v e l o f t h e thermodynamik cycle, which i s maintained a t a c e r t a i n l e v e l b y a p r e s s u r e r e g u l a t i n g v a l v e . The s c r a p e r s c r a p e s t h e o i l f r o m t h e r o d d u r i n g t h e upward s t r o k e and pumps t h e o i l f i l m a l w a y s e x i s t e n t d u r i n g t h e r e t u r n s t r o k e back. H e r e w i t h sometimes a small gas l e a k appears t o g e t h e r w i t h an o i l 1 eak. Several models o f pumping r i n g o p e r a t i o n have been d e a l t w i t h i n t h e l i t e r a t u r e . S i m p l e m o d e l s C1,21 h a v e b e e n d e v e l o p e d t o p r e d i c t t h e p u m p i n g r i n g p e r f o r m a n c e t o some e x t e n t . M o r e complex a n a l y s e s C 3 , 4 , 5 1 y i e l d v a l u e s f o r pumping c a p a c i t y t h a t a r e i n b e t t e r a g r e e m e n t with the rather l i m i t e d experimental date a v a i l a b l e . They a l s o p r o v e d t h a t i t i s p o s s i b l e t o solve t h e equations by using e i t h e r the f i n i t e d i f f e r e n c e o r t h e f i n i t e e l e m e n t method. The U n i v e r s i t y o f T o l e d o ( M i c h i g a n , USA) i s a l s o w o r k i n g on t h i s SUbJeCt; t h e y d e c i d e d t o u s e t h e f i n i t e d i f f e r e n c e m e t h o d . To o b t a i n c o m p a r a b l e r e s u l t s we u s e d t h e f i n i t e e l e m e n t m e t h o d f o r t h e same m o d e l u n d e r t h e same c o n d i t i o n s .
h
Lo
i n i t i a l clearance seal length f l u i d pressure v a r i a t i o n of f l u i d pressure stroke time d i s p l a c e m e n t ( v e c t o r : ur,uz) v i r t u a l displacement c y l i n d r i c a l co-ordinates
n
.. o_'
boundary domain s t r a i n (vector: E ~ , E f l u i d dynamic v i s c o s i t y Poisson's r a t i o stress (vector: G ~ , G ~ domain angular v e l o c i t y gradient operator stands f o r transpose of
~
,
E
~
,
o_ D d
E f_
h
s t r a i n shape f u n c t i o n e l a s t i c i t y matrix rod diameter seal thickness Young's modulus force (vector: fr, f z ) i l u i d f i l m thickness
0
scraper pumping
u Figure 1
The S t i r l i n g e n g i n e .
~
, G ~ , T ~ ~ )
1.1 N o t a t i o n
B
~
~
)
570 2
THE MODELLING
2.1
Geometry
The e l a s t i c i t y m a t r i x 0, l i n k i n g t h e s t r a i n E and t h e s t r e s s e s .L, i s f o r an i s o t r o p i c material :
OF THE PROBLEM
The g e o m e t r y o f a n o r m a l l y u s e d pumping r i n g i s p r e s e n t e d i n F i g u r e 3. A150 t h e p r e s s u r e s , n o r k i n g on t h e d i f f e r e n t s u r f a c e s , a r e m e n t i o n e d i n i t . S i n c e t h e t h i c k - n a l l e d end o f t h e r i n g does n o t u n d e r g o h a r d l y any d i s t o r t i o n , n e h a v e chosen t h e p r o b l e m d e p i c t e d i n F i g u r e 4 as a model f o r o u r c a l c u l a t i o n s . I n t h a t model t h e r i n g i s c u t o f f j u s t b e f o r e t h e t h i c k - w a l l e d end and a t t h a t c u t a l l t h e d i s p l a c e m e n t s a r e p u t t o z e r o . ;he i n c r e a s e i n t h e i n n e r d i a m e t e r of t h e r i n g i s taken i n t o account by p r e s c r i b i n g t h e f i l m p r e s s u r e a t t h e b o u n d a r y P, (0 b a r ) .
G
G
4'
Z
r
'e
-
- Dc
(3)
rz
with:
cycle kopseol mean cycle pressure
The e q u i l i b r i u m e q u a t i o n s a r e :
1 - re ing e
oil seal
Figure 2
2.2
and:
- -1
pumping r i n g
crank case
(5)
r
scraper
r
I,J
Assembly o f a pumping r i n g & s c r a p e r .
Stress analysis
By symmetry t h e t w o components o f d i s p l a c e ~ n e n t s i n any p l a i n s e c t i o n o f b o d i e s o f r e v o l u t i o n a l o n g t h e i r a x i s o f symmetry d e f i n e c o m p l e t e l y t h e s t a t e of s t r a i n and, t h e r e f o r e t h e s t a t e o f s t r e s s . So, f i r s t o f a l l , ne l o o k f o r t h e r e l a t i o n b e t w e e n d i s p l a c e m e n t s and s t r a i n . From Z i e n k i e n i c z Cbl ne have: R O O DIAMETER fi2,ooo m m
Figure 3
=
Bu_=&
Geometry o f t h e pumping r i n g .
From t h e s e e q u i l i b r i u m e q u a t i o n s we can d e r i v e t h e " Ga 1 e r k in " - e q u a t ion s :
with:
-
(G
n + z n )6ur r r r z z
an
& =
I n
f r 6ur r d r d z
r ds,
ds,
=
571 z
n + G n = r z r z z
-
pi
on
ri
-
and
p,
on
r,,
On a l l t h e b o u n d a r i e s t h e d i s p l a c e m e n t s u u a r e f r e e , e x c e p t f o r P, where ur = uSo t h e f i n a l f o r m o f t h e G a l e r k i n - e q u a t i o n s t h e e l a s t i c problem i s :
-
I
p o Lur r dz
-
I
pi
Lur r dz
r, ‘
ROO
-
I
p tur
(10)
and
0. for
r dz
r2
DIAMETER /12.000
(11)
F i g u r e 4 d x i s y m m e t r i c model u s e d f o r t h e c a l c u l a t i ons. and:
2.3
F i l m Pressure a n a l y s i s
The R e y n o l d s e q u a t i o n , a c o m b i n a t i o n o f t h e e q u i l i b r i u m e q u a t i o n and t h e e q u a t i o n o f mass c o n s e r v a t i o n , r e l a t e s t h e p r e s s u r e and t h i c k n e s s of the o i l f i l m :
I
f Z 6uz r d r dz
n
H u l t i p l i c a t i o n w i t h Lp and i n t e g r a t i o n t o z a l o n g P, g i v e s :
or:
I
0
-
6p
a
(- ha
2
t
5 1 uh)dz
t
I
ah 6p d: at
= 0
P a r t i a l i n t e g r a t i o n and t h e f a c t t h a t Lp = 0 a t b o t h ends o f t h e f i l m (p, = 0 and p, = 60 b a r ) g iv e s t h e ” G a 1e r k in “ - eq ua t i on : (ct6ut
+ G,Lu,I
r ds,
ds,
=
an (7) Since t h e f l u i d f i l m cannot s u s t a i n a n e g a t i v e p r e s s u r e o f any p r a c t i c a l s i g n i f i c a n c e , t h e o i l must become c a v i t a t e d . The p r o b l e m o f d e t e r m i n i n g t h e f r e e boundary s e p a r a t i n g t h e l u b r i c a t e d and t h e c a v i t a t e d r e g i o n h a s been s t u d i e d b y o t h e r s . We have c h o s e n t h e a p p r o a c h o f t h e v a r i a t i o n a l i n e q u a l i t y C71. The s o - c a l l e d ”Reynolds” boundary c o n d i t i o n i s r e a l i z e d by t h e non-negativity constraint:
Hence:
p z , 0
I
2.4
f_T.L& r d r dz
n
Numerical s o l u t i o n :
B o t h G a l e r k i n - e q u a t i o n s ( 1 1 ) and, ( 1 4 ) d e s c r i b e o u r p r o b l e m . I t e r a t i o n on t = t i s g i v e n by:
Ue h a v e a d i s t r i b u t e d l o a d i n g on t h e b o u n d a r i e s ( F i g u r e 4 1 and no f o r c e f_:
~n
r r
-
(15)
p,
+ T
n = - p o o n r m - p o n r 2 ; r z z O3
= 0 on P ,
,
and
-
pr = 0 on P,
n
572
-
pntl
6u
r dz =
Vpn VLp dz
r2
-
I
t
r2
po Lur r dz
-
1
p,
L u r r dz
(16)
r,
rO
and:
or:
hntl Vpn(ihn)'
VSp dz
t
r2
Equation (17) is linearized in h n+l and p n t l
by:
With a modified Cranck-Nicolson scheme (midpoint rule):
and:
So we obtain for the first term of equation
(17):
)
Vp"'
our coupled set Galerkin-equations becomes:
VLp dz =
r2
-
I
-
I
p:tl
Su
r dr =
hn+l vpn+l vLp dz r2
po Lu
dr dz
-
I r,
and:
I
-Vp"'
VLp dz
t
rs
Vpn 3(hn)'
2
VLp dz
t
p, S u
r dz
(22)
573 In o r d e r t o c o u p l e t h e " R e y n o l d s " - e q u a t i o n t o the "elasticity"-equation a line element with t h r e e u n k n o w n s in t h e n o d e s i s used. T h i s element is placed against t h e innner boundary of t h e p u m p i n g r i n g , r2. T h e u n k n o w n in t h e n o d e s on t h e b o u n d a r y a r e t h e d i s p l a c e m e n t s u r l u z a n d the film pressure p. S e e A p p e n d i x I.
5.
T h e mesh and t h e form o f t h e w h o l e c r o s s section of t h e pumping ring at a certain time h a v e b e e n d e p i c t e d in F i g u r e 5. T h e b e n d i n g of t h e r i n g is e v i d e n t . T h e p e r i o d i c d e f o r m a t i o n
tI
RESULTS
-----,
Ik"6'h
With t h e e x c e p t i o n of t h e u n i f o r m c l e a r e n c e r i n g p r o p o s e d i n C81 all p r e v i o u s p u m p i n g r i n g m o d e l s required no clearance between t h e ring and t h e rod during t h e r e t u r n stroke. For r i n g s with z e r o c l e a r a n c e o r an i n t e r f e r e n c e fit, t h i s assumption i s certainly valid. However, a ring that b e c o m e s worn during o p e r a t i o n o r that initially has a poor fit d u e t o manufacturing errors would have a positive clearance that affect i t s pumping capacity. We h a v e i n v e s t i g a t e d t h e c a s e :
L
Seal l e n g t h Hod d i a m e t e r Seal t h i c k n e s s C1 e a r a n c e 01 1 v 1 s c o s 1 t y Young s m o d u l u s Poisson s ratio Speed Stroke
[i
11
w
s
""8
Figure 5 Finite element mesh and form o f the ring at a certain time. due t o a sinussoidal velocity is presented only a s t h e d i s p l a c e m e n t s a l o n g t h e b o u n d a r y P,an P, (see Figure 6 ) . The different curves correspond with t h e numbers in t h e velocity diagram. The periodic pressure distribution along t h e s a m e b o u n d a r y i s g i v e n i n F i g u r e 7.
= 8 m m = 12 mm
d = 1 m m h o = 8 y.m 11 = 0.0?78
E
PY.P,"g
rc/m2
= 5.27-10 N/m2 = 0.44 = 151.8 r a d / s = 46 m m
t
t
I
z
LILT.
4
LE-Z".
t
t
5
.
- .0
*
4
.
. .
,
\ \ \ \
- 2 . d
2 LILI".
+
LILT..
-"I N
t
-2.5,
i
-1.m 5.:1
,
,
6.0
+
'
.
'
b
7
t
i
. .
a -8.i 0 m 1.00 n m ¶a0: '(000 1100 YI ' YOI 1- 8 am0 ' m b 0
m
,I*".
1.00
nm
¶a0
'(000 1100 YI YOI
1-
am0
+
F i g u r e 6 D e f o r m a t i o n of t h e b o u n d a r y a s a f u n c t i o n of t i m e .
LL-ZII.
r,
and
r,
+
F i Q u r e 7 P r e s s u r e profile of t h e f i l m a s a f u n c t i o n of time. '
574
References Ill
C21
C31
143
C51
Cbl
C71
C81
c91
Cl0l
K U Z H A , D.C., ' A n a l y s i s o f Pumping R i n g s ' , ASHE J . o f Lub. Tech. 1971, 93, 287-292. EUSEPI, H.W. e t a l . , 'An A n a l y t i c a l and E x p e r i m e n t a l I n v e s t i g a t i o n o f an E l a s t i c Puaping R i n g ' , Proc. o f t h e N i n t h I n t . Conf. on F l u i d S e a l i n g 1981, 219-235 ( B H R A F l u i d E n g i n e e r i n g , UK). ZULL,L. and KETTLEBOROUGH, C.F., 'An Elastohydrodynamic A n a l y s i s o f T r a n s i e n t Pumping R i n g O p e r a t i o n ' , ASNE J. o f Lub. Tech. 1975, 97, 195201. S M I T H , P.J. and KEITH, T.G., ' S i m u l a t i o n o f an O i l Pumping R i n g Seal f o r a S t i r l i n g Engine', S i m u l a t i o n Aug. 1980, 49-60. JONG, E.A. d e e t a l . , ' A F i n i t e Element C a l c u l a t i o n o f t h e Elastohydrodynamic Behaviour o f a R e c i p r o c a t i n g S e a l ' , Proc. o f t h e N i n t h I n t . Conf. on F l u i d S e a l i n g 1981, 403-412, (BHRA F l u i d E n g i n e e r i n g , UK.) ZIENKIEWICZ, O.C., 'The F i n i t e Element H e t h o d ' , N c G r a w - H i l l Book Company L t d . U.K., 1977. ROHDE, S.H. and HcALLISTER, G.T., 'Variational Formulation f o r a class o f F r e e Boundary Problems A r i s i n g i n Hydrodynamic L u b r i c a t i o n ' , I n t . J . o f Engng. S c i . 1975, 13, 841. ETSION, I . , ' A n a l y s i s and D e s i g n o f a U n i f o r m - C l e a r a n c e Pumping R i n g Rod S e a l f o r t h e S t i r l i n g E n g i n e ' , ASRE J . o f Hech. D e s i g n 1981, 103, 67-72. 'Sepran user manual', SEGAL, G., I n g e n i e u r s b u r e a u SEPRA, L e i d s c h e n d a m The N e t h e r l a n d s , 1984. LEEUWESTEIN, A., ' E i n d i g e Elementen Rethode v o o r de R e y n o l d s v e r g e l i j k i n g ' , L a b o r a t o r y f o r H a c h i n e e l e n e n t s and Tribology, Technical University " D e l f t " , 1982.
6p. f . I 1 So t h a t : 2 1 pi i=l
i-1
a+. a t
I r
. ax
h 1 1dx + 124 a x
it1
1
I Kfl ah
dx =
r
with: x2 - x
f,=--
,
- x,
x,
x - x, = x 2 - xi
f,
and:
f r o m which f o l l o w s :
The c o u p l e d R e y n o l d s - e l a s t i c p r o b l e m i n R, be w r i t t e n w i t h (Galerkin-equations):
can
APPENDIX I D e r i v a t i o n o f a l i n e a i r l i n e element The R e y n o l d s e q u a t i o n i n R,
r u n s as f o l l o w s :
I& r - I -
1 ~
from which f o l l o w s t h e G a l e r k i n - e q u a t i o n :
u
n af . (hn)'
dlL ax
a+
n+l 1 dx h ax
dx+
ax
t
r
I
ah
j = 1m.i
dx = 0
;fj
r
With t h e h e l p of a m o d i f i e d Crank-Nicolson scheme ( m i d p o i n t r u l e ) and
The f i n i t e e l e m e n t a p p r o x i m a t i o n i s t h e n : ,l
d! = h n t l - h n ; hgntl dt At
hntl
t
2
hn
.
575 we get,:
Rewriting the l a s t equation yields: a+.
2
(hn)
ax
n 3f
dx
r
dhg""
dx t
r
r +
%$
( h s + dhg')'
t
I
r hn+l r f
j dx =
r
1%
n 3f.
(hs +dhgn)*dhgn
3
r
dx
t
r
r
a+ ax
dx
w i t h p ( x ) = ;(x)
=
or: +
-
(hgn)2(hg"t'
I r
h5
1
t
?*dhto
f . dx J
n af .
%2
hgn)
dx
t
r
L
n t l af >dx ax ax
n a
t
%2
J
r
y
r i=l
t
x2
f,
xt
and
-
x
-and
=
-
f,
X I
p. f.; 1
1
x - x, = 7 xz x,
-
3f1 -1 a+2 = -and - = ax x Z - x1 ax
1 ___ x 1 - X,
in w h i c h p e r t i m e s t e p f o r t h e f i r s t Newton s t e p hg" = h t o and p
w i t h hg" = h 5
9
t
Now t h e above e q u a t i o n can b e w r i t t e n as f 01 lows:
= pto dhg"
hS = i n i t i a l h e i g h t
S pg"'
t
dhg = d i s p l a c e m e n t
and hg
'" -
hgn = dhgntl
-
where
dhg n
The e q u a t i o n becomes:
5 .
n t l af
$-dx
( h s + dhg')'
t
r
+
I
1
(2*dhg
ntl
r
-
2*dhto)
f . dx = 0 J
t h e new h t o p e r t i m e s t e p is 2 r h g
n+ 1
t h e new p t o i s
2*pgnt+l
t h e new dhto
2*dhgnt1
is
.
1,1-1
-
hto PtO
-
dhto
(H
t
L
t
N ) dhg""
= R + V
t
1
576 The e l e m e n t m a t r i x now becomes:
I-1,i-1
he element m a t r i x can be w r i t t e n as
"1
I
1 I
a.
1,1-1
*I
L '1-1 , i - 1 t
'1-1,1-1
'i-i,i-1
1-1,1-1
t
l'*' Vl
'iIi-1
II
The e l e m e n t v e c t o r i s : *I
ml,l-l t 5. 191
t
nl,l-l
v and w a r e t h e unkown d i s p l a c e m e n t s w i t h
= v:
dhg""
pg
n+l
= p
For the e l a s t i c i t y p a r t t h e f i l m pressure functions as external load.
A =
-
[ r
pgntl
f j r dx
191
t 7-l111-1
1-1,1-1
51,1-1
a. .
1,1-1
5.
1.
l1,I-l
I
PI
5 .
.
111
SESSION XIX MACHINE ELEMENTS (1) - RING OILED BEARINGS Chairman: Professor W.O. Winer
PAPER XIX(i)
Thermal network analysis of a ring-oiled bearing and comparison with experimental results
PAPER XIX(ii) Performance characteristics of the oil ring lubricator - an experimental study
This Page Intentionally Left Blank
579
Paper XIX(i)
Thermal network analysis of a ring-oiled bearing and comparison with experimental results D. Dowson. A.O. Mian and C.M. Taylor
Synopsis The thermal analysis of plain journal bearings is a complex matter and this may be particularly true for bearings of the self-contained type. The lubricant feed temperature is generally not known for self-contained bearings and is mainly a function of the capacity of the bearing to dissipate the heat generated within the lubricant film. A further difficulty arises from the fact that the thermal environment may be different for specific applications of a particular design. The purpose of this paper is to demonstrate a thermal network analysis for predicting the performance of self-contained journal bearings. Theoretical predictions have been made for the performance of a 110 mm diameter ring-oiled journal bearing and a comparison undertaken with experimental results for a projected bearing pressure Of 1 m / m 2 and an average ambient temperature of 23OC. 1.
INTRODUCTION
self-contained ring-oiled journal bearing is shown in Figure 1 , illustrating the main features of the bearing. For clarity the lubricant reservoir or oil sump, in which the lower portion of the ring was immersed, and the shafdjournal to which the lubricant (oil) was delivered hy the rotation of the ring are not shown.
A
Possibly the earliest publication on the performance of ring oiled bearings is that of Karelitz (1) in 1930. The parameters involved in the delivery of oil by the ring to the journal have been detailed more recently by Heshmat and Pinkus (2). The accurate prediction of the oil delivery by a ring o f a specified design under given operating conditions is still an interesting and difficult task. It was expected that the test bearing (shown in Figure 1 ) would be starved due to the small delivery of oil by the ring to the journal. The performance of the test bearing was therefore calculated in terms of a starvation parameter (B /R) which could be determined if the oil defivery to the journal (equal to the side-leakage flow) was known. It was evident that under thermal equilibrium conditions, all the heat generated within a self-contained bearing must be dissipated to its surroundings. The heat dissipation capacity may therefore restrict the operating range of self-contained bearings. Hence a consideration of this heat dissipation capacity must form an integral part of the analysis when predicitons of performance are made. The complex geometry of many self-contained bearings makes the use of detailed analyses, such as those based on the use of finite elements, a major task involving considerable
computational effort. Furthermore, the analysis for dissipation of heat must be linked to an analysis of the fluid film characteristics to obtain consistent solutions for the bearing performance. In this study a thermal network model was used to model the dissipation of heat from the fluid film to the ambient surroundings through network elements. The network elements were connected thermally at nodes. This technique has been reported earlier by Baudry ( 3 ) who applied the technique to a ring-oiled bearing, although it was necessary to use an average film temperature as a reference to obtain solutions for the node temperatures and bearing performance. Kaufman, Szeri and Raimondi (4) also reported the use of such a thermal analysis to predit the performance of a disc lubricated bearing, employing an iteration to balance the heat generation and the heat dissipation. Details of the network model were not presented. With the thermal network analysis presented here, a finite difference technique was employed to obtain solutions for starved, finite width partial arc bearings consistent with the thermal network model. Solutions were obtained for the performance of the test bearing operating in the thermal environment provided by the experimental apparatus shown in Figure 2, allowing a direct comparison between theoretical and experimental results under equilibrium conditions to be made. 1.1
Notation
A
area.
Af b
adjustment factor for moving the starvation boundary circumferentially. axial width of starvation boundary.
580
Te
effective film temperature.
Tf
oil feed temperature to hearing o r oil sump temperature.
Tin bearing inlet temperature.
Tmax maximum bearing temperature. U
surface velocity of journal.
ys
axial location of starvation boundary, measured from the edge of the bearing.
0
circumferential co-ordinate, measured from the position of maximum film thickness. angular location of hearing inlet. 0
Figure 1
dcOu
angular location of starvation boundary with the Couette flow assumption.
dr
angular location of reformation boundary.
p
lubricant density.
v
kinematic visocisty
E
eccentricity ratio.
A ring-oiled hearing
axial width of journal bearing.
R R 0
axial width at a line inlet to the bearing assumed to be full of oil ( o r "groove" width).
C
radial clearance
C
specific heat capacity of lubricant.
D
journal diameter.
G
thermal conductance.
h
lubricant film thickness.
hO
hC;3V
H H
lubricant film thickness at inlet. lubricant film thickness at rupture. bearing power loss.
oil
k Q ,
heat carried away by the lubricant. thermal conductivity. flow at line inlet to bearing.
Qsideside-leakage flow.
T
temperature.
Figure 2 2.
The experimental apparatus
AN ANALYSIS FOR STARVED, FINITE WIDTH PARTIAL ARC BEARINGS
The test bearing contained a bush, split horizontally, with two oil grooves located in a plane 90' to the vertical load applied to the shaft. A portion of the upper half of the bush was removed to accommodate the ring. Oil starvation at the inlet groove is an inherent feature of this type of bearing. The bearing characteristics will be determined almost entirely in the lower half of the bush (excluding the circumferential extent of the grooves) and a finite width partial arc bearing
581
analysis is therefore appropriate.
In order to generate the dimensionless variables required for the thermal network analysis, a dimensionless form of the Reynolds' equation was solved using a finite difference approximation technique (5). An isoviscous lubricant was assumed.
In addition to the lubricant supplied to the inlet of the bearing arc by the ring, it has been shown ( 6 ) that the oil leaving the trailing edge of the arc may adhere to the journal and he conveyed to the inlet. By assuming that an inlet axial width (B ) was full of lubricant, a reformation boundary'at the converging inlet would be formed, as shown in Figure 3. The fol1owii:g assumptions were used initially to locat'e the reformation boundary. p . 1
't
-
0
The notation for determining the reformation boundary
All the lubricant leaving the end of the bearing adhered to the journal and was recirculated to the inlet ($= Q0). The circumferential pressure gradients in $ 1 the inlet region ($ < $ were negligible i.eo Couef?k? flow & $!I': Rupture of the film occurred at the ($=T) position, the flow leaving this region being carried on the journal to the bearing inlet. Assumptions (ii) and (iii) were required in order to simplify the analysis such that the reformation boundary could be determined (a priori) for a given bearing geometry together with a value of starvation parameter ( R /B). Referring to Figure 3, a gross flow continuity equation was written for the circumferential flow at the reformation boundary €or half the bearing width at an angular location ($cou)
-
B-b * L
J B-Bo
n
L
h = c(l+ 8cosI$ )
Also,
Equation ( 3 ) Amplied that the location of the reformation boundary was not a function of the eccentricity ratio or (B/D). This was in general agreemetlt with the theoretical and experimental results presented by Miranda (7) who found that the reformation angle was a weak function of the eccentricity ratio and (B/D) for a gauge supply pressure of zero at the inlet groove (where B = "groove" width). A physical explanation ?or this may he found by considering that for decreasing values of the eccentricity ratio, the convergence of the inlet film would be reduced, tending to increase (I$ ) , whereas the recirculating flow would be inzreased and operate to decrease The net result would be to maintain a ($r). constant shape for the starvation boundary €or fixed values of (Bo/B) and ( $ o ) . A similar argument can he applied €or increasing eccentricity ratio.
0;
----,,-Reformation boundary with Couette flow assumption True locations of reformation and rupture boundaries Figure 3
where y was measured from the edRe of the bearing.
B-b n
L
It was evident, however, that the pressure gradients in the converging inlet region would be positive in the circumferential direction, and the rupture boundary would be located downstream of the ($=TI position. Hence the true location of the reformation houndary would he upstream of the value indicated by equation (3), s o an adjustment factor (A f) was used to reposition the boundary upstream of the location indicated by equation (3), nearer its true position. %ou-
Af=
$
$0
- $ o
Thus,
From equations (3) and ( 4 ) , the value of
(A ) was estimated by using the results for the reformation angles quoted in ( 7 ) for (B /B) in the range 0.189-0.8, and for a range ofo The values eccentricity ratios of 0.2 - 0.8. of (A ) were found to be in the range 1.2 - 1.4 f s o (Af=1.3) was chosen as an initial approximation and kept constant for simplicity. The dimensionless variables required for the analysis could thus be calculated and listed in a data file for a given bearing geometry (B/D, arc span, direction of load) and for a given value of the starvation parameter A graph plot of predicted (B / R ) . chgracteristics for the test bearing is shown in Figure 4 . A curve fitting technique was then employed to facilitate the interpolation of variables such that an "effective" film temperature could be used to calculate the duty parameter (the Sommerfeld number) and hence (E), (Qin), (Qside), (H) and (I)). Fuller
582
details of the partial arc bearing analysis may be found in reference (5) where data for a wide range.of bearing parameters is also presented in Appendix D.
'Ii\ 4t
-1 -2
I
\*Hg-H*/yq+" .!
.
b .
I.,.
I. U . .
h.".,.,
+-i 222.. 0-
...I
"I...
.11..1
Figure 5
The thermal network model for the test bearing
approach offered considerable flexibility, allowing a network model for a particular type of bearing to be constructed with greater ease. Changes in dimensions, material properties and the surface heat transfer coefficient could also be made more easily. .lI.
.a . I
.,
.I .6 ~ E r n L r l C i t "P.Ll0
.2 .3
.7
.a .?I
1.1
1
.a
.t
.1
.3
.,
.CC.n(rlClL"
.a
.6 .7 "LIO
.a
.I 1 .a
Conductance
Descriprlon of heat f l o w
(C)
.a . I
Figure 4
.2
.3 .4
.I .6
.7
.a .s 1.0
Predicted characteristics of the test bearing
3. THE THERMAL NETWORK MODEL In self-contained bearings all the heat generated by viscous action within the lubricant film must be dissipated to the surroundings. In this study it was assumed that all the heat was dissipated through network elements to the ambient surroundings under thermal equilibrium conditions. The thermal environment for the test bearing was defined by (a) the air temperature, (b) the temperature of the metal in contact with the housing of the test bearing and (c) through considering that the net heat input to the bearing through the shaft close to the support bearings was zero. A direct comparison between the theoretical predictions and experimental measurements could thus be made. The thermal network model devised for the test bearing, consisting of seventeen elements and fourteen nodes, is shown in Figure 5. The number of elements used in this model was judged to be consistent with the accuracy of the assumption of constant viscosity in the fluid film and the accuracy with which data such as the surface heat transfer coefficient could be estimated. Each element in the thermal network model was considered to consist of one of six simple element types indicated in Figure 6 . This
Figure 6
Element type definitions
For each element, it was possible to relate the heat flow (H) through an element and the temperature difference (AT) across it by using the thermal conductance (G):
H
=
GAT
(6 1
In the electrical analogue the thermal conductance, heat flow and temperature are equivalent to the electrical conductance, current and voltage respectively. The formulae for the thermal conductance of elements of type 2 and I, given in Figure 6 were derived from Fourier's law of heat transfer,
583
dT dx
H=-kA
(7)
The expression for an element of type 1 was taken from reference (8) and was a modification of equation (7) using geometrical factors. Invoking Newton's law of cooling, H = a A AT
(8)
enabled the conductance of a type 3 element to be established. A formulae for the surface heat transfer coefficient ( a ) with a rotating surface, derived empirically and listed in reference ( 9 1 , was used to estimate the conductance of an element of type 5. The empirically derived values of this surface heat transfer coefficient may include a component due to heat loss by radiation. The evaluation of a conductance element of type 6 for the convection of heat from the fluid film to the oil sump may be obtained straightforwardly. For a starved bearing, the o i l flow delivered by the ring may be assumed to be equal to the side-leakage flow as shown in F o r design purposes, it was Figure 7(b). desirable to estimate the maximum bush temperature. Referring to Figure 7(a) a heat balance for the oil entering and leaving the bearing arc region was employed ( 5 ) to yield the following equation for the inlet temperature Figure 7 (Tin)
u
Tin
=
Te
"oil - -
A representation o f the major assumptions for the heat balance for the bearing
Qin' A heat halance for the recirculating and supply oil (5) in conjunction with equation ( 9 ) renders,
Te
=
Tf
+
Hail -
(10)
Qs ide' Assuming the effective temperature to be the mean of the inlet temperature (Tin> and the maximum temperature (Tmax ), the latter could be estimated, as,
T max
=
2Te
- Tin
(11)
Clearly a more sophisticated estimate of maximum temperature, based upon experimental evidence as it becomes available, will in due course be possible.
4. THE THERMAL NETWORK ANALYSIS Using the electrical analogy, it was possible to obtain a solution for the thermal network model presented in Figure 5 for the unknown temperatures (analogous to the voltages) using Kirchoff's current law. Kichoff's current law states that the sum of currents flowing into a node is zero, where currents flowing away from a node are negative, and currents flowing into a node are positive. Thus it was possible to write an equivalent heat balance at each node with an unknown temperature. At node 1, for example,
A set of linear equations containing the unknown temperatures could thus be obtained, [MI [TI = [RHS]
where
(13)
[AA]
=
a coefficient matrix constructed from element conductances.
[TI
=
a column matrix ( o r vector) of unknown temperatures.
[RHS] = a column matrix including heat sources and sinks.
A solution for the unknown temperatures could be obtained if all element conductances, heat sources and sinks were known, [TI
tAA1-l [RHS]
(14)
A thermal network analysis solution incorporating test bearing operating characteristics (for a set value of (B /B)) consistent with the thermal network moael for the dissipation of heat could therefore be obtained. A computer program was developed for this purpose based on the following procedure:
An effective temperature for the fluid film (T was assumed. The bearing power loss an8 side-leakage flow could then be determined and all element conductances calculated. A solution for all node temperatures could be obtained as indicated by equation ( 1 4 ) . A new estimate for the
584
effective temperature (T ' ) was thus possible (according to t& thermal analysis described previously) which was consistent with the heat dissipation for the thermal network model. (iii) If the estimate for the effective temperature used in (i) differed from that determined from the solution of equation (14) (outside a tolerance for the temperature of O.O5OC), a new estimate for the effective temperature (T = (T +Te')/2) was adopted. Steps ( i ) a d (iiIewere repeated in this iteration process until convergence was achieved. This procedure was found to be very stable, requiring approximately 8-10 iterations € o r convergence. Predictions for the performance of the test bearing could be made for the given operating conditions of load and speed with a specified thermal environment. The operating characteristics, including the eccentricity ratio, the attitude angle, inlet and side-leakage flow, bearing power l o s s , maximum temperature, temperatures at selected locations (nodes) and the heat flow through selected elements could be listed. These solutions were obtained rapidly, requiring approximately 0.2 CPU seconds on an Amdahl 580 computer for one operating condition. Furthermore, the effect of changes in selected parameters such as the clearance ratio, o i l viscosity grade, air temperature, surface heat transfer coefficient, shaft speed and load could be studied interactively
.
5.
A COMPARISON BETWEEN EXPERIMENTAL MEASUREMENTS AND THEORETICAL PREDICTIONS
An experimental apparatus designed to measure the performance of a 110 mm diameter ring-oiled bearing has been detailed in reference [5]. The test bearing specification and operation was as follows, journal diameter = 110 mm nominal clearance ratio (c/R) = 0.0016 ring inside diameter = 190 mm self-aligning bush B/D = 0.7 for lower half of bush, B/D=0.18 for upper halves. IS0 VG32 oil as lubricant (v100~C=5.4c~t) bearing arc span = 130' load direction = vertically downwards standard ring immersion = 25% of inside diameter. The thermal network analysis predict ons for a projected bearing gressure of 1MN/m and an air temperature of 23 C were used in the comparison with experimental results.
3
Although the thermal network analysis was not sufficiently detailed to allow a direct comparison for temperatures at each thermocouple position adopted, the comparisons shown in Table 1 for a shaft speed of 50 Hz indicate an excellent agreement between the predictions and experimental results not only for temperatures but also for other operating characteristics, A value for the starvation parameter of (Bo/B = 0.115) was used such that the side-leakage flow (or the oil delivered by the ring to the
journal) was approximately the same as that measured experimentally.
~ n l e tF
~ O W( m 3 / s )
Eccentricity Ratio Power Lass
-.ring
oil
(lo
sump Temperature
(OC)
Inlet ccoove Tcmprature
(Oc)
outlet croove Temperacure, Effective Temperacure (OC)
0.87
0.79
580
511
66.1
57.5
75.5
80.8
83.7
86.9
Temperature Difference ACrosa Upper U f e Joints ('C)
Table 1
A comparison between theory and experiment for a rotational frequency of 50 H a specific loading of 1 MNlmf'and an air temperature of 23OC.
From Table 1 it can be seen that a substantial portion of the flow entering the hearing arc was predicted to originate from the recirculating flow from the end of the bearing arc carried over by the shaft. A study of the dimensionless data indicated that this would be true at high starvation (low values of (R /B)) o r at low values of the eccentricity rati8. The bearing was estimated to be 87% starved f o r this operating condition. The predicted power loss was somewhat lower than that which was measured experimentally. This may be attributed to the neglect in the analysis of the contribution to the power l o s s from the upper half of the bush, the groove inlet and outlet regions and the churning of the oil by the ring. The lower predicted housing temperatures, however, were consistent with the lower predicted power loss. The predictions and experimental measurements for the eccentricity ratio and the side-leakage flow are shown in Figures 8 and 9 respectively. The predictions are presented for different values of the starvation parameter (B /B) where (Bo/B=l.O) represents a flooded inle? region (0% starvation) and (~~/B=0.05)represents high starvation (approximately 93%). In Figure 8, the eccentricity ratio is predicted to he sensibly constant for a given value of (B,/B). This may be interpreted by considering the Sommerfeld number which is a function of the eccentricity ratio. For a given load and bearing geometry, therefore, the eccentricity ratio would be constant i f an increase in shaft speed produced a decrease in
585
lubricant film in the context of the surrounding bearing and pedestal. This is not normally the case for currently available plain bearing design procedures (e.g. lo).
(b)
Determination of the temperatures of the bearing structure and the heat dissipation characteristics.
Both these aspects involve potential factors which may limit the range of operation of a ring-oiled bearirig.
Figure 8
The predicted and measured eccentricity ratio at a specific load of 1 MN/m2 1.111
,KO
(iii) Predictions of the performance of a bearing using the model have been compared with experimental data from a 110 nun shaft diameter ring-oiled bearing. Excellent agreement: has been demonstrated. Fuller details of the experimental apparatus and much wider empirical data will he presented in a future publication. The purpose of the present paper has been to identify the thermal network analysis and its application. (iv) The analysis as it stands requires the input of a starvation parameter (R /B) representing the width of full flu% film at the inlet groove location. For the purpose of the present study this has been determined from experimental evidence. This will be a requirement until a satisfactory analysis of the viscous lifting process of lubricant from the sump and the transfer of lubricant from the ring into the bearing clearance space is developed.
Figure 9
The predicted and measured side-leakage f ow at a specific load of 1 MN/m
3
(v) The thermal network model has proved a reliable and flexible t o o l which may be used quickly and with confidence to assess the influence of a wide range of design and enviromental variables upon the bearing performance.
7. viscosity such that the product of the dynamic viscosity and the shaft surface speed was constant. This appears to he very nearly the case and is confirmed by the experimental results. It may also be deduced that for a constant eccentricity ratio, side-leakage flow would be a linear function of the shaft rotational frequency, as shown in Figure 9. The starvation for the bearing can he seen to increase from about 50% to 86% for an increase in shaft speed from 15 Hz to 50 Hz. Fuller details of the cpmparison between theoretical preditions of the test bearing performance and experimental measurements may be found in reference (5). 6.
The study described was undertaken as part of a Science and Engineering Research Council Cooperative Award in Science and Engineering scheme. The authors are pleased to acknowledge the financial contribution of Michell Bearings-Vickers plc and the technical contribution of a number of the company's staff to the project. APPENDIX 1 References (1)
Karlitz, G.B. "Performance of oil-ring bearings", Trans. ASME., Vol. 5 2 , 1930.
(2)
Heshmat, H. and Pinkus, 0. "Experimental study of stable high-speed oil rings", Trans. ASME, J. of Tribology, Vol. 107, 1985.
CONCLUSIONS
(i) A thermal network analysis suitable for the prediction of the performance of a ring-oiled bearing has been presented. This has linked curve fitted data for the lubricant film characteristics to an analysis for the dissipation of heat from the bearing through conductance elements modelling the thermal paths.
"Some thermal effects in (3) Baudry, R.A. oil-ring journal hearings", Trans. ASME, Vol. 67, 1945. (4)
(ii) The model developed enables, (a)
ACKNOWLEDGEMENT
Prediction of the performance of the
Kaufman, H.N., Szeri, A.Z., and Raimondi, A.A. "Performance of a centrifugal disk-lubricated bearing", Trans. ASLF., Vol. 21, 1978.
586 (5)
Mian, A.O. "The performance of ring-oiled bearings", Ph.D. thesis, University of Leeds, Dept. of Mechanical Engineering, U.K., 1986.
(6)
Heshmat, H. and Pinkus, 0 . "Mixing inlet temperatures in hydrodynamic bearings", Trans. ASME, J. of Tribology, Vo. 107, 1985.
(7)
Miranda, A.A.S, "Oil flow, cavitation and film formation in journal bearings including an interactive computer aided design study", Ph.D. thesis, University of Leeds, 1983.
(8)
ESDU Item No. 78028 "Equilibrium temperatures in self-contained bearing assemblies. Part 111 : Estimation of thermal resistance of an assembly''. Engineering Sciences Data Unit, London, 1979.
(9)
ESDU Item. No. 78029 "Equilibrium temperatures in self-contained bearing assemblies. Part IV : Heat transfer coefficient and ioint conductance''. Engineering Sciences Data Unit, London, 1979. 84031 "Calculation methods for steadily loaded axial groove hydrodynamic journal hearings", ESDU International Ltd., London, 1984.
(10) ESDU Item No.
587
Paper XIX(ii)
Performance characteristics of the oil ring lubricator - an experimental study K.R. Brockwell and D. Kleinbub
Results are presented from an experimental investigation of oil rings ranging in size from 86 am to 127 nun inside diameter, running on a 57 mm diameter shaft at speeds up to 3600 revfmin. The start up and steady state operating characteristics of a number of ring designs are presented and found to be dependent on ring bore geometry, oil viscosity and, to a lesser extent, ring submersion level. Ring design is also important from the standpoint of oil delivery to the bearing, and an optimum cross section is recommended from the results of a parametric study. Furthermore, it is shown that the lubrication distribution arrangement (grooving) of the journal bearing influences the oil flow characteristics of the ring lubricated bearing system. 1
INTRODUCTION
Many rotating machines, including motors, fans, pumps and turbines use self contained, hydrodynamic bearing systems to support and locate the rotating components. Such bearing assemblies are commercially attractive (1) because of their lower cost, increased reliability and ease of maintenance, compared with bearings having external lubrication systems. Consequently, demand for them has increased i n recent years. A self contained bearing assembly typically contains a number of extra components such as a heat exchanger and lubricant circulation system. New (1) divides lubricant circulation systems used in current bearing designs into two categories, ie. 'pressurised' and 'non pressurised' devices. Non pressurised types usually have a fixed disc or a loose ring. The oil ring lubricator is popular because of its simplicity, reliability and low cost and was applied initially in low speed machinery at least 100 years ago ( 2 ) . Its range of use has now been considerably extended, so they are now used where shaft surface speeds are as high as 20 mfsec (3). Studies carried out in the 1930's and 40's (2,4,5,6,7) established a basic understanding of ring behaviour. They showed that the rotation of the ring is dependent on the propulsive force between the rotating shaft and the ring. Later studies (8,9) derived dimensionless correlations of ring speed and oil delivery for the different regimes of operation. In particular, Lemmon et a1 (9) obtained quantitative expressions for ring behaviour as functions of oil characteristics, ring geometry and ring immersion level. However the effect of the bearing itself on the oil feed mechanism was not investigated. Heshmat et a1 (10) incorporated the bearing in their study and concluded that most oil ring lubricated bearings operate, to some degree, under starved conditions. The purpose of this experimental program was to extend the scope of oil ring studies and to observe the effect of ring and bearing design on oil flow through the bearing clearance space. Initially, a study of ring geometry was conducted in the absence of the bearing, which the ring would normally lubricate. It will be seen that
the ring passes through several regimes of operation, and also that there is a considerable delay before the ring reaches its maximum speed, after start up. T.n the second phase of the experimental program, a bearing was incorporated in the test rig and the rate of oil discharge from the bearing clearance space was measured. From a parametric study involving several ring designs, it will be seen that for a given bearing there is an optimum ring configuration (from the point of view of oil delivery and ring stability). Furthermore, it will be shown that the bearing lubrication distribution arrangement (grooving) is important from the standpoint of the oil flow through the bearing.
2 EXPERIMENTAL PROGRAM 2.1
Test equipment
As stated in the introduction, the objective of the first section of the work was to determine the start up and steady state behaviour of a number of oil rings of differing design. The test rig constructed for this purpose is shown in It shows the 3.5kW variable speed Fig. 1. electric motor which drove the 57mm diameter shaft at speeds up to 3600 revfmin. Also shown are the oil reservoir, the plexiglass splash cover, the adjustable weir used to control the oil level in the reservoir and the data acquisition system. Fig. 2 shows the arrangement of the four 3 m diameter nylon pins used to axially The clearance locate the ring on the shaft. between the head of the pins and the side of the ring was lmm on each side, ie. it was large in order to minimise viscous drag effects. The system developed to measure the acceleration of the ring at start up, as well as the ring velocity during steady running conditions, consisted of an emitter and a phototransistor detector, both mounted in a specially manufacTwo such tured aluminum housing (Fig. 3). identical systems were used to monitor ring and shaft speed. They were mounted on the rig so that they were approximately 5mm from the moving surfaces. These surfaces were painted with a number of equally spaced marks using matt black
588
F u r t h e r d e t a i l s of t h e b e a r i n g c o n f i g u r a t i o n and o p e r a t i n g c o n d i t i o n s are g i v e n below:
paint.
Rearing:
Diameter = 5 7 m , Length = 6 4 m , D i a m e t r a l c l e a r a n c e = 0.14mm, Ring s l o t width = 16mm.
Operating Conditions:
Load = 200N Speed v a r i a b l e from 0 t o 3600 revfmin.
T a b l e 1 shows t h e c r o s s s e c t i o n of t h e d i f f e r e n t r i n g s . A l l r i n g s were f a b r i c a t e d from SAE 660 bronze, a f t e r Heshmat et a1 ( 1 0 ) who showed t h a t o i l d e l i v e r y by t h e bronze r i n g is about 10 p e r c e n t h i g h e r t h a n f o r r i n g s manufactured from The i n t e r n a l d i a m e t e r of b r a s s o r muntz metal. t h e r i n g s ranged between 8 6 m and 127m.
F i g . 1 V i e w of t e s t r i g study.
-
1 Emitter 2 Aluminum body 3 Nylon i n s e r t
r i n g speed
4 Leads 5 Detector
4 Q P
S i g n a l s from t h e d e t e c t o r were f e d t o a p u l s e h e i g h t comparator i n which p u l s e s above a c e r t a i n t h r e s h o l d l e v e l were shaped i n t o h i g h l o g i c l e v e l s i g n a l s , and a n y t h i n g below was cons i d e r e d low l o g i c l e v e l . Using s p e c i f i c i n s t r u c t i o n s i s s u e d t o t h e d a t a a c q u i s i t i o n system, a n a l o g d a t a from t h e p u l s e h e i g h t comparator was collected. Thus, f o r a p a r t i c u l a r t i m e i n t e r v a l and sampling r a t e , v e l o c i t y was c a l c u l a t e d knowing t h e s p a c i n g of t h e b l a c k marks p a i n t e d on t h e r i n g and s h a f t s u r f a c e s .
Fig. 3
2.2
Schematic of o p t i c a l s e n s o r .
T e s t procedure
Experiments t o d e t e r m i n e optimum r i n g c o n f i g u r a t i o n from t h e s t a n d p o i n t of r i n g speed were c a r r i e d out u s i n g t h e a p p a r a t u s shown i n Fig. 1. test procedure adopted was to first The s t a b i l i s e t h e t e m p e r a t u r e of t h e o i l in t h e r e s e r v o i r and t h e n t o a d j u s t t h e o i l l e v e l t o g i v e t h e r e q u i r e d r i n g submersion l e v e l . Steady s t a t e t e s t s were u s u a l l y commenced a t 100 revfmin and t h e r o t a t i o n a l speed of t h e r i n g was determined u s i n g t h e o p t i c a l tachometer s y s t e m . The s h a f t speed was t h e n i n c r e a s e d i n s t e p s by
1 E l e c t r i c motor 2 Support b e a r i n g s 3 Shaft 4 O i l reservoir
Fig. 2
5 Bearing 6 Loading mechanism 7 Loose r i n g 8 Graduated v i a l s
Ring a x i a l l o c a t i o n arrangenient.
Fig. 4 shows a s c h e m a t i c of t h e modified t e s t r i g used t o measure t h e amount of o i l It d e l i v e r e d t o t h e b e a r i n g by t h e oil r i n g . shows t h e b e a r i n g , t h e o i l r i n g l u b r i c a t o r and o i l r e s e r v o i r , t h e s h a f t assembly and t h e l o a d i n g O i l d i s c h a r g e d from each end of t h e mechanism. b e a r i n g was c o l l e c t e d in g r a d u a t e d v i a l s , so t h a t t h e flow r a t e could be determined. The t e s t b e a r i n g ( B e a r i n g I ) , which is shown in F i g . 1 4 , had a c i r c u l a r bore w i t h two a x i a l g u t t e r w a y s , Each each 5 7 m long x 3 m deep a t t h e j o i n t . g u t t e r w a y extended 15' e i t h e r s i d e of t h e j o i n t Also, a t t h e t o p of t h e b e a r i n g was an line. a x i a l groove 3 7 m l o n g x 1 3 m wide x 2m deep.
Fig. 4
Schematic o f t e s t rig-oil delivery studies.
589
means of the variable speed drive and the ring speed measuring procedure repeated up to 2500 revlmin. Start up tests were limited to the 1 2 7 m size ring with plain (Type 2) and grooved (Type 4) bore surface configurations. Ring acceleration was measured for sudden shaft starts The time taken for the (up to 2000 revlmin). shaft to reach this speed was 0.5 sec. Experiments with higher shaft speeds were abandoned because of splashing of oil on the optical sensor. Ring immersion levels for both the start up and steady state tests varied between 5 and 25 percent of the ring inside diameter. Lubricants used ranged from a light hydraulic oil ( I S 0 VGl5) to a heavy gear oil (IS0 VG220), providing a range of working viscosities of between 18 centistokes (cSt) and 700cSt. Ring ype Q
Ring Cross Section
Description
plain bore.
I" 87d '
2
y K 7 A 6 . 4 f k 14.34? r14.31.J m 6 .
3
4
f
Rectangular section with relieved sides, plain bore.
Rectangular section, square grooved bore*
3 730'
5
Rectangular section with relieved sides, square grooved bore*
A
L a 6 . 4 fb14.34f
3
Trapezoidal section, square grooved bore*
RING SPEED STUDIES
Lemon et a1 ( 9 ) state that the most important factor affecting oil delivery of the ring is the speed of the ring itself, but they point out that at higher speeds, centrifugal forces fling the oil from the ring before it can be delivered to the bearing. Heshmat et a1 (10) give a detailed account of the behaviour of high speed oil rings and suggest that there are a number of regimes of operation through which the ring passes, as its speed increases. The first two regimes are well documented. At low shaft speeds (Regime l ) , the friction between the ring and the shaft is sufficient for the ring to be driven without any significant amount of slip between them. As the speed of the shaft continues to rise, slippage occurs and ring speed drops, but with a further rise in shaft speed, a hydrodynamic film is established between the ring and shaft and there is again an increase of ring speed with shaft Heshmat et a1 (10) have shown speed (Regime 2). that unstable ring motions, both in the plane of rotation (Regime 3) and in conical and translatory modes of vibration (Regime 4) occur with further increases in shaft speed, which significantly reduces the oil delivery to the bearing. 3.1
p12.81
4
speed was then increased in steps, repeating the procedures outlined above, to a maximum shaft speed of 3600 revlmin. Bearing temperature was monitored with a thermocouple mounted in the wall of the bush, approximately 2mm from the bearing surface and in line with the applied load.
Ring steady state behavlour
In this first part of the study, the effect of several ring design and ring operating parameters on ring speed and stability were examined. As stated earlier, this work was carried out in the absence of the bearing, which the ring would normally be lubricating. The factors investigated were: a
Rectangular section with relieved sides and bore. ~-
a
Oil viscosity Immersion depth Ring bore configuration
Ring sizes are shown in Table 2 below. ~
*3 grooves, 1.5mm wide x 1.5mm deep 1401127 Table 1
Ring cross section details
To determine optimum ring configuration from the standpoint of oil delivery, the experimental Tests apparatus shown in Fig. 4 was used. involved the use of the 114mm and 127mm rings, each with a variety of cross sectional shapes and bore configurations. Immersion levels ranged between 10 and 20 percent of the ring inside diameter and the oil in the sump ( I S 0 VG46) was maintained at 40°C. The experimental procedure was similar to that described above. After the chosen ringbearing combination had been mounted in the test apparatus with the oil collectors in place, the oil temperature was stabilised and the ring submersion level was adjusted. Tests commenced at 500 rev/min and after the bearing temperature had stabilised, ring speed was noted and side flow from the bearing was determined by collecting oil in the graduated vials. The journal
* *b=6.4
* Table 2
Ring details
-
Ring speed investigation
3.1.1 Oil viscosity. As the temperature of the oil increases, so its viscosity decreases. Past investigators (2,4,5,6,8 and 9 ) have studied the effect of changes to oil viscosity on ring operating speed. Broadly speaking, it has been shown that the speed of the ring is insensitive to changes in oil viscosity, and data
590
illustrating this point was presented by Lemmon et a1 (9). Ring speed data for the 114mm ring (Type 1) and 1 2 7 m ring (Type 2 ) , both having plain bores, are plotted in Fig. 5. The results indicate that ring speed is insensitive to oil viscosity, but this only seems to be true for the thinner oils. Higher viscosity oils do give rise to substantial changes in ring speed. In the case of the 127mm ring, the difference in ring speed between the 18cSt oil and the 700 cSt oil is of the order of 40 percent at higher shaft speeds.
Shaft speed, revlmin. 0
400
a
--. E m
m- 0.
ar a
M
c 0.
2
0
0 700 cSt
-
300
200
;0.
Shaft speed, rev/min.
1
100
~
0
205 cSt A 90 cSt
1.2
0.8
0.4
Shaft speed, mlsec.
0 50 cSt
Fig. 6 Low speed operation, 114mm plain ring, 20% immersion.
A 18 cSt
0.5
altered by substantial changes in oil level (between 10 and 20 percent of ring inside diameter). Data for the 86mm and 102mm plain rings (Type 11, for immersion levels which range between 5 and 25 percent of the ring inside diameter are plotted in Fig. 7. The small decrease in ring speed associated with increased Typically, a ring submersion is confirmed. change in ring speed of 10 to 15 percent occurs
127mm ring (Type 2) U
ar
v)
2 0 m al
1.5 M
E:
d
/
1
/
/
;
i
s
r
'-9lz e
mf: m 0
1
I
15% 20%
"
0.5
* 0
See Fig. 6 I
Fig. 5
I
I
I
Ring speeds for different oil viscosities, 20% immersion.
The transition from a 'no slip' (Regime 1) to a 'slip' (Regime 2) mode of operation for the 114mm plain ring is clearly evident in Fig. 6 . The extended region of 'no slip' operation obtained with the low viscosity oils, is particularly apparent. Overall, the relationship between oil viscosity and ring speed is complex with the thinner oils extending the range of 'no slip' operation of the ring. In marked contrast, the high viscosity oils give rise to ring slippage even at the lowest shaft speeds. The opposite effect is clearly the case (Fig. 5) in Regime 2 where the thicker oils give rise to higher ring speeds. Previous workers 3.1.2 Depth o f immersion. (2,5,8 and 9) have illustrated the decrease in ring speed associated with increased ring submersion. Lemmon et a1 (9), however, states that oil delivery from the ring is not significantly
. i l l
v)
"-
1.
m al
ar a M
c 1
2
0
"
~~
0
2
4
6
Shaft speed, misec. Fig. 7 Ring speeds for different immersion levels, 90 cSt viscosity oil.
8
59 1 i n Regime 2. A t lower s h a f t s p e e d s , changes t o t h e r i n g ,immersion l e v e l have a n even smaller e f f e c t on r i n g o p e r a t i n g s p e e d , and can be considered i n s i g n i f i c a n t . Fig. 7 shows t h a t l a r g e r i n c r e a s e s i n r i n g speed can be a t t a i n e d w i t h a n immersion l e v e l of j u s t 5 percent. However, such a low l e v e l would be u n a c c e p t a b l e i n a p r a c t i c a l a p p l i c a t i o n , because t h e r i n g must be s u f f i c i e n t l y below t h e s u r f a c e of t h e o i l a t a l l times, t o e n s u r e t h a t Furthert h e i n n e r . s u r f a c e is always c o v e r e d . more, a t h i g h e r s p e e d s , i t was n o t i c e d t h a t t h e o i l u n d e r t h e r i n g was d i s p l a c e d s i d e w a y s , f u r t h e r r e d u c i n g t h e immersion l e v e l . This e f f e c t w a s a l s o n o t i c e d by Heshmat e t a 1 ( 1 0 ) .
3.1.3 Ring b o r e c o n f i g u r a t i o n . All p a s t i n v e s t i g a t o r s a g r e e t h a t some form of r o u g h e n i n g o r g r o o v i n g of t h e r i n g b o r e i n c r e a s e s t h e l i m i t of o p e r a t i o n w i t h o u t s l i p , and a l s o i n c r e a s e s t h e More r e c e n t l y , speed of t h e r i n g i n Regime 2. Heshmat e t a1 ( 1 0 ) showed t h a t t h e optimum g r o o v e d e p t h from t h e p o i n t of view of oil d e l i v e r y is
1.5mm. To v e r i f y e x p e r i m e n t a l l y t h e e f f e c t of g r o o v i n g on r i n g p e r f o r m a n c e , a 1 2 7 m r i n g , 1 4 . 3 m wide, was machined w i t h 3 c i r c u m f e r e n t i a l s q u a r e g r o o v e s e a c h 1 . 5 m wide x 1 . 5 m deep (Type 4 ) . F i g . 8 compares d a t a f o r t h i s r i n g w i t h t h a t o b t a i n e d f o r a p l a i n b o r e (Type 2) r i n g f o r a 20 p e r c e n t immersion l e v e l . It can be s e e n t h a t t h e grooved r i n g r u n s a t s u b s t a n t i a l l y h i g h e r s p e e d s ; i n f a c t , i n Regime 2 , t h e i n c r e a s e is a b o u t 80 percent. F u r t h e r m o r e , t h e r e g i o n of no s l i p o p e r a t i o n is e x t e n d e d by more t h a n 100 p e r c e n t . A l s o t h e Type 4 r i n g w a s found t o , b e a v e r y s t a b l e d e s i g n o v e r t h e r a n g e of s p e e d s t e s t e d . Data f o r t h e 1 2 7 m r e l i e v e d r i n g (Type 6) is a l s o p l o t t e d on F i g . 8 and shows t h a t t h i s d e s i g n is i n f e r i o r t o t h e Type 4 d e s i g n a t h i g h e r s h a f t speeds. However, t h e Type 6 d e s i g n d o e s e x t e n d t h e Regime 1 mode of o p e r a t i o n .
A Type 2 L Type 4
. 1.5
W Type 6
0 0
2
4
6
a
S h a f t s p e e d , m/sec. Fig. 8
E f f e c t of b o r e c o n f i g u r a t i o n on 1 2 7 m r i n g s p e e d , 20% immersion, 90 c S t viscosity oil.
F i g . 9 shows t h e development of t h e oil s h e e t on t h e u p s t r e a m s i d e of t h e 127mm grooved r i n g , f o r a r a n g e of s p e e d s up t o 2500 revfmin. As t h e speed i n c r e a s e s , s o t h i s s h e e t becomes more pronounced, u n t i l a t h i g h e r s h a f t s p e e d s , the sheet breaks down through centrifugal effects. The t h i c k r o p e of oil bounding t h e o u t s i d e edge of t h e s h e e t is c l e a r l y e v i d e n t .
Fig. 9
Development of o i l s h e e t a t d i f f e r e n t s h a f t speeds.
592
Further tests performed on the 102mm ring with modified bore surfaces gave confirmation of the superior performance of the circumferentially grooved ring. A V grooved configuration and a knurled configuration were also better than the plain bore (Type 1) design from the standpoint of ring speed, but there was evidence of instability, even at low shaft speeds. This unstable motion was typical of that found in Regime 3 (lo), with the ring oscillating in the plane of rotation. The knurled ring was particularly poor in this respect, and also exhibited rocking and translatory modes of vibration at low shaft speeds. 3.2
Ringlshaft speed relationship
-
regime 2
When the full film mode of operation (Regime 2) is reached, ring speed begins to increase with increasing shaft speed. Results obtained from this study have confirmed that ring speed also increases with higher oil viscosity. Lemmon et a1 (9) suggested that ring speed was also a function of the diameter of both the shaft and the ring, and went on to propose that those factors could be related by comparing a Reynolds Number for the driving action of the shaft with the Reynolds Number for the ring. This leads to the following relationship:
where k l and k 2 are constants. Accordingly, experimental results from the 8 6 m , 114mm and 127mm plain rings with oil viscosities which ranged between l8cSt and 700cSt have been used to determine the Reynolds Number for the ring and shaft, and these are plotted in Fig. 10 for the case of 20 percent immersion. Close agreement is evident with the results of Lemon et a1 ( 9 ) .
A al
z5
0 127mm ring
P
0 A
86mm ring 114mm ring
Id
Oil lifted from the oil reservoir as a film on the surface of the oil ring, is delivered to the shaft from the inside surface of the ring by a squeezing action as the ring passes over the rotating shaft (4). Use is also made of scrapers to remove oil from the top and sides of the ring, which can increase the flow of oil delivered to the bearing. However, ring behaviour at start up is also important. Rotating machinery driven by an electric motor can accelerate to full speed in a matter of a second, or less. The ring will take somewhat longer to reach its full operating speed, so the bearings may initially be starved of oil. To investigate this matter, start up tests were conducted on the 127mm grooved ring (Type 4 ) with three different oil viscosities. The.shaft accelerated from rest to 2000 rev/min in 0.5 sec. The results which are plotted in Fig. 11 clearly indicate that oil viscosity has a strong effect on ring performance, with the thinner oils causing a slower increase of ring velocity. For example, with the 18cSt oil, ring speed was still increasing after 4 sec. of operation, whereas with the 700cSt oil, steady running conditions were reached in under 2 secs. Fig. 11 also shows run up times obtained from the 1 2 7 m plain (Type 2) ring, which are similar to those obtained from the Type 4 ring. Thus, grooving of the ring bore seems to have little effect on ring start up characteristics. Fig. 12 compares the run up time of the Type 4 ring for immersion levels of between 5 and 25 per cent of ring inside diameter. It can be seen
1
18 cSt 90 cSt
1
7
m
'0 4
103
-
h p?,
(5%-25% immer
M
e
-a a,
~
102
1 o3
1o4
105
Shaft Reynolds Number Fig. 10 Reynolds Number relationship for plain bore oil rings, 20% immersion. 3.3
1
Ring start up behaviour
Past studies (8,9,10) have established many of the details of ring behaviour during steady running conditions, as a function of the bearing operating conditions, the characteristics of the oil and the geometry of the ring and shaft. The limiting speed of ring and disc lubricated bearNevertheless, ings have also been defined (11). in spite of its limited operational envelope, the oil ring lubricator continues to be popular, mainly because of its simplicity and reliability.
Type 4 ring
0
0
1
2
3
4
Time, secs. Fig. 11
Start up characteristics of 127mm plain and grooved bore rings, 20% immersion.
5
593 t h a t changes t o t h e submersion l e v e l of t h e r i n g have only a small e f f e c t on t h e r i n g a c c e l e r a t i o n characteristics.
ng O D / I D Number
r
A
5% immersion
*
2
15%
. l
*
4
*b = 3.3,6.4,12.8
5
*c = 1.6,3.3
*
6 Table 3
,. 0
1
2
3
4
4
Start up characteristics of 127mm ring (Type 4 ) f o r different immersion levels, 90 c S t viscosity oil.
RING OIL DELIVERY STUDIES
Because of i t s s i m p l i c i t y , t h e o i l r i n g l u b r i c a t o r remains p o p u l a r even though i t h a s a l i m i t e d envelope of o p e r a t i o n . However, because o n l y a p a r t of t h e o i l l i f t e d by t h e r i n g from t h e o i l reservoir is delivered t o t h e bearing clearance s p a c e , b e a r i n g s l u b r i c a t e d by l o o s e r i n g s o f t e n o p e r a t e w i t h some d e g r e e of o i l s t a r v a t i o n . T h i s may n o t be d e t r i m e n t a l t o b e a r i n g h e a l t h a t r e a s o n a b l e s h a f t s p e e d s , however, f o r h i g h speed o p e r a t i o n , o i l s t a r v a t i o n w i l l cause increased b e a r i n g t e m p e r a t u r e s , h i g h e r power l o s s e s and, i n a s e v e r e case, b e a r i n g s e i z u r e . 4.1
Ring d e t a i l s investigation
-
o i l delivery
5
Time, s e c . Fig. 12
*b=6.4
Ring p a r a m e t r i c s t u d y
I n t h i s second phase of t h e e x p e r i m e n t a l program, t h e b e a r i n g and r i n g assembly was t e s t e d as shown in Fig. 4. The o i l d i s c h a r g i n g from each end of t h e b e a r i n g was measured a c c u r a t e l y in t h e manner d e s c r i b e d above. T h i s flow r e p r e s e n t s t h e amount of o i l d e l i v e r e d t o t h e b e a r i n g by t h e r i n g , under s t a r v e d c o n d i t i o n s of o p e r a t i o n . The o i l r i n g p a r a m e t e r s i n v e s t i g a t e d w i t h r e s p e c t t o t h e i r e f f e c t on o i l d e l i v e r y are as follows: Ring weight Bore c o n f i g u r a t i o n Ring c r o s s s e c t i o n
Details of r i n g s i z e s a r e shown i n T a b l e 3 below. Data from t h i s r i n g p a r a m e t r i c s t u d y is p l o t t e d i n Fig. 13. The e f f e c t of changing r i n g weight on o i l d e l i v e r y is i l l u s t r a t e d i n Fig. 1 3 ( a ) f o r t h e case of t h e 114mm r i n g (Type 4 ) . This was accomplished w i t h r i n g s of d i f f e r e n t r a d i a l thickness (b) ie. 3.3mm, 6.4mm and 12.8m. Data collected from t h e s e rings i n d i c a t e s t h a t t h e optimum r i n g t h i c k n e s s is 6.4m, although t h e apparent d i f f e r e n c e i n o i l d e l i v e r y c h a r a c t e r i s t i c s between t h i s r i n g and The l i g h t e s t r i n g ( b = t h e o t h e r two w a s s m a l l . 3.3) r a n a t t h e h i g h e s t s p e e d s , r e a c h i n g 2 m/sec (335 rev/min) a t a s h a f t speed of 3600 rev/min; however, i t s motion was less s t a b l e t h a n t h e heavier rings. The h e a v i e s t r i n g ( b = 12.8) was
t h e s l o w e s t of t h e t h r e e r i n g s t e s t e d , r e a c h i n g 1.45 m/sec (243 rev/min) a t 3600 rev/min, but it was s t a b l e up t o 3600 rev/min. The 6.4mm t h i c k r i n g w a s a l s o s t a b l e up t o t h i s speed. Tests t o compare t h e performance of t h e 127mm r i n g w i t h d i f f e r e n t b o r e c o n f i g u r a t i o n s showed i n Fig. 1 3 ( b ) t h a t t h e grooved Type 4 r i n g produced t h e l a r g e s t o i l flow. The Type 2 r i n g ( p l a i n b o r e ) a l s o performed w e l l , e x h i b i t i n g o i l d e l i v e r y c h a r a c t e r i s t i c s only marginally i n f e r i o r t o t h o s e of t h e grooved r i n g . The o i l d e l i v e r y of t h e Type 6 r i n g dropped beyond 1000 rev/min, e x h i b i t i n g poor o i l d e l i v e r y c h a r a c t e r i s t i c s a t h i g h e r s h a f t speeds. T h i s s t u d y showed t h a t t h e Types 2 and 6 r i n g s both o p e r a t e d a t a speed of 0.8 m/sec (120 r e v l m i n ) a t a s h a f t speed of 3600 revlmin. This is under h a l f t h e speed of t h e Type 4 r i n g , which reached 1.8 m/sec (271 rev/min) a t 3600 revlmin. Thus, h i g h r i n g speed is n o t n e c e s s a r i l y a t r u e p o i n t e r t o good o i l d e l i v e r y c h a r a c t e r i s t i c s . Heshmat e t a1 ( 1 0 ) h a s shown t h a t r i n g c r o s s s e c t i o n h a s a n i m p o r t a n t i n f l u e n c e on r i n g performance. For a t r a p e z o i d a l shaped r i n g , t h e optimum s i d e a n g l e was found t o be 30°, regardless of r i n g d i a m e t e r o r i n s i d e groove geometry. Heshmat e t a1 ( 1 0 ) a l s o comments t h a t as t h e s i d e a n g l e approaches z e r o i.e. a s t r a i g h t s i d e d r i n g , t h e d r a g on t h e r i n g is t h e n a t a maximum and t h e ring operates e r r a t i c a l l y . Accordingly, t o examine t h e e f f e c t of r i n g shape on o i l d e l i v e r y , 114mm r i n g s were f a b r i c a t e d w i t h Type 3,4 and 5 c r o s s s e c t i o n s . Data p l o t t e d in Fig. 1 3 ( c ) shows t h a t t h e Type 4 r i n g ( b = 6.4) performed as w e l l as t h e Type 5 r i n g The Type 3 w i t h a s i d e l a n d w i d t h of ( c ) 1.6mm. r i n g w i t h s t r a i g h t s i d e s performed less w e l l , p a r t i c u l a r l y a t higher s h a f t speeds. Ring Types 3 and 5 ( c = 3.3mm) b o t h e x h i b i t e d e r r a t i c b e h a v i o u r and r a n a t a much reduced speed of 0.4 m/sec (67 r e v l m i n ) a t a s h a f t speed of 3600 rev/min. I n comparison, r i n g Types 4 and 5 ( c 1.6mm) b o t h reached 1.8 m/sec (302 r e v l m i n ) a t 3600 rev/min. Erratic b e h a v i o u r of t h e Type 3 and 5 ( c = 3.3mm) r i n g s w a s a t t r i b u t e d t o an e x c e s s i v e l y h i g h l e v e l of d r a g between t h e s i d e of t h e r i n g and t h e c l o s e s t w a l l of t h e s l o t , c a u s i n g low r i n g speed o p e r a t i o n . Occasional a c c e l e r a t i o n of t h e s e r i n g s t o a h i g h e r o p e r a t i n g speed was s e e n t o be accompanied by t h e i r moving t o t h e c e n t r e of t h e s l o t . T h i s e f f e c t was n o t i c e d by Heshmat et a1 ( 1 0 ) .
-
594
Shaft s p e e d , rcv/min. 0 3
1000
3000
2000
4000
I
I
I
1
A b=6.4mm
2
1
0
3 A Type 4 , b=6.4mm u?
8 Type 2
I 0 4
,
2
.
13b)
-
-
Fig. 14 Bore configuration of the two journal bearings.
Bore profile 127mm ring
v m
It can be seen that changing the oil distribution arrangement has a considerable effect on the bearing oil flow, ie. at higher shaft speeds, Bearing 2 oil flows were 50% higher than those measured with Bearing 1. Lower temperatures recorded in the wall of Bearing 2 gave further indication of the improved oil flow characteristics of this bearing design. Fig. 15 includes a curve showing the theoretical full flow requirement of the bearing. It can be seen that the total side flows measured fell short of this requirement, indicating that the bearing was operating under partially starved conditions. This confirms the findings of Heshmat et a1 (10). o i l viscosity was 46cSt.
01
'1
1
-
O
L
E
" 3 0 d
Cr,
r 2 .
13c) - C r o s s section 114mm ring A Type 3 8 Type 5. c=1.6mm 0 Type 4, b=6.4mm
5-
-
iD
1 '
I
0 i
X
3-
A 1 2 7 ring, # 2 bearing A 114 ring, #2 bearing 0 127 ring, #1 bearing 114 ring, 111 bearing 0 Oil f l o w requirement (12
U
0 1
I
4
0
I
8
I 12
a, vl
--. "E
Shaft speed, m/sec. Fig. 13 Optimum ring configuration from the standpoint of oil delivery. #1 Bearing, 20% immersion, 46 cSt viscosity.
-
3
2i
1-
0
4.2
Bearing study
To assess the importance of the bearing lubricant distribution arrangement (grooving) on the o i l flow characteristics of the ring lubricated bearing system, tests were conducted on two journal bearings of differing design. Bearing 1 is described i n section 2.1. Bearing 2 was similar to Bearing 1 i n all respects, but with the inclusion of a circumferential groove 29 mm wide x 2mm deep, machined in the top half of the bearing. The two bearings are shown i n the photograph of Fig. 14. Fig. 15 gives plots of the comparative performance of the two bearing designs, with 114m and 1 2 7 m rings of the Type 4 configuration. Immersion level for those experiments was 20 percent of the ring inner diameter, and the sump
4
8
12
Shaft speed, m/sec. Fig. 15 Oil f l o w for the two bearing configurations, Type 4 ring, 202 immersion. ( ( 5 3 ) Bearing temperature, OC) 5
CONCLUSIONS
a)
It is confirmed that the ring passes through several regimes of operation as shaft speed increases. At low speeds the sliding between the ring and the shaft was immeasurably small. Hydrodynamic oil films formed at higher speeds lead to slippage between the ring and journal. Consequently, the
595 circumferential speed of the ring is normally only about 15-30 percent of shaft speed, depending on ring design and operating conditions. Modification of the ring bore can increase ring speed substantially. Circumferential grooving appeared to provide maximum ring performance and stability; other modified bores performed less well and some caused unstable motion and low operating speeds in the second regime of operation. Variations in ring immersion level have only a small effect on the start up and steady running behaviour of the ring. Sump oil viscosity has a greater effect, particularly on the run up time of the ring when accelerating from rest. Tests on plain rings with a wide range of operating conditions have confirmed the validity of the Reynolds number relationship between shaft and ring. A parametric study of several ring designs showed that the optimum ring cross section from the point of view of oil delivery is either the trapezoidal cross section or the modified rectangular cross section, each with a side land width of approximately 1.5mm. Larger land widths created undue drag and erratic ring operation. The recommended ring bore configuration is one with circumferential grooves, 1.5m wide x 1.5mm deep. Rings operating at higher speeds do not necessarily possess better oil delivery characteristics. Other ring and bearing parameters often influence the flow pattern more. The oil flow characteristics of the ring lubricated bearing are affected not only by ring design but also by the grooving in the In this study, one bore of the bearing. bearing design has been shown to be superior to another, but further work is needed to establish an optimum bearing grooving arrangement. 6 ACKNOWLEDGEMENT Acknowledgement is due to the National Research Council Canada for their permission to publish the results of this study. The authors are also indebted to Dr. C. Dayson for his help in the preparation of this paper. Dedication This paper is dedicated to Mr. Fred Hildreth who designed the test equipment used in this study and who passed away earlier this year soon after retiring from the National Research Council Canada.
References
NEW, N.H. 'Self contained bearing assemblies with hydrodynamic lubrication', Proc. Instn. Mech. Engrs. Seminar-Self contained bearings and their lubrication, November 1984. KARELITZ, G.B. 'Performance of oil ring bearings; Trans. ASME., 1930, 52, 57-70. KEUSCH, W. 'Self contained slide bearings for large rotating machines', Proc. Instn. Mech. Engrs. Seminar-Self contained bearings and their lubrication, November 1984. BAILDON, E. 'The performance of oil rings', General discussion on lubrication, Instn. Mech. Engrs., Group 1, 1937, 1-7. BAUDRY, R.A. and TICHVINSKY, L.M. 'Performance of oil rings', Mechanical Engineering, 1937, 59, 89-92. BAUDRY, R.A. 'Some thermal effects in oil-ring journal bearings', Trans. ASME., 1945, 67, 117-122. HERSEY, M.D. oil rings', 59, 291.
'Discussion of performance of Mechanical Engineering, 1937,
OZDAS, N. and FORD, H. 'Oil transfer and cooling in ring-oiled bearings', Engineering, August 26, 1955, 268-271 and October 21, 1955, 570-573. LEMMON, D.C. and BOOSER, E.R. 'Bearing oil ring performance' , Trans. ASME., Journal of Basic Engineering, 1960, 82D, 327-334. HESHMAT, H. and PINKUS, 0. 'Experimental study of stable high-speed oil rings', Trans. ASME., Journal of Tribology, 1985, 107, 14-22. NEALE, M.J. 'Tribology Handbook', Section A6-Ring and disc fed journal bearings, 1973. (Butterworths. London). Engineering Sciences Data Unit, Item Number 84031 'Calculation methods for steadily loaded axial groove hydrodynamic journal bearing', 1984.
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SESSION XX MACHINE ELEMENTS (2)- CAMS AND TAPPETS Chairman: Professor W.O. Winer PAPER XX(i)
Mixed lubrication of a cam and flat faced follower
PAPER XX(ii)
Elastohydrodynamic film thickness and temperature measurements in dynamically loaded concentrated contacts: eccentric cam-flat follower
This Page Intentionally Left Blank
599
Paper XX(i)
Mixed lubrication of a cam and flat faced follower D. Dowson, C.M. Taylor and G.Zhu
SYNOPSIS Considerable attention has been focussed upon the mechanical power losses in internal combustion engines in recent years. Broadly speaking these may be of the order of 15% of the fuel energy. Most of these losses can be attributed to the friction associated with the lubricated machine elements in the engine, namely piston ring assembly, bearings and valve train components. Development of the understanding of the operational characteristics of dynamically loaded engine bearings and piston rings is more advanced than that of the valve train, particularly the cam and follower. The work described in the present paper represents part of a continuing study of cam and follower lubrication characteristics. It is widely thought that the cam and follower contact operates with a degree of surface interaction for virtually the whole of its cycle. As an initial step towards the understanding of surface topography influences, a rigid surface model for the mixed lubrication of a cam and follower is developed in this paper. Results are presented to demonstrate the effect of surface roughness and its distribution upon nominal film thickness, the load carried by the asperities and power loss. The proportion of the total load carried by the asperities and the proportion of power loss associated with asperity contact are detailed. The influence of lubricant viscosity and camshaft speed upon nominal film thickness during the cycle are also discussed. 1.
INTRODUCTION
Conventional wisdom suggests that in the internal combustion engine the bearings enjoy hydrodynamic (or elastohydrodynamic) lubrication conditions whilst the piston rings operate with fluid film or mixed lubrication experiencing a significant period of effective separation from the liner due to the lubricant film developed. However, it is believed that the cam and follower contacts suffer boundary lubrication with almost continuous surface contact during the operational phase. Evidence is beginning to emerge that cam and follower contact may not be quite as severe as usually envisaged, due possibly to micro-elasto-hydrodynamic lubrication. In this paper an analytical model for the mixed lubrication of a cam and flat faced follower is presented and results indicating the influence of surface roughness detailed. This is the first such analysis for a cam and follower and the assumption that the bounding solids are rigid has been made (elastic deformation of the asperities is of course The authors are proceeding with a considered). mixed lubrication analysis incorporating elastohydrodynamic effects and the early indications are that the effect upon film thickness compared with the predictions of the present model is significant. This would be expected but it is still important to lay the foundation of rigid surface analysis in order to provide a benchmark for future studies. The model developed uses approaches for rough surface hydrodynamic lubrication and the
contact between rough surfaces developed by other workers. Analyses for the lubrication of the piston ring seal which are analagous to that given here have been undertaken (1,2). The investigation has concentrated on the effects of surface roughness height and the distribution of roughness between the cam and follower. Results for the variation of nominal film thickness, power loss and load carried by asperity contacts are presented. 1.1
Notation
a
follower acceleration
A
apparent area of contact
AC
e
real area of contact eccentricity (scrub radius)
-
Figure 1
1
E
equivalent Young‘s modulus
FA
friction force due to asperity interaction
F2’F5/2
integrals for Gaussian height distribution
h
nominal film thickness (= ho
hT
hT
I.ocal film thickness (= h
+
+
rS1
average film thickness
hO
nominal minimum (central) film thickness
H
film thickness ratio (h/o)
2 x /2Rc)
+ rS2)
600
L
Cam lift mean hydrodynamic pressure
P PA
mean asperity pressure
pX
hydrodynamic load component in x direction - Figure 1
P
hydrodynamic load
RC
instantaneous radius of curvature of nominal contact point on cam
R b
cam base circle radius
U
surface velocity
Y
U
e
-
Figure 1 (= W ) H
mean entraining velocity of lubricant
"r
variance ratio
W
total load
W
asperity load component
wH
hydrodynamic load component (=
A
lo'
pY)
l -
cam width
W XY
Y
coordinate axes
-
Figure 1
Figure I
Cam and Follower represented by a cylinder and a plane
rate of change of shear stress with pressure
(a)
6
asperity radius of curvature
(b)
6
roughness amplitude
11
asperity density
IJ
lubricant viscosity
0
composite roughness height
T
hydrodynamic surface shear (11)
T
shear due to asperity interaction
a
General view of cam and follower contact Geometry of the surfaces in the vicinity of contact point of cam and follower
analysis being valid for a general cam profile in contact with a flat faced tollower. The mean entraining velocity is given by,
-
ue = [Rc
equation
+?]$
where the instantaneous radius of curvature is, A
TO
Ts
boundary film shear stress
$x,4s,4f '4fS' fp'
n
camshaft rotational frequency
2.
-
surfaces
-
de a =-Sl dt
2.1 Cam and Follower Kinematics and Loading Figure 1 shows the contact between a cam and flat faced follower and the detailed geometry of the surfaces in the vicinity of contact with rough surfaces. The cam is designated component number 1 and the follower number 2. The kinematical analysis of this arrangement has already been presented in detail ( 3 1 , such
(3)
The absolute velocity of the follower in the x-coordinate direction (u ) is zero whilst that 2 the cam is, of the point of contact on
Figure 1
THEORETICAL BASIS
(2)
S12
and the acceleration of the follower is,
factors in average flow model
Subscripts (1,2)
R c = Rb + L + &
shear stress constant
u1 = (Rc
-
-$]Q
(4)
Details of the determination of the loading on cam and follower are also presented in reference ( 3 ) . This involved taking a balance between inertia and spring forces with frictional effects and structural flexibility and damping neglected. 2.2
Hydrodynamics of Rough Surfaces
The one dimensional form of the averaged
601
Reynolds equation developed by Patir and Cheng (4,5) was adopted in the determination of the hydrodynamic effects associated with the contact. This equation takes the form,
that the maximum error hetween the numerical results and new formulae is less than 1%. 2.4
Total Load
The total load acting on unit width of the cam (W) at any instant during the cycle was taken to be the sum of the hydrodynamic component (WH> obtained by integration of the pressure distribution obtained from the solution of equation ( 5 ) - and the asperity contact force <WA)
-
The pressure and shear flow factors ($x and $s) enable the average lubricant flow to be related to averaged quantities, such as mean pressure (p) and nominal film thickness (h). They are obtained from numerical experiments and full details of the values adopted for calculations carried out as part of the present paper may be Only isotropic surfaces found in ( 4 ) and (5). (surface pattern parameter (y) = 1) were considered due to the limitation of the asperity interaction model adopted. The roughness distribution between the surfaces was described by the variance ratios
where
(0)
is the composite rms roughness and
( 0 , 0 ) the standard deviations of the roughness
ambligudes. 2.3
Asperity Interactions
The asperity interaction model used in the present analysis was developed by Greenwood and Tripp ( 6 ) . The model assumes a Gaussian distribution of asperity heights and constant radius of curvature of the symmetric asperities. For the purpose of the study carried out here elastic asperity deformation was assumed for which the asperity contact force is given by
.
w = wH + wA 2.5
(10)
Friction Force and Power Loss
The total friction force acting on the surfaces involves three components due to (a) the mean viscous shear flow (b) the local pressure on asperities and (c) the force due to pressures acting on the nominal geometry. In their calculations associated with piston ring performance the authors of references (1,2) omitted consideration of the third of these components, which for rigid surfaces may not be valid. It is also important for rigid surfaces to include the contribution of terms arising from pressure effects, whereas for elastohydrodyanmic conditions these may well be negligible compared with effects due to viscous sliding. The analysis here has incorporated all three components. The x- d i rection stresses on the cam (Figure 1) are given by (51,
U(u2-u1) T
The real area of contact (A ) is related to the apparent area of contact (A7 through the relationship
a
=
-
(Of h (mean viscous shear)
(local pressure on asperities)
- Pdh
Tc dx (pressure on nominal geometry) The integrals of the Gaussian distribution (F ) are detailed in reference ( 6 ) . The total contact extent was divided up into small regions of nominal constant height enabling the following interpretations,
p"
A F5/2
]:[
=
jF5/2 -m
[$]
and
T~ = T
+ T ~ T~ +
(11)
The determination of shear stress factors and $ ) has been achieved by numerical f The ($ ) factor has also been simulation p5). evaluated by Patir and Eheng ( 5 ) ,
dx
.m
Rohde (1) has provided curve fitted formulae for the asperity contact functions (Fn). These lacked adequate accuracy particularly in the region of (h/a
In their calculations ( E ) was arbitrarily set to (u/lOO) and analytical integration was possible using a polynomial density function for (f(y)) which closely approximated to a Gaussian distribution. The present authors have used the same analysis as that detailed by Patir and Cheng. The small film thickness ( E ) represents the value below which a hydrodynamic shear stress is taken not to exist, and for which
602
Cam action angle Base circle radius Maximum lift Cam nose radius Cam width Spring stiffness Spring mass Bucket and valve mass
boundary friction is assumed. Results -9 are presented here for a value of ( E = 10 m); a physical interpretation is that this represents the approximate thickness of a boundary layer of molecules. Small variations in the value of ( E ) were found to be insignificant. When the total x-direction stress on the cam ( T ~ )exceeded the shear strength of the boundary film, the latter was adopted in computing friction force. The boundary film shear stress was taken to be given by, (12)
The total friction force associated with asperity interaction was estimated from the integration of the shear stresses over the real contact area, assuming that the shear strength of the surface film was given by, T
A
=-r o
+
Full details of the variation of important parameters such as load, lift, follower acceleration, lubricant entraining velocity, eccentricity (scrub radius) and cam radius of curvature at the nominal contact point can be found in (3). The following operating conditions and values were chosen as a reference case. Cam rotational frequency (Q) Equivalent Young's Modulus of both surfaces (El) Lubricant viscosity (p) Composite roughness height of both surfaces ( u ) Variance ratio (V r12 r ' = Ratio ( ~ 1 8 ) Product (QBU) Shear stress constant Po) Rate of change of shear stress with pressure (a)
@A
where p is the asperity pressure. Integration A over the real area of contact gives the friction force due to asperity contact as, FA =
T~
Ac
+
aW
A
In evaluating the total torque and power loss associated with the cam care must be taken. The total load acting due to normal mean pressure may be equated to a force system acting at the instantaneous centre of curvature (Figure (1)). This will assist in determining cam torque for the case of rigid surfaces. 3.
NUMERICAL ANALYSIS
The method adopted for the determination of the cyclic variation of film-thickness between cam and follower has been well established (e.g. 1). An initial value of the nominal central film thickness (h ) at some instant during the cycle was estimates, and the normal velocity (dh /dt) determined such that applied load balancedothe sum of the asperity and hydrodynamic components. A time step could then be marched out according to the Eulerian method such that, h
(t
+
St) = ho (t)
+
bt. dho
dt
It was clearly necessary to proceed beyond one complete cycle to achieve convergency. The starting value of nominal central film thickness had little influence upon convergency, this being achieved with only a few steps after the complete cycle. The criterion for convergence was that the nominal central film thickness should be within 0.1% of its value on the previous cycle at some point. 4.
RESULTS
The results presented here relate to the four power polynomial cam detailed in reference (3) where the lift characteristic may be found. Important data relating to the arrangement is given below,
60' 18mm 9.4 mm 13.1 mm 12.0 mm 38 kN/m 0.045 kg 0.15 kg
50 Hz
2 . 3 ~ 1 0pa ~~ 0.1 Pas
0.2 0.5
!im
0.0001 0.056 2x10 Pa
0.08
The load at the cam nose at this rotational frequency was 240N and it should be noted that there was a cam/follower clearance on the base circle of the cam, with a ramp between the base circle and the action profile which extended f60° from the nose. Results are presented in Figures 2-10. Cam angle is taken as the abscissa variable and the following parameters plotted. (a)
nominal minimum film thickness
(b)
load carried by the asperities (and the percentage this represented of the total)
(c)
power loss and the percentage of this due to asperity contact.
The influence of variable composite surface roughness, lubricant viscosity, cam rotational frequency and distribution of roughness between surfaces are investigated. 5.
DISCUSSION
Figure 2 shows the variation of the nominal minimum film thickness during the cam cycle for the reference case and a series of other composite surface roughnesses. The reference case, with a composite surface roughness of 0.2 pm, reflects a typical surface finish to be found i n practice. The predictions for the smaller values of roughness have been included because the theoretical approach adopted is more sound at the values of nominal minimum film thickness to composite surface roughness (H) obtained. This aspect will be discussed later. The results are all for the case of an equal distribution of roughness between the surfaces and the case of smooth surfaces was computed by adopting an extremely small composite roughness height of 0.001 pn~.
603
(iii)A period in the vicinity of contact o f the cam nose where the minimum film thickness is small and fairly constant, corresponding to a region where the entraining velocity and cam radius are small and vary only slowly.
Figure 2
The effect of composite surface roughness height on minimum film thickness Smooth surface Composite surface roughness 0.01 um -- Composite surface roughness 0.02 pm +Composite surface roughness 0.04 pm --9- Composite surface roughness 0.20 )-lm --&- Composite surface roughness 1.00 um
-
__-_--
-_
The effect upon nominal film thickness of the introduction of sur€ace roughness into the analysis film thickness is clearly evident from Figure 2. For the reference case the nominal minimum film thickness at the nose increases by about 6 5 % with respect to the smooth surfaces prediction whilst ths increase approaches 200% for a composite surface roughness of 1.0 urn. It should be noted that the film thickness ratio (H) decreases from about 0.17 to 0.1 with the increase in composite surface roughness from 0.2 to 1.0 um. This serves as a reminder that the basis of the analysis undertaken is brought into question at such low values of film thickness ratio. The curve fitted data of Patir and Cheng ( 4 , s ) obtained by numerical experiment is limited to a film thickness ratio For the case with a composite surface of 0.5. roughness of 0.04 urn the value of film thickness ratio (H) at the nose is about 1.0. Particular care must be taken, therefore, in interpreting data for the two larger values of composite surface roughness adopted. The load carried by the asperities and the proportion this represents of the total load is detailed in Figure 3.
The film thickness variations of Figure 2 exhibit the well known features, namely, ( i ) Two portions of the cycle where substantial thicknesses of lubricant film are generated. These correspond to the rising and falling flanks of the cam where the entraining velocity of the lubricant into the contact and the contact effective radius of curvature are large. (ii) Two regions of extremely small film thickness at times where the entraining velocity of lubricant into the contact drops to zero for an instant. On the basis of quasistatic analysis the minimum film thickness at these two points would be zero,.however,here the squeeze film analysis which has been undertaken reveals the protective action of squeeze film lubrication. Although the film thickness values predicted at these two points are extremely small (about 0.03 pm for the reference case) it is interesting to note that for all the cases depicted the film thickness on the rising flank as the nose of the cam moves onto the follower is greater than that as the nose leaves the falling flank. This is in agreement with the results according to the smooth, rigid surface analysis presented in reference ( 7 ) but contrary to the elastohydrodynamic analysis presented in the same paper. Since the rough surface analysis presented here is for rigid bulk components there is a consistency in the cornpa ri son.
Figure 3
The effect of composite surface roughness height on load carried by the asperities and the percentage this represents of the total load. __--_.Composite surface roughness 0.2 g m -------Composite surface roughness 1.0 pm
Only the reference case and details for a
604
composite surface roughness of 1.0 pm are presented. This is because for smaller roughnesses the load carried by the asperities becomes extremely small. The total load at the nose is 240N and that carried by the asperities is about 3N and 12N for the composite surface roughnesses of 0.2 um and 1.0 um respectively. Thus the asperity load is only a small percentage of the total load, a situation which has been noted for the piston ringlcylinder Indeed the present geometry liner contact (1). is significantly less conformal than that of the piston ring and cylinder liner contact considered by Rohde and the proportion of load taken by the asperities is somewhat less. At times of significant squeeze film action the percentage of load carried by the asperities increases but still only represents about 8% of the total load for the highest composite surface roughness. The small load carried by the asperities is a direct consequence of the non-conformal geometry of the cam and follower contact. It is interesting to contrast the predicted small percentage of the total load carried by the asperities with the significant increase in nominal minimum film thickness from the smooth surface case, which for the reference case (0 = 0.2 3m) was about 6 5 % . This is as a result of the small value of the pressure flow factor (Ox) at the low values of film thickness ratio (Hi. Physically this implies a considerable restriction to the lubricant flow (which is only in the direction of surface motion for the analysis here) and a consequent enhancement of nominal minimum film thickness for the same applied load. In Figure 4 the power l o s s and the proportion of this due to asperity contact are presented. Data for the smooth surface situation and composite surface roughnesses of 0.01, 0.02 and 0.04 pm are given (again with the The same isotropic roughness on both surfaces). results for the higher values of composite surface roughness have been omitted. This is because predictions which were inconsistent with physical reality, in which power loss decreased with increasing roughness, were made. This resulted from the curve fitted data for ( 0 proving unsatisfactory at the low values of film thickness ratio (H) for which it was not strictly applicable. For a composite surface roughness of 0.04 pm the power loss at the nose position was some 35 watts, an increase of about 50% from the smooth surface case. The proportion of this increase due to asperity interaction is extremely small, as was the proportion of load carried by asperity contact. Once again this reflects the low degree of asperity interaction because of the nonconformal geometry coupled with the larger nominal film thickness. The latter affects the factor (4 ) of the mean viscous shear and the asperity foading in the direction of surface motion due to local pressure (see section 2.5) causing the significant increase in power loss. The increase is almost entirely derived from hydrodynamic effects. Figures 5 and 6 show the effect of the variation of lubricant viscosity and camshaft speed upon the nominal minimum film thickness during the cycle. Apart from the applied
.m
-m
-a
-60
Figure 4
-50
-40
-50
-W
-a -10 a 10 a P Can s n g h ldsgreee from n o d
-P
-10 -10 o 10 10 P Cam angle ( d o g r r s from n o d
-P
w
50
a
m
w
50
a
m
The effect of composite surface roughness height on power loss and the percentage of this connected with the asperities.
-Smooth surface ----- Composite surface roughness 0.01 pm ---- -Composite surface roughness 0.02 um .+Composite 0.04 pm
surface roughness
variables other factors remain as for the reference case detailed previously, with the composite surface roughness taken as 0.2 pm. The-results of these two figures are physically consistent. A five-fold increase in lubricant viscosity causes an increase in the nominal minimum film thickness at the nose by a factor of about 3.8 whilst a reduction in cam rotational frequency from 50 to 16.7 Hz is reflected in a drop in this film thickness from 0.058pm to 0.011pm. A consideration of the effect of the distribution of roughness between the cam and the follower is presented in the remaining Figures. The results detailed thus far have related to situations with the same isotropic roughness on both surfaces. In such situations the second term on the right hand side of Reynolds equation (equation (5)), representing fluid transport due to sliding in a rough bearing, is not influential as regards generation of the mean pressure since ($ ) becomes zero. If, however, the surfacesShave different isotropic roughnesses this term becomes important hecause there is a net fluid transport resulting from the combination of sliding and roughness. The shear flow factor ( $ s ) depends not only upon the film thickness ratio (H) and the orientation of surface roughness, but also on the roughness heights of
605
complicated by the fact that the lubricated contact is moving through space. The entraining velocity of lubricant into the contact changes direction whilst the sliding velocity with respect to the contact does not. Care is needed therefore in specifying the appropriate sign to the velocities and shear flow factor (@s) when solving Reynolds equation. Two cases showing the effect of isotropic surface roughness distribution upon the nominal lubricant minimum film thickness are presented in Figures 7 and 8. In Figure 7 the composite surface roughness is kept fixed at 0.2 pm and in addition to the reference case, where both surfaces have the same roughness, results for a smooth cam and rough follower and smooth follower and rough cam are presented. The data of Figure 8 is for a fixed composite surface roughness of 0.04 pm with otherwise the same situation. The contrast between the relative predictions of minimum film thickness at the nose dependent upon roughness distribution for the two cases is apparent. Figure 5
The effect of lubricant viscosity on minimum film thickness. -Lubricant viscosity 0.010 Pas -----Lubricant viscosity 0.025 Pas -Lubricant viscosity 0.050 Pas
-- -_
.Id a
I :
:
:
:
:
:
;
:
:
:
:
o ID m 35 C m q l o ldogree. from m d
- m - d o - - s o a . n a - - l o
.
.
.
.
o
50
m
m
Figure 7
The effect of distribution of roughness between cam and follower surfaces on minimum film thickness. -Same
Figure 6
The effect of camshaft rotational speed on minimum film thickness. -Camshaft rotational speed 3000 r.p.m. (50.0 Hz) Camshaft rotational speed 2000 r.p.m. (33.3 Hz) -.-_- Camshaft rotational speed 1000 r.p.m. (16.7 Hz)
-----
the individual surfaces (5). Here we shall restrict ourselves to cases of isotropic roughness.
In the case of the cam and follower situation the influence of fluid transport resulting from sliding and roughness is further
------.
roughness on cam and fol lower -Smooth cam and rough follower 0.2 pm 4mooth follower and rough cam 0.2 pm
For the composite surface roughness of 0.2 pm, Figure 7 shows that the case of a smooth cam and rough follower would result in a higher nominal minimum film thickness at the nose than for equal roughnesses on the surface. With a smooth follower and rough cam the converse is true, the film thickness being less than for the case of the same roughness or cam and follower. The trend of this variation is reversed for a fixed composite surface roughness of 0.04 pm as may he seen on Figure 8. The difference between the two cases is
606
explained by the way in which the shear flow factor ( $ ) varies with the film thickness ratio (H). A s TH) is reduced (from say 3 or 4 ) the value of (0 ), or more strictly its modulus, increases. 'However, it reaches a peak and then decreases rapidly towards a value of zero. This is because the number of asperity contacts and the real area of contact increases and as the total contact area approaches the apparent contact area the shear flow factor must tend towards zero since it reflects a mean flow. The restrictions on the data at low values of film thickness ratio which have already been mentioned must here be borne in mind.
A further interesting feature of the results is that the trend of the influence of the distribution of surface roughness upon minimum film thickness at the nose is reversed on the flanks at locations around f40° from the nose. This is because although the direction of relative sliding remains the same, the direction of lubricant entrainment into the contact has altered. The effect of the shear flow factor through the second term on the right hand side of the Reynolds equation (equation (5)) is therefore changed in sense.
The effect of the distribution of roughness between surfaces upon load carried by the asperities and the percentage this represents of the total is shown in Figure 9 for a fixed composite surface roughness of 0.2 pm. The influence is small and this would be expected from the results for the case of the same roughness on the cam and follower presented already (Figure 3 ) . The trend of the variations is consistent with that which would be expected from the graph of minimum film thickness shown in Figure 7 . With a smooth cam and rough follower, for which there was an increase in nominal minimum film thickness at the nose, Figure 9 shows there is a corresponding reduction of the load carried by the asperities. The converse is true for the smooth follower and rough cam. .Id
2
if
f7 L
2
k .Id
Figure 8
i
The effect of distribution of roughness between cam and follower surfaces on minimum film thickness.
r
-Same
roughness on cam and follower ------Smooth cam and rough follower 0.04 pm ---- Smooth follower and rough cam 0.04 pm
For the case of a €ixed composite surface roughness of 0.2 Um the modulus of ($ ) exhibits an increasing trend with increasing fflm thickness ratio (H) at the values of (H) in the region of the nominal minimum film thickness at the nose. The sign of ($s) then depends upon the distribution of roughness. With a composite surface roughness of 0.04 pm the minimum film thickness ratio at the nose is of order unity and the appropriate values of ($s) are around the turning Point on the ($ -H) curve. The two situations result in differ& trends reflecting the tendency of the fluid in the valleys of the rough surface either to help or to impede the transport of lubricant in the gap between the components. The changes in the nominal minimum film thickness at the nose due to the variation in distribution of isotropic roughness investigated are not insignificant. For the fixed composite roughness height of 0.2 Um (Figure 7 ) the increase with a smooth cam and rough follower is 30%, whilst the decrease with a smooth follower and rough cam is 21%.
Figure 9
The effect of distribution of roughness between surfaces on load carried by the asperities and the percentage this represents of the total load.
-Same roughness on cam and follower ------Smooth cam and rough follower 0.2 pm -----Smooth follower and rough cam 0.2
urn
In the final figure, Figure 10, the influence upon power loss and the proportion
607
associated with asperity contact of the distribution of roughness is presented. As has already been discussed the contribution of asperity friction to the overall power loss is small. The data presented is for a fixed composite surface roughness of 0.04 pm and the situations of both a smooth cam and rough follower and a smooth follower and rough cam cause an increase in power loss at the nose in comparison with the case of equal distribution of roughness. However, the contribution to the power loss of asperity friction as a percentage of the total is greater for the smooth cam and rough follower than the case of equal distribution of roughness, whilst the opposite is true for the smooth follower and rough cam. A simple explanation of these observations is difficult because of the complexity of the various influences upon friction and power loss but the extremely small contribution of asperity interaction to the total power loss will once again be noted.
h
situation in which elastohydrodynamic lubrication plays an important role. A consequence of this is that for realistic surface roughness values of (say) 0.2 I-lm the predicted film thickness ratio (H) is less than 0.5 at the nose position. The data of Patir and Cheng is only valid for film thickness ratios greater than 0.5 and hence caution must be exercised when interpreting results for such situations. Ry using values of composite surface roughness less than those likely to be encountered in practice the authors have created a situation for which the model used is valid. Analysis of the mixed luhrication o f a contact in which values of film thickness ratio less than 0.5 are encountered is extremely difficult. For such a situation the deformation of asperities would be expected to influence the nominal geometry and this and other factors at the moment means that such cases have not yet satisfactorily yielded to analysis. In the analysis presented here fixed values of the quantities (0/4) and (rl80) have also been taken and a further investigation of the influence of these parameters based upon experimental data would be valuable. In addition no results have been presented for the case of non-isotropic roughness. Whilst the analysis of Patir and Cheng for the hydrodynamics of rough surfaces can accommodate non-isotropic roughness, the Greenwood and Tripp model for the contact of rough surfaces cannot. It is interesting to note, however, that despite this, Rohde ( 1 ) did investigate non-isotropic roughness, using the same model for the piston ringrcylinder liner contact. Despite the limitations of the results presented here they do provide a useful assessment of the influence of surface roughness upon cam and follower lubrication. The development of the model to consider conditions of elastohydrodynamic lubrication will be a valuable step forward and the authors are at present involved in such an advance. The preliminary results are encouraging and full details will form the subject of a future publication. 6.
CONCLUSIONS
A mixed lubrication analysis of a cam and flat Figure 10
The effect of distribution of roughness between surfaces on power loss and the component of this connected with the asperities. roughness on cam and follower ------Smooth cam and rough follower 0.04 pm -._ Smooth follower and rough cam
faced follower has been presented. For the case of the same isotropic roughness on both surface it has to be shown that, (a)
An increase in composite surface roughness results in an increase in nominal minimum film thickness
(b)
The proportion of the load carried by the asperities is small because of the nonconformal nature of the contact
(c)
For small values of the composite surface roughness the power loss increases with increasing roughness. The asperity component of power loss is small the major effects being hydrodynamic
-Same
-_
0.04
urn
The study of mixed lubrication in a cam and flat faced follower presented here has considered a model in which the bulk solids are taken to be rigid although elastic deformation of asperities has been considered. A s a result of this, predicted values of the nominal minimum lubricant film thickness are substantially less than would be expected for the practical
With an equal distribution of isotropic roughness between the cam and follower surfaces it has been demonstrated that the trend of
variations is influenced by the magnitude of the compos.ite surface roughness. For a larger fixed composite roughness the situation of a smooth cam and rough follower will result in a larger nominal minimum film thickness at the nose than for the case of the same roughness on both surfaces. However, for a smaller fixed composite surface roughness the situation can be reversed. The analysis has been undertaken for the case of rigid bulk components. This has resulted in the prediction of small values of nominal film thickness and the ratio of this to surface roughness height for which in some cases the data used has been extrapolated beyond the limit for which it was established. Whilst this is recognized as a limitation the results presented will be of value and interest to designers. A study is in hand to extend the analysis to cover the elastohydrodynamic lubrication of the cam and follower with considerations of surface roughness. 7.
ACKNOWLEDGEMENT
The authors are pleased to acknowledge the technical and financial support provided by the Ford Motor Co. Ltd. (Dunton Research and Engineering Centre) to studies of cam and follower lubrication in the Department of Mechanical Engineering at the University of Leeds. APPENDIX 1 References ROHDE, S.M., 'A mixed friction model for dynamically loaded contacts with application to piston ring lubrication', Proceedings of the 7th Leeds-Lyon Symposium on Tribology - Friction and Traction, p 262, 1981, Butterworths. RUDDY, B.L., DOWSON, D. and ECONOMOU, P.N., 'The influence of running-in of the twin land type of oil-control piston ring upon long-term engine oil consumption', Proceedings of the 8th Leeds-Lyon Symposium on Tribology - The Running-In Process in Tribology, p 162, 1982, Butterworths. DOWSON, D., HARRISON, P. and TAYLOR, C.M., 'The lubrication of automotive cams and followers', Proceedings of the 12th LeedsLyon Symposium on Tribology - Mechanisms and Surface Distress', p 305, 1986, Butterworths. PATIR, N. and CHENG, H.S., 'An average flow model for determining effects of threedimensional roughness on partial hydrodynamic lubrication', Trans ASME, J. Lub Tech, 100 (l), p 12, 1978. PATIR, N. and CHENG, H . S . , 'Application of average flow model to lubrication between rough sliding surfaces', Trans ASME, J. Lub. Tech, 101 (21, p 220, 1979. GREENWOOD, J.A. and TRIPP, J.H., 'The contact of two nominally flat rough surfaces', Proc. I.Mech.E., Vol. 185, p 625, 1970-71.
[7] REDEWI, M.A., DOWSON, D. and TAYLOR, C.M., 'Elastohydrodynamic Lubrication of line contacts subjected to time dependent loading with particular reference to roller bearings and cams and followers, Proceedings of the 12th Leeds-Lyon Symposium on Tribology - Mechanisms and Surface Distress, p 289, 1986, Rutterworths.
609
APPENDIX 2 New curve fitted formulae for the asperity contact functions (F,) asperity heights
Let H
=
for a Gaussian distribution of
h/o d 1exp( d2Ln(H1 -H)+d3 [Ln(Hl -H) ]2,
for
H 2 2.0
for 2.0 < H < 3.5
[Ln(H2-H)] 2,
for 3.5 2 H < 4.0 4.0 2 H
Ilo.o
flexp(f2Ln(Hl-H)+f3[Ln(Hl-H)]
F512(H)
=
f4exp(fsLn(H2-H)+f7[Ln(H2-H)]
2 2
)
for
)
for 2.0 < H < 3.5
f7(H3-H)
H1
=
9.0
for 3.5 5 H
4.0
4.0 2 H
8.0 -40 0.14704 x 10 2 0.70373 x 10 2 -0.12696 x 10 H2
H 2 2.0
H3
=
=
4.0
dl
=
d2
=
d3
=
d4
=
0.74104 x
d5
=
2 0.30813 x 10
d6
=
f6
=
d7
=
0.88123 x
f7
=
0.11201
d8
=
0.21523 x 10
f8
=
0.19447 x 10
fl =
0.11755 x
f2
0.67331 x
=
loL
2 f3 = -0.11699 x 10 -20 f4 = 0.15827 x 10 2 f5 = 0.29156 x 10
-0.36470 x 10
-0.29786 x 10
The table below compares the values of Fn according to the numerical results, Rohde's curve fits (1) and the new formulae.
I
Numerical
I
Rohde [l]
New Formulae
0.0
0.50000
0.61664
0.65501
0.54874
0.50217
0.61831
0.5
0.20978
0.24038
0.23519
0.20579
0.20934
0.23956
1.0 0.07544
0.08055
0.07661
0.07040
0.07559
0.08060
1.5
0.02289
0.02285
0.02212
0.02149
0.02307
0.02300
2.0
0.00578
0.00542
0.00547
0.00568
0.00578
0.00542
2.5
0.00120
0.00106
0.00111
0.00124
0.00120
0.00106
3.0
0.00020
0.00017
0.00017
0.00021
0.00020
0.00017
3.5
0.00003
0.00002
0.00002
0.00003
0.00003
0.00002
4.0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
I
I
This Page Intentionally Left Blank
611
Paper XX(ii)
Elastohydrodynamicfilm thickness and temperature measurements in dynamically loaded concentrated contacts: eccentric cam-flat follower H. van Leeuwen, H. Meijer and M. Schouten
This paper describes some results of local film thickness and temperature measurements in an eccentric cam-flat follower contact by means of miniature vapour deposited thin layer transducers. Complex transducer patterns can be realized by employing photolithography, allowing local measurements in axial direction. A full film will develop at relatively low speeds. At high speeds chemical reaction layer formation starts. Film thickness and temperature at both sides of the contact differed appreciably, thus invalidating the assumption of line contact. Under high loads and misalignment a constriction in the film thickness, typically for EHD contacts, appears at the heavy load side. Only after supporting the follower on a self-aligning elastic hinge, a line contact condition could be attained. Temperature variations of the follower surface were found to be moderate. The transducers worked well and have a satisfying life expectancy. 1
INTRODUCTION
Research in full film lubrication of concentrated contacts under steady state conditions has a long history. Many examples can be found in the literature, e.g. Martin ( 1 1 , Grubin ( 2 1 , Blok ( 3 ) , Dowson and Higginson ( 4 ) , Herrebrugh (S), Hamrock and Dowson (61, and Ertel (7). This resulted in the well established approximative formulae for minimum and central film thickness of line and elliptical contacts, which proved to be a valuable part of the designer's toolkit. In all these cases constant load, entrainment velocity and curvature are assumed. These presumptions are very often satisfied in full film lubricated concentrated contacts. Hence, a quasi-stationary analysis will often yield good results. A graphical survey of these solutions is presented in Fig.1 . However, sometimes concentrated contacts experience variations in load, tangential surface velocity, or curvature, too large to admit a quasi-steady approach. In addition to entrainment, squeeze action helps in fluid film formation. After many investigators of nonsteady state engine bearings, concentrated contacts having both entrainment and squeeze action will be designated "dynamically loaded". This adjective is preferable to the connotation 'instationary", which has a different meaning in fluid mechanics. The problem of normal approach of deforming bodies, without entrainment, has received attention from some authors, the first one being Christensen ( 8 ) . Herrebrugh (9) presents an elegant method for the isoviscous squeeze film between two elastic cylinders. Vichard (10) gives a comprehensive analysis of the case of combined entrainment and squeeze action, using a GrubintErtel type of approach. Although this paper is published 15 years ago, it is the most authoritative in this field. Holland's ( 1 1 ) method, even though it has been corrected in details by Haiqing et a1 ( 1 2 ) , suffers from the elementary error that, under
conditions of EHD lubrication, Reynold's equation is no longer linear in the pressure, hence making the superposition principle invalid. Therefore, it will lead to erroneous results. Oh ( 1 3 ) solves the moderately loaded case, using the promising complementarity method, but needs unrealistic high load frequencies to obtain any marked squeeze effect . A quasi-steady analysis will often suffice, but will definitely fail when the entrainment velocity momentarily equals zero. Piston ring-cylinder liner and (many) camfollower contacts show this behaviour. Therefore, they are fine examples of dynamically loaded concentrated contacts. It can be expected that computer simulations of dynamically loaded concentrated contacts will soon be able to predict the behaviour more accurately. Thus, there is a need for an independent verification by detailed experimental results. Theoretical models yield film thickness, pressure, and temperature distributions in space as a function of time. Experiments result in distributions in time as a function of position in space. Experiment and theory can only be compared if the data are converted into the same way of representation. Therefore, many experiments with a high resolution are needed. The film thickness plays a critical role from a designer's point of view. However, the literature is rather scarce on this point, as can be concluded from the following brief survey. Pure squeeze flows in highly loaded point contacts are studied by Paul and Cameron ( 1 4 1 , ( 1 5 1 , and by Safa and Cohar ( 1 6 ) . The former ones study local film thickness, using optical interferometry, and deduce viscosity behaviour from this, while the latter ones obtain pressure data as a function of position and
612
time by means of small vapour deposited transducers. Combined entrainment and squeeze effects are found in piston ring-cylinder liner and cam-follower contacts. It is surprising to learn that most experiments on piston rings are performed on working engines, or motored test rigs made from stripped down engines, whilst the bulk of reports on cam mechanisms describe experiments with specially designed test rigs. Examples of piston ring experiments can be found in (17)-(23). The rings in these references have a diameter of the order of 100 mm, except the crosshead diesel engine (231, which has a cylinder bore of 570 mm. The ring width is of the order of 2.5 to 10 mm. The mode of the lubrication is often intermediate elastohydrodynamic, which means that loads are moderate (24). Pressure measurements are carried out in (17), (18) and (231, using piezoelectric transducers of 0.25 or 0.20 mm. effective diameter. In (19) and (20) the film thickness is monitored by electrical oil film resistance. Other techniques employed in film thickness experiments are capacitance measurements (17), (la), (191, and (211, or inductance measurements (22), (23), using a proximity probe. Transducer dimensions are 0.25-0.5 mm in the former, 2-4 mm in the latter case.
thickness and pressure distributions in the piston ring case, and some global film thickness and local pressure and temperature measurements in the cam-follower case. Much more experimental data are needed, in particular on film thickness distributions. Thus it was thought important to carry out local film thickness and temperature measurements in a cam-flat follower contact. This paper describes initial results obtained with an eccentric cam.
b c C
E Fr F gE gF gu gs gv g h:in h
In (10) and (25)-(34) cam-follower experiments are described. In all test rigs the cam radius was of the order of 50 mm, while the cam width was of the order of 10 to 20 mm. Film thickness is estimated from global capacitance in ( l o ) , (25), (30) and (31), and from overall contact resistance in (26). Local film thickness measurements, employing the 0.25 mm dia. gauge described in (la), are reported in (29). The loads are rather low. This paper also shows pressure measurements obtained with a 0.25 mm effective dia. piezoelectric transducer. Local pressures and temperatures can also be obtained using miniature thin film transducers, see (271, ( 2 8 1 , (32) and (33). These transducers have a minimum width of 10 vm. To find local cam surface temperatures, an infrared scanning system was employed in (341, with a spot size of 0.45 mm. Steinfuhrer (35), and Coy and Dyson (36) describe cam-follower test rigs, designed to simulate the peculiar vanishment of entrainment effects of many cam mechanisms. In (35) the average film thickness is found by an indirect measurement of capacitance, while (36) describes profile measurements of cam and follower, leading to the conclusion that hydrodynamic effects are important, even when the estimated film thicknesses are much smaller than the surface roughness. Piston ring experiments are troublesome because of the many degrees of freedom of the ring, and the number of transducers which can be mounted, thus limiting the number of transducer positions. The mode of lubrication is intermediate or medium elastohydrodynamic. A cam-flat follower geometry allows the transducer to be positioned at any wanted location (see section 31, thus permitting many measurements. The mode of lubrication is medium till full elastohydrodynamic, or mixed film. The literature provides some local film
NotatiQn
1.1
H
1 m
Hertzian semi-contact width spring stiffness in cam-follower test rig capacitance reduced modulus of elasticity load reference load elasticity number load number speed number squeeze number pressure-viscosity number entrainment number minimum film thickness dimensionless minimum film
_ _
thickness h=h(gp,gv) dimensionless minimum film thickness H=H(g ,gF ) contact length
reciprocating mass in camfollower test rig fluid film pressure P R resistance R reduced contact radius tr time temperature T rolling speed U pressure viscosity coefficient a dynamic viscosity fl e dimensionless time, B=wt; at maximum lift e=o relative film thickness, A=hmin/o A o maximum Hertzian pressure N/m2 oHz composite RMS surface roughness value m- 1 w rotational speed S 2
FULL FILM LUBRICATION IN CAM MECHANISMS
Due to surface roughness, many concentrated contacts operate in the mixed film lubrication region. A fluid film in a concentrated contact is an extremely stiff element, which can, after some running in, result in full film lubrication conditions (361, even at elevated temperatures (37). Anyway, as long as the contacting asperities share only a small portion of the total load, full film considerations can still provide reliable design directives. As the experiments to be carried out are needed to verify full film theoretical results, precautions have to be made to assure that the theoretical assumptions are met as close as possible. The most important influential variable is the oil viscosity. Hence a high viscosity oil was selected, and experiments were carried out at room temperature. This will at the same time prevent additive effects, if
613
formulated oils are used. The speed should be sufficiently high to ensure seperation of the contact surfaces. Very high speeds have to be avoided to prevent high film temperatures, hence viscosity thinning and surface film formation, see Section 4.2. 3 TEST RIG The test apparatus used in this study has been described elsewhere (27),(28), so only brief details will be given here. A horizontally mounted camshaft 1 (see Fig.2) lifts a pistonlike follower body 2 in a horizontal plane. This follower is supported by a double hydrostatic bearing 3. A load cell 4 measures the normal load in the contact. Follower, support bearing and load cell are mounted in one unit, which can traverse across the cam in a vertical plane by means of a precision linear bearing, connecting it to the steady support. This motion is indicated in Fig.3 by a vertical arrow. By doing so, it is possible to bring every point on the follower into contact with the cam. Thus it is possible to investigate the complete cam cycle with one single transducer; the transducer is fixed on the follower. During normal operation, each point on the follower contacts two points on the cam, see Fig.3, if it is located in the sliding area. This is a specific feature of cam mechanisms. Hence, two times per cycle a transducer will detect a signal. In Fig.3 corresponding cam and follower positions are marked. A transducer located at position 1 on the follower will produce a signal as long as the cam base circle is in contact. At position 2 it will yield a time dependent signal when points 2 and 3 on the cam contact the follower, and so on. The camshaft is supported in precision single row angular contact ball bearings. As the cam is not crowned, alignment with the follower is essential. Precautions were taken to realize a very good alignment. A check of the footprint with Prussian blue indicated a misalignment of less than 2 pm over the contact length. Just as in many modern automotive engines, it was decided to employ follower plates, which can be changed much easier than a complete follower. Because of their compact size and simple shape they are easy to manufacture and fit in most evaporation jars. To assure full film lubrication, a copious supply of oil is necessary. Former experiences with a pin and disc machine learn that jet lubrication will be convenient. An extreme pressure gear oil was chosen because of its high dynamic viscosity and high pressureviscosity coefficient. A quasi-steady calculation of the camfollower contact learns that film thicknesses of the order of 1 pm and even smaller can be expected. This necessitates a filter which will arrest all particles larger than 314 of the minimum film thickness. Absolute submicron filters do not permit a sufficiently large fluid flow in main stream applications, so it was decided to put it in a bypass, see Fig.4. The main stream contains a large hydraulic filter 4 . From there the stream is divided over three parallel flows, i.e. the double hydrostatic bearing 2, the absolute filter 3 and the jet 1. Eventually all fluid will arrive
at a large sump with a capacity of about 50 1. Oil quality can be checked by means of sampler 5. Some numerical data of the test rig are given in Table I. Maximum load Speed minimum maximum nonuniformity Cam
width radius eccentricity roughness (RMS) material hardness
Follower reciprocating mass plate roughness (RMS) ditto incl. transducers material hardness
2000 N 2.45 5: 34.5 s 1 %
17.5 * l0:;m 30.49 *lo3 m 4.09 *10--6m 0.12 *10 m 16 Mn Cr 5 E HRC 60
2.51 kg O.OI*10-6m 0.07*10-6m X155CrVMo 12 1 HRC 60
Spring stiffness
5.68 * 104N/m
Filters hydraulic absolute
3.0 * lO:$o 3.0 * 10 m
Oil SAE classification 85W-140 viscosity (30 C) 0.61 Ns/m press.visc.coefficient (30 C) 2.81 * 10-8m2/N
Table I. Some numerical data of the camfollower test rig.
“1-1
Unless it is stated otherwise, experiments are carried out at 5.00 s shaft speed and 750 N 235 N constant load. Fig.5 shows a detail of the test rig. The camshaft is removed to allow a view on the follower plate with transducers. 4 LUBRICATION MODE Before any local measurements are carried out, it is necessary to make certain that full film conditions can be attained. Within the regime of full film lubrication a subdivision can be made into parched (381, starved, and fully flooded lubrication. The latter can be subdivided again into 4 subregimes (61, EHD lubrication (designated by VE in Fig.1) being one of them. 4.1 Global film thickness measurements Full film conditions in the cam-follower contact can easily be checked by a measurement of the Ohmic resistance or the capacitance of the film. For this, the follower plate is insulated by a 5 mm. perspex layer. In addition, the follower is insulated from the test rig mass by the hydrostatic bearing, and at the load cell and the guide blocks of the follower body. First, the electrical insulation at the
614
camshaft ball bearings was checked by measuring the capacity over these bearings. A custom modified SKF Lubcheck Mk2 11'7s employed. At 750 N (average load) and 2.5 s oil film breakdown occurred at the risjng flank of the cam. At there was no breakdown. speeds over 3.00 s Next, the resistance of the oil film during a cam cycle was determined, using a voltage divider. This device has several settings and gives a maximum of 20 mV over the contact. Fig.6 gives a result. The noise in Fig.6a emenates from a 50 Hz interference source, which could not be located. It follows from Fig.6 that the film resistance is in betweeen 3.6 kQ and 10 kQ. Therefore, full film conditions actually exist during a cam cycle. Actually, an entire seperation of cay and follower already prevails at 3.00 s shaft speed, because the resistance is more than 360 Q. Eventually, global capacitance measurements were carried out, using the Lubcheck capacitance monitoring system. A typical result is presented in Fig.1. The film thickness values in this and following Figures is calculated by Dyson et al's (39) method. At maximum follower lift the number of cycles (time base) starts, so 8=0. 4.2 Surface laver formation and runninq-in Full film theory does not take account for the formation of thick surface layers. At high sliding speeds a chemical film may form (401, which will set an upper limit to the speed. Indeed a sudden rise in the global film thickness could be detected. If a worn sliding area of the follower plate is used, resulting in a low A value (relative film thiclpess), shaft surface films will form even at 5 s speed within 1800 s . When a fresh part of the plate surface is used, reaction films can after 2500 till develop at speeds over 13 s 5000 s . See Fig.8b. In this Figure surface layers are formed after about 700 s , probably initiated by a short period of slightly increased speed. If these layers are formed on the follower surface, an even number of capacitance jumps is needed, see Section 3. In addition, when employing an eccentric cam, they 0. have to be located symmetrically around €I= This is not the case. Consequently, the surface layers are on the cam and not on the follower. A possible explanation is the high content of Cr in the follower plate material (41). The development of a surface layer is a dynamic phenomenon, in that the layer can change in position and extent in time, see Fig. 8c. Several times a reaction film was initially formed on the end of the rising flank, next grew in extent (not in thickness), and moved slowly to the beginning of the returning flank. This is indissolubly related to changes in the surface roughness during running in. In most cases the reaction layer was removed after coasting down to a lower speed (Fig.8d) or coming to a complete stop. If it is assumed that the reaction layer has the same dielectric constant as the oil, the layer thickness can be estimated. In Fig.8b and c the reaction film has a thickness of the order of 0.4 pm, which is the same as found by Georges et a1 ( 4 0 ) , and Johnston et a1 (41). Georges et a1 (40) use surface profile and resistance readings to detect chemical films formed under pure sliding and high speed, while
Johnston et a1 (41) employ optical interferometric and resistance measurements to investigate films developed under high sliding. The capacitance method can give quantitative information on film thickness, while the resistance method is only qualitatively. In addition, it can be used in engineering situations (lubricated steel-steel contacts), which gives it an advantage over interferometric methods. Chemical films can only develop at elevated temperatures, typically over 80 C. Therefore, the temperature should be kept low. If a new sliding area is used, film breakdown does not occur. A sliding area which has been used before should be avoided. In that case, cam and follower have to run-in again with a high initial roughness, leading to high local temperatures and hence surface layer development. It is shown up here that anti-wear additives are effective in cam mechanisms. In this respect, this work is an extension of Johnston et al.'s ( 4 1 ) investigations to more hostile conditions. After running-in no film breakdown could be detected. If the load is increased to 1250 250 N, and the _speed decreased to the minimum speed of 2.45 s , there are still no asperities in contact. Coasting down from these adverse conditions results in Fig.9. 4.3 ouasi-statipnarv -ul
.
.
.
m Very often entrainment velocities are high and contact lengths are short, resulting in a very short passage time of the fluid in the contact. A lubricant molecule will hardly 'feel' any variations during its passage through the contact. Gu (42) presents a qualitative criterion for the applicability of quasistationary analyses at lubrication problems. A contact can be considered stationary, if the passage time is much smaller than the contact duration time. But his concept of contact duration time is not well defined, and in the case of cam mechanisms it could be infinity. Nevertheless, it can be concluded that many concentrated contacts which seem to be dynamically loaded at first sight, show a quasi-steady behaviour. An eccentric cam, having an eccentricity much smaller than the radius, could be one of them. An initial study into the behaviour of dynamically loaded concentrated contacts has been completed recently and will be published later ( 4 3 ) . It is an improvement of Vichard's analysis ( 1 0 ) . The computer programme of ( 4 3 ) was used to obtain film thickness versus time plots for an eccentric cam-flat follower contact, for two sets of operating conditions. Figs.10 and 1 1 show results for 750 5 250 N, and 1250 2 250 N, respectively. The dimensionless film thickness behaviour can be described by two dimensionless numbers, viz. squeeze number:
.
entrainment number:
*3
615
The squeeze number can be considered as a quantifiquation of Gu‘s ( 4 2 ) criterion. The reference film thickness h* is the steady state film thickness under reference speed and reference load conditions. Figs. 10 a and 1 1 a show the quasi-stationary film thickness h*, Figs. 10 b and 1 1 b present the uast / behaviour d?namic of the minimum film thickness hmin 1 h*, and Figs. 10 c and 1 1 c the deviations in minimum film thickness from the quasi-steady solution, all as a function of time 8. Fig. 10 b and 10 c show that this problem is an initial value problem. From these plots it can be concluded that deviations are within 0.3 % . Therefore, an eccentric cam profile, which is only an introduction to more complex shapes, can be considered as quasistationary. Note that the squeeze effect increases the load capacity under normal approach (0.5<0<1.0), but acts in an opposite way under detaching conditions (0<8<0.5). If the conditions in a lubricated concentrated line contact may be treated as steady state, the film thickness can also be found from dimensionless charts as provided by Hoes (44) or Johnson (45). The representation by Hoes lends itself pre-eminently to curvefitting purposes, while Johnsons map is more appropriate to ascertain the physical effects playing a role. In Fig. 12 the cam cycle is marked in the Johnson map. The medium load cycle is depicted by 1-1, the high load cycle by 2-2. Symmetric cams will always be represented by lines rather than loops. The crosshatched parallelogram is the area investigated by Dowson and Higginson (4). The denser shaded part represents the intermediate EHD lubrication region. It follows from this Figure that the medium load case (750 +_ 250 N) is in the intermediate region, whereas the high load case (750 +_ 250 N) is just in the EHD region. The numbers in Fig. 12 are defined by Johnson as F2 )1/2 elasticity number: gE = ( 2 ‘I,uErl Rr
film thickness
h
hmin . F nouRrl
=
The representation in both maps (44,451 has the drawback that varying load and/or speed will affect all three dimensionless groups, including the film thickness. For that reason Fig.1 was drawn, which is purely another representation of the same data. The numbers in Fig.1 are already i9dicated by Johnson (45): a E F )-1/2 load number gF = ( r
Rr 4 3
speed number gu =
(
a Er ‘I0
)’I4
Rr film thickness H = (aEr)2 hmin n
r.
r Fig. 13 shows the two cam cycles indicated by 1-1 (medium load), and 2-2 (high load), respectively. The Dowson and Higginson region is again crosshatched.
There is another advantage in this representation. If the composite surface roughness is known, relative film thickness values for A can be represented by lines. And so it is possible to indicate critical A values. In Fig.13 A cc indicates A=l for the eccentric cam. As tfie cam cycles are located at A-values an order of magnitude higher, it must be possible to have full film lubrication. The experiments described in Section 4.1 confirm this. 5
INSTRUMENTATION
A brief historic review of the development in vapour deposited transducers for lubricated concentrated contacts can be found in (32). Since then Frey ( 3 3 ) and Baumann (46) completed their experiments with a cam-flat follower and a 2 disc machine. They measured pressure and temperature distributions using thin film microtransducers. There are only a couple of papers on local film thickness measurements by means of these microtransducers, and they all concern 2 disc experiments. Experiments with dynamically loaded contacts seem to be welcome. 5-1 Thin film microtransducers
The microtransducers used in highly loaded lubricated contacts must meet some stringent requirements. These include: - high resolution - fast transient response, RC-time - small interference with substrate surface topography - small perturbation of stress and temperature patterns on the substrate surface - high sensitivity to changes in parameters to be measured - applicability on heat treatable steels - good adhesion to the substrate - low wear (sufficiently long life) Generally, vapour deposited transducers can meet these requirements. Almost all investigators in this field favour SiOx as an insulating layer. But the electrical, mechanical and thermal properties differ substantially from steel, thus affecting the temperature in the contact. A material with more steel-like properties is A1203, and should therefore be preferred. Titanium is used for temperature transducers, because it has a very low pressure sensitivity. As capacitance transducers require only that the material be a good conductor, the same materials can be used as for the temperature transducer. The only difference is in the geometry used. The electrical resistance R can be made in any arbitrary value. Owing to the processing equipment a value of about 1 kP was chosen. The conductor pattern should have a much lower resistance, and is therefore made of gold. To keep the parasitic capacitance C low, and to avoid pinholes , the conductor paEt8rn area is kept small. The terminals of the conductor Pattern are connected to a print board by 20 vm dia. gold wires. Fig. 14 gives a schematic drawing of a thin film transducer. The transducers can be made by (a) vapour deposition through a mechanical mask, or (b) by vapour deposition followed by laser beam cutting (16) or by a photolithographic process.
616
In this case photolithography was adopted. It is a very versatile technique. All layers are deposited in one sequence, without breaking the vacuum of the evaporation jar. The layers are partially removed by a selective etching technique, using a photomask. What remains are the conductor and transducer patterns. If desired, a protective coating can be deposited on top of these layers after etching. Photolithography enables complicated transducer shapes and the manufacture of many transducers together on the same substrate. Flexible photomasks will easily accomodate to the substrate curvature. Fig.15 shows an example of the geometry of the transducers used in the experiments of section 6. The temperature transducer (top two in Fig.l5), can be positioned at any wanted location. The upper capacitance transducer (bottom two in Fig.15) necessitates a rotating contact from the camshaft to the fixed world. The lower capacitance transducer gets around this problem, but has a lower resolution. 5.2
Sisnal processinq
In most cases a Wheatstone bridge, or a compensation circuit is used when small changes in Ohmic resistance are to be measured. In this case a new design was used, in which a reference resistance is not needed (Fig.16). A constant electric current of 1 mA flows through the transducer, which corresponds to a certain voltage drop. The difference with an externally adjustable reference voltage is amplified by a broadband instrumentation amplifier (50 x). High quality, low noise operational amplifiers are used, resulting in a large bandwidth and a high slew rate. To obtain maximum sensitivity of the dynamic signal, the reference voltage can be adjusted for small changes in resistance due to operating temperature variations. The absolute temperature can now be deduced from the reference voltage. The capacitance is measured by an SKF Lubcheck Uk I1 capacitance monitoring system (47). The high frequency output signal is sampled by a digital transient recorder. In most experiments a sample rate of 0.2 ms. was used. In film thickness geometry studies a sgqlple rate of less than 10 p s is needed (at 5 s shaft speed). 6
MEASUREMENTS
In earlier experiments (32) it was found that under the same operating conditions large differences in the local temperature can occur. At the same spot and point of time the temperature increase could well be 1.5 C, but also 15 C . A slight misalignment in the camfollower contact was thought to cause different lubricant conditions across the contact. The transducer layout (Fig.15) permits temperature measurements at different axial positions, allowing capacitance measurements at the left and right side of the contact. Thus it is possible to investigate the assumption that the contact may be considered as a line contact. 6-1
Capacitance measurement under sisalisnment
Fig.17 offers a first impression of the film
thickness formation. The capacitance transducers used in this experiment are both positioned 3 mm under the center line, one at the left, the other at the right side. The most salient feature is the difference between the left and right capacity variations. The largest minima occur shortly after maximum lift, where about 0.2 pm (at the left) and 0.85 p m (at the right) minimum film thickness occurs, indicating a misalignment of about 5 * rad. The other extrema are approximately 0.4 pm, and 2.2 pm, respects~~ly, suggesting a misalignment of about 10 rad. Hence in spite of all precautions, there are no line contact conditions. Obviously, the left side of the contact is more heavily loaden. This was also justified by wear scars at the same side. There are many experiments supporting the diagnosis of misalignment. Fig.18 is the result of another experiment, where a new follower plate has been used. Here the left side transducer indicates a minimum film thickness of about 0.4 pm at maximum follower lift. What happens when the load is increased, can be seen in Fig.19. The minimum film thickness drops to a value of about 0.3 pm at 1500 N. A perfect line contact is difficult to realize, but it should be pursued. Therefore a self-aligning elastic hinge support was designed, which is located between follower body and plate, see Fig.20. All results from the next section are obtained using this device. 6-2
Capacitance and temperature measurements
Due to time constraints it was not possible to carry out film thickness measurements as in Fig.17, that is with two transducers located along the same line. Instead of this, two opposing temperature transducers (see Fig.15) were positioned at 4.20 mm and 5.20 mm under the center line, resulting in Fig.21A and El. The minimum film thickness at the right is about 0.7 pm, at the left 0.8 pm for the first extrema, and about 0.9 pm (right) and 0 . 8 pm (left) for the second extrema in Fig. 21 A , and 1.1 pm (right) and 1.0 Vm (left) in Fig.21 B. Obviously, the second conductor wire is located at the point of maximum contact sweep on the follower plate. The conclusion is near that the alignbent in the contact is much better now at 5 10 rad. This should be supported by temperature measurements. Fig.22 was obtained under exactly the same conditions as Fig.18, except that the elastic hinge support is mounted between follower plate and follower body. The minimum film thickness is about 0.5 pm. Eventually Fig.23 was obtained. This Figure demonstrates the possibility to scan the complete cam cycle using only one transducer. Figures 24, 25, and 26 show some comparisons between left and right side temperatures on the follower plate. In Fig.24 and 25 the long transducer type was used, in Fig.26 the short type (Fig.15). The cam cycle temperature is scanned in Fig.27. 7 DISCUSSION From Fig.17 it can be concluded-that misalignment of the order of 10 rad occurs
617
This may not seem large, but it should be viewed on the very thin film thickness scale. The Figures presented in this paper could lead to the conclusion that the left side of the follower plate is always on the heavy load side. Many other experiments falsify this. The misalignment problem can only be alleviated by a self aligning support. Another feature of the local capacitance measurements from Fig.17 are the parasitic capacitances between transducer and substrate, and between substrate and cam. This results in a low shaft speed frequency capacitance variation, as obtained in global capacitance measurements, see Fig.7. These parasitic capacitances can be made smaller by employing a thicker insulating layer. Under increased loads the film thickness profile develops a constriction at the heavy load side, as can be found in Fig.19. This film thickness dip is well known in steady state EHD lubrication. It is important, because the film thickness minimum will often be found here. This reduction in film thickness would not have occurred if the cam-follower contact were perfectly aligned. Without the elastic follower plate support, maximum temperature variations were found to be 16.4 C at maximum lift on the left (heavy load) side, measured with a long type transducer. A short type, having its sensor more outwards, would definitely have detected a higher temperature.
under misalignment conditions. At these temperatures reaction layers formation can be expected (41). However, the temperature increase at minimum lift is far too low to explain the measured values. It could be that starvation occurs. This will be investigated later by local pressure measurements.
After mounting the elastic support, alignment was improved by an order of magnitude. Temperature differences in axial direction are low. It should be kept in mind that Figs.24 and 25 were obtained with a long type transducer, and Figs.26 and 27 with a short one. The distance between two opposing transducers is about 6.25 mm for the long type, and 11.25 mm for the short type. Hence the maximum temperature change of 14.5 C is probably much lower than the corresponding value without a self-aligning device. Note that in Fig.27 the temperature increase is maximum at about 1.80 mm before maximum lift. Under isothermal conditions the minimum film thickness can be estimated by employing empirical formulae, like the Dowson and Higginson expression (4). At 40 C follower plate temperature it can be found that the steady state minimum film thickness amounts about 1 . 8 pm at maximum lift, and about 1.6 pm at minimum lift. This is in contrast with the experimental findings. The measurements show that (1) the film thickness is much smaller, and (2) the film thickness at maximum lift is always smaller than the film thickness at minimum lift. It is known from steady state EHD lubrication that the oil inlet temperature plays a decisive role in film thickness calculations. First, the inlet position needs to be known as a function of time, and next temperature measurements must be carried out at that location. From Fig.27 it can be concluded that the contact temperature at maximum lift rises about 12 C, and at minimum lift about 1C. At 55 C and maximum lift the steady state minimum film thickness will drop to 0.8 pm. Hence temperature effects cannot remain out of it, but also they cannot explain the difference comprehensively. A 60 C temperature rise at maximum lift is needed to obtain film thicknesses as low as 0.2 pm, which can develop
Most follower plates did show some wear after some time. Occasionally no wear was observed, after 15 hrs. of running, which is sufficient for laboratory applications.
Capacitance field deviations and conductor pattern effects may result in capacitance reading errors. The photolithographic technique allows for complicated transducer shapes. Therefore these problems can be tackled by employing guard shields and conductor patterns, permitting 3-point or 4-point measurements. So far, transducer signals are stored and reproduced directly. Signals are acquired at a certain spot as a function of time. They need processsing to be able to compare them with numerical simulations, i.e. a representation at a certain point in time as function of position. This automatization will be realized in the near future.
An eccentric cam is not the best example of a real dynamically loaded concentrated . contact. This study is rather an introduction to the experimental techniques for this type of contact, than an exhaustive experimental investigation. Many automotive cams have a momentarily zero entrainment effect. This will be the subject of a forthcoming paper.
8 CONCLUSIONS
Vapour deposited transducers were used in an eccentric cam-flat follower contact. Temperature and film thickness measurements were carried out as an introduction to experiments in dynamically loaded concentrated contacts. If the contact has some misalignment, mixed film conditions may prevail at the heavy load side. The rise in film temperature can become high, leading to chemical reaction layer formation. The capacitance measuring technique allows an examination of the extent and thickness of these layers. Good alignment was obtained after incorporating a self-aligning elastic hinge support. The measured temperature variations are not high, but need to be included in film thickness calculations. Probably starvation occurs, because calculated film thicknesses are still too high. The transducers worked well and have a satisfactory life expectancy. Further experimental studies, incorporating film thickness, temperature, and pressure measurements in a cam-follower contact having momentarily zero entrainment action, are needed. 9
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the contribution of L.Kodde, for his advice on instrumentation, and R.A.F.KL)nig,for his help in the manufacture of the transducers. This
618
work was partly carried out under a contract of the Dutch Foundation for Applied Sciences STW which is appreciated very much. References
MARTIN,H.M. 'Lubrication of gear teeth", Engineering (London), 1916, U , pp. 119121 and p.527. GRUBIN,A.N. "Fundamentals of the hydrodynamic theory of lubrication of heavily loaded cylindrical surfaces", in: Investigation of the Contact of Machine Components', by Kh.F.Ketova (ed.), Central Scientific Research Institute for Technology and Mechanical Engineering (TsNITMASh), Book No. 30, Moscow 1949 (D.S.I.R. Translation No. 337), pp. 115-166. Note: this reference is almost identical to ref.(7). BLOK,H. Discussion. Gear Lubrication Symposium, Part I, The Lubrication of Gears, Journal of the Institute of Petroleum, 1952, 38, pp.673-683 DOWSON,D., and HIGGINSON,G.R. Elastohydrodynamic Lubrication, SI (2nd) edition, 1977 (Pergamon, Oxford) HERREBRUGH,K. 'Solving the Incompressible and Isothermal Problem in Elastohydrodynamic Lubrication Through an Integral Equation', Journal of Lubrication Technology, ASME Trans.1968, 90, pp.262-270 HAMROCK,B.J., and DOWSON,D. Ball Bearing Lubrication, 1981 (Wiley New York) ERTEL,A.M. 'The Calculation of Hydrodynamic Lubrication of Curved Surfaces under High Loads and Sliding Motion "'(in German) Fortschritt-Berichte der VDIZeitschriften, Vol.1, No.115, edited by 0.R.Lang and P.Oster11984, 92 pp. Note: This work is believed to be Ertel's Ph.D.Thesis, TsNITMASh, 1945, published by his advisor Grubin (2). CHRISTENSEN,H. "The oil film in a closing gap", Proc.Roy.Soc.Lond., Series A, 1962, 266, pp.312-328 HERREBRUGH,K. 'Elastohydrodynamic Squeeze Films Between Two Cylinders in Normal Approach", Journal of Lubrication Technology, ASME Trans. 1970, 2, pp. 292302. (10)VICHARD,J.P. "Transient effects in the lubrication of Hertzian contacts", J. Mech. Engng. Sci., 1971, 11,pp. 173-189 (11) HOLLAND,J. "Instationary EHD Lubricants", (in German), Konstruktion, 1978, 30, pp. 363-369 (12) HAIQINC,Y.,XIAZE,Z.,and YUXIAN,H. "The computation of unsteadily loaded EHL film thickness and other lubrication parameters of cam-tappet pairs of IC engines and analysis of their performance', in: Developments in Numerical and Experimental Methods Applied to Tribology, Paper VII(iii), Butterworths,London, 1984, pp.171-181 (13) OH,K.P. "The Numerical Solution of Dynamically Loaded Elastohydrodynamic Contacts as a Nonlinear Complementarity Problem", Journal of Tribology, Trans. ASME, 1984, 106,pp. 88-95 (14) PAUL,G.R., and CAMERON,A. "An absolute high-pressure microviscometer based on refractive index', Proc.R.Soc.Lond.A., 1972, 331, pp.171-184 (15) PAUL,G.R.,and CAMERON,A. "The ultimate
shear stress of fluids at high pressures measured by a modified impact microviscometer", Proc.R.Soc.Lond.A., 1979, 365, pp. 31-41 (16) SAFA,M.M.A., and GOHAR,R. 'Pressure Distribution Under a Ball Impacting a Thin Lubricant Layer*, Journal of Tribology, Trans.ASME, Series F, 1986, lea, pp. 372376 (17) HAMILTON,G.M., and MOORE,S.L. "The lubrication of piston rings. First paper: measurements of the oil-film thickness between the piston rings and liner of a small diesel engine', Proc.Instn.Mech.Engrs., 1974, 188,pp.253261 (18) MOORE,S.L.,and HAMILTON,G.M. 'Ring pack film thickness during running in*, in: The Running-in Process in Tribology, Paper VII(ii), Butterworths, Guildford, U.K., 1982, pp. 153-161 (19) PARKER,D.A., STAFFORD,J.V., KENDRICK,M., and GRAHAM,N.A. 'Experimental measurements of the quantities necessary to predict piston ring-cylinder bore oil film thickness, and of the oil film thickness itself, in two particular engines*, in: Piston Ring Scuffing, Mechanical Engineering Publications, London, 1976, pp. 79-98 (20) CLOVER,M.F., and LYNN,F.A. "Factors affecting piston ring scuffing during running-in", in: Piston Ring Scuffing, Mechanical Engineering Publications, London, 1976, pp. 45-59 (21) FURUHAMA,S., ASAHI,C., and HIRUMAIM. "Measurement of Piston Ring Oil Film Thickness in an Operating Engine", ASLE Trans.,1983, X I pp. 325-332 (22) DOW,T.A., SCHIELE,C.A., and STOCKWELL,R.D. "Techniques for Experimental Evaluation of Piston Ring-Cylinder Film Thickness", Journal of Lubrication Technology, Trans.ASME, 1983, pp.353-360 (23) TODSEN,U. 'Investigation of the tribological system piston-piston ringcylinder liner", (in German), VDIForschungsheft, No.628, 1985 (24) DOWSON,D., RUDDY,B.L., and ECONOMOU,P.N. "The elastohydrodynamic lubrication of piston rings", Proc. R. SOC. Lond., Series A, 1983, 386, pp. 409-430 (25) VICHARD,J.P., and GODET,M.R. 'Simultaneous measurement of load, friction, and film thickness in a cam-andtappet system", in: Experimental Methods in Tribology, Proc.Instn.Mech.Engrs., 1967-68, 182, pp.109-113 26) G O M I Y A , K ., KAWAMURA,M., and FUJITA,K. 'Electrical Observation of Lubricant Film Between a Cam and a Lifter of an OHV Engine", SAE Paper No.780930 27) SCHOUTEN,M.J.W. 'Elastohydrodynamic Lubrication, Interim Report' (in German), FKM-Forschungsheft N0.40~1976 28) SCHOUTEN,M.J.W. 'Elastohydrodynamic Lubrication, Final Report', (in German), FKM-Forschungsheft Nr.72, 1978 (29) HAMILTON,G.M. "The hydrodynamics of a cam follower', Tribology International, 1980, 2,pp. 113-119 (30) SPIEGEL,K. 'Contribution to EHD Lubrication of Cam-Follower Pairs',(in German), Ph.D.Thesis, Clausthal University of Technology, Germany, June, 1982 (31) SMALLEY,R.J., and GARIGLI0,R. "The
m,
619
Role of Tappet Surface Morphology and Metallurgy in Cam/Tappet Life", in: Tribology of Reciprocating Engines, Butterworths, Sevenoaks, 1983, pp. 263-272 SCHOUTEN,M.J.W.,van LEEUWEN,H.J., and MEIJER,H.A. "The Lubrication of Dynamically Loaded Concentrated Hard Line Contacts: Temperature and Pressure Measurements", AGARD Conference Proceedings, No.394, San Antonio, TX, April 22-26, 1985, AGARD 1986 FREY,D. 'Pressure, Temperature and Load Measurements in an Instationary EHD Contact', (in German), Ph.D.Thesis, Karlsruhe University, Germany, 1985 BAIR,S., GRIFFIOEN,J.A., and WINER,W.O. "The Tribological Behaviour of an Automotive Cam and Flat Lifter System", Journal of Tribology, Trans. ASME, 1986,
100, pp. 478-486
STEINFUEHRER,G. "The Oil Film Thickness at Entrainment Velocity Change. A Contribution to the Tribology of Instationary Motions", Ph.D.Thesis,Hannover University, 1978 COY,R.C., and DYSON,A. "A Rig to Simulate the Kinematics of the Contact Between Cam and Finger Follower", Lubrication Engineering, 1983, 39, pp. 143152
WATKINS,R.C. 'A new approach to the deviation of viscosity in lubricated contacts", in: Eurotrib 85, Vol .I, Paper 2.9, Elsevier, Amsterdam, 1985 KINGSBURY, E. "Parched Elastohydrodynamic Lubrication", Journal of Tribology, Trans. ASME, 1985, 107, pp. 229-
Fig.2 Cam-follower test rig. See text
233
DYSON,A., NAYLOR,H., and WILSON,A.R. "The measurement of oil-film thickness in elastohydrodynamic contacts", Proc.Instn.Mech.Engrs. 1965-1966, 180, pp. 119-134
213
GEORGES,J.M., TONCK,A., MEILLE,G., and BELIN,H. 'Chemical Films and Mixed Lubrication", ASLE Trans., 1983, 26, pp.
114
293 - 305
JOHNSTON,G., CANN,P.M., and SPIKES,H.A. 'Phosphorus anti-wear additives: thick film formation and its effect on surface distress", in: Global Studies of Mechanisms and Local Analyses of Surface Distress Phenomena, 12 th LeedsLyon Symposium (1985) (42) GU.A. 'Elastohydrodynamic Lubrication of Involute Gears', Journal of Engineering for Industry, Trans. ASME, 1973, 95,pp.
Fig.3 Cam-flat follower measurement principle. See text
1164-1 170 (43) LUYTEN,C.T.P.M. "The dynamic behaviour
of heavily loaded EHD lubricated line contacts", M.Sc.Thesis, Eindhoven University of Technology, Hay 1986 (44) MOES,H. Discusion, Proc.Instn.Mech.Engrs., 1965-1966, 180. pp. 244-245 (45) JOHNSON,K.L. "Regimes of elastohydrodynamic lubrication", J.Mech.Engng.Sci., 1970, 12, pp.9-16 (46) BAUMANN,H. 'Pressure and temperature measurements by vapour deposited thin layer transducers in an EHD line contact' (in German), Ph.D. Thesis, Karlsruhe University, Germany, 1985 (47) HEEMSKERK,R.S., VERMEIREN,K.N., and DOLFSHA,H. 'Measurement of Lubrication Condition in Rolling Element Bearings", Trans. ASLE, 1982, 29, p p . 519-527
Fig.4 Oil filtering circuit. See text.
620
lo-'
lo-)
lo-)
lo"
10"
lo1
10'
loa
9"-
Fig.1. Dimensionless film thickness chart for the steady state case
1
0
0.0
Fig.5 Cam-follower test rig detail; camshaft removed.
'I
1
1
.o
e-
2.0
Fig.6 Cam-follower contact resistance, (a) at 3.6 kQ, 100 kHz LPF; (b) at 3.6 kQ, 100 Hz LPF
621
t I
1
I
h I
I
( M I
I
11)
20
eFig.7 Global capacitance monitor reading at 150 f: 250 N, 5 s-l
1.0 0.8 0.6
b
1.o 0.8 0.6
C
0.8 *. 0.6
d
-+ 2 .o
1.0
0.0
3.0
e+ Fig.8 Reaction layer developmept on the cam surface, (a) at 11.43 s-', after 600 s; (b) after a short q r i o d at 14.00 s , after 700 s; (c) at 13.45 s , after 1500 s; after 500 s at 5.42 s , 2000 s total elapsed time
h
I
. . . . 0.0
I
I
I
I
I
3.0
4.0
4.75
. _ . . .
1.o
2.0
0Fig.9 Global capacitance monitor (film thicknegs) reading under coasting down conditions, initial load 1250 & 250 N, speed 2.47 s
622
t
'15
hquast
h*
110
105
1 00
0 95
0 PO
085
0
02
OL
06
08
10
Fig.10 Thqretical film-shickness curves for 750 i 250 N load and 5 5-l shaft speed, g =6.28 10 , gw=1.58 10 , (a) quasi-stationary solution, (b) dynamic solution, (c) de8iation from steady state
Fig.11 Thqretical film-shickness curves for 1250 2 250 N load and 5 s-l shaft speed, g =8.11 10 , gw=3.41 10 , ( a ) quasi-stationary solution, (b) dynamic solution, (c) devsation from steady state
Fig.14 Schematic drawing of a thin film transducer, (1) substrate, (2) adhesive << <( layer, (3) insulating layer, (4) transducer, (5) conductor pattern, (6) protective layer (optional)
Section A - A
Fig.15 Transducer geometries. Upper two: temperature transducers, lower two: >> >> capacitance transducers.
1
LO
L
*
J
J-t 0.00S.
-
o.o*o
coin
623
Fig.12 Dimensionless film thickness chart after Johnson (45) showing two cam cycles 1 - 1 CC CC and 2-2
Fig.16 Amplifier (schematic), used in the temperature experiments. RT: temperature transducer; V reference voltage R
Fig.13 Dimensionless film thickness chart showing cam cycles 1 - 1 , and 2-2
624
t
t
h
h (em)
0
1.0
0.0
0.1
8Fig.17 Film thickness at left (-1 and right (-.-.-) side of the cam-follower contact. Transducer position 3.00 mm under center line.
rig.18 Film thickness at left side of the contact. Transducer position at center line (maximum lift)
t
t I
h lrml
I
w~
0.0
~0.33 0.1
eFig.19 Film thickness at left side of the contact. Transducer position at center line (maximum lift). (a) 750 250 N, (b) 1000 5 250 N , (C) 1250 250 N.
to
0.0
IR
9-
and Fig.21 Film thickness at left (-1 right (-.-.-)side of the contact. Temperature transducer position (a) 4.20 mm, (b) 5.20 mm under center line
t I
0.5 prn
Fig.20 Elastic hinge support and follower plate
1 0.0
0.1
eFig.22 Film thickness at right side of the contact. Transducer position at center line (maximum lift)
la
a-
and Fig.Lq Temperature at left (-1 right ( - . - . - ) side of the contact. (( (( Transducer position 4.30 mm under center line. Long transducer geometry.
625
t
h
0.7 0.6
0.7 '0.6
0.5
0.5
0.4
I
.nr
I
I "*-
Fig.23 Film thickness at right side of the contact. Transducer positions are (a) 4 . 0 5 m, (b) 2.10 mm, (c) 0.10 mm under, and (d) 3.90 mm over center line
00
1.0
ao
110
0
9-
Fig.25 Temperature at left (-) and right (-.-.-I side of the contact. Transducer position 3.80 mn under center line. Long transducer geometry.
t
T fC)
Fig.26 Temperature at left (-) and right ( - . - . - I side of the contact. Transducer position 0.20 mu under center line. Short transducer geometry.
55
50
45
40
35
0.0
1.0
Fig.27 Temperature at right side of the contact. Transducer positions are (a) 3.80 mm, (b) 2.20 mm, (c) 0.20 mm under, and (d) 1.80 mm, (el 3.80 mm over center line. Long transducer
This Page Intentionally Left Blank
SESSION XXI MACHINE ELEMENTS (3)- ROLLING BEARINGS Chairman: Dr. C.M. Taylor
PAPER XXl(i)
Study of the lubricant film in rolling bearings; effects of roughness
PAPER XXl(ii)
The prediction of operating temperatures in high speed angular contact bearings
PAPER XXl(iii) Study on lubrication in a ball bearing
This Page Intentionally Left Blank
629
Paper XXl(i)
Study of the lubricant film in rolling bearings; effects of roughness P. Leenders and L. Houpert
This paper reports an experimental and analytical study of the lubricant film formation of deep groove ball bearings and spherical roller bearings under full film and marginal lubrication conditions. A capacitive technique was used for the film thickness measurements: the experimental results were compared to calculations based on the standard EHD formulae. Using this method, it was found that even under what is presently considered as very poor lubrication conditions, full lubricant separation occurred in the rolling element/ring contacts of the test bearings. Furthermore, these separating films were equal to the ones expected from smooth EHD theory. An explanation of this phenomenon can possibly be found in elastic conformity of the asperities in the rough contacts or elastic asperity flattening. 1
INTRODUCTION
The lubricant film formation in bearing contacts is one of the major factors influencing rolling bearing performance. This study determines the lubricant film formation of rolling bearings that operate under marginal lubrication conditions and at heavy loads. Under such conditions, early bearing fatigue and bearing wear are normally expected. The lubricant film formation was deduced from the measurement of the electrical capacitance of the bearing using a "Lubcheck" instrument in a similar way to that reported in Ref. 1 by Heemskerk, Vermeiren and Dolfsma. To assess qualitatively the output from Lubcheck, a computer program was developed to calculate the bearing capacitance as a function of bearing geometry and operating conditions such as shaft speed, load and lubricant properties. It was originally thought that under conditions of marginal lubrication ( low A ) , there would be cnntinuous metal-to-metal contact between the surface asperities of rolling elements and rings. Such contact would lead to continuous shortcircuiting of Lubcheck and to measurement of "infinitely high" capacitances. It was therefore intended to first determine the lubricant film formation and properties under conditions of full film separation (high A ) and then to extrapolate these findings to the low A reg ion. However, it became clear, that also under the applied marginal lubrication conditions, a fully separating lubricant film was present between bearing rolling elements and raceways. This led to some interesting
hypotheses and conclusions on the effects of roughness on lubricant film thickness in rolling bearings. 2
EXPERIMENTAL ARRANGEMENT
The test bearings were 6309 deep groove ball bearings (45 mm bore diameter) and 22220 CC spherical roller bearings (100 mm bore diameter). The bearings were specially produced to have well defined surface roughnesses on the working contact surfaces, with a minimum of variation within one bearing. Average RMS surface roughness values of new test bearings are given in Table 1. Here also are given the composite RMS roughnesses ( u 1 in the contacts, defined by: 2
Rq ring
2 +
1
Rq rolling element
whereby Rq is the RMS surface roughness. The surfaces of the 6309 rings were stone honed, resulting in a longitudinal roughness pattern, and the balls were polished. The inner rings of the 22220 CC bearings were also stone honed with a longitudinal pattern, while the outer rings had a cup-ground surface, which gives a cross-pattern. For the experiments on the deep groove ball bearings, the XMY-100A friction measuring rig was used, which has been described in Refs. 2 and 3 by Houpert and Leenders. On this rig, test parameters such as shaft speed, operating temperature, radial and axial bearing loads are very well controlled; the bearing torque is measured using hydrostatic bearings. The experiments on the spherical roller bearings were performed on a modified SKF R3 endurance test rig, described by
630
Leenders in Ref. 4. On this rig, the test bearings were mounted on the free ends of the test machine shaft, which was driven from a continuously variable electric motor. The test bearings were loaded using hydraulic cylinders and lubricated from a circulating oil system with high performance oil filtering. On both test rigs, the temperatures of the inlet and the oulet oil and of the test bearing inner and outer ring were measured using thermocouples. The electrical capacitance of the test bearings was measured from the bearing outer ring to the inner ring using a Lubcheck instrument (Ref. 1). The Lubcheck and thermocouple signals were taken from the rotating shaft using a slipring. A cross-section of one of the test positions of the R3 test rig is shown in Fig. 1 , in which the thermocouple positions are also indicated. The tests were conducted using pure radial loads up to 26 kN for the deep groove ball bearings and up to 140 kN for the spherical roller bearings. Half of the ring circumference was therefore loaded. 3
ELECTRICAL CAPACITANCE CALCULATION
A computer program was developed to calculate the electrical bearing capacitance as a function of bearing geometry and operating conditions. In this program, the ring/rolling element contacts were considered as a set of series and parallel capacitors as shown in Fig. 2. The magnitude of each of the capacitors depends on the thickness of the separating lubricant film, on the geometry of the (elastically deformed) surfaces in contact and on the dielectric properties of the lubricant. The approach to calculate the capacitance of each of the contacts was similar to the one described by Dyson, Naylor and Wilson in Ref. 5. Further details are shown in the Appendix. The tests were conducted in the EHD lubrication regime, and to calculate the lubricant film thicknesses in the loaded zones of the bearings, the well-known formula of Hamrock and Dowson was used (Ref. 6).
where H , U, G I W and k' are the classical dimensionless central film thickness, speed, material, load and ellipse ratio parameters. The film thicknesses were corrected using the thermal correction factor of Wilson (Ref. 7). Roughness correction factors as calculated by Patir and Cheng, Ref. 8 were not considered. For the capacitance calculations, each of the loaded rolling element/raceway contacts was divided into five zones: the inlet zone, the EHD zone, the outlet zone and
two side zones of the contact. (see Fig. 3). The inlet, outlet and side capacitances were calculated using line contact approximations, taking into account the elastic deformations of the surfaces. The EHD zones were considered to be a capacitor consisting of two parallel plates, with an area of that of the Hertzian contact ellipses. The di-electric constant of the lubricant was assumed to be pressure dependent. (see the Appendix). It was found that, especially for the relatively small 6309 deep groove ball bearings, also the capacitance of the unloaded zone contributed significantly to the total bearing capacitance. The capacitance of the unloaded zone was calculated by approximating the undeformed surfaces of the rolling element/ring contacts by paraboloids, and by taking into account the real separation 6 (see Appendix) between the rolling element and the ring. 6 is a function of the maximum Hertzian deflections in the bearing, and of the bearing play in mounted condition. It was assumed that compared to the inner ring/rolling element contacts, the outer ring/rolling element contacts of the unloaded zone have such a high capacitance that they could be considered metallic contacts for the calculations. This is because the film thicknesses on the outer ring (in the unloaded zone) are 2 or 3 orders of magnitude smaller than the ones at the inner ring. Together with the stray capacitance of the system, the total capacitances of the loaded and unloaded zone(s) were added up to give the total bearing capacitance at the particular set of operating conditions. Note that surface roughness effects were not taken into account in the capacitance calculations; the bearing surfaces were assumed to be perfectly smooth. 4
CONFORMANCE BETWEEN MEASURED AND CALCULATED CAPACITANCES:
To verify the calculation method of section 3, experiments were first conducted on 6309 deep groove ball bearings, which have a relatively simple geometry compared to the 22220 CC spherical roller bearings. A 6309 has 8 balls and therefore 16 ball/ring contacts, while a 22220 CC has two rows of 19 rollers each (76 ball/ring contacts); furthermore, a 22220 CC has land riding cages and a guide ring. Fig. 4 shows the measured and calculated bearing capacitances plotted versus shaft speed for a 6309 deep groove ball bearing operating at a test load of 18.8 kN The bearing is lubricated with Shell Turbo 68 oil, with an inlet temperature of 36'C and a flow rate of 0.8 l/min. Calculated lubricant film thicknesses of the most heavilyloaded inner ring/ball contact (HCI) are also shown. The test bearing outer ring temperature was first stabilised
631
at 40°C at a shaft speed of 1500 rpm, after which a speed sweep was made from 0-6000 rpm. During the sweep, which took less than one minute, the measured inner ring side face temperature increased from 4 0 ' to 54°C (Table 2). These measured temperatures were used for the calculations to estimate the effective lubricant viscosity in the inlet zones of the EHD contacts. Fig. 4 shows a very good conformance between measured and calculated capacitances, which gives confidence in the method and, more specifically, in the lubricant film calculations used. The Lubcheck measurements clearly indicated full film separation in all the contacts, which was confirmed by the calculated film thicknesses. The composite roughness of the ball/ring contacts was 0.05 pm (Table 1 1 , which means that even at a shaft speed of 500 rpm there is full film separation with a calculated A > 4 ( A is the ratio of lubricant film thickness to composite RMS surface roughness). It is interesting to note that to the total bearing capacitance of 600-1100 pF, the stray capacitance contributes 230 pF and the unloaded zone 200 pF. Fig. 5 shows the measured and the calculated capacitances of a 22220 CC spherical roller bearing as a function of shaft speed at two test loads: 50 kN and 140 kN. The test bearings were lubricated with Shell Turbo T68 oil, with an inlet temperature of 49°C and a flow rate of 2.2 l/min. The test bearing temperature was first stabilised at an outer ring temperature of 58°C at a shaft speed of 1500 rpm. At the test load of 140 kN,the "bulk" inner ring temperature was measured to be 20°C higher (for thermocouple positions, see Fig. 1). These temperatures were used as effective temperature for the film thickness calculations. Then, similar to the experiments with the 6309 bearings, a speed sweep was made from 0 to 2500 rpm in less than one minute. Here, the stray capacitance and the unloaded zone capacitance are less than 10% of the total bearing capacitance. It is clear from Fig. 5 that also here excellent agreement was found between measured and calculated capacitances. At a shaft speed of 2500 rpm, the calculated film thickness of the most heavily loaded inner ring/ roller contact (HCI, see Fig. 5 ) was 0.35 pm, corresponding to A = 2.5. For the outer ring/roller contacts, the calculated A value was close to 1. At these A values it was expected that the Lubcheck signal would show some high capacitance "spikes", indicating moments of simultaneous metal-to-metal contact of an outer ring/roller/inner ring contact. However, hardly any spikes were observed at 2500 rpm. At 500 rpm, where AIR = 1 and AOR i 0.5, ,'itwas expected that Lubcheck would show metal-to-metal contact more than
50% of the time. Here, although some spikes were measured, the percentage metal-to-metal contact time was far less than 10%. Furthermore, the calculated capacitance, for which smooth surfaces were assumed, conformed very well to the measured capacitance. No effect was observed in the measurements of the bearing cages and guide rings. Based upon the above findings, it was decided to repeat the experiments, but now under more extreme marginal lubrication conditions. 5
222200 CC SPHERICAL ROLLER BEARINGS OPERATING UNDER MARGINAL LUBRICATION CONDITIONS
To create marginal lubrication conditions with the 22220 CC test bearings with given roughnesses on the surfaces (Table 1 1 , the following steps were taken: By regulating the oil flow, the outer ring operating temperature was increased from 58" to 75°C. The Shell Turbo 68 oil was replaced by the less viscous TT 9 oil. (TT 9 is a mineral oil of the same family as Turbo T 68;' TT 9 has only 9 cST at 40°C.) The inlet oil temperature was increased from 45' to 65"C, thus reducing the effective viscosity in the inlet zones of the EHD contacts. Table 3 shows the principal test condition, that was now set. The experiments run under the above conditions showed very strong running-in effects. This is illustrated in Fig. 6 where the measured bearing capacitances were recorded for a 22220 CC test bearing. To ensure recording of the running-in process, new, unrun bearings were mounted, after which the bearinqs were warmed up to approximately 60 with preheated inlet oil. Then the test rig was started, the first speed sweep was made from 0-2500 rpm and the thermocouple and Lubcheck signals were recorded. With the first sweep (Fig. 6 run 1 1 , the measured capacitance remained extremely high, which indicated continuous full metallic contact between the rollers and the inner and outer rings. After approximately 15 minutes of running at 1500 rpm, the second speed sweep (run 2 ) resulted in a clear decrease in capacitance at speeds exceeding 1000 rpm. Calculations indicated that the inner ring/roller contacts probably already had "lift-off", while the outer ring/roller contacts were still running with full metallic contact, due to the relatively high outer ring roughness (Table 1 ) . The third sweep (run 31, after 4 hours of running at the principal test condition of Table 3, indicated an even thicker lubricant film, but still the Lubcheck signal contained a lot of "spikes". After another three days of testing (run 41, the running-in process was found to be fully completed, and Lubcheck indicated a fully separating
-
632
lubricant film between all inner ring/ roller/outer ring contacts. In Fig. 7, the data of run 4 are represented in a different way, and calculated capacitances and film thicknesses (HCI) are also shown here. Excellent agreement was found between calculated and measured capacitances. At 1 5 0 0 rpm, the calculated film thickness HCI was now 0.07 pm, which leads to an initial Avalue of the inner ring contacts of 0 . 5 at this speed. Even at 5 0 0 rpm ( A initial inner ring = 0 . 2 5 ) there was a fully separating film between the roller/ring contacts. The outer ring/roller contacts operated here at an initial A value of 0 . 2 5 . Furthermore, the calculations indicated, that the thicknesses of these films were very similar to those calculated with the assumption of perfectly smooth surfaces. Also here the "bulk" inner ring temperatures were used for the calculations. Visual inspection after testing showed that the raceway surfaces of both the inner and the outer rings were considerably smoother, although a roughness pattern was still visible. The rollers seemed unchanged. These observations were confirmed by Talysurf 4/Talydata 1 0 0 0 roughness measurements, taken before and after testing. The measurements were taken across (axial) and along (circumferential) the raceways. A summary of the principal measurement results is given in Table 4 , where the initial and after running-in parameters are shown. The RMS surface roughness of the contacting ring surfaces decreased somewhat due to running-in; for the inner ring/roller contacts, the RMS composite roughness decreased from 0.14 to 0 . 1 2 pm, and for the outer ring/ roller contacts, a decrease was observed from 0.30 to 0.20 pm. With lubricant film thicknesses in the order of 0.07 pm at 1 5 0 0 .rpm, the "run-in" composite roughness values led to A values of 0 . 5 8 for the inner ring and 0.35 for the outer ring/roller contacts, and 100% of metallic contact could therefore still be expected, but PCT (Percentage of Contact Time, see Ref. 1 ) was close to zero. This means that fully separating lubricant films seemed to exist under actual low A operating conditions. The clear decrease in RMS slopes indicated that smoothening of the surfaces had taken place, probably by plastic deformation of the high frequency/short wavelength roughness profiles. It is not completely clear why the rollers were hardly affected by running-in. Possibly this was influenced by the roller finishing which also involves plastic surface deformation. Similar tests with 6309 deep groove ball bearings with artificially rough inner rings (RMS composite roughnesses of 0 . 5 pm instead of 0 . 0 5 pm) also showed strong running-in and the
formation of separating lubricant films under actual low A conditions. In. Ref. 9 a similar phenomenon was reported by Leenders et a1 for grease lubricated 60 mm bore diameter spherical roller bearings. The bearings were operated under marginal lubrication conditions and also here Lubcheck measurements indicated the formation of separating lubricant films. After continuous operation of 2 0 days or more, no roller wear could be detected, which illustrates the good performance of these bearings as a result of the formation of a separating EHD lubricant film. 6
DISCUSSION
The good agreement found between calculated and measured capacitances gives confidence that the method described can be used for quantitative lubricant film thickness assessment in rolling bearings. The separating lubricant films that were found to exist in run-in bearing contacts operating under marginal lubrication conditions (down to A = 0 . 1 ) cannot fully be explained. The Lubcheck measurements and the posttest investigations showed that it is not a matter of an electrically insulating chemical layer that had formed on the surface asperity tops in the contacts. An explanation is offered by the "conforming film" hypothesis. A conforming film is defined here as a lubricant film that follows the larger wavelength or low slope composite roughness pattern while remaining parallel. Such films are discussed in Ref. 1 0 and are schematically shown in Fig. 8 . The surfaces run in to a degree that they can elastically "follow" the composite asperities, which may only be possible for asperities of a relatively long wavelenqth. The shorter wavelength asperities disappear during running-in, e.g. by plastic deformation. After running-in, there is no longer substantial metallic contact and thus the full contact load is carried by the lubricant film. The experiments showed that the thickness of such a film may be the same as that which can be expected to form with perfectly smooth contacting surfaces. Similar conclusions were found by Houpert and Hamrock, Ref. 1 1 , where EHL calculations were performed on nonsmooth surfaces. Fig. 9 , for example, shows the steady state film thickness and pressure distribution calculated on a roller having a tranverse bump. Because in the calculations all deformations are put on one surface only (instead of being shared between the two surfaces), the bump seemed completely flattened, leading to a parallel and horizontal film separation. In a later study, elastic calculations and EHL calculations performed on a 3-D contact having longitudinal roughness will lead to
633
similar conclusions, as shown by Tripp et a1 in Ref. 12. If relatively rough surfaces are in contact, running-in must first take place before separating lubricant films can form. There is some evidence that in the present experiments when one of the surfaces was relatively rough (i.e., ring) and the other relatively smooth (i.e., ball), the critical roughness wavelength is approximately 100 pm. Note that this critical wavelength value could also be converted into a critical slope value. It was measured after testing that the roughness with wavelengths >lo0 pm was not significantly influenced by the running-in process, while shorter wavelength roughness had partly disapeared, mainly by plastic deformation due to the high asperity slope. Using the concept of "functional filtering" for the surface roughness measurements (Ref. lo), a cut-off length of 25-100 pm seems a good choice here instead of the cut-off of 800 pm that is generally used. Such low cut-off lengths would lead to the measurement of much lower and therefore possibly more realistic roughness values from a A point of View. Furthermore, from the pressure point of view, low wavelength and high slope features should not be omitted since they will cause large local pressure. Plastic deformations are then expected, leading to possible metallic contacts. There are indications that the ability of contacting surfaces to gradually smoothen during running in, such that a separating lubricant film can be formed, is an important contact performance factor. When the contacting surfaces have high initial roughnesses with high composite asperity slopes, surface distress may occur, which can lead to premature bearing failure. 7 CONCLUSIONS Based upon measurement and calculation of the electrical bearing capacitance, a method has been developed to quantitatively assess the lubricant film formation in the rolling element/ ring contacts of rolling bearings. Using this method it was found that in both deep groove ball bearings and spherical roller bearings, the lubricant film thicknesses could be predicted with a workable accuracy by standard EHD film thickness formulae. The actual "bulk" ring temperatures were used here to estimate the effective surface temperatures in the inlet zones of the contacts and not the outer ring outside surface temperatures often used. In the "high load" experiments of this study, temperature differences of 1 5 " or more were commonly found between the outer ring outside surface temperature and the inner ring bulk temperature. Fully separating EHD lubricant films were found to form in bearing
contacts under marginal lubrication conditions. The magnitude of these films was the same as the film thickness for smooth surfaces. The runningin of the surfaces played an important role, but even the fully run-in surfaces had composite roughnesses several times higher than the calculated lubricant films. Fully separating lubricant films under actual marginal lubrication conditions cannot readily be explained. One hypothesis seems interesting enough to investigate further. It states the concept of a "conforming" lubricant film when the local pressures do not exceed the elastic-plastic limit. The concept of "functional filtering" when measuring surface roughnesses, developed by Sayles et a1 (Ref. 101, seems applicable here. For cases such as this study, the generally used cut-off length of 800 pm should possibly be reduced to a cut-off length of
The authors would like to thank Dr. I.K. Leadbetter, Managing Director of the SKF Engineering & Research Centre B.V., for his kind permission to publish this paper, and Mr. H. Dolfsma and Mr. H. van Engelenburg for their technical assistance. References
( 1 ) HEEMSKERK, R.S., VERMEIREN, K.N. and DOLFSMA, H. "Measurement of lubrication condition in rolling element bearings", ASLE Transactions, Vol. 25, October 1984. ( 2 ) HOUPERT, L. and LEENDERS. P. "A study of mixed lubrication in modern Deep Groove Ball Bearings", Proc. 11th Leeds-Lyon Symposium, 1984. ( 3 ) HOUPERT, L. and LEENDERS, P. "A theoretical and experimental investigation into rolling bearing friction", Proc. of the EUROTRIB '85 Conference, Vol. 1 . (4) LEENDERS, P. "Endurance testing in practice", Ball Bearing Journal 217, October 1983. ( 5 ) DYSON, A., NAYLOR, H. and WILSON, A.R. "The measurement of oil film thickness in Elastohydrodynamic contacts", Proc. Inst. Mech. Engs., 1965-66, Vol. 180 Pt 3B. (6) HAMROCK, B.J. and WWSON, D. " Isothermal elastohydrodynamic lubrication of point contacts, Part I11 - Fully flooded results", ASME Journ. of Lubr. Tech., Vol. 99, 1977.
WILSON, A. R. "An experimental thermal correction factor for predicted oil film thickness in elastohydrodynamic contacts", Proc. of the 5th Leeds-Lyon Symposium on Tribology, 1978. PATIR, N. and CHENG, H.S. "Effect of surface roughness orientation on the central film thickness in EHD contacts", Proc. 5th Leeds-Lyon Symposium, 1978. LEENDERS, P., HUISKAMP, B. ., BROCKMULLER, U. and HEEMSKERK, R. "New developments on R2F testing of lubricating greases", submitted to NLGI Spokesman, 1986. ( 1 o 1 SAYLES, R:s., DESILVA, G.M. s., LEATHER, J.A., ANDERSON, J.C. and MACPHERSON , P B "Elastic conformity in Hertzian contacts", Tribology International, December 1981. ( 1 1 ) HOUPERT, L. and HAMROCK, B.J., "Elastohydrodynamic calculations used as a tool for studying scuffing", Proc. 12th Leeds-Lyon Symposium, 1985. (12) TRIPP, J., HOUPERT, L., IOANNIDES, E. and LUBRECHT, T. "Dry and lubricated contact of rough surfaces", submitted to the I.M.E. Conference, 1986.
..
Loaded zone First, the extent of the loaded zone was determined by calculating the loaded zone parameter ( € 1 by: E =
6 max
26max
+
ACOSC~
where 6max is the total Hertzian deflection of the bearing outer ring relative to the inner ring;- A is the radial bearing play in mounted condition (taking into account the radial bearing play, shaft and housing fits and thermal expansion effects);-a is the contact angle. It can be shown that under radial load conditions, the loaded zone in the bearing extends over the angle $1 (see Fig.2): (61
$1 = 2 arccos(1-2~)
For the capacitance calculations, the contacts in the loaded zone were divided into five zones: the inlet, outlet, Hertz and 2 side zones (see Fig.3). The total capacitance of a loaded IR contact, for example, is now: CIR(n) = CIR(")inlet
+
CIR(n)Hertz
APPENDIX +
BEARING ELECTRICAL CAPACITANCE CALCULATIONS As shown in Fig.2, the ring/rolling element contacts can be considered as a set of series and parallel capacitors. The stray capacitance of the system is another parallel capacitor. The total capacitance of the n-th inner ring/ rolling element/outer ring contact (C(n)) can now be written as: CIR(") C(n) =
. COR(") '
(3)
CIR(") + COR(")
where C I R ( ~ )and C O R ( ~ )are the capacitances of the n-th inner and outer ring/rolling element contact. The total bearing capacitance (Ctot) is now:
where i is the number of rolling element rows and z is the number of rolling elements per row. All the experiments reported in this paper were carried out with purely radial bearing loads. Under such loading conditions, the bearings have a loaded and unloaded zone. For the calculation of the bearing capacitance, the contacts in the loaded and in the unloaded zone were treated differently.
CIR(n)outlet
+
2CIR(n)side
(7)
where the right-hand size terms are the contributions to the total contact capacitance of each of the five zones. The inlet, outlet and side capacitances were determined in a very similar way, as reported in Ref.4, using line contact curve-fitted approximations. The relative di-electric constants of the oil were taken here as Er = 2.65 in the inlet and side zones, and as = 3.00 in the Hertzian zone (pressure effect). As the outlet is mostly filled with air, the di-electric constant here was Er = 1. The EHD areas of the contacts were considered to be capacitors consisting of two parallel plates with areas equal to the Hertzian contact ellipse area, and a separation between the plates, as given by the local lubricant film thickness (Eqn.(2)):
where E o = 8.85 pF/m, a(n) and b(n) are the (untruncated) Hertzian contact ellipse dimensions, and hc(n) is the local, thermally corrected EHD lubricant film thickness of the n-th contact. The local rolling element loads were calculated with:
635
where $ varies between -$I and $1, and Qmax is the load of the most heavily loaded bearing contact: F, Qmax = 4.37* z i cosa
(10)
where FR is the radial bearing load.
*
4.37 is the value corresponding to E = 0.5 for point contacts.
Unloaded zone In the unloaded bearing zone, the rolling elements and the rings were considered to be undeformed. Furthermore, it was assumed that the rolling elements were contacting the outer ring during running due to centrifugal effects. This led to Eqn.(ll) for the minimum gap ( 6 1 between the rolling elements and the inner ring:
where $ is the positional angle of the rolling element. The (EHD) film between the rolling element and the outer ring was normally orders of magnitude smaller and was neglected in Eqn.(ll), as was the contribution of the ( relatively high) outer ring/rolling element capacitance to the total contact capacitance in the unloaded zone. The gap h(n) between the undeformed surfaces of the n-th rolling element/inner ring contact was now approximated by:
where Rx and Ry are the reduced. radii in x and y direction. The capacitance of the n-th rolling element in the unloaded zone is now:
Ry Rx
I I 0
0
dxdy (13) RX
26$Rx+y2
-+
x2
RY
Similar to Ref.5, it was assumed here that under operation these contacts are filled for 75% with oil. The total bearing capacitance can now be calculated by adding the contributions of the loaded and unloaded bearing zones, using Eqn.(4).
636
TABLE 1: RMS roughness values of working surfaces of new test bearings (measured with a cut-off length of 0.8 mm).
----0.05
I----I-
0.05
0.01
0.05
0.05
0.28
0.11
0.14
0.30
TABLE 2: Numerical data for Fig.4
------ ___---- -- ----------CALCULATED CALCULATED CENTRAL SHAFT INNER RING LUBRICANT MEASURED SPEED TEMPERATURE VISCOSITY CAPACITANCE CAPACITANCE FILM THICKNESS HCI (urn) (pF) (“C) (cst) (pF) ( rpm)
~ - - - ----
40 40 41 45 49 54
2000 4000 4990 5890
68 68 65 54 45 36
1 1
------
1100 760 630 600 600 600
---------
976 742 608 550 557 579
_ I _ _ _ -
0.23 0.35 0.50 0.61 0.60 0.56
- ----
TABLE 3: Principal marginal lubrication test condition for spherical roller bearing 22220 CC.
TABLE 4: Summary of surface roughness measurements before and after testing.
ELEMENT
---IR OR Rollers
-____
0.11 -0.11
-
0.19 1.9-1.9
637 Inlet zone
zone - Hertzian --
Thermocouple to measure the inlet oil temperature
I
/
\
Outlet zone
- -
I
Rollinglelement
C J
I
\
I b I
I Ring I
I
Fig. 3 . D i f f e r e n t zones considered i n a n EHD c o n t a c t ; ( t h e 2 s i d e z o n e s a r e n o t shown).
1 8' I I
I
h
LL
a
v
i
Fig. 1. The R 3 t e s t r i g u s e d f o r t e s t i n g spherical r o l l e r bearings.
0
c
0 0
'.
.
Calculated C
.- .-.-.-.-.-.-.-.- .-.-.-._._._.-._._. .-.-.-.-._.-.___. - .Measured C
t::
-0.6 n 0.6 .00..55
HCI
Radial load Fr
$. c
n0.4 0.4
.
'.
Stray 0 0
z
0
Shaft speed (rpm
----
Fig. 4 . Measured a n d c a l c u l a t e d c a p a c i t a n c e i n t h e deep groove b a l l bearing.
I n n e r r i n g L MECHANICAL
I
Electrical contact OR ELECTRICA L
Fig. 2. Mechanical and e l e c t r i c a l d e s c r i p t i o n of t h e b e a r i n g .
E
.0.3 0.3 3 .0.2 0.2 5 -0.1 0.1 I I I 1000 2000 3000 I 4000 I 5000 I 6000
638 0 0 0
0 0
A8 II 2
!w E u
II
I1
h
cI
LL Q
LL Q
Y
Y
Measured (140 kN)
<\
4-
0 c
0
4
0
.I-
0
'\
Calculated
0
0 0
0
0
0
z
II II
e
II
Calculated (50 kN)
C
z
C
10
0 0
560
4
I
I
1000 1500 2000 2 Shaft speed ( rpm I=====-
500
10'00 1dOO 2d00 2500 Shaft speed ( rpm )=====-
Fig. 7. Measured a n d c a l c u l a t e d capacitance f o r the spherical r o l l e r bearing i n t h e l o w f i l m thickriess range.
F i g . 5. Measured a n d c a l c u l a t e d capacitance i n t h e spherical r o l l e r bearing.
2a
> Rolling element
h
0 0
z X
U
,"
Ring
1°1
"0 1
"Conforming" film
Fig. 8. The " c o n f o r m i n g " f i l m a s 1.0 proposed i n r e f
mm
.
10
1000
1500
2000
2500
3000
Shaft speed ( rprn I--
F i g . 6. C a p a c i t a n c e measured d u r i n g running i n . 1 : s t a r t i n g t i m e ; 2: a f t e r 1 5 m i n u t e s ; 3: a f t e r 4 h o u r s ; 4 : a f t e r 3 d a y s .
1.50-
6
1.25-
a C
1.00
.75.
.
c a
P
d L
.50*
.25-
Y) II)
1
L
a
0
.-
-.25-
-.54;
1
I;
Undeformed bumpd'
-2
-1
0
X-Coordinate x/b
.<; r ' 1
:I
-
1
Pig. 9. E l a s t i c a l l y d e f o r m e d bumb a s calculated in ref. 11
.
639
Paper XXl(ii)
The prediction of operating temperatures in high speed angular contact bearings R. Nicholson
The prediction of heat generation and final operating temperature of high speed angular contact bearings is of considerable importance in order to control operating clearances in service, as well as providing the basis for an understanding of the local lubrication parameters. This paper develops a technique whereby the heat generated within the bearing can be estimated, and hence track temperatures assessed.
1 INTRODUCTION
The requirement to operate large, high speed gas turbine engines for the aerospace market poses several challenges to the bearings and oil system designer. Oil scavenge temperatures are of vital importance in two respects. Firstly, the high peak temperatures experienced severely limit the oil types available, and stretch those oils to their temperatures of 500K are not 1imits : uncommon. Secondly, the oil must be cooled. At times, this extracted heat can be useful. For example, fuel pre-heating on cold days can be essential. More generally, however, it represents a l o s s in efficiency and the encumberance of large and heavy air coolers hindering the airflow. For the bearing itself, whilst absolute temperature levels are not critical - mainshaft bearing tool steels are tempered at around 820K - the race differential temperatures are critical. The designer must ensure adequate control of bearing clearance at all points of the flight envelope. For instance, the expected life from a bearing can easily vary by as much as 50% depending upon running clearance (Fig. 1).
1.0
to use. Within Rolls-Royce, Astridge and Smith (2) adopted a different approach, attempting to assess the contributions due to separate elements within the bearing design, but only appropriate to roller bearings. Ford and Foord ( 3 ) and Dominy ( 4 ) added to this by consideration of slip within roller bearings. This paper represents an attempt to continue these studies, but applied to angular contact bearings, and to progress to the estimation of race temperatures.
2 ANALYTICAL METHOD The task of estimating the heat generation has been approached following the method of Astridge and Smith. The sources of heat generation in the bearing chamber have been assumed to be:(a)
rolling friction at each ball/race contact. This is described by Crook ( 5 ) as being proportional to the central oil film thickness, and so is a function of load, speed, temperature and material properties.
:: ; EXPECTED L I F E R P T I O
(b)
0.4.
0.2.
BEARING DIAMETRAL CLEARANCE
0.05 FIG. 1
0.10
0.15
0.20
0.25
EXPECTED L I F E R A T I O V S DIAMETRAL CLEARANCE
The heat generated by bearings has been the subject of much work over the years, but it's probably true to say that Palmgren's work (1) remains the most widely used overall. It has considerable appeal because it is so easy
cage inner/outer location diameters. These are taken as fully flooded concentric plane bearings. It is quite possible to extend this at a later date to include eccentricity terms. It seemed fair to assume that the inner cage land would always be flooded because of the oil feed arrangement most commonly used, but this assumption is less certain at the outer cage diameter. However, as the cage losses are small, and unless the clearance at the cage outer d;ameter becomes very small, then the change in torque from fully flooded to partially flooded would be small in comparison with the total losses.
640
(c)
ball/cageset windage. Theodorsen and Reiger ( 6 ) suggest that the windage power loss of a cylinder may be expressed as
where W
=
power loss
Cf
=
friction coefficient
r
=
cylinder radius
1 = cylinder length w
=
angular velocity
p = density
Essentially, this equation, in appropriate units is used in the model. However, a difficulty arises with choosing a suitable expression of the density. If the bearing chamber is flooded, with all of the ball cavity filled with oil, then oil density is appropriate. However, in a good chamber design, flooding should not occur and the ball eavity will contain an air/oil mist. It might reasonably be expected that effective density = Kx(mass flow rate of oil)x(oil density).
The force balance is checked using the relative velocities generated by the real outer race speed, the new cage speed, and the real inner race speed, and including the additional force Jhich has arisen because of the sliding on the inner race ie. the difference between the real and the imaginary inner race speed. Nett heat flow into or out of each bearing race from the surroundings must be input to the program. This is because the thermal condition of a bearing will vary significantly from installation to installation. For example, bearings in test rigs often have a nett heat loss to the test box and environment, whereas the same bearing in an engine may well have a nett heat in flow. The difference between the two cases can be of the same order as the heat generated within the bearing. The heat transmission mesh used is shown Linear heat transmissions are in Fig. 2 . Heat flow assumed from the points shown. areas and path lengths are calculated from basic bearing geometry data.
In this model, K has been set to 0.36. This gives an effective density of (0.05 x oil density). at .068 kg.s-l oil flow rate, which does not seem unreasonable. (d)
(e)
ball spin power loss. Outer race control has been assumed, as it covers the great majority, if not all, of our bearings. Hence, the conventional spin terms are only included at the inner race/ball contact. sliding power loss. A routine is included to test the force balance in the bearing. This checks to see if there is sufficient force at the inner race driving points to overcome the outer race resistances. If it is insufficient, gross sliding will take place at the inner race contact. The external evidence of this will be a change in cage speed from epicyclic to approach the outer race speed. This again assumes outer race control and that therefore sliding takes place at the inner race contacts. In order to generate this slip, the program increments the cage speed towards the outer race speed. It achieves this by using an imaginary inner race speed which steps towards the speed of the outer race. At each step a bearing geometry routine, after Harris ( 2 1 , recalculates the contact angles, relative velocities and new cage speed appropriate to this imaginary race speed.
KEY SOURCES OF HEAT HEAT TRANSFER PATH TEMPERATURE LOCATIONS
FIG. 2
T Y P I C A L HEAT TRANSFER MESH
The simultaneous equations governing the temperature distribution are readily generated. It should be noted that these are not linear equations because the oil flow distribution around the ball is unknown. This results in terms containing local mass flow and temperature. The following approach was chosen. (a)
assume an oil flow distribution within the bearing
(b)
gince the individual sources of heat are calculated and nett heat flows are inputted, then the expected scavenge temperature may be calculated
(c)
solve the simultaneous (now linearisedl equations, with the scavenge temperature as one of the unknowns
(d)
compare the expected with the "solved" scavenge temperature
The program repeats (a) to (d) with several oil flow combinations registering the best fit with scavenge temperature.
64 1
The program also iterates on bearing clearance, altering the thermal growths at each loop until it is thermally stable. Fluid properties within the bearing are calculated at the temperature appropriate to its location. Hence, for example, the oil viscosities at the cage bore are set to values which are different from those at the cage outer diameter.
This had the effect of varying the race temperatures, bearing clearance and heat to oil. Comparisons of measured and predicted values of outer race temperature and heat to oil are shown in figs. 3-6. Figs.3 and 4 show the datum case with no cooling flows, but varying the shaft speed. Figs.5 and 6 show
CORRELATION WITH EXPERIMENT
3
Initial correlation was achieved using a 260 min pitch circle diameter angular contact ball hearing. A pair of similar bearings were
loaded against each other on a single shaft in Shaft speed and input torque measurements provided a knowledge o f total power consumed by the test box. This was compared with measurements of heat to oil deduced from mass flow rate and temperature measurements of individual lubricating oil feeds and scavenges. Thermal conduction through test box panels was also measured, and was found to be small on this rig - less than 5% of total power. The test bearing was mounted with oil passageways in the outer race housing and in the bore of the shaft. Heat could be extracted from the races into the oil and hence measured. Careful design and monitoring of both feed and scavenge flow rates ensured that no mixing took place between cooling and lubricating flows. Thus, the rig could be run at constant speeds, load, and lubricant flow rate whilst varying the heat extraction via the cooling passageways. a well instrumented test box.
FIG.5 PREDICTED VS MEASURED HEAT TO O I L 2 6 0 MM BEARING, 2 5 0 0 - 1 0 0 0 0 R.P.M. SHAFT SPEEDS 0-5 KW INNER,O-2.8 KW OUTER RACE HEAT EXTRACTIONS 0.02-0.07 KG/S LUB. O I L FLOWRATE
PREDICTED PACE rEMPERATURE
200
/
180
160
140 1 0 1 H E A T TO O I L
I
MEASURED RACE TEMPERATURE 140
///
TEST VALUES
-
6 '
PREDICTED VALUES
180
14
1
PREDICTED HEAT TO O I L KW
A
12' 2 0 0 DUTER RACE
A
rEMPERATURE
10.
hb
'C 180
I
TEST VALUES
b
14C
* .
4
X
b
b *
121
2 '
,
FIG. 4
4000
6000
8000
OUTER RACE TEMPERATURE VS SHAFT SPEED-DATUM 260
l
h
*- 4
KEY I
140 MM BEARING
*
1 9 5 MM
A
2 6 0 MM
1
.
..
mb SHAFT SPEED R.P.M
2000
A A
& *
6'
- PREDICTED VALUES
X
A t
X
C 220
all the results obtained in this particular test series. They cover shaft speeds of 2500 - 10000 rpm, inner and outer race heat extractions of up to 5 and 2.8 kW respectively, and lubricating flow rates of 0.02 to 0.07 kg.s-l. Following these successful correlations, additional data from earlier bearing tests was examined. Figs.7 and 8 demonstrate the effectiveness of the
HEAT TO O I L VS SHAFT SPEED-DATUM 2 6 0 M M BEARING
160
200
F I G . 6 PREDICTED VS MEASURED OUTER RACE TEMPERATURE 2 6 0 MM BEARING, 2 5 0 0 - 1 0 0 0 0 R.P.M. S H I F T SPEEDS 0 - 5 KW INNER,O-Z.8 KW OUTER RACE HEAT EXTRACTIONS 0 . 0 2 - 0 . 0 7 KG/S LUB. O I L FLOWRATE
/
FIG. 3
160
M M BEARING
MEASURED HEAT TO O I L
KW
10000 F I G . 7 PREDICTED VS MEASURED HEAT TO O I L 1 4 0 . 1 9 0 . 2 6 0 MM BEARINGS. 2 0 0 0 - 1 7 5 0 0 R.P.M. SHAFT SPEED 3 AND 7 . 5 CS NOMINAL V I S C O S I T Y AT 1 0 0 C 8 . 9 - 2 7 KN A X I A L LOAD, 0 . 0 3 - 0 . 0 8 5 KG/S FLOYRATE
642
(3) 180
The Effect of Elastohydrodynamic Traction Behaviour On Cage Slip In Roller Bearings, ASME, JOLT, Vo1.96, No.3(1974).
f TEMPERATURE -2
(4)
160
140'
I
b
KEY
A '
I
1 4 0 MM BEARING
+
1 9 5 MM
I)
P
2 6 0 MM
.
A
(5)
140
160
180
200
F I G . 8 PREDICTED V S MEASURED OUTER RACE TEMPERATURE 1 4 0 . 1 9 0 . 2 6 0 MM BEARINGS. 2 0 0 0 - 1 7 5 0 0 R . P . M . SHAFT SPEED 3 AND 1 . 5 CS NOMINAL V I S C O S I l Y AT 1 0 0 C 8 . 9 - 2 ) KN AXIAL LOAD. 0 . 0 3 - 0 . 0 8 5 K G l S FLOWRATE
method for bearings of 140, 195 and 260 mm pitch circle diameter, for the following range of conditions: axial load
8.9 - 27 kN
shaft speeds
2000
oil viscosity
3 and 7.5 CS nominal at 100°C
oil flow rates 0.03
4
- 17500 rpm - 0.085 kg.s-'
CONCLUSIONS
The analytical technique has been shown to correlate well with measured values of heat to oil and race temperature for a range of conditions applicable to gas turbine installations. Further work is necessary to define the effects of scavenge efficiency and oil feed arrangement as they are likely to affect the effective density within the bearing.
5 ACKNOWLEDGEMENT The author wishes to thank J R Ward and R J Moss of Rolls-Royce plc for their contributions to the experimental data presented here and to Mr Ward's helpful insights, which reported data by itself does not necessarily convey.
6 REFERENCES (1)
HARRIS, T A Rolling Bearing Analysis, 1966 (Wiley, New York).
(2)
ASTRIDGE, D G and SMITH, C F Heat Generation in High Speed Cylindrical Roller Bearings, Elastohydrodynamic Lubrication 1972 Symposium, p83-94.
CROOK, A W The Lubrication of Rollers IV Measurements of Friction And Effective Viscosity, Phil.Trans.R.Soc. London, Ser A, 255,, 281-312 (1963).
MEASURED RACE TEMPERATURE ' C 120
DOMINY, J A The Minimum Lubrication Requirements of High Speed Roller Bearings, PhD Thesis 1980, Rolls-Royce plc.
b
120
FORD, R A J and FOORD, C A
(6)
THEODORSEN, T and REGIER, A Experiments On Drag Of Rotating Discs, Cylinders And Streamline Rods At High Speeds. NACA TR793 1944.
643
Paper XXl(iii)
Study on lubrication in a ball bearing T. Fujii, M. Ogata and Y. Shimotsuma
Seizure damage sometimes occurs suddenly in bearings being operated stably at high speeds even though they ostensivly demonstrate fluid lubrication characteristics. The purpose of this research is to obtain basic materials to study the cause of this. Hereupon, we first of all investigated the position of oil film breakdown on the rolling surface using ceramic balls; and secondly, we quantitatively measured the oil film breakdown phenomena at the micro asperities using a DC current circuit and a capacitor.
1 INTRODUCTION The prevention of seizure damage to rolling bearings being operated under high speed revolution and the obtaining of high speed stability is an extremely important theme, Bearings which were being operated under seemingly stably high speed revolution sometimes undergo sudden seizure damage. Seizure damage occuring in rolling bearings can be divided into two main types. The first type of damage occurs at the rolling surface between the rolling element and the bearing ring. The second type of damage occurs at the sliding surface between the retainer and the rolling element, and between the retainer and the bearing ring guide surface. Opinions regarding the sliding surface seizure damage have already been reported in part. This paper will discuss oil film formation and oil film breakdown in the rolling surface which is a cause of rolling surface seizure damage. Due to the establishment of the elastohydrodynamic theory ( 2 ) , oil film formation in the rolling surface can now be fairly accurately estimated. However, the oil film thickness formed in the meantime is in most cases an extremely low value of 1 Urn or less. Accordingly, considering the surface roughness and the magnitude of surface waviness of the rolling surface due to machining it is difficult to expect perfect fluid lubrication at the rolling surface. Moreover, fitting misalignment of the bearing complicates this problem even further. That is, the EHL oil film formed at the rolling surface in most cases is broken down instantaneously at the tips of the asperities. To accurately ascertain the occurrence frequency and states of oil film breakdown is a key point in defining the causes of seizure damage at the rolling surface. The oil film breakdown phenomena that occur instantaneously at the micro area of the rolling surface asperity tips is measured by forming a DC electric current between both surfaces ( 3 ) - ( 4 ) . In this paper, the oil film breakdown phenomena is treated quantitively as the oil film breakdown ratio E X by using an RC circuit to charge the capacitor with a quantity of electricity via the oil film breakdown area on
the micro asperity tip. Hereupon, the changes in the oil film breakdown ratio E% are investigated using an angular contact ball bearing under varying revolution speeds, thrust loads, supplied oil temperatures, and outer ring fitting misalignment conditions. Ceramic rolling elements possessing electrical insulation characteristics are used in order to investigate the changes in the oil film breakdown phenomena during a single revolution of a rolling element. The purpose of this research is to ascertain the oil film formation and breakdown characteristics in rolling surfaces, and to obtain basic materials for stable, high-speed operation of rolling bearings.
2 EXPERIMENTAL PROCEDURE 2.1 Test bearing and lubrication oil
di=25 mm d0=62 mm
B =17 mm
a =15
Fig. 1
Test bearing and fitting m isalignment
644
For this test, the bearing used is JIS 4-class angular contact ball bearing #7305. The shape and dimensions are as shown in Fig.1. The bearing is fitted with a phenol resin machined cage and is machined to high-precision. Also, the lubrication oils used are turbine oils VG32 and VG56 containing no additives. The temperature of the lubrication oil is controlled by a separate tank during circulation, and is supplied at a rate of 1.44~10-~l/sec.
2.2 Test apparatus The test apparatus and the electrical measuring circuit are shown in Fig. 2. The main shaft driving the test bearing is coupled by a plastic spline coupling to the shaft belt-driven by a motor provided with a planetary gear step- less reduction gear. A thrust load is applied to the test bearing by an angular contact ball bearing of same model number and a coil spring installed at the other end. The test bearing is lubricated by oil jets supplied via the 8 fine holes in its circumference. The fitting misalignment of the test bearing outer ring is adjusted by using the fine adjustment screw to slightly rotate the housing within the horizontal plane. A s can be seen from Fig. 1, outer ring fitting misalignment B o rad can be ascertained by measuring 6 mm using a dial gage of precision 0.001 mm installed at the inner shaft. For the installation error, there exists a bearing ring circumferential angular position where the contact pressure between the bearing ring and the rolling elements, and the angle of contact is greatest. This angle is measured to be&= 0 rad, and the rotary direction of the retainer is treated as an angle of plus value.
1
R
In order to ascertain the oil film formation state in the test bearing rolling surface, an electrical circuit which utilizes the inner and outer rings as poles is included. I n order to provide electrical insulation, bakelite bushes are inserted between the inner ring and rotary main shaft, and between the outer ring and the housing, respectively. In order to obtain a connection between the inner ring and the electrical circuit, the rotary main shaft is provided with slotted holes in the axial direction in which lead wires and mercury slip ring A are used.
2.3 Measurement of the oil film breakdown position In the test bearing shown in Fig.2, all of the rolling elements are replaced with ceramic balls except for one steel ball. Next, in the electrical circuit shown in Fig. 2, switch S is set to B, and contact S2 is connected to contact B. This forms the electrical circuit in order - battery - mercury slip ring A - inner r i n g steel ball - outer ring - resistor - capacitor battery. The transmission current state is measured on the oscilloscope and the FFT. When oil film breakdown has occured at the rolling surface during bearing operation, the ceramic balls are electrically insulated. Therefore, electrical current flows only when oil film breakdown has occurred between the steel ball and the raceway surface. A magnet sensor is installed at the inner ring and retainer, and this is used to ascertain the circumferential angular position when oil film breakdown has occurred. This steel ball for detection is magnetized,
1
/\
from outer ring
,
Fig. 2 Teat apparatus and electrical measuring circuit
1
645 and rotational behaviour of this steel ball is measured by the search coil and slip ring B imbedded into the retainer. 2.4 Measurement of oil film breakdown ratio
As already explained earlier in this paper, the oil film breakdown state at the rolling surface can be measured when switch S 1 in the electrical circuit, shown in Fig. 2, is set to B and switch S z is set to B. Capacitor C and resistor Ro are included in order to quantitatively measure this oil film breakdown state at the rolling surface. Firstly, when switches Si and S2 are connected to A, a circuit which includes resistor R and capacitor C is formed, The circuit at this time is set to a metallic contact state in which an oil film is not formed at all at the rolling surface. The characteristic curve of the charge to the capacitor at this time is shown by curve A in Fig. 3 . Opposed to this, curve B shows the characteristic curve of the charge to the capacitor when switch S1 is set to B and the charge is passed through oil film resistance Rx of the rolling surface. As shown in expression ( 1 ) below, the oil film breakdown ratio is defined by taking the time ratio up to when the voltage stabilizes to a constant voltage from voltage rise in curves A and B.
0
tS
td time s
Fig. 3
C h a r a c t e r i s t i c curve o f charge t o capaci t e r
100 H
w
90
.$) 80 U
m $4
-0
70
60 50
n
Namely, oil film breakdown ratio E is shown to be 100%when the rolling surface undergoes metallic contact, and is shown to be 0% when both surfaces are separated by a fluid film. The capacity of the capacitor was switched in accordance with the lubrication state of the rolling surface. Also, capacitor set voltage E l is expressed as E l < O.l*Eo s o that the charge characteristic curve can be linearly approximated, When measuring the oil film breakdown state electrically, the occurrence of electric discharge phenomenon via thin oil films is a point which must be taken into consideration. Using the test 100 90 O N
w
80
0 .rl
u
70
$4
5
60
2
50
c
40
5
30
m
D
.r(
w
4
20
.r(
0
10 0
-
E
40
v.(
30
rl .rl
4 d
0
20
10 0 10-1
100
101
circuit voltage E V
Fig. 4
E l e c t r i c discharge phenomena v i a t h i n o i l films
646
apparatus, bearing and electrical circuit used in this research, as shown in Fig. 4 , the shaft speed and thrust load were varied to investigate the influence of electric discharge. From this diagram, oil film breakdown ratio E indicates a contact value within the range of the applied circuit voltage 0.1 V 2 EoS 1 V. At Eo> 1 V, the tendency for E to suddenly increase, namely the tendency for the transmission current to increase, and this shows that the phenomenon of electric discharge via thin oil films occurs. The oil film breakdown ratio is the ratio between the quantity of electricity flowing through the rolling surface with oil film resistance R, over a fixed amount of time, and the quantity of electricity flowing through the short circuit with resistance Ro over a fixed amount of time. Accordingly, oil film breakdown ratio E indicates a different value depending on the setting of resistance R,,. Fig. 5 shows the relation between oil film breakdown ratio E obtained from a test where resistance R,, is altered, and circuit The smaller the current when passed current I,. through the short circuit and when resistance Ro is increased, the greater value the oil film breakdown ratio indicates even if the oil film formation state is the same. Conversely, is resistance Ro is set to a fixed value, it is clear that the mutual test results can be compared. Further, as the oil film breakdown frequency is extremely high, increases in the impedance by the capacitor can be ignored. Accordingly, hereupon, the test was carried out with the applied voltage set to Eo = 0.5 V, with fixed resistance set to Ro = 15 kR, and capacitor capacity set in two stages at C = 4 7 , 1000 pF.
Fig. 6
3 3.1
RESULTS and DISCUSSIONS Measurement of oil film'breakdown position
In Fig. 6 , Wr and WC indicate the inner ring revolution and retainer revolution signals, respectively; and the signal-to-signal distance represents a single revolution. The rotary angular position that generates signal S of W, represents Jl, = 0 rad, the maximum angle of contact ~1 of the rolling element at the point of outer ring fitting misalignment. wb indicates rotational behaviour, and the peak-to-peak distance represents a single rotation. Wf indicates an oil film formation state at the rolling surface; electrical resistance value Rx = 0 indicates a state of metallic contact occurrence, and R, = m indicates a state whereby both surfaces are completely separated by the oil film. The rotational speed of the rolling elements seen in the figure varies in a single retainer revolution. At position Jlc = 0 rad shown S, the magnitude indicates large rotational speed, and it is apparent that considerable oil film breakdown is occurring simultaneously at that position. When this high-load drive area is escaped from, the rotational speed of the rolling elements suddenly decreases thus causing considerable sliding to occur at the rolling surface (5). However, it is apparent that satisfactory oil film formation is maintained due to the surface pressure being low even during slipping. Figs. 7 and 8 show the relation between rolling element movement and the oil film breakdown position when the outer ring fitting misalignment Bo = 0 . 7 ~ 1 0 - ~ 2, . 4 ~ 1 0 - ~is varied,
R e l a t i o n between o i l f i l m breakdown wave and b a l l motion ( B o = ~ . O X ~ O rad, - ~ n = 2000 rpm, P t = 98 N)
f o r f i t t i n g misalignment
647
n = 500 rpm, Pt = 196 N
n = 2500 rpm, Pt = 196 N
n = 5000 rpm, Pt = 196 N
n = 2500 rpm, Pt = 588 N
n = 5000 rpm, Pt = 588 N
I n = 500 rpm, Pt = 588 N
Bo
misalignment
Fig. 7
0.7x10-3 rad
R e l a t i o n between o i l f i l m breakdown wave and b a l l motion f o r f i t t i n g misalignment
n = 500 rpm, Pt
=
196 N
n
=
2500 rpm, Pt = 196 N
n = 5000 rpm, Pt = 196 N
I n = 500 rpm, Pt = 588 N
n = 2500 rpm, Pt = 588 N
I n
misalignment
Fig. 8
=
I =
I
I
J
I
I
I
I
5000 rpm, Pt = 588 N
BO
=
2.4~10-~ rad
Relation between o i l f i l m breakdown wave and b a l l motion f o r f i t t i n g misalignment
I
648
In either of the two figures, the upper row (1)( 3 ) shows the thrust load Pt constant at 196 N, and the lower row (4)-(6) shows Pt constant at 588 N. The left column (1) and (4) indicates the case of shaft speed n = 500 rpm, the center column ( 2 ) and (5) the case of shaft speed rpm value n = 2500 rpm, and the right colum ( 3 ) and (6) the case of shaft speed n = 5000 rpm. In Fig. 7, (1) was observed at a low speed of n = 500 rpm and a low load of Pf = 196 N. At points S-1, S-2 and S-3 where $C = 0 rad is exhibited, considerable oil film breakdown occurs, and the rotational speed of the rolling elements also is high. However, at the vicinity of $c = 3 ~ / 2rad slightly before QC = 0 rad, delay in the rotational speed of the rolling elements can be seen along with satisfactory oil film formation at the rolling surfaces. When the load is increased to Pt = 588 N , the range in which considerable oil film breakdown occurs is extended, and only slight oil film formation at the vicinity of $c = 3 / 2 remains. Although, in the case of both loads, the oil film formation state exhibits a trend towards improvement when the speed is increased, it can be seen that oil film breakdown occurs near to points S-1, S-2 and S-3 indicating maximum surface pressure at the rolling surfaces. In Fig. 8, (1) was similarly observed at a low speed of n = 500 rpm and a low load of Pt = 196 N. Considerable oil film breakdown occurs at the center of point S $c = 0 rad, and in the vicinity of $c = 3 ~ / 2the rolling elements exhibit rotational slipping, and the results are
as shown in Fig. 7 in that a satisfactory o i l film is formed. However, the results differ as follows; considerable oil film breakdown occurs at position JlC = 1~ rad, the rolling elements exhibit considerable rotational slipping near to Jlc = T / 2 before Jlc = TT rad, and at that position oil film formation can be seen. Similarly, the oil film formation state exhibits a trend towards improvement accompanying an increase in speed. Although the positional relation between rotary angular position Jlc and oil film formation/breakdown accompanying this increase in speed, demonstrates seemingly unclear tendencies, it is clear that in the meantime they possess a mutual relation. 3.2
Quantative measurement of lubrication states by oil film breakdown ratio
From the waveform of the oil film breakdown we can ascertain the oil film breakdown position during a single revolution of the rolling element. Also, we can ascertain the fixed tendencies of the oil film formation state regarding fitting misaligment, load, and speed. However, when the load is not small, and the speed has been increased, the electrical resistance becomes almost infinite, and as can be seen from the waveform it becomes almost linear. In such a state, it is difficult to catch minute oil film breakdown occurring instantaneously at the tips of the asperities. Hereupon, the oil film breakdown ratio is used to quantitatively measure the oil film breakdown state together with minute
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shaft speed 0 100 rpm 0 500 rpm 0 1000 rpm 0 6000 rpm
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0
20
40
60
80
100
120x102
thrust load Pt N
Fig. 9 Oil film breakdown ratio load Pt
E
versus thrust
-rl-
' 0
I
I
20
40
60
80
oil temperature T "C
Fig. 10 Oil film breakdown ratio temperature T
E
versus oil
649
M
w 0
.rl
0
m
-0
% w
P 4 E ?I 'u
4
0
10" o i l film parameter
Fig. 11 Oil film breakdown r a t i o
E
versus o i l film parameter A
oil film breakdown occurring instantaneously at the tips of the micro asperities. Fig. 9 shows the changes in the oil film breakdown ratio in relation to thrust load P t , and that the supply oil temperature is constantly maintained at T = 20 "C. In the case of any one of the revolution speeds, oil film breakdown ratio E exhibits a tendency to increase accompanying load increase. Further, Fig. 10 shows changes in the oil film breakdown ratio in relation to the supply oil temperature when the thrust load is kept constant at Pf = 490 N . Regardless of the revolution speed in this instance, the oil film breakdown ratio E exhibits a tendency to increase accompanying a drop in the supply oil temperature. These tendencies materialize even under different set conditions. However, repeatability of absolute values is difficult due to the state of the rolling surfaces over the passage of operation, and other additional conditions. The results shown in Fig. 11 were obtained by varying the thrust load, shaft revolution speed and supply oil temperature under other conditions. In the area I of oil film parameter A < 1, the fact that it is an area of boundary lubrication which includes parts of continual metallic contact is shown. In the area II of 1 6A < 4 , E follows a tendency to decrease accompanying an increase in A , and shows the fact that it is an area of mixed lubrication in which fluid lubrication greatly contributes to the load distribution ratio. Regionm of A 2 4 indicates E S O % , shows satisfactory oil film formation, and a state close to fluid lubrication. In the above, the lubrication state of the rolling surfaces against the oil film parameter A was investigated into using the oil film breakdown ratio E. In the kame tendency as another result ( 6 ) , in accordance with the theory of oil film breakdown it is evident that the oil film formation state improves accompanying an increase in oil film parameter A, However, we would like to bring your attention to the fact that although regionlU indicates E e! O X , it indicates E + 0% in spite of an increase in oil We would further like to, in film parameter p,
.
A
10-
oil film parameter A
F i g . 12 Oil film breakdown r a t i o f i 1m parameter A
E
versus o i l
650
particular, investigate oil film breakdown ratio E more precisely in connection with the occurrence of seizure damage to rolling surfaces in the light-load, comparatively high-revolution range Fig. 12 shows in the area 2 4 , the general view is that E decreases together with an increase i n A if lubrication theories are applied. However, in the figure, E exhibits the tendency to remain at the same level in relation to an increase in A , and that value exhibits great scattering. The range A 2 4 shows high speed and light load, and is an area that clearly demonstrates fluid lubrication characteristics if the friction coefficient of the bearing is measured. This scattering of E , is deduced as a potential cause of sudden seizure damage in rolling bearings. Fig. 13 shows the changes in oil film breakdown ratio E in relation to the changes in shaft revolution speed n. Area I shows the boundary lubrication area of numerous metallic contact parts, and area II a mixed lubrication area of a high load distribution ratio caused by fluid oil film. Aream shows, in spite of value E being low, an increase in the oil film breakdown ratio that accompanies an increase in speed can be seen in area These are the results obtained after precisely analyzing phenomena E = 0 and E = 0 in rangem in Figs. 10 and 11. It is deduced that this is one of the potential causes for sudden seizure damage in rolling bearings at high speed.
m.
.
4
CONCLUSIONS
Two tests were carried o u t in order to establish a basis for shedding some light upon the oil film breakdown mechanism of ball bearings under operation. Within the scope of these tests, the following results were obtained. Firstly, in order to check the oil film breakdown position, an angular contact ball bearing, into which one steel ball for detection and nine ceramic balls were inserted, was used. As a result, under operation the ball bearing underwent slipping in the low-load rotary angular area. However, the oil film formation state was satisfactory at this position. If anything, considerable oil film breakdown occurred at the low-slippage high load distribution area. Secondly, a DC electrical circuit using a capacitor and resistance, including oil film resistance at the bearing rolling surfaces, was formed. The oil film breakdown ratio was used to quantitatively measure the oil film breakdown phenemenom that occurs instantaneously at the micron-level. As a result, the phenomenom whereby oil film breakdown increases together with an increase in the shaft speed was ascertained. This was deduced to be a potential root cause of seizure hindering stable high-speed operation of bearings.
5 ACKNOWLEDGEMENTS We would like to express o u r gratitude to Mr. Hideya Wakabayashi of Mitsubishi Electric Co., Ltd for his extensive efforts, and to Mr. Heihachiro Inoue and Mr. Shinichi Kudo of Kansai University for their kind help.
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References 10
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t
FUJII, T., OGATA, M. and SHIMOTSUMA, Y. 'Oil film formation between ball bearing cage pockets and rolling elements', Proc. JSLE Int. Tribo. Conf., Tokyo, 1985, 597-602 DOWSON, D. and HIGGINSON, G.R. 'Elasto-hydrodynamic lubrication' , Pergamon Press, 1966 COURTNEY-PRATT, J.S. and TUDOR, G.K. 'Analysis of the lubrication between the piston rings and cylinder wall of a running engine', Proc. Instn. Mech. Engrs. 1946, 155, 519-527. OGATA, M., MATSUDA, T., FUJII, T. and SHIMOTSUMA, Y. 'Research on ball bearing lubrication - oil film breakdown and.rolling - sliding friction', Tech. Rep. Kansai Univ., 1979, 20, 1-10. KAWAKITA, K. and ARIYOSHI, S. 'Actual ball behaviours and the effect of the shaft spped on ball motion in a radial ball-bearings', Proc. JSLE Int. Trib. Conf., Tokyo, 1985
579-584.
F i g . 13 O i l f i l m breakdown r a t i o
speed n
E
versus s h a f t
HEEMSKERK, R.S., VERMEIREN, K.N., and DOLFSMA. H. 'Measurement of lubrication condition in rolling element bearings', ASME Trans., 1981, 2, 4, 519-527
SPECIAL LECTURE Chairman: Professor F.T. Barwell
Continuity and dry friction: An Osborne Reynolds approach
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Continuity and dry friction: An Osborne Reynolds approach Maurice Godet and Yves Berthier
While Tribology is known t o regroup all sciences concerned with contacts, and particularly lubrication and wear, it may come a s a surprise at a Symposium celebrating t h e Centenary of Reynolds' Equation t o devote t i m e t o Dry Friction and attempt t o look at that subject with a Reynolds or Continuity approach. The purpose of t h i s paper is t o suggest that Flow and Continuity play a s important a r o l e in Lubrication and in Dry Friction and that much can b e learned by applying Reynolds' teaching t o t h i s new area.
I ) I NTRODUCT I ON
2) LOAD CARRYING
Thick film lubrication is studied by Mechanical Engineers while Friction and Wear phenomena a r e explained by Material Scientists (1,2). T h e approaches used in each subject a r e t h u s so very different that it s e e m s unlikely t o bring them together other than under t h e common name A closer look however sugQf Tribology. gests that both possess great like-ness and that it might b e worth-while finding out what they have in common, and looking into t h e possibilities of formulating, even loosely at start, a common approach. The like-ness is that both subjects are concerned with t h e load-carrying capacity of mechanisms and with friction, i.e. with forces. It is commonly accepted that Mechanics prevail when thick oil films a r e present but that Materials Science takes over when t h e film is broken. Thus, in lubrication, forces a r e accounted for from equilibrium and continuity analyses, while in "dry friction" t h e s a m e forces a r e "justified" by material composition. This is of course unsapoint of tisfactory from a mechanical view as n o theory links composition t o forces. T h e purpose of t h i s paper is t o suggest that Flow and Continuity play a s important a role in Lubrication and in Dry Friction end that much can b e learned by applying Reynolds' teaching t o t h i s last field. Some of t h e ideas advanced here have already been presented on different occasions (3-5) but it seemed fitting on this occasion t o show that Reynolds' thought is not limited t o t h e lubrication side of Tribology.
2.1
)
t h e mechanical view
It is well known ( 6 ) in thick f i l m lubrication, that loads can b e generated through different mechanisms (wedge, squeeze, stretch and external pressures). The velocity field which is obtained by solving simultaneously, with strinqent boundary conditions, fluid constitutive laws and t h e equations of equilibrium is introduced in t h e equation of continuity t o yield t h e Reynolds' equation. Thus, physically, that equation is nothing more than a flow conservation relation. Most textbooks ( 7 , 8 ) illustrate how continuity can lead t o load-carrying and t h i s point will not b e discussed further here. Note however that in t h i s analysis:
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t h e material (the oil for instance) is identified by its rheology (constitutive laws), relative surface velocity is accomodated across, t h e film, t h e film possesses its own velocit y field and dynamics.
Further, for a given fluid, load and friction can b e calculated for mechanisms whose dimensions and running conditions a r e defined precisely. Thus this analysis can b e used in t h e design process, a s the information produced is quantitative. The method is efficient. 2.2) t h e material view
The t o n e is different a s t h e words a r e not t h e same. T h e quantitative load carrying concept is abandoned and the qualitative notion of surface protection introduced (9). Solutions are sought among coatings, films , screens, identif i ed by thei r composi ti on and suggested
on the basis of past t h a t i n t h i s approach:
-
-
experience.
Note
the material (film, coating, identified transfer f i l m etc.) i s by it s chemi c a l composi t i on r e l a t i v e v e l o c i t y accomodation i s n o t discussed, films a r e looked upon as static identities, s t r o n g l y adherent to t h e r u b b i n g surf aces.
,
If, for given materials, l o a d and f r i c t i o n cannot be c a l c u l a t e d , available i n f o r m a t i o n can be i n t e r p o l a t e d w i t h c a r e and used w i t h g r e a t c a u t i o n i n t h e design process. The method i s q u a l i t a t i v e , i t s efficiency depends on t h e d i s t a n c e over which i n t e r p o l a t i o n i s conducted.
2.3) p a r t i a l c o n c l u s i o n s One way o f b r i n g i n g b o t h p o i n t s o f view together i s t o note t h a t a l l loadc a r r y i n g m a t e r i a l s must possess:
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a chemical composition a r h e o l o g i c a l behaviour which can lead t o one o r more c o n s t i t u t i v e 1awe
which c o n s t i t u t i v e laws can be i n t r o d u c e d as above i n t h e equations of e q u i l i b r i u m t o y i e l d , w i t h s a t i s f a c t o r y boundary conditions, displacement fields which i n t u r n must satisfy conservation conditions. I f p e r t i n e n t , t h i s approach would l e a d t o a g e n e r a l i s e d t h i n f i l m mechanics lutheory which would a p p l y as w e l l t o b r i c a t i o n and d r y f r i c t i o n . While the presence and mode of action o f a load-carrying m a t e r i a l (the o i l ) a r e r e s p e c t i v e l y accepted and understood i n lubrication, the situation i s n o t as c l e a r i n " d r y f r i c t i o n " as w i l l be shown below.
2.4)
terminology
For b r e v i t y i n the discussion t h a t follows, the load carrying materials, w i l l be known whatever be t h e i r nature, as " t h i r d bodies" and t h e r u b b i n g solids w i l l be c a l l e d " f i r s t bodies".
3 ) THE CONTACT
3.1
)
1u b r i cated c o n t a c t s
EHD p i c t u r e Figure 1 i s a classic.al ( 1 0 ) . A dvnamic view of t h e s u b j e c t would have shown t h e o i l being d r i v e n i n t o t h e i n l e t bv t h e moving s u r t a c e s and flowing accross t h e c o n t a c t u n t i l i t s e x i t . F i r s t body s u r f a c e s a r e completely separated by or the o i l f i l m and show no wear t r a c e s any form o f degradation. Here. t h e t h i r d body, which i s the load-carrving oil film, provide5 f u l l p r o t e c t i o n . This is t h e c l a s s i c a l three-bodv c o n t a c t .
.3.2) drv f r i c t i o n Glean s u r f a c e s a r e known to adhere t o .each o t h e r ( 1 1 ) and t n e bond formed must be broken t o r r e l a t i v e motion t o occur. Most t i r s t displacements. waves escepted i 12). r e s u l t i n l o c a l des-tructions o f t i r s t body surtaces and t h u s i n t h e f o r m a t i o n of wear p a r t i c l e s . F i a u r e 2 i131 i l l u s t r a t e s what happens t o a hard s t e e l surface rubbinq aqainst glass a f t e r a l i m i t e d number o f s h o r t strokes. Scratches a r e noted. p a r t i c l e s a r e detached bv one mechanism o r another. and because of scale factors (fig.3), these particles are trapped a t l e a s t momentarilv i n t h e verv c o n f i n e d space o f t h e c o n t a c t . Wear debris. or wear p a r t i c l e s which torm r a pidly alter the nature of the contact which g r a d u a l l y chanaes from a two t o a three-bodv c o n t a c t as i n EHD.
3.3) r o l e o f wear d e b r i s As noted above, the presence ot debris a l t e r s the nature o t the contact. The l o a d - c a r r v i n a r o l e o f d e b r i s w i l l be i l l u s t r a t e d below. However. f o r c l a r i t v :
1 ) t h e a b r a s i v e r o l e (14) ot
partiot cles which accounts f o r some the wear, i s acknowledqed but w i l l n o t be discussed i n d e t a i l here. 2 ) loose i s o l a t e d debris (fia.4) which e s t a b l i s h d i s c r e t e bridaes also between first bodies w i l l n o t be discussed e i t h e r . Further. i t w i l l be assumed i n the following d i s c u s s i o n t h a t a drop i n wear r a t e corresponds t o an i n c r e a s e i n loadc a r r y i n g capaci t v . Debris a c t i o n has been demonstated i n a s e r i e s o f experiments (15.16) conducted on a r i g i d p i n and disc machine (fia.5) u s i n g a w h i t e c h a l k s t i c k and a liqhtlv f r o s t e d disc. P i n s of d i f f e r e n t s e c t i o n s were tested. Trace f o r m a t i o n was followed throuqh the semi-transparent disc by p l a c i n g a m i r r o r a t 45 degrees. Tracers o f coloured c h a l k were used when necessary. The behavigur noted with chalk, which was chosen because i t produces thick traces. was also obtained with other m a t e r i a l s which ranged from metals t o plastics. Continuous and oscillating t e s t s were r u n (17). The c o n c l u s i o n s reached proved t o be q u i t e qeneral and w i l l t h e r e f o r e be presented as such. They i n dicate that:
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debris flow within the contact. the speed of p a r t i c l e s i s variable.and i s r a r e l v equal t o t h a t of t h e disc. d i f f e r e n t l e v e l s o f separation. o r 1odd-carrvi nq capaci t y ( f 19.6) of were observed alona t h e c o n t a c t . a s o l i d t r a c e i s l e t t behind t h e contact (tig.6). I n continuous r o tation. some o+ t h e d e b r i s trom t h e t r a c e i s r e c i r c u l a t e d ( fia. 7 ) ,
655
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t h e rest is removed from t h e contact bv a prow effect. The trace is completelv destroved inside of t h e contact. New and old debris mix t o form t h e new trace. d u e t o debris recirculation the r a t e of wear decreases, or t h e load-carrying capacity increases. over t h e first f e w rotations sugqesting that equilibrium conditions exist. a wedge cut at contact entrv significantly increases load-carrving capacity b v increasing t h e rate of debris recirculation t h u s reducing t h e rate of formation of new debris Cfi9.8). load-carrving capacitv varies with pin orientation (fig.9). The wear rate of a rectanqular pin drops if t h e longer s i d e is parallel t o t h e direction of motion. t h e normal load distribution is altered b v t h e presence of debris (18). contacts a r e not always full (fiq.4), o r a r e starved.
None of these point5 taken individually prove that non-viscous third bodies have load-carrying capacitv but t h e sum of t h e evidence indicates a stronq convergence between t h e results presented here and lubrication predictions. Indeed, if each point is examined separately. o n e concludes that in dry friction a s in lubrication:
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third-bodv flow is present third-body film thickness varies alonq t h e contact a trace is left behind t h e contact which mixes with i ncomi ng "lubricant" equilibrium conditions exist entry conditions modify load-carrvinq capacity t h e load-carrying capacity depends on specimen orientation. Recall than t h e load-carrying capacity of a plain slider bearing increases with t h e square of its length in t h e direction of motion and only linearlv with its width. t h e pressure distribution between t w o first-bodies can b e siqnificantlv altered bv t h e presence of a third-bodv. starvation esists.
Recall also that a parrallel between w a n u l a r and EHD traction (19) was noted in quite unrelated studies.
Further all these r u n s considered tests in which at least initiallv. the third-bodv was produced bv normal feed (1.e. from t h e first bodies themselves) and not tanoentiallv a s in lubrication. It is important at this point t o determine i t normal feed is a condition for "apparent" load-carrvinq. To answer this question a series of tests ( 2 0 ) using
artificial third-bodies were run. For instance artificial iron oxides powders, were run in steel against steel contacts, t o see if t h e presence of these powdered oxides modified t h e process. No significant differences were noted a s soon a s equilibrium conditions were reached further strengthening t h e evidence in favour of t h e generality of t h e process. If a s sugqested, powders have loadcarrvi ng capacities, the protective mechanism of powdered solid lubricants, could b e better undersyood. The behaviour noticed with these lubricants was indeed very similar t o that described above for both natural and artificial third-bodies. Figure 10. which gives t h e variation in wear rate measured in a fretting test run with and without solid lubricant. clearly shows that t h e presence of solid lubricant postpones t h e t i m e at which wear starts. bv insuring a fairlv complete separation of t h e first-bodies. Wear is seen t o begin when t h e lubricant is completelv eliminated.
3.4) load-carrying
tests
A s noted earlier, t h e results presented indicate that load-carrying exists if one:
. admits .
that a decrease in wear is indicative o f an increase in loadcarrying capacity accepts t h e evidence concerning particle flow in t h e visualisation ex per i men t s.
While apparentlv convincing. t h e arqument is still indirect and could be strengthened bv a n experiment that would produce direct proof of load-carrving capac i t v. Direct proof of load-carrving is obtained when tor a qiven first-body separation. a load is transmitted by a third-bodv. Unfortunately, unlike oils, powders a r e not transparent and t h e direct transposition of t h e well known interferometric techniques t o dry friction studies is not possible. A substitute 121) consists in imposing mechanically a separation between first-bodies o n e of which is a fixed instrumented inclined plane. t h e other a horizontal slider (fiq.11). A uniform powder bed is deposited on t h e slider and driven inside of t h e convergent formed by t h e inclined plane and t h e slider. The reaction on t h e slider can be measured with a dynamometer. Indications concerning t h e nature of t h e flow of t h e powder was obtained by disposinq in grid pattern small (0.5 mm) lead balls in s o m e of t h e beds tested. Grid deformation could b e monitored by taking X ray pictures at different points along t h e slider course (fig.12). Clearly, t h e scale of t h e contact is multiplied a s t h e thickness of t h e bed is of t h e order of t h e centimeter and the inclined plane is 20 cm long.
656 Fiqure 13 shows t h e variation in load-carrying c a p a c i t y along t h e s t r o k e obtained w i t h chalk and g l a s s powders and i l l u s t r a t e s t h e t y p e of r e s u l t s than can be produced w i t h such a device. Without qoinQ i n t o t h e d e t a i l s o f t h e operation, the load-carryinq c a p a c i t y was seen to vary with:
1)
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t h e materials tested
- humidity 2 ) - s l i d e r speed
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plane angle bed t h i c k n e s s
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t h e n a t u r e and roughness o f s u r f aces
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3)
the
kurther, powder properties are moditied d u r i n q t h e t e s t by t h e runninq conditions. respectively F i q u r e s 14 and 16 show representative velocity fields. Continuous v e l o c i t y p r o f i l e s are illustrated 14 w h i l e r u p t u r e f r o n t s are in tiqure presented i n f i g u r e 15. F i g u r e 16 shows an SEM p i c t u r e i l l u s t r a t i n g t h e r a t c h e t The formed b y successive r u p t u r e f r o n t s . a n a l y s i s o f a l l r e s u l t s show that, with both m a t e r i a l s t e s t e d , v e l o c i t y accomodation i s achieved throuqh a t l e a s t three mechanisms:
1 ) continuous
velocity i n lubrication) 2) rupture .3) s l i p a t t h e boundary.
fields
(as
Powders a r e t h u s capable ot i n s u r i n g l o a d - c a r r y i n g capacity. Caution must of course be e x e r c i s e d b e f o r e t h e conclus i o n s drawn from t e s t s o f t h i s n a t u r e can be e x t r a p o l a t e d t o s i t u a t i o n s i n which the s c a l e o f events i s so very d i f f e r e n t than t h a t found i n contacts. Neverthel e s s . as d e b r i s i s always present i n contacts and t a k i n g i n t o account all the converging evidence presented here, i t i s most l i k e l y t h a t f r i c t i o n and wear ids lubrication) are governed b y t h i r d - b o d y rheology and dynamics. The s i t u a t i o n found w i t h powders i s however more complicated because o f the three p o s s i b i l i t i e s offered for velocity accomodation which can occur simultait very difficult to neously, making p r e d i c t t h e e f f e c t o f t h e parameters l i s t e d above which govern l o a d - c a r r y i n g capacity. Let us l o o k a t one of these two extra possibilities.
3.5) boundary c o n d i t i o n s One o f t h e g r e a t s t r e n q t h of lubrication theory i s t h a t i t can r e l y on a very powerful boundary c o n d i t i o n between tirst and t h i r d bodies which s t a t e s t h a t i s "no s l i p a t t h e wall" between there the o i l f i l m and t h e b o r d e r i n g surfaces. While that condition has occasionallv been questioned ( 2 2 ) , i t has proved c o r rect even under h i q h shear conditions, and i s one of t h e b a s i c assumptions o f Reynolds theory.
partiAdhesion between powders o r cles (23.24) and s o l i d s i s a d i f f i c u l t s u b j e c t which w i l l n o t be discussed here. Elementary t e s t s show however how very differently powders and surfaces can react. Six powders and three surfaces were chosen. Small elongated heaps of powder were deposited on t h e s u r f a c e or A f l e x i b l e metal r u l e (fig. substrate. 17) was drawn across t h e heap s i m u l a t i n g t h e passage o f a s l i d e r bearing. After one passage, t h e s u b s t r a t e i s l i f t e d and s l i g h t l y shaken t o e l i m i n a t e non-adhering powder. Three g r a p h i t e s (Table I), one natural with a wide s i z e d i s t r i b u t i o n two synthetic w i t h d i f f e r e n t p a r t i c l e sizes, 0.5 t o 2 microns and 50 t o respectively 60 microns, two i r o n oxides (hematite and magnetite a l s o w i t h wide size spectra) and chalk powder were used i n these t e s t s and rubbed a g a i n s t a s l a b o f plywood, and two drawing mats o f d i f f e r e n t roughness. 18 p r e s e n t s t h e 3 steps of Figure the tests. Figures 18 a,b and c show r e s p e c t i v e l y t h e o r i g i n a l heaps, t h e same heaps after t h e passage o f t h e metal rule, and what i s l e f t on t h e s u b s t r a t e after l i f t i n g and light shaking. It clearly shows t h a t t h e adhesion to the substrate i s variable. Great d i f f e r e n c e s a r e noted between powders o f identical composition b u t o f d i f f e r e n t g r a i n size. This simple observation suggests t h a t p a r t i c l e adhesion can vary d u r i n g a wear test, i n which p a r t i c l e s detached from f i r s t bodies a r e ground t o s m a l l e r sizes. F i g u r e 17 i s a sketch o f what i s observed w i t h one o f t h e powders which c o i l s up as the metal r u l e t r a v e r s e s t h e heap. This does n o t concern boundary conditions alone b u t a l s o shows t h a t i n t e r n a l s t r e s ses a r e induced i n t h e b i a x i a l compacting process. Clearly, t h e boundary c o n d i t i o n s i n three-body dry f r i c t i o n are much more d i f f i c u l t t o express than i n l u b r i c a t i o n . A p a r a l l e l a n a l y s i s o f powder rheology, which would consider b o t h t h e chanqes i n properties brought about by non homogeneous compacting and t h e d i f f e r e n t forms of velocity adaptation would lead t o similar conclusions. Nevertheless, the differences observed i n t h e modelling should n o t h i d e t h e b a s i c s i m i l a r i t i e s i n t h e mechanics of b o t h s u b j e c t s which are centered on l o a d - c a r r y i n g and flow.
4) THIRD-BODY
DYNAMICS
Each f l o w problem has a source, a purpose and a s i n k . I n wear, t h e source i s the first-body, t h e purpose i s t h e load-carrying, and t h e s i n k i s t h e contact exit. 4.
I)
t h e source
The source produces t h e wear p a r t i c l e s and t h e mechanisms o f wear particle product ion :
-
adhesion abrasion f a t i g u e etc.
657 a r e w e l l known. The o n l y d i f f e r e n c e w i t h classical wear t h e o r y i s t h a t we do not t a l k here o f wear by adhesion, abrasion, fatigue etc. but o f p a r t i c l e production by one of these mechanisms. 4.2)
t h e purpose
The purpose as mentioned above i s load-carrying. I t i s governed by t h e a b i l i t y o f t h e third-body t o remain i n t h e contact and accomodate t h e v e l o c i t y d i f ference between f i r s t - b o d i e s . 4.3)
the sink
The sink drains t h e p a r t i c l e s that a r e n o t r e c i r c u l a t e d b u t which a r e e l i m i nated from t h e contact. It i s obviously governed by c o n t a c t shape, machine dynamics ( v i b r a t i o n s ) etc.
5 ) CONCLUSION Focussing on c o n t i n u i t y , flow and load-carrying concepts i n " d r y f r i c t i o n " has helped our understanding o f t h e subject enormously. As soon as t h e first p r o o f s were gathered, i t was c l e a r that the b e s t guidance was g i v e n by f o l l o w i n g p o i n t by p o i n t t h e l o g i c and assumptions of t h e l u b r i c a t i o n t h e o r y and t h i s , to our mind, j u s t i f i e s our r e f e r e n c e t o a Reynolds approach. T h i s framework i s a l s o useful t o measure how much more complex the elementary d r y f r i c t i o n process i s when compared to t h e elementary lubrication process c h a r a c t e r i s e d p o s s i b l y by a p l a i n bearing analysis. T h i s complexity, we have seen comes from t h e f a c t t h a t :
. .
. .
third-bodies generated in dry f r i c t i o n experiments s u f f e r large changes i n mechanical properties for small changes in mechanical c o n d i t i o n s , and a l s o small changes i n t h e environment. t h e y can accomodate v e l o c i t y grad i e n t s i n d i f f e r e n t ways. f i r s t and t h i r d - b o d y boundary conditions a r e not defined i n a sat i s f a c t o r y manner. contacts a r e n o t always f u l l , and t h e v o i d p a t t e r n i s n o t necessaril y steady, making i t d i f f i c u l t t o use c o n s e r v a t i o n equations.
The p i c t u r e suggested here i s not altogether encouraging as t h e distance between t h e presentation of the subject It i s and i t s s o l u t i o n i s considerable. hoped t h a t t h i s d i s t a n c e w i l l n o t discourage Mechanical Engineers t o e n t e r this field i n f o r c e b o t h on experimental and t h e o r e t i c a l grounds. I t i s suggested t h a t at least i n t h e next few years, very c l o s e cooperation w i t h open minded mater i a l s c i e n t i s t s i s necessary t o advance.
fie t er enc e s
DOWSON D. i3. and H I G G I N S O N G. Fr. "Elastohvdrodvnamic L u b r i c a t i o n " Ferqamon Press London 14Lm BOWDEN F.P. and TliBUfi: D . " l h e trict i o n and l u b r i c a t i o n ot s o l i d s " Vol. 1 and 2 . Clarendon. Oxford GODET M. "Fondements mecaniques de l a T r ib o l oqi e" Mecani que, Hater i dux. 1972 E l e c t r i c i t C " 1 'Usure T . 2 . 34-44, GODET M.. PLAY D., BERTHE D. "An attempt t o provide a u n i f i e d treatment of T r i b o l o g y through load carryinq, c a p a c i t y , t r a n s p o r t , and continuum mechanics" ASLEIJOLT April 1980, Vol. 102. 153-164. GODET M. "The t h i r d - b o d y approach: a WEAR, 100 mechanical view o f wear" i1984) 437-452 BAKWELL F. , "Etearing Systems'' Oxford U n i v e r s i t y Press, London 1979. Chap. 4,5 and 6. CAMERON A. "Principles of Lubricat i o n " Lonqmans Green 1966 S Z E K I A. Z. " T r i b o l o g v " Hemisphere Pub l i s h i n q Corporation. Washington 1980 BUCKLEY D.H. "Surface effects in adhesion, f r i c t i o n . wear and l u b r i c a t i o n " E l s e v i e r 1981. Chap. Y and 10. DALMAZ G . and GODET M. "An apparatus for t h e simultaneous measurement of load. t r a c t i o n . and - f i l m t h i c k n e s s i n lubricated sliding point contacts" Triboloqy. June 19.72, 111-117. M A U G I S D. "Adherence of s o l i d s " in "Microscopic Aspects of adhesion and L u b r i c a t i o n " Ed. GEORGES J.M. Elsev i e r 1982 M. PROGRI H.. VILLECHAISE B., GODET "Houndarv conditions in a two-bodv contact formed by a r e c t a n g u l a r polvurethane s l a b pressed a g a i n s t an a r a l d i t e plane" ASME/ J - o f T r i b . Vol 197, n'1,138-141,1985. COLOMBIE Ch. "Usure i n d u i t e sous pet i t s dCbattements" ThQse Ecole Cent r a l e de Lyon 19 Septembre 1986. MISRA A. and FINNIE I. "An experimental studv o f t h r e e body abrasive wear" WEAR. 85 (1983) 57-69 and GODET M. "Visualization PLAY D. ot chalk wear" i n "The wear o-f nonm e t a l l i c m a t e r i a l s " Ed. D.D.Dowson et a l . MEP London 1978. PLAY D. and GODET M. " S e l f - p r o t e c t i o n of h i o h wear m a t e r i a l s " ASLE Trans. Vo1.22. n ' l . 1979, 56-64 BERTHIER Y. and PLAY D. "Wear mechanisms i n o s c i l l a t i n q bearinqs" WEAR. 75 (1982) 369-387 BERTHIER Y. " E f f e t du comportement du patroisieme corps sur l ' u s u r e des liers secs en mouvement alternatif" Th@se I N S A de Lvon. 9 J u i l l e t 1982. p - 141. GENTLE C. R. PAUL G. R. .and CAMERON A. "Some evidence o f g r a n u l a r behavi our i n elatohydrodynamic t r a c t i o n ' ' ASLE Trans. Vo1.23, 2. 155-162. COLOMBIE Ch, BERTHIEH Y.. FLOQUET A. V I N C E N T L., GODET M. " F r e t t i n g : Loadcarrying capacity of wear debris'' J.of Trib.Vol 106. A p r i l 1984. 194-
.
202.
658 BERTHIER Y., ref.18. Annexe AII, p. 187 22) HOZEANU L. and SNARSkY L. "Physicochemical c h a r a c t e r i z a t i o n of. surface rouqhness and i t 5 t r i b o l o q i c a l i m p l i cations" i n " S u r f a c e Kouqhness Ef fects i n L u b r i c a t i o n " Ed. DOWSON e t al. MEP 1978. 23) SIMBO G., ASALAWA S., SOGA N . "Measurement o f adhesion f o r c e of powder particles bv powdered tensile s t r e n a t h method" J. SOC. Mater. Sc. Japan 1968 n@ 17 540-544. 24) STRIJBUS S. "Powder-wall friction" Powder Technoloqy. n"18, 1977, 209214.
21,
Fig. 3 : Contact scale (relative size of wear particles)
Fig. 4 . Discrete bridges formed by isolated rubber debris
Fig. 1 : EHD contact
Fig. 5
. Pin a n d disc machine
Fig. 2 : Damage progress during the f i r s t hundred strokes
Fig.6 : Different levels ot separation (view through glass disc)
659
Fig.7
C o n t a c t trace after passes ( p i n removed) :
10
load Fig. 9 : E f f e c t of pin orientation
on wear
unlubricated allowed hardened steels allowed hardened steels with solid I ub ri cant
Fig, 0 : Tangentiel f e e d e f f e c t The t r n c e i s trapped by the wedge Fig. 10 ..Solid lubricant ( C F x ) effect inclined plane
-
I
c--(
Fig. 11
:
Powder load carrying device
F2
c1 powder
size (pm 1 specific surfacearea m2. gr -1
chalk CaCO3 1-4
magnetite Fe3 01,
20- 30
-
2
Table I
F1 hematite FeP03
10 cm
I ref. 21 1
G3
natural graphite
G2
G1
synthetic graphite
synthetic graphite
5- 10
15-30
50-60
12
1 .1
2
: Powders tested
0.5
-2
25
660 Lead balls
hclincd DhnC
\I
powdrr- bed
he 2 hS
Fig. 14: Continuous velocity profile ( V x )
IWf.211
inclined plane
disphcement Fig. 15 : Rupture fronts in powder beds Irei. 21 I
\
X - Ray tilm
Fig. 12
:
Velocily gradient determination
in powder beds
h d p carrying
I ref. 21 I
1 15
dP ,-Ra=
6.5 pm m
--
20
I
stroke length Fig. 13: Load versus stroke length Id.211
(m)
661
powder : graphite
40 pm
1
a ) original
heaps
cross section A A Fig. 16 : Ratchet formed by successive rupture fronts
blafter passage of t h e rule
Y
I
I
1st body
Fig. 17
5.
Powder
powder
- wall
boundary conditions
c ) a f t e r lifting and light shaking
Fig. 18 : R u l e t e s t
This Page Intentionally Left Blank
WRITTEN DISCUSSIONS AND CONTRIBUTIONS
This Page Intentionally Left Blank
665
Written discussions and contributions
DISCUSSION SESSION I
-
KEYNOTE ADDRESS The method to be applied to this problem, which seems to work well for the other examples given, is based on Eq. (1) of the chapter:
'Osborne Reynolds' A. Cameron. Emeritus Professor H Blok (The Netherlands).
m-1 xP The author rightly brought out that Beauchamp Tower, around 1883, did not "invent" but "discovered" hydrodynamic lubrication. Now, it can be found out from ancient logbooks of piston steam bearings that even long before this discovery at least, albeit far from all, their bearings had, during successive inspections after running in, shown no or barely any signs of progressive wear. S o , judging from posterior insight, Mother Nature had already come to the rescue through both "inventing" and applying hydrodynamic lubrication all by herself. At that time engineers were not even aware of Mother Nature's llself-assignedll task of performing what properly was their job, let alone why more often than not she was unsuccessful. After all, even for the distant future engineers are well advised to work with her hand in hand, especially through design that is well adapted to her needs as to putting her in a better position to help them in optimizing from their various points of view. Dr J H Tripp (SKF, The Netherlands). The following is a discussion on the third of the puzzles stemming from the 1886 Reynolds' c cos paper, concerning the integration of (1 XI-" which was needed to solve the journal bearing. In his verbal presentation, though apparently not in his manuscript, Cameron has suggested that Reynolds may have consulted Todhunter 11) for the answer, one of the standard works then available on integral calculus. Having used the same reference myself [ 2 ] , however, it comes as no surprise that, if Reynolds indeed failed to find the integral during the tripos exam, subsequent searching through Todhunter would scarcely have helped towards the result.
-
The integral appears as an example to be worked at the end of Chapter 111 on reduction methods for integration, where on p. 51 we find: Example 19 If @(n)
=
J
(1
+
-n
c cos x)
dx, shew (sic) that
=
ax
= XP xm/m
xp xm/m -
where X denotes x
-
(bnp/m)
+
(p/m)
I
I
xm XT1(dX/dx)dx
xm+n-l xp-l dx
(1)
b x n.
Todhunter proceeds to derive five more forms of this "formula of reduction" [31, even extending the range of usefulness of the method by exploring the substitution z = sin X. None of this, however, appears immediately helpful in tackling Example 19, since the integrand is not manifestly of the form in Eq. (1). In attempting to bring it to such a form, the required result persistently drops out as one of the intermediate steps, indicating that even if Eq. (1) can be applied, it is not the easiest method for this case. Indeed, from the forms of the reduction equations, we might speculate Reynolds simply arrived at the premature and mistaken conclusion that some kind of a series expansion was the best to be hoped for. With hindsight, perhaps, the integration is not too severe and may be achieved by several routes. As an example, one proof requiring only a few easy steps is given in the Appendix. Thus, the task of integrating powers of the gap function for a journal bearing ought not to have presented any great difficulty, especially with men of the calibre of Maxwell and Lord Rayleigh around. As Cameron broadly hints, however, Reynolds may well have had reasons for wishing to work out the result privately for himself! REFERENCES
[I]
Todhunter, I., "A Treatise on the Integral Calculus", (Macmillan and Co., London), 4th ed., 1874.
[2] Kindly supplied by Professor Cameron. [3]
Curiously, this 4th edition of Todhunter still misprints the fourth of the six formulae of reduction, a circumstance that can hardly have strengthened Reynolds' confidence in his endeavours.
APPENDIX
666 If Y = (1 + c cos x), the identity (Y Y*
- 2 Y + (1
-
2 c
they 0(nJ = jY-n dx and c cos x can be written
= =
2 -c2 sin x
(A)
By direct differentiation it is clear that cd (Y-"+'sin
x)/dx
=
c2 (n-1) Y-" sin2 x -*l - Y
+
y-n+2
that multiplying Eq. (A) by (n-1) Y-" and substituting for the right side from Eq. (8) yields so
=
- cd (Y-*l
sin x)/dx
+ Y-n+2 -
y-n+l
(C)
Collecting terms together (n-2) Y-"+~
-
(2n-3) y-*'
+
(1-c2) (n-I) Y-"
Integration of Eq. (D) now leads to (11-21 0(n-2)
- (211-3) 0(n-1) + (1-c2) (n-l)@(n) =
- c sin x Y-*'
which upon re-arrangement is the required form. This result, nowadays routinely included in standard tabulations, cannot of course be used to obtain 0(1) necessary to start the recursion. Todhunter however already treats this case as early as Chapter 1, where it appears as Example 14 on page 15. Reply by Professor A Cameron (Cameron-Plint Ltd., Cambridge). Professor Blok is, of course, quite right. Fitted bearings had been used for many years on train axles and often must have run on hydrodynamic oil films. In fact Tower used them, as his research took place in London's Metropolitan Railway depot at Edgeware Road. It could well have been that the variable performance of these things caused the Institution to commission Tower's research. The reasons behind this I discussed in reference (7) of the paper. The importance of Tower's work was the totally unexpected discovery of a high pressure oil film supporting the load. It is kind of Dr Tripp to have taken the time to study Todhunter and point out that the integration of dx/(a+bcose)" does not follow from the reduction formulae given in Chapter 3. Reynolds would have had to differentiate example No 19 to shew (the Oxford dictionary allows show and shew!) that the integral was correct.
-
I have checked Todhunter's books and have looked at the 2nd edition of 1862, the 4th of 1874 and the 6th and final one of 1880. The Ist, 3rd and 5th of 1878 I have not been able to find. Todhunter died at 6 Rrookside Cambridge on the 1st of March 1884 (14)*. The misprint Dr Tripp noticed in equation 4 of art 30 in the 4th
*
further references can be found at the end of the author's reply.
edition was neither there tn the 2nd nor in the 6th edition. Though all the pages and equations are the same in most editions, slight variations in spacing show that every new edition must have been reset. Professor Booker reminded me of his Technical Brief (25) where he references Todhunter's 1880 book as well as one by Williamson, which I had not seen. I have found all 7 editions of this, "An Elementary Treatise on the Integral Calculus" in the Cambridge University Library. It was published by Longmans Green in 1875, 77, 80, 84, 88, 96 and 1906. In all of them the table of contents refers to those integrals. In Chapter 1, section 18 the integration of dO/(a+bcosO) is discussed and in Chapter 3 the reduction of dx/(a+bcosx)" is explained. He also gave a 'further method using the transformation (a+bcosx) = (a+b) cos2 ( X / 2 ) + (a-b) sin2 ( X / 2 ) which leads to the use of tan (x/2). These were sections 70 and 71 in the first edition and 74 and 75 in all subsequent ones. Reynolds therefore had examples available for him, had he wished to look. In ref. (25) Professor Booker also mentioned that Sommerfeld himself never gave the well known "Sommerfeld substitution". In fact he used a reduction formula with tan (x/2). He continued doing so right up to his last book (26) in 1947. Booker quotes Hardy (27) as giving the integration in 1905 using the standard tan (x/2) method. I find that Hardy commented "A more elegant method is to use the transformation (a+bcosx) (a-bcosy) = a2-b 2 * It is interesting how, in this subject, ascriptions get transposed. Petroff's law was actually given by Sommerfeld who had, it seems, misread Petroff's very lengthy paper (see ref 4). It could also be that Roswall, who was the first, it appears, to have used the "Somm rfield Substitution" (l+@o&) (I-Ecosy) 4 in his book (with no provenance) = I-E to have found it in Hardy's text, which was at that time very well known. Perhaps the "Sommerfeld Substitution" should be called the "Hardy substitution".
I am indeed grateful to the discussors for raising the points they have which encouraged me to study them in detail. New References [24] Isaac Todhunter. In Memoriam. Cambridge Review. 1884. March 5, 12, 19. Also Dictionary National Biography. 1884. 914. [25] Booker, J.F. A Table of the .Journal Bearing Integral Jour. Basic Eng. Trans. ASME (D). 1965. 87. 533-535. 261 Sommerfeld, A. Vorlesungen uber theoretische Physik. 1947. Vol. 2, Mechanik der deformierbaren medien. Section 36. 244-252. 271 Hardy, G . Integration of functions of a single variable. Cambridge Tracts in Mathematics and Mathematical Physics. 1905. No. 2. 44 (iii).
667
[28] Boswall, R.O. Lubrication, 1928. SESSION I1
-
The Theory of Film Longmans.
HISTORY
'Michell and the Development of Tilting Pad Bearings' J E L Simmons and S D Advani. Mr A M Mikula (Kingsbury Inc., Philadelphia, U.S.A).
I was very interested in Messrs. Simmons' and Advani's paper and how they documented not only A.G.M. Michell's contribution to the science of tribology, but also to the industrial introduction of tilting pad bearings to Europe. However, notwithstanding their fine job, I feel there is one point that must be clarified for the sake of accuracy. During the discussion of the formation of the Michell Bearing Company, credit for the invention of the tilting pad bearing is mistakenly attributed to A.G.M. Michell. I say mistakenly because in a paper published in 1929 (Ref. I ) , A.G.M. Michell acknowledges that "Professor Kingsbury's work was commenced a few years earlier". Finally, I would like to commend Messrs. Simmons and Advani for this informative paper. Reply by Dr J.E.L. Simmons (University of Durham, U.K.) and Mr S.D. Advani (Vickers plc Michell Bearings, U.K.).
-
The authors thank Mr. Mikula for his comments. The story of the parallel and independent inventions of Albert Kingsbury and A.G.M. Michell is covered in some detail by Dowson ( 4 ) . Mr. Mikula is quite correct in reminding us that Kingsbury commenced his experimental work several years before Michell's theoretical analysis and patent were published in 1905, although Kingsbury's first publications on the subject were not made until a number of years after those of Michell.
It was not our intention to bring this well-established history into question but rather to trace the story from Michell's invention through the early industrial applications to the formation of the Michell Bearing Company. The equivalent material for Kingsbury and the company he founded is set out in his paper of 1950.(18). From this it is clear that although Kingsbury was able to date his invention back to laboratory work first carried out in 1898, the first successful industrial applications are clearly those credited to Michell and described in our paper. For his part Kingsbury describes two attempted installations in the years before 1910 by the Westinghouse Electric Company which for different reasons did not come into service. Successful shop tests were carried out at Westinghouse in the period 1910-11 and it seems that the first successful field installation for which Kingsbury was responsible was a thrust bearing supplied in 1912 to the Pennsylvania Water and Power Company for use in an hydroelectric plant. On re-reading Kingsbury's account of these early years it is interesting to note that he
himself was not clear at first why his centrally pivoted bearings operated successfully. In describing experimental work by his students at Worcester Polytechnic Institute he states, ' I . . . . , the good operation with central supports for the shoes was not explained by the theory of Reynolds, who assumed constant viscosity in the oil. It was explained in later years by H.T. Newhigin, then Manager of the Michell plant, who suggested that, because of the continuous friction bearing, the oil became less viscous as it passed in the film from the leading edge to the trailing edge of each shoe; and this appears to be the main explanation". Of course, we now know the situation is more complex than this and that thermal and mechanical bending of the pad are also significant factors in the operation of many tilting pad bearings. SESSION IV
- THRUST BEARINGS
(1)
'Three Dimensional Computation of Thrust Bearings' C.M.McC. Ettles Emeritus Professor H. Blok (The Netherlands). In view of the comparatively low modulus of elasticity of the usual coating materials constituting the rubbing surfaces on tilting pads, it is agreed that the bimetal effect on the thermal distortion is in general only of secondary importance. However, through a non-rubbing coating on the --side of pads, and consisting of a fairly rigid material having a comparatively high coefficient of thermal expansion, designers may at least partly compensate for the usually deleterious bending deformation which is caused by the pressures generated hydrodynamically in the lubricant film. Already some 25 years ago this expedient proved its worth in the discusser's country, The Netherlands, i.e. in redesigning medium-size thrust bearings running at high speeds and thus being highly affected thermally.
-
Reply by Dr C.M.Mc.C. Ettles (Rensselaer Polytechnic Institute, New York, U.S.A.). The method of counteracting thermal distortion described by Professor Blok is most ingenious. A problem with bimetallic shoes (or in this case, tri-metallic) is whether to bond the materials over the whole surface (as is the case with a babbitt layer) or to allow €or expansion. Overall bonding will give a shear stress at the interface, which might be quite large if the added layer is used to substantially compensate for deflection. SESSION VI
-
ELASTOHYDRODYNAMIC LUBRICATION (1)
'Solving Reynolds Equation for EHL Line Contacts by Application of a Multigrid Method', A.A. Lubrecht, G.A.C. Breukink, H. Moes, W.E. ten Nape1 and R. Rosma. Emeritus Professor H. Blok (The Netherlands). In a sentence introducing their formula (15) the authors state (quote), "The dimensionless parameters describing the E.H.L. contact problem were first described by Moes [lo]". Like many other investigators the authors
668
must have overlooked the paper, "Die Theorie der hydrodynamischen Schmierung unter besonderer Beruecksichtigung physikalischer Erweiterungen", published by W. Peppler several years earlier (V.D.I. Berichte, Vol. 20, V.D.I. - Verlag, Dusseldorf, West Germany, 1957).
By using this formula, which can be easily programmed on a pocket calculator, designers can now calculate a minimum film thickness estimate, without having to worry about the operating conditions. One formula fits the whole map, which seems very elegant.
In discussing the figure on page 13 of his paper W. Peppler points out that it was reproduced from an unpublished memorandum of the present discusser. Now, in s o far as the results depicted in the figure concerned were not attributed to others, they date hack to work the discusser did at the then Royal Dutch/Shell-Laboratory "Delft" in 1944. The three major results then obtained, but never published by the author himself, are the following ones of which the first two are reflected in that figure.
Can the authors give some idea of the accuracy in the so-called VR-regime (rigid surfaces, piezoviscous fluid), which is important for e.g. cone roller hearings, and in the very high load part of the regime investigated by Dowson and Higginson, where Hooke (1978) finds an initially positive slope in the Johnson (1970) map? In the VR regime, film thickness is higher than predicted by the Grubin formula, I believe. And in the very highly loaded VE regime, film thickness is a bit lower than predicted by the Dowson and Higginson formula, o r the Crubin/Rrtel approximation. The formula proposed by the authors is most welcome and will be a very useful design tool.
First, by suitahly combining dimensional analysis with the complete analytical formulation of the isothermal E.H.L. problem it was proved that, without loss of generality, results in terms of minimum film thickness can be expressed through a set of only three dimensional groups. In fact, once only one such "triple" set has been derived, quite a variety of interdependent such sets can readily be established f o r the different evaluational and correlational purposes that one may have in mind. The set used in the figure concerned, and also that in the "Moes" E.H.L. Chart (see, e.g., the authors' figures 1, 2 and 5), belongs to that variety of "triple" sets. Second, a comparatively simple method was developed for arriving at a first approximation to the limiting curve for E.H.L. conditions where the flow in the film may be considered not only isothermal hut even isoviscous, i.e. in that the effects of pressure on viscosity may be ignored. The curve thus obtained was indeed depicted in Peppler's, or say the discusser's, figure. Third, a rigorous method for evaluating the same limiting curve boiled down to an integral equation. In 1968 it was K. Herrebrugh who published his skilful numerical evaluation of the same equation. Remarkably enough it then turned out that, at least up to about M=5 (see the authors' nomenclature), the discusser's approximate curve came fairly close to Herrebrugh's rigorous one. In any case, the discusser agrees fully with naming the present limiting curve after the latter investigator. Mr H J van Leeuwen (Eindhoven University of Technology, The Netherlands). The multigrid method seems to be very powerful in ehd line contacts and elliptical contacts. It is appreciated that calculation times can be reduced by several orders of magnitude. In this discussion I would like to pay attention to the Moes curve fit of all obtained numerical data. This curve fit shows an inherent smooth merging of three asymptotes, viz the ErtelIGrubin solution for heavily loaded deforming contacts, the MoesIHerrebrugh solution for elastic/ "mbel isoviscous contacts, and the MartinIGu solution for inelastic/isoviscous contacts. This is possible by using the dimensionless groups H', L and M, a representation which lends itself to curve fitting very well.
Reply by Mr A.A. Lubrecht, Dr G.A.C. Breukink, Dr H. Moes, Dr W.E. ten Nape1 and Professor R. Bosma (Twente University of Technology, The Netherlands). The authors thank Professor H. Blok for his comments and they agree that perhaps another name should be given to the film thickness plot and the dimensionless parameters used, honouring also the early contributors in this field. The authors thank ir M. van Leeuwen for his comments and they would like to answer that the accuracy of the numerically calculated minimum film thickness values was better than 5%, except for the L=25 calculations. The fit of formula (15) is believed to be better than 10% and it is exact for the asymptotes. F o r those who want to use thi.s equation as a design tool, the simplifications used should be pointed out i.e. isothermal flow and, very important, fully flooded conditions have been assumed.
SESSION IX
-
ELASTOHYDRODYNAMIC LUBRICATION (4)
'Transient Oil Film Thickness in Gear Contacts Under Dynamic Load'. A.K.
TIEU and J. WORDEN.
Mr H.J. van Leeuwen (Eindhoven University of Technology, The Netherlands).
It seems that Vichard's 1971 paper is becoming a classic in the transient EHD lubrication camp. This discussion is rather a short commentary than a question. The crucial point in Vichard's paper is the (redundant) boundary condition, that the reduced fluid film pressure be equal to the inverse value of the pressure-viscosity coefficient not at the entrance of the Hertzian contact zone, but in the centre. Under stationary conditions his equations merge into the ErtelIGrubin equations. By putting the houndary condition in the centre rather than at the inlet, the additional load due to the
669 squeeze effect will he much larger, and consequently, the damping will he higher (the film thickness is much slower in changing to another value).
Reply by Professor D. Dowson and Mr Z.M. Jin (Institute of Tribology, Department of Mechanical Engineering, The IJniversity of Leeds).
The question of vibrations in a set of involvute gear wheels has been studied before. I helieve that in 1981 Wang and Cheng published two papers on this subject in ASME's Journal of Applied Mechanics. Their analysis includes Vichard's approach for the fluid film behaviour, and it was combined with a Finite Element analysis for the bending and torsion in the gears.
We are grateful to Professor Murakami for his interest in our paper on the micro-elastohydrodynamic lubrication of synovial joints. The effective film parameter (A) is normally defined as the ratio of the minimum film thickness to the composite roughness average (1 4a) for the two bearing solids. In microelasto-hydrodynamic lubrication, the effective composite roughness average varies throughout the conjunction and, in the case of the dynamic conditions considered in the present paper, with time. However, in general terms we found the effective film parameter (A) to be in excess of 10 during the periods of each cycle governed by entraining motion. When squeeze-film action dominated the situation the surface roughness partially recovered its initial form and the effective film parameter fell to about 2. We would stress, however, that it is the absolute value of minimum film thickness, rather than the film thickness ratio alone, that is important under these circumstances.
Reply by Dr A.K. Tieu and Mr J. Worden (University of Wollongong, Australia). We thank Dr van Leeuwen for his comments. In his paper, Vichard correctly assumed that at the centre of contact, the reduced pressure Q approaches its asymptotic value 1/G, the inverse of pressure-viscosity coefficient. From an inspection of the expression for Q under transient condition, Q is not constant in the area of contact, but it changes from inlet to the centre. F o r steady-state case, an analysis in 'Principles of Lubrication' (A Cameron, Longmans 1966) shows that the difference of the reduced pressures Q at the inlet and at some distance inside the contact zone is small, hut not zero as implied by the discusser. If the integration is carried out at inlet, then appropriate boundary condition of Q (different from 1/G) should be used. The ASME paper by Wang and Cheng came from Ref ( 1 1 ) in the paper. In their paper, the dimensionless load includes the gear root radius, whereas here the relative radius of curvature of the gear pair was used as per Ref (l), which results in higher applied loads, and more significant squeeze film damping. This paper considers in isolation the effects of different dynamic loads on the gear film thickness, and in particularly it can apply to those loads resulting from gear tooth chattering with excessive backlash. The frequency and level of impact loads were not considered in Ref (11). An aim here is to arrive at a simple criteria based on the parameters Xp, xe, which indicates readily without an involved analysis, whether arbitrary applied loads would have any significant influence on the film thickness.
Professor Murakami's second point refers to the possible role of enhanced viscosity and hence film thickness associated with the gel formation on cartilage surfaces. He no doubt has in mind the earlier indications of the mechanism of boosted lubrication (Dl) and this is incorporated in some of the analysis presented in the paper. We find that it is necessary to invoke the 'surface viscosity' concept in order to predict realistic values of the coefficient of friction. The powerful role of boosted lubrication is evident in the preliminary results presented in the paper. We are now undertaking a more detailed study of the combined role of microelastohydrodynamic and boosted lubrication actions in synovial joints. Our findings indicate that boosted lubrication leads to substantial improvements in the predicted film thickness and theoretical coefficients of friction which are still small, but just within the range of experimental results recorded in the literature. References
SESSION XI1
-
BIO-TRIBOLOGY n [ l ] Dowson, D., Unsworth, A. and Wright, V.
(1970), "Analysis of 'Boosted Lubrication' in Human Joints", J. Mech. Engrg. Sci., Vol. 12, NO. 5, pp 364-369.
'An Analysis of Micro-elastohydrodynamic Lubrication in Synovial Joints under Conditions of Cyclic Loading and Entraining Velocities'
-
BEARING DYNAMICS (2)
D. DOWSON and Z.M. JIN.
SESSION XV
Professor T Murakami (Kyushu University, Japan). Your application of micro-elastohydrodynamic lubrication theory to natural synovial joints provides strong support for fluid film lubrication in natural joints under walking condition. (1) Could you show some data on the effective film parameter which is defined as the ratio of minimum film thickness to effective roughness after deformation? ( 2 ) Have you investigated the influence of highly concentrated gel formed on articular cartilages in the concave area on the effective film parameter under thin film conditions?
'Oil Film Rupture Under Dynamic Load? Reynolds Statement and Modern Experience' O.R.
LANG
Mr H.J. van Leeuwen (Eindhoven University of Technology, The Netherlands). The experiments carried out at Karlsruhe University on dynamically loaded hearings can be very helpful in testing different approaches for calculations of journal orbits, film
670
pressures, etc. I understand that these experiments confirm the results of the Holland-Lang method.
I would like to know whether the HollandLang method makes use of different boundary conditions compared to the Holland method, as it was published in 1959. Are there any other differences? If the boundary conditions are Reynolds type, viz. Vp=o, p=o at the outlet, for both pressure components P (pure wedging action) and (central pure squegzing action), a simple %perposition is no longer allowed because this boundary condition is not linear. The Reynolds equation is linear in p, however, and therefore methods like the mobility method or the impulsewhirl angle method are correct from a mathematical viewpoint: they fulfil the boundary condition at the outlet hecause only one free boundary exists (for the pure squeezing action). Two free boundaries, as arise in the Holland approach, will never coincide, and hence the problem is overspecified. For example, at the outlet, where vp=p=o, it is only valid to state that Vpd = -Vp and V Pd = -pv.- Requiring vpd = pd = 0 and Vpv = pv - o will lead to a smooth pressure profile, but also to results which are in error, expecially at high loads. Therefore, the Holland approach of 1959 will lead to erroneous results. Reply by Dr.-Ing. West Germany).
O.R. Lang (Daimler-Benz AG,
The 1959 solution of Holland used an analytical solution for pure wedge and pure squeeze for an infinitely long bearing, approximated to finite length by the assumption of a parabolic pressure distribution. The Holland-Lang solution uses numerical solutions for pure wedge and pure squeeze for finite length with Reynolds' boundary conditions for both independently with high accuracy by a refined mesh, especially in the region of high pressures and the outlet. This is p-esented in the doctoral thesis of Butenschon, University of Karlsruhe 1976. The formulation of Reynolds' boundary conditions in terms of pressures p = o and Ap = o is not the primary physical one, this is continuity within the total pressure development. Circumferential flow and the two different types of velocity profiles develop independently as long as the gap is filled. Dynamically loaded bearings are overfeed and the oil inlet is chosen according to the state of the art. The point is not a pure mathematical correct solution, but a physical one, concerned with pressure development, which is given in Holland-Lang as well as in the mobility method or impulse-whirl angle method. An alternative physical theorem is that of minimum potential energy, which Swift used to assure Reynolds' boundary conditions for wedge action. Minimum potential energy means highest load capacity and minimum friction. Theoretical superposition of combined wedge and squeeze with Reynolds' boundary conditions yields significantly smaller pressures, lower load capacity and higher friction than the Holland-
Lang superposition. The experiments at Karlsruhe University confirm this in displacement and pressure extent. Moreover, nearly twenty years of practical experience in the field of dynamic loading gives much more security than a one-sided mathematical viewpoint, which is not the unique one. There is no mathematical rule against superimposing independent solutions of a linear differential equation, as long as the physical boundary conditions are fulfilled. This is common in vibration, thermodynamics or stress-strain calculations. SESSION XIX BEARINGS
-
MACHINE ELEMENTS (1)
-
OIL RING
'Performance Characteristics of the Oil Ring Lubricator. An Experimental Study' K.R. BROCKWELL and D. KLEINBUB Emeritus Professor H. Blok (The Netherlands) The authors' Figure 10 shows their success in deriving a fairly strict non-dimensionalized correlation in terms of only two dimensionless groups, 1.e. the two Reynolds numbers, Rering and Re However, judging from other figureih@l%; have not been non-dimensionalized so far, the use of merely two such dimensionless groups cannot very well yield an equally small correlational scatter. The reason may well be sought in the fact that in ring-oiling at least a few more physically influential quantities have in general to be accounted for than the five represented in the two Reynolds numbers. These five quantities are: the circumferential speeds of the ring and the shaft, V and v , the viscosity and density of the oil, 6 and P: and some "scaling" length, L , for which the discusser suggests the d%meter of the shaft. So, as to the dynamics of the ring the authors have considered only the viscous and the inertial forces. However, in ring-oiling three more kinds of forces will have to be accounted for, that is, the gravitational and the surface-tensional ones as they occur in the various oil flows, and the load imposed by the weight of the ring on its entraining oil film. These forces can be represented by introducing three more physically influential quantities, that is, the gravitational acceleration, g, the surface tension, 0 , and the unit load, W, as referred to the unit width of the rubbing surface.
Now, the eight physically influential qualtities thus introduced in total, will yield a complete set of three more, that is five True, the dimensionless groups, N through N authors' two Reynolds numbers 1 coula be included therein but the discusser prefers the following set from which, after all, their two Reynolds numbers can be reconstructed, if so desired.
.
N1=(P/ug)'I4.Vr;
N2=(P/0g)1'4.Vs;
N.,=(g/u
3 P ) 1/$. (1 a,b,c)
(ld,e>
671
Accordingly, the non-dimensionalized relationship sought can be cast into the form,
where the function F is to be determined by the present, "separational" technique of correlation. Further, the symbol Sh stands collectively for all the shape factors, including the ratio L /L between the inner diameter of the ring 5nd"Ehe "scaling" length, and the relative depth of immersion which here is also referred to Lsc. Some major advantages accrue from the "separational" techniques involved. First, three of the eight physically influential quantities, that is a, P, and g vary so slightly with respect to the others that they may be considered to remain constant, or very nearly so, both in experimentation and in actual ringoiling practice. Second, and consequentially, we are left with only five quantities that in general will vary appreciably. These are the quantities indicated by the symbols, V , etc., that appear separated from each other fn that each has been assigned its own dimensionless group in which no other such variable quantity appears. Moreover, in each such group of the set N1 through N5 the variable quantity thus separated is multiplied by some power product merely of constant quantities, that is, P , g and u . Last but not least, each such power product may be conveived as the reciprocal of a constant "referential" quantity with which the variable quantity concerned may be compared, or say, measured. It may now readily be seen that the "separational" character of relationship ( 2 ) allows for a very simple kind of nondimensionalization of correlations like the one possible in Figure 10, i.e. those where one single family of curves depicts the relationship sought between Vr, V and 0. In fact, by virtue of the (near-) constincy of the aforementioned "referential" quantities, the qualitative trend of the authors' curves will be preserved when replotting in terms of N 1 , N2 and N3, respectively. However, as a warning to users of such correlations one should specify the constant numerical values of the two remaining groups, the "loading" number N4 and the Moreover, the geometry of "scaling" number N,. the elements releva.nt to the ring-oiling process, as represented in the general relationship ( 2 ) by the collective symbol Sh, should preferably be specified by a sufficiently detailed drawing in which the depth of immersion is also to be indicated. A similar "separational" technique may well prove successful also in condensing the work of other investigators in ring-oiling, such as the little known but extensive work in the doctorate thesis, "Oelmengenmessungen in Ringschmierlagern", submitted by K. Mueller to the Technische Hochschule of Berlin, Germany, in 1931 (re-issued in the Versuchsergebnisse der T.H. Berlin, lO.Heft, Munich and Berlin, 1931). In so far as, such as in Mueller's work, rates of delivery of the oil, Q, are to be correlated, one may conveniently employ the present, "separational" technique again. That is, instead of the above-defined dimensionless
ring speed V , the group N1 rela in group N = (Pf gs/~s)Fpe.Q may be intfoduced. The lat8er group may be conceived as just another function of the remaining four physically significant groups N2 through N5, and, of course, of the relevant geometry as it is expressed by the collective symbol Sh. Mr. H. Heshmat (Mechanical Technology Inc., New York, U.S.A.).
I have read with great interest and enjoyment Keith Brockwell and Kleinbub's Experimental Work on Oil Ring bearings. I was pleased to find out that, they were able to verify some of the results that we reported in Ref. 10. There are two matters of fact which might be of relevance to the article. It is still not clear whether test bearing L/D ratio has any effect on the ring operational performance, speed, oil delivery and etc. Furthermore, what is optimum bearing configuration in relation to ring operation? Also of interest is that what is the limitation of oil sump temperature, in particular, cold start-up conditions. 1i.e. T~ < - o Oc). In the light of the above considerations, further valuable work would involve modifying the tester to give flow rates for more relevant design parameters. Reply by Mr. K.R. Brockwell and Mr. D. Kleinbub (National Research Council of Canada, Vancouver, Canada). The authors would like to thank both Professor Blok and Dr. Heshmat for their interesting and valuable comments. Professor Blok correctly points out that the authors have considered only two dimensionless groups; namely the Reynolds Numbers of the ring and shaft. The authors believe that this simple correlation was first proposed by Lemon et a1 (1) in their paper on oil ring performance, when they showed that ring and shaft speeds, ring and shaft diameters and oil viscosity could be represented by the equation:
They went on to show that for different levels of ring immersion, the ring speed varies inversely on an average as the 0.2 power of the ring immersion. Fig. 10 shows that the results from this latest study are in close agreement with the results of Lemon et a1 (1). Professor Blok suggests that the forces acting on the ring can be represented by five dimensionless groups, N through N , and the authors are grateful fok the discuzser's comments concerning the form into which the non-dimensionalized relationship may be cast. It is of interest to note that Needham (2) listed the forces acting on the ring, dividing them into those which assist motion, those which oppose motion and those necessary for ring equilibrium. Needham ( 2 ) developed equations which allowed him to predict the film
672
thickness betweeen the ring and shaft, but he took the analysis no further than this. The authors agree with the comments of Dr. Heshmat. Over the past five decades, much work has been done to establish an optimum ring configuration from the standpoint of ring speed and oil delivery. However, optimum bearing configuration remains unclear and further testing is required to estahlish guidelines for improved bearing designs.
[n21 Rair, S. and Winer, W.O., Surface Temperatures and Glassy State Investigattons in Tribology, NASA Contractor Rpt. 3162, July 1979.
9
10
8
It is clear there is still much work to be done.
c
[l]
Lemmon, D.C. and Rooser, E.R. "Bearing oil ring performance", Trans. ASME, Journal of Basic Engineering, 1960, 82D, 327-334.
7'-
a t
1aS-1
EFFECTIVE VISCOSITY (FROM MODEL)
6
0"
5: 5
A
[2] Needham, F. 'Oil ring operation', M.Sc. Thesis, The University of Leeds, 15 August 1969. CONTRIBUTIONS SESSION VIII
(2 1
2
0 FALLING BODY VISCOMETER
0
0 HIGH STRESS VISCOMETER
- ELASTOHYDRODYNAMIC LUBRICATION
A
STRESS-STRAIN APPARATUS
-2
'Elastohydrodynamic Lubrication of Point Contacts for Various Lubricants' G. DALMAZ and J.D.
80
40 PRESSUREAPSI
CHAOMLEFFEL
Dr. S . Bair and Professor W.O. Winer (Georgia Institute of Technology, Atlanta, U.S.A.). The authors are to be commended for their contribution toward confirmation of the Hamrock and Dowson film thickness equations and for the limiting shear stress rheological model. It should be noted that the viscous term
Figure D1. Viscosity pressure isotherm (60C) for 5P4E by indicated methods o f measurement. Lines of constant shear rate predicted from model (D2). SESSION X
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LURRICANT RHEOLOGY
General Contribution Professor J Jakobsen (Technical IJniversity of Denmark, Lyngby). in the limiting shear stress model that we proposed (5,6) is not anti-symmetric about zero shear stress and should be replaced, Ref [Dl], with 'I
L
-1
WO
r [TL]
so that we get
It is interesting to note the similarity between the authors' Figure 5, a plot of mean apparent viscosity VS. pressure and our Figure D1 of Ref [D2], which is a plot of the same quantities but measured in three different rheometers. References [Dl] Cecim, R. and Winer, W.O., 'Lubricant Limiting Shear Stress Effect on EHD Film Thickness", ASM?? Journal of Lubrication Technology, Vol. 102, April 1 9 8 0 , p. 214.
Non-Newtonian Properties at Extremely High Shear Stress of some Widely Used Lubricating Oils Investigations on liquid lubricant rheological behaviour at high shear stress can he conducted with capillary flow when the ratio of length (L) over diameter (D) is small. L/D = 1 yields a level of shear stress of m a . This approaches the average shear stress level in sliding, moderately loaded elastohydrodynamic (EHD) contacts. ((11, Figures 3 , 4, 5). Presumably, the stresses in the inlet to the contact - creating the EHD film thickness - are generally depicted at this level. Inlet and exit corrections (2), ( 3 ) , ( 4 ) , (5), (6), ( 7 ) as well as shear heating corrections (81, ( 9 ) , ( l o ) , (71, (1) can be applied to the capillary flow at appropriate pressure and temperature levels ( I ) , (111, (12), (13). The figures of ( I ) , (121, ( 1 3 ) do outline the thermal corrections. The well established Rabinowitsch analysis for Non-Newtonian liquids in capillary tubes presented in (14) - should also be considered. The material liquid characteristics
673 applicable to many lubrication conditions of interest can thus - through 1) inlet-exit corrections, 2) thermal corrections, and 3 ) the Rabinowitsch corrections - be derived from measurements as corrected capillary flow curves. Pressure-viscosity properties and flow Re number limitations do set an upper boundary for the useful shear stress range to approximately 2 MPa. Measurements are performed with Re < 1 in order to ensure that validity of end corrections (15),
Measurements of a mineral, base oil (Figure 3 ) show significant shear thickening properties at stress levels above 10 kPa (17). Viscosity did increase with a factor of about 4 , when shear stress was raised to 1 MPa. The spread of the measured points from a continuous flow curve Viscosity Pa s
(16).
Investigations of a synthetic test oil, Figure 1, in the high pressure, extreme-high shear capillary viscometer (12) show Newtonian behaviour at varying pressure levels.
Pa. 10
I
1 104 LID:
a s pm:
102 MPO
-
1
a1
10
,v
'
10 5
106
Shear stress
MPn
T
Pa
Figure 3 A mineral, base oil. Significant shear thickening effect is found above 10 kPa shear stress. The spread of the measuring points may indicate a minor change from liquid state. LID = 1, 25 C , Pressure: z .1 MPa. (17).
Figure 1 Flow curves from a synthetic test oil in the high pressure, extreme-high shear capillary viscometer. The figure shows the thermal correction at some pressure levels and L/D ratios for the test oil, 25 C , Pm = pressure. (12). Capillaries of different T,/D ratios are employed. Figure 2 and Figure 1 of ( 1 3 ) give for a synthetic lubricant of the same type and basic viscosity - Newtonian behaviour at about 0.1 m a . Deviation from constant, shear independent viscosity is caused by dissipation in the capillary ( 7 ) and yields the experimental thermal correction for the test o i l .
indicates a departure from a pure liquid state. A blend of the base oil with a VI-improver, (Figure 4 ) (11.5% W) - modelling widely used,
visco-static oils - retains the shear thickening characteristics at extremely high stress, however with a reduced shear stress sensitivity. Viscosity II Pa s
1
Viscosity II Pa s
I t
.1
tI 104
105
Shear stress
.1
I 104
105
Shear stress
106 T
106 T
pa
Figure 4 A blend of the mineral oil of Figure 3 with a VI-improver (11.5% W). The blend is assumed a model Oil of the visco static type oil widely used in combustion engines. L/D = 1, 25 C, Pressure: .1 m a . (17).
Pa
Figure 2 Calibration flow curve (17) with a synthetic lubricant of same type and viscosity as the test o i l of Figure 1. LID = 1, 25 C, Pressure: = . I MPa.
The VI-improved blend appears also to decline from a pure liquid state. A traction oil, Figure 5, has shear thickening properties and also too shows sign of changing from liquid state at the highest stresses.
674
At shear stresses less than 4 kPa, Figure 6 , two dimethyl silicone o i l s show Newtonian properties. Non-Newtonian, strong shear thinning behaviour is found in the range 4 kPa. Deviation from Newtonian characteristic increases with the degree of polymerization (viscosity level). viscosity Pa s
ll
at the surface is a primary film determining factor in the physical function of the viscosity pump effect in the inlet to EHD bearing areas. The real material flow curve characteristic (viscosity as a function of pressure, temperature, and shear stress) should be applied to analytic, numerical approaches for determination of film thickness and separation in EHD contacts as well as in thick film bearings. References Jakobsen, J. and Winer, W.O., 'High Shear Stress Behaviour of Some Representative Lubrications', ASME Journal of Lubrication Technology, Vol. 9 7 , July 1975, pp
.1
479-485.
Sampson, R.A. 'On the Stoke's Current Function', Phil. Trans. Roy. SOC., A 1 8 2 , 1891, pp 449-518. Bagley, E.B., 'End Corrections in the Capillary Flow of Polyethylene', Journal of Applied Physics, 2 8 , No. 5 , 1957, pp
.01
105
104
106
Shear stress
'I
Pa
Figure 5 A traction oil. The flow curve shows significant shear thickening. LID = 1 , 25C, Pressure: a .1 MPa ( 1 7 ) .
624-627.
Weisberg, L.H., 'End Corrections for Slow Viscous Flow through Long Tubes', The Physics of Fluids, Vol. 5 , No. 9 , Sept. 1962, pp 1033-1036. Ashino, I. 'Entry Flow in a Circular Pipe', Trans. Japan SOC. Eng., No. 302, 1969.
Viscosity TI Pa s
Happel, J. and Brenner, H., 'Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media', Prentice-Hall, Englewood Cliffs, N.J., 1965.
Jakobsen, J. and Winer, W.O., Dissipative Heating Effects and End Corrections for Viscous Newtonian Flow in High Shear Capillary Tube Viscometry, ASME Journal of Lubrication Technology, Vol. 97, July 1975, pp 472-478. Gerrard, J.E. and Philoppoff, W. 'Viscous Heating and Capillary Flow', 4th International Congress of Rheology, Paper 103
104
Shear stress
105 T
106
Pa
Figure 6 Two dimethyl silicone oils. The flow curves show Newtonian behaviour for stress below E 4 kPa. Strong shear thinning effects are seen at stresses above 4 kPA. Heating effects can be expected to be significant only above stresses of 100 kPa. Both materials remain liquid. Deviation from Newtonian properties increases with degree of polymerization. LID = 54.2 and 5 . 3 3 , 37.78 C, Pressure = . I m a . ( 1 1 ) . The flow curves are presented with the independently variable shear stress along the abscissa axis for ease of application, as stress
51, 1963.
Gerrard, J.E., Steidler, F.E., and Appeldorn, J.K. 'Viscous Heating in Capillaries: The Adiabatic Case, ACS Petroleum Division Meeting, Chicago, Ill, Sept. 1964. Gerrard, J.E., Steidler, F.E. and Appeldorn, J.K., 'Viscous Heating in Capillaries: The Isothermal-Wall Case', ACS Petroleum Division Meeting Atlantic City, N.J., Sept. 1965. Brinkman, H.C. 'Heat Effects in Capillary Flow 1', Applied Science Research A2, 1951, pp 120-124. Andersen, H., Silicone Oliers Reologi ved Extreme Forskydningsspaendinger, (Rheology
675
of Silicone Oils at High Shear Stress) Report, Department of Machine Elements, The Technical University of Denmark, 1975. Larsen, P.C., Konstruktion og Afprdvning of Hdjtryksciskometer ti1 Maling ved Hbje Forskydningsspaendinger,M.Sc. Thesis, Department of Machine Elements, The Technical University of Denmark, 1977. Jakobsen, J., Hansen, P.K., Larsen, P.C. and Peitersen, J. 'Elastohydrodynamic Film Thickness Measurements and High Shear Viscometric Investigations of Lubricants'. Written Discussion, Proceedings, Performance and Testing of Gear Oils and Transmission Fluids, The Institute of Petroleum, London, 21-23 October 1980, pp 206-2 10. Van Wazer, J.R., Lyons, J.W., Kim, K.Y. and Colwell, R.E. 'Viscosity and Flow Measurements', 1963. Johansen, F.C., 'Flow Through Pipe Orifices at Low Reynolds Numbers', Proc. Roy. SOC. A 126, 1930, pp 231-245. Bond, W.N., 'The Effect of Viscosity on Orifice Flows', Proceedings of the Physicists Society, 33, 1921, pp 225-230. Bond, W.N., 'Viscosity Determination by Means of Orifices and Short Tubes', Proc. Phys. SOC. 34, 1922, pp 139-144. Petersen, L., 'Optisk Bestemte Trykfordelinger i Elasthydrodynamiske Punktkontakter', M.Sc. Thesis, Department of Machine Elements, The Technical University of Denmark, 1982. SESSION XX1 BEARINGS
-
MACHINE ELEMENTS (3)
-
ROLLING
'Study of the Lubricant Film in Rolling Bearings; Effects of Roughness', P. LEENDERS and L.G. HOUPERT Dr G.T. Nielsen (Technical University of Denmark, Lyngby) Surface Fils Generated in Rolling/Sliding EHD Contacts INTRODUCTION Investigations at the Technical University of Denmark have elucidated that the surfaces of ball bearings and roller bearings before and after use do contain surface layers of various compositions. These compositions are completely different from the composition of the bulk of the bearing material (AISI 52100). The test specimens used in the experiments were ordinary standard bearing components. An Auger Electron Spectroscopy ( A E S ) analysis of the test specimens before use in the experiments, showed that the composition of the specimen surfaces were quite different from one supplier to the next. After the running-in experiments AES analysis showed that the composition of the test specimens could be related to running-in
parameters. The thickness of surface films found on test specimens were in the order of 10 to 100 nanometres. The surface films found on unused specimens contained sulphur, carbon, oxygen, chlorine and iron. During running-in the lubricant used was a sulphur containing paraffinic oil, and the surface films found on the test specimens after the running-in tests contained sulphur, carbon, oxygen and iron. Literature The compositions of surface films on steel have been studied by several authors. A representative list is given here (1-12). From the previous works in this field it is evident that a correlation exists between the composition of steel surfaces, EP-additives and the running-in parameters. Levine and Peterson (1) have investigated the generating process of sulphur containing films on steel surfaces, and the effect of these on the coefficient o f friction. The films were created by submerging heated steel specimens into a mixture of cetane and elemental sulphur. Levine and Peterson found that a layer of iron sulphide was formed on the surface, and the friction could be related to the thickness of the films. Bisson et a1 (2) have investigated the influence on wear and scoring of sulphur and chlorine containing surface films. They found that the films were beneficial for less wear, and that the coefficient of friction employing chlorine was less than using sulphur. Godfrey has analysed surface films generated with commercial EP-additives. He found that sulphurized mineral oil promotes oxidation of steel under EP-conditions, whereas sulphur was a minor constituent but necessary for high load carrying capacity. Buckley (3) has used AES in studying the influence of chlorine containing hydrocarbons on the surfaces of steel in pure sliding. He concludes that the sliding friction will increase the amount of chlorine in the surface, and states that the mechanical parameters do effect the surface concentration of chlorine and the friction. Also, he states that there is an optimum load for surface coverage and minimum friction. Baldwin ( 4 ) has studied steel surfaces containing sulphur and phosphor with X-Ray Photoelectron Spectroscopy and found that a chemical reaction between sulphur and iron took place at the surface during a pin on disc test. Further, he concludes that metal sulphides are beneficial to the boundary lubrication process. Bird and Galvin (5) have examined films formed on steel in immersion and rubbing tests. The additives used were elemental sulphur and some commercial sulphur/phosphor containing additives. The method of analysing used was ESCA. In the rubbing tests they found that the surfaces were covered with a larger amount of metal sulphide than in the immersion tests. They also found iron sulphide on specimens ruhbed with commercial additives, but no iron phosphate. Phillips et a1 (6) have studied the influence of zinc dibutyldithio-phosphate (ZDP) on the composition of sliding steel surfaces. AES combined with argonion sputtering was used to get the depth profiles of iron, carbon, oxygen, sulphur and iron of the worn surfaces. Approximately 10 times as much sulphur was found in the surfaces using ZDP compared to cases where no ZDP was used. Furthermore, no
676
simple relation between carbonloxygen and load seemed to appear. McCaroll et a1 ( 7 ) found similar relations using AES, and Debies and Johnston (8) - through SAES - show that smooth areas of sliding steel surfaces contain more sulphur than areas of the same surface where more heavily scoring was traceable. Schumacher et al. ( 9 ) have investigated the film thickness as a function of the concentration of additive, and they found an optimum film thickness of load carrying capacity in a Reichert test apparatus. Previous works in this field at the Technical University of Denmark (10-11) have confirmed that steel surfaces are related to other factors than the bulk of the material; namely load, the presence of a lubricant, additives, rolling/sliding velocity, asperities, coefficient of friction.
In order to create a standard surface for the experiments, an investigation of different grindinglfinishing processes were carried out. This led to the result that only a most radical grinding process was capable of removing chlorine and sulphur satisfactorily from the surface. The process employs grains of alumina (0.05 micrometre) spread on clear paper as grinding/finishing media. Furthermore, it was found that the grinding edges and the grinding
Rel I
Test Specimens The test specimens used were ordinary ball/ rolling bearing elements. The main part of the test rig comprised an outer ring of a spherical bearing (97,6 mm diameter) and a steel ball (82 mm diameter) rolling/sliding against the ring. Figure la and lb show the depth profiles of sulphur, carbon, oxygen, chlorine and iron as they appeared on the specimens received (originals) from two different manufacturers. From the curves (Fig la and lb) it is evident that the process used by the manufacturers created surface films on the steel materials through the grinding and finishing of the elements.
1
I
Lb MIN
Rel.
I
0.5
Figure la AES depth profiles in the surface of test specimens ( A I S I 52100 steel) shipment I. The units on the abscissa is sputtertime in The units on minutes (1 min. is approx 0.4 nm). the ordinate is the relative Auger intensity. Values for sulphur and chlorine are multiplied by 5.
Figure lb AES depth profiles in the surface of test specimens (AISI 52100 steel) shipment 11. Units as figure la. Values for sulphur and chlorine as in la. grains should not be used more than once in order to carry away properly the ground steel particles from the steel surface. Figure 2 shows, the depth profiles of S, C1, C, 0 and Fe. It can be seen from the depth profiles that only a minor amount of S and C1 remains in the surface film. Instead, as expected, a large amount of oxygen is present in the surface. The surface roughness was after the grinding in the order of 0.1 (Ra). The first running-in test was performed with a medically clean paraffinic oil without dissolved elemental sulphur. This was done in order to prove whether any sulphur from the bulk of the material would diffuse to the surface during a test run with full EHD conditions. The result was in accordance with baking tests in the UHV chamber, showing no surface segregation occurring below approx. 250 deg. C . , and thus in accordance with Taylor (12) who arrived at similar findings. Figure 3 , shows the depth profiles of the elements in the surface films after the running-in period. The load was 1450 Newtons, the rolling velocity 17 m/s, the running-in time 180 minutes and the tractional force on the area of contact 2,5 Newtons. The depth profiles are very similar to the profiles from the standard surface
677
Figure ( 2 ) , and, therefore, it can be concluded that no sulphur did segregate from the bulk of the material towards the surface. Ra values were in the order of 0.1
CI
-S I
Figure 2
L'O MIN
THE STANDARD SURFACE
Figure 3 AES depth profiles in surfaces which have been run-in under fully lubricated EHD conditions with sulphur containing o i l .
Rel. I
AES depth profiles of the test specimens after
the grinding with the alumina grains. The values for sulphur and chlorine is multiplied by 5.
The 4 0 min. on the abscissa is approx. equal to 16 nanometer or to 40 atomic layers. Running-in with sulphur containing oil Fully lubricated EHD conditions The lubricant used in the film generating process was made from medical clean paraffinic oil with elemental sulphur dissolved at 100 deg. C. The oil was thereafter cooled to room temperature and decanted. During running-in full EHD conditions were established. The minimum film thickness calculated according to the Dowson and Hamrock formula was always in excess of twenty to thirty times the sum of the maximum asperity heights. T e number of contact 3! events were varied from 3.10 to 5.106 and the tractional force on the contact between 2.5 and 7,9 Newtons. The normal force was varied from 550 Newtons to 1 4 5 0 Newtons. Figure 4 and 5 give a representative picture of the surface €ilms generated on the steel specimens under the conditions mentioned. The depth profiles clearly indicate that the surfaces of the steel specimens change composition during fully lubricated EHD conditions. SEM micrographs did not indicate any wear or change in the surface topography, and further running-in did not seem to change the depth profiles.
L'O MIN
AES depth profiles of sulphur ( x 5 ) Figure 4 and iron in the surfaces of two test specimens. The running-in conditions being: Normal load 1 4 5 0 Newtons, rolling velocity 17 m/s and running-in time 7 5 minutes. The numbers in brackets ( 2 , 5 and 6 , l ) refer to the tractional force in Newtons acting on the contact during the running-in.
678
Running-in with sulphur containing oil Asperity contact conditions Laboratory tests were also conducted under starved EHD conditions in order to specify whether an asperity contact situation during operation would influence the surface films. In these experiments the steel balls and the rings were only wetted with so minor quantities of sulphur containing oil that asperity contacts were enabled. The slip velocity was kept constant at three different values (7, 15 and 35
Rel.
I
-
conditions had a thinner sulphur containing film compared to the surfaces run-in under fully lubricated EHD conditions, and the amount of sulphur in the film seemed to be more well defined. The depth profiles of carbon and oxygen from surfaces where a sulphur containing lubricant was used during runningin have been omitted in Figure 4, 5 and 6 . This has been done in order to simplify depth profiles of sulphur and iron since the depth profiles of carbon and oxygen did not seem to be related to the running-in parameters in a simple manner. Chlorine was only found in insignificant amounts in the first few atomic layers in these surfaces.
Rel.
I
0.5
Figure 5 AES depth profiles of sulphur (x5) and iron in the surface of a test specimen run in under the same conditions as the two shown in Fig. 4. The tractional force is 7.9 Newton. percent of the rolling velocity). Higher values of the slip than 20 percent of the rolling velocity lead to initial scoring of the test surfaces at the start of the running-in period. During the running-in the normal load was 1450 Newtons and the rolling velocity was as before 17 m/s. The number of contact events was of the order of 5 ~ 1 0 ~ Figure . 6 shows the depth profiles of sulphur in the surface of the specimens after the running-in tests. The layers containing sulphur appear here to be thinner than in the tests with fully lubricated EHD conditions, and SEM micrographs showed that the tops of the asperities were slightly deformed o r worn. To the naked eye the steel surfaces appeared unchanged. Further running-in did not seem to alter the depth profiles although there is evidence, that a prolonged running-in period (11) will result in a depth profile showing no iron in the uppermost layers of the surface. Such a profile might be found on a scoring resistant surface after an extremely long period of operation with combined rolling and sliding. The surfaces
-
run-in under starved EHD
I
Lb MIN
AES depth profiles of sulphur (x5) Figure 6 and iron in the surface of a test specimen run in under starved EHD conditions. The normal load 1450 Newtons, rolling velocity 17 m/s and running-in 90 minutes. The slip was kept constant at 7% of the rolling velocity Conclusion Deriving from the shape of the depth profiles it is evident that steel surfaces in a rolling andlor sliding situations do create surface layers with an elemental composition differing from that of the bulk of the material. The surface layers will not necessarily be detectable with a Scanning Electron Microscope and certainly not have any visual changes to the naked eye. References [l]
Levine, E.C. and Peterson, M.B.: 'Formation of Sulphide Films on Steel and Effect of such Films on Static Friction'. NASA Technical Note 2460.
.
679
Bisson, E.E., Johnson, R.L. and Swickert, M.A.: 'Friction, Wear and Surface Damage of Metals as affected by Solid Surface Films'. Research Review, Paper 31, 1957. Godfrey, D.: 'Chemical Changes in Steel Surfaces during Extreme Pressure Lubrication'. ASLE Trans. Vol. 5, 1962. Buckley, D.H.: 'Friction Induced Surface Activity of some Simple Organic Chlorides and Hydrocarbonds with Iron'. ASLE Trans. Vol. 17, 1973. Baldwin, B.A.: 'Chemical Characterization of Wear Surfaces Using X-Ray Photelectron Spectroscopy. Lub. Eng. Vol. 32, No. 3, 1975. Bird, R.J. and Galvin, G.D.: 'The Application of Photoelectron Spectroscopy to the Study of E.P. Films on Lubricated Surfaces', Wear, Vol. 37, p. 143, 1976. Phillips, M.R. et al: 'The Application of Auger Electron Spectroscopy to Tribology'. Vacuum Vol. 26, No. lO/ll, 1977. McCarroll, J.J. et al: 'Auger Electron Spectroscopy of Wear Surfaces'. Nature, Vol. 266, 1977. Debies, T.P., Johnston, W.G.: 'Surface Chemistry of some Antiwear Additives as Determined by Electron Spectroscopy'. ASLE Trans. Vol. 23, No. 3. 1979. [lo] Schumacher, R. et al: 'Auger Electron Spectroscopy Study on Reaction Layers formed under Reichert Wear Test Conditions in The Presence of Extreme Pressure Additives'. Tribology Int. Dec. 1980, p. 331. [ll] Nielsen, G.T.: 'Dry Rolling Contact Friction of Physically, Chemically Modified Surfaces'. M.Sc. Thesis. Department of Machine Element, Technical University of Denmark (in Danish) 1980.
'Friction of Highly Loaded [12] Kvistgaard, E.: Contact Areas in Dry Rolling with Traction and Sliding'. M.Sc. Thesis. Department of Machine Element (in Danish) 1982. [13] Taylor, N.J.: 'Auger Electron Spectrometer as a Tool for Surface Analysis (Contamination Monitor). Jour. of Vac. Sc. and Tech. Vol. 6, No. 1, 1969. [14] Hamrock, B.J. and Dowson, D.: 'Isothermal Elastohydrodynamic Lubrication of Point Contacts Part I - IV', Jour. of Lub. Tech., et ASME Paper No. 76-Lub-30, 1976. [15] Nielsen, G.T.: 'Surface Films with Extreme Load Carying Capacity on Hard Steels', Ph.D. Thesis. Technical University of Denmark (in Danish) 1984.
General Contribution Dr. J.A. Dominy (Rolls-Royce Ltd., Derby, U.K.). The discussion on this session has raised the
question of the different approaches to research - particularly theoretical research betweeen industry and the academic world. At Rolls-Royce Mr. Nicholson and I see ourselves in the role of an interface between the original work being carried oiit and published in the literature on the one hand and the requirements of the deigner at his drawing board on the other. The work that we do is often, by nature, a reduction of the original research into a technique which is quick and econmical in computer terms, is user friendly in the widest sense and only requires data that the designer has available. The paper that Mr. Nicholson has presented in this session is typical of our approach. Traction is, to my mind, a classical example where the academic institutions have done excellent work in developing understanding and models of the echanisms but where the models that are available require input data that is not readily available to the designer.
It is unusual for an industry or company to be in a position to justify personnel in such an "interface" position and they must rely on the output of the research estahlishments directly or commercial institutions such as ESDU International Ltd. I do not suggest that the "long-haired" research is unnecessary since the gleam in the academic's eye today will become the design method of 5, 10 or even 15 years time. EHL itself is an example of this. However, it does seem to me that, generally, there is a hole in the literature. There is a need for research institutes to present their own work in the way that, say, the ESDU International Ltd., already do if it is to be of more direct use to the designer and if they are to prevent the feeling of irrelevance that the industrial designer often has towards the literature available to him.
This Page Intentionally Left Blank
68 1
13th LEEDS-LYON SYWOSIUn ON TRIBOLOGY FLUID F I W LUBRICATION
- OSBORNE REYNOLDS CENTENARY
8th-12th SEPTEMBER 1986 LIST OF AUTRORS TITLE RlllIB --
AFPILIATION/ADDRESS
DR.
G.M. Adled*)
The University of Edinburgh, Department of Mechanical Engineering, The King's Buildings, Edinburgh, EH9 3JL, U.K.
MR.
S.D. Advani
Michell Bearings plc Scotwood Road, Newcastle upon Tyne. NE15 6LL. U.K.
DR.
A. Artiles
Mechanical Technology Inc. 968 Albany-Shaker Road, Latham, N.Y. 12110, U.S.A.
PROF. F.T. Banvell
26, Woodland Way, Brighton, BN1 8BA.
DR.
University of Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteur Pineau, 86022 Poitiers, Cedex, France.
A. Benali(*)
PROF. D. Berthe
DR.
Y. Berthier
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France. Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batirnent 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
DR.
P. Bezot
Universite de Nice, Laboratoire Physique Matiere Condensee, (UA 190), Pare Valrose, 06034, Nice, France.
DR.
A.G.
Blahey
Esso Petroleum Canada, P.O. BOX 3022, Sarnia, Ontario, Canada N7'T 7MI
DR.
D.A. Boffey
The University of Edinburgh, Department of Mechanical Engineering, The King's Buildings, Edinburgh, EH9 3JLy U.K.
TITLE NAME --
AFFILIATION ADDRESS
MR.
R. Boncompain(*)
Institut National des Sciences Appliques , Laboratoire de Mecanique des Contacts, Batiment 113, 20 Avenue Albert Einstein, 69621 Villeurhanne, Lyon, France.
DR.
A. Bonifacie(*)
Enset, Dakar, Senegal.
PROF. R. Bosma,
Twente University of Technology, Department of Mechanical Engineering, (Tribology) , Postbox 217, Enschede, The Netherlands.
DR.
R. Boudet(*)
Lahoratoire de Genie Mecanique, Toulouse, France.
Ir.
F. Bremer
Hederlandse Philips Bedrijven B.V. CFTISAQ 2100, 5600 MD Eindhoven, The Netherlands.
DR.
G.A.C.
Twente University of Technology, Department of Mechanical Engineering, (Tribology) , Postbox 217, Enschede, The Netherlands.
MR.
D.E. Brewe
National Aeronautics and Space Administration, Lewis Research Center, Cleveland, Ohio 44135, U.S.A.
MR.
K.R. Brockwell
National Research Council of Canada, 3650 Wesbrook Mall, Vancouver, British Columbia, V6S 2L2, Canada.
Breukink(*)
PROF. R.H. Buckholz,
Columbia University, Department of Mechanical Engineering, New York 10027, U.S.A.
PROF. A. Cameron
Cameron-Plint Ltd., 39 Maids Causeway, Cambridge. CB5 8DE. U.K.
682
TITLE NAME -DR.
DR.
AFFILIATION/ADDRESS
J.P. Chaomleffel(*)
R.J. Chittenden
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20 Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France. The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng Leeds LS2 9JT. U.K.
.
DR.
J.W. Coburn(*)
PROF. G. Dalmaz
I.B.M. Almaden Research Center, Department K33/802, 650 Harry Road, San Jose, CA 95120, U.S.A. Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Vil leurbanne, Lyon, France.
AFFILIATION/ADDRESS
TITLE NAUE --
IR.
J.G. Fijnvandraat Philips Research Laboratory, Prof. Holstln. 2, 5600 JA, Eindhoven, The Netherlands.
DR.
M. Fillon(*)
University of Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteur Pineau, 86022 Poitiers, Cedex, France.
MR.
R. Firoozian(*)
The University of Liverpool, Department of Mechanical Engineering, P.O. BOX 147, Liverpool. L64 3BX. Merseyside, U.K.
DR.
C.G. Floyd
Michael Neale 6 Associations, 43 Downing Street, Farnham, Surrey, GU9 7PH. U.K.
PROF. J. Frene
University of Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteut Pineau, 86022 Poitiers, Cedex, France. Kansai University, Faculty of Engineering, 3-35 Yamate-cho 3 chome, Suita-shi, Osaka 564, Japan.
PROF. D. Dowson
The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K.
PROF. T. Fujii
PROF. J.L. Duda
Pennsylvania State Univ. Department of Chemical Engineering, 160 Fenske Laboratory, University Park, PA 16802, U.S.A.
MR.
W.W. Gardner
Waukesha Bearings Corporation, Box 798, Waukesha, WI 53187, U.S.A.
DR.
D.T. Gethin
University College Swansea, Singleton Park, Swansea, SA2 8PP
MR.
C. Giannikos(*)
Columbia University, Department of Mechanical Engineering, New York 10027, U.S.A.
PROF. H.G. Elrod,
DR.
C.M.McC
Ettles
14 Cromwell Court, Old Saybrook, CT 06475, U.S.A. Rensselaer Polytechnic Institute, Mechanical Engineering Department, Troy, N.Y. 12181, U.S.A.
DR.
H.P. Evans(*)
University College, Department of Mechanical Engineering, Newport Road, Cardiff. CF2 1TA. U.K.
DR.
P.D. Ewing(*)
Imperial College of Science and Technology, Department of Mechanical Engineering, Tribology Section, Exhibition Road, London. SW7 2BX. U.K.
PROF M. Godet
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
DR.
P.T. Gorski(*)
Reliant Electric Co., Greenville, U.S.A.
MR.
D.W.F.
YARD Ltd., Charing Cross Tower, Glasgow, G2 4PP. U.K.
MR.
R. Hall,
Goslin,
The University o f Leeds, Department of Mathematics, Leeds LS2 9JT, U.K.
683 TITLE NAllE --
AFFILIATION/ADDRESS
PROF. B.J. Hamrock,
DR.
D.J. Hargreaves(*)
DR.
H. Hashimoto
PROF. K. Hayashi,
Dr.
MR.
DR.
DR.
DR.
DR.
G.J.J.
C. Hesse-Bezot(*)
K. Hirasata(*)
P.L. Holster(*)
C.J. Hooke,
University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, Hongo, Runkyo-ku, Tokyo 113, Japail.
The Ohio State University, Department of Mechanical Engineering, 206 West 18th Avenue, Columbus, Ohio 43210-1107, U.S.A.
PROF. Y. Hori,
Queensland Institute of Technology, Department of Mechanical Engineering, George Street, GPO Box 2434, Brisbane, Queensl.and, Australia 4001. Tokai University, Department of Mechanical & Production Engrg. Faculty of Engineering, 1117 Kitakaname, Hiratsuka-shi, Kanagawaken, Zip Code 259-12, Japan.
DR.
L.G. Houpert
SKF Engineering 6 Research Centre, B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands.
DR.
T.Y. Hua(*)
DR.
T. Huang(*)
Shanghai university of Technology, Research Institute of Bearings, 149 Yanchang Road, Shanghai, China. Tsinchua University, Beijing, China.
DR.
I. Hussla(*)
I.B.M. Almaden Research Center, Department K33/802, 650 Harry Road, San Jose, CA 95120, U.S.A.
DR.
K. Ikeuchi
Kyoto University, Department of Mechanical Engineering, Faculty of Engineering, Kyoto University, Kyoto 606, Japan
Osaka Industrial University, 83 Inari-Onmaecho. Fukakusa, Fushimi-ku, Kyoto, Japan.
van Heijningen, TH Delft, Department of Mechanical Engineering, Laboratory for Machine Elements & Tribology, P.O. Box 5039, 2600 GA Delft, The Netherlands.
H. Heshmat,
TITLE NAME --
PROF. D.B. Ingham(*)
Mechanical Technology Inc., 968 Albany Shaker Road, Latham, New York 12110, U.S.A.
MR
Universite de Nice, Laboratoire Physique Matiere Condensee, (UA 190), Parc Valrose, 06034, Nice, France.
.
The University of Leeds, Department of Applied Mathematical Studies, Leeds LS2 9JT, U.K.
R.W. Jakeman
Lloyd's Register of Shipping, 71 Fenchurch Street, London EC3M 4BS. U.K.
MR.
Z.M. Jin,
Osaka Industrial University, 83 Inari-Onmaecho, Fukakusa, Fushimi-ku, Kyoto, Japan.
The University of Leeds, Institute of Tribology, Dept. of Mech. Eng. Leeds. LS2 9JT. U.K.
DR.
Nederlandse Philips Bedrijven B.V. CFT/SAQ 2100, 5600 MD Eindhoven, The Netherlands.
Hans L. Johannesson, Chalmers University of Technology, Division of Machine Elements, S-412, 96 Goteborg, Sweden.
MR.
J.H. Johnson
The University of Birmingham, Department of Mechanical Engrg. South West Campus, P.O. Box 363, Birmingham. B15 2TT. U.K.
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN, U.K.
DR.
G.J. Johnston(*)
Imperial College of Science and Technology, Department of Mechanical Engineering, Tribology Section, Exhibition Road, London SW7 2BX, U.K.
684
TITLE HAlIE --
AFPILIATION/ADDlUZSS
DR.
C.G.M.
MRS.
Elisabet Kassfeldt Lulea Technical University, S-95187, Lulea, Sweden.
DR.
G.
Karami,
PROF. E.E.
DR.
MR.
DR.
MR.
DR.
DR.
Klaus(*)
K. Kleinbub(*)
Dr.-Ing.
DR.
Kassels(*)
O.R. Lang
F.L. Lee(*)
R.T. Lee(*)
P. Leenders(*)
H.J. van Leeuwen
A. Leeuwestein
W. Li(*)
TH Delft, Department of Mechanical Engineering, Laboratory for Machine Elements & Tribology, P.O. Box 5039, 2600 GA Delft, The Netherlands.
Shiraz University, Department of Mechanical Engineering, School of Engineering, Shiraz, Iran. Pennsylvania State University, Department of Chemical Engineering, 160 Fenske Laboratory, University Park, PA 16802, U.S.A.
TITLE NAME --
AFFILIATION/ADDRESS
DR.
P.J. Lidgitt,
Ministry of Defence (PE), DGME, ME 233, Block B, Room 90, Foxhill, Bath, Avon. BA1 5AB. U.K.
DR.
S.C. Lid*)
Pennsylvania State University, Department of Chemical Engineering, 260 Fenske Laboratory, University Park, PA 16802, U.S.A.
MR.
A.A. Lubrecht,
Twente University of Technology, Postbus 217, 7500 AE Enschede, The Netherlands.
MR.
Q. Mao(*)
Reijing Institute of Technology, China.
PROF. H. Marsh,
The University of Durham, Department of Engineering, South Laboratories, South Road, Durham DH1 3LE.
National Research Council of Canada, 3650 Wesbrook Mall, Vancouver, British Columbia, V6S 2L2, Canada.
DR.
J.B. Medley,
Daimler-Benz AG, Abt. EZBM, Postfach 60 02 02, D 7000, Stuttgart 60, West Germany.
University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, N2L 3G1, Canada
DR.
J.O. Medwell,
Pennsylvania State University, Department of Chemical Engineering, 160 Fenske Laboratory, University Park, PA 16802, U.S.A.
University College Swansea, Department of Mechanical Engineering, Singleton Park, Swansea SA2 8PP
DR.
H. Meijer(*)
The Ohio State University, Department of Mechanical Engineering, 206 West 18th Avenue, Columbus, Ohio 43210-1107, U.S.A.
Eindhoven University of Technology, Department of Mechanical Engineering, WH 04-102, NL-5600, MB Eindhoven, The Netherlands.
MR.
A.O. Mian,
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
F.R. Mobbs
The University of Leeds, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
DR.
H. Moes(*)
Twente University of Technology, Department of Mechanical Engineering, (Tribology), Postbox 217, Enschede, The Netherlands.
SKF Engineering 6 Research Centre, B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands. Eindhoven University of Technology, Department of Mechanical Engineering, WH 04-102, NL-5600, MB Eindhoven, The Netherlands. Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhoven, The Netherlands. Columbia University, Department of Mechanical Engineering, New York 10027, U.S.A.
PROF. A. Mori,
Kyoto University, Department of Mechanical Engineering, Kyoto 606, Japan.
685
TITLE NAHE --
AFFILIATION/ADDRESS
Columbia University, Department of Mechanical Engineering, S.W. Mudd Bldg. Room 236, New York 10027, U.S.A.
DR.
D.W. Parkins
Cranfield Institute of Technology, College of Manufacturing, Building 50, Cranfield, Bedford, MK43 OAL. U.K.
Ms.
A. Perlman(*)
Columbia University, Department of Mechanical Engineering, S.W. Mudd Bldg. Room 236, New York 10027, U.S.A.
DR.
M.R.
I.B.M. Almaden Research Center, Department K33/802, 650 Harry Road, San Jose, CA 95120, U . S . A .
DR.
J.F. Pierre,
CERMO, 21 Rue Pinel, 75013 Paris, France.
DR.
0. Pinkus(*)
Mechanical Technology Inc., 968 Albany Shaker Road, Latham, New York 12110, U.S.A.
MR.
B.S. Prabhu(*)
Indian Institute of Technology, Madras, India.
MR.
T.G. Rajaswamy,
Senior Engineer, Tribology Laboratory, Corp. R. & D., Rharat Heavy Electricah Ltd., Vikas Nagar, Hyderabad, 500593, India.
MR.
T Muralidhara Rao(*)
Tribology Laboratory, Corp. R. & D., Bharat Heavy Electricals Ltd., Vikas Nagar, Hyderabad, 500593, India.
Rolls-Royce plc, P.O. Box 31, Derby, DE2 8BJ. U.K.
DR.
G.S. Ritchie,
Universite de Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteur Pineau, 86022 Poitiers, Cedex, France.
GEC Research Ltd., Engineering Research Centre, Whetstone, Leicester, LE8 3LH, U.K.
MR.
J.A. Ritchie,
Kansai University, Faculty of Engineering, 3-35 Yamate-cho 3, Suita, Osaka 564, Japan.
The University of Leeds, Department of Applied Mathematical Studies, Leeds. LS2 9JT. U.K.
MR.
C. Rodwell
GEC Large Machines Ltd., Special Contracts Div. Rushton Avenue, Thornbury, Bradford, BD3 852.
DR.
W.Y. Saman(*)
Scientific Research Council, Baghdad, Iraq.
H. Mori(*)
Kyoto University, Department of Mechanical Engineering, Kyoto 606, Japan.
MR.
P.G. Morton,
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN, U.K.
DR.
Military Technical College, Baghdad, Iraq.
J.E. Mottershead, University of Liverpool, Department of Mechanical Engineering, Liverpool, L69 3BX.
PROF. E.A. Muijderman(*)
Nederlandse Philips Bedryven B.V. CFT/SAQ 2100, 5600 MD Eindhoven, The Netherlands.
PROF. T. Murakami,
Kyushu University, Department of Mechanica1 Engineering, Faculty of Engineering, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812, Japan.
DR.
MR.
MR.
W.E. ten Napel(*)
S. Natsumeda,
R. Nicholson.
PROF. D. Nicolas
MR.
MR.
AFFILIATION/ADDRESS
PROF. Coda H.T. Pan,
DR.
PROF. N. Motosh(*)
TITLE NAME --
M. Ogata
N. Ohtsuki
Twente University of Technology, Department of Mechanical Engineering, (Tribology), Postbox 217, Enschede, The Netherlands. The University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113 Japan.
Kyushu University, Department of Mechanica 1 Engineering, Faculty of Engineering, 6-10-1 Hakozaki, Higashi-ku, 812 Japan.
Philpott
686
AFFILIATION/ADDRESS
TITLE NAME -DR.
M.D. Savage,
DR.
G .E
The University of Leeds, Department of Mathematics, Leeds LS2 9JT, U.K.
of Waterloo, . Schneider(*IUniversity Department of
-TITLE NAME
AFFILIATION/ADDRESS
PROF. M. Tanaka
DR.
C.M. Taylor
Mechanical Engineering, Waterloo, Ontario, N2L 3G1, Canada. DR.
M. Schouten(*)
MR.
N.P. Sheldrake
MR.
Y. Shimotsuma(*)
Simmons
Eindhoven University of Technology, Department of Mechanical Engineering, WH 04-102, NL-5600, MB Eindhoven, The Netherlands. The University of Leeds, Institute of Tribology Department of Mechanical Engineering, Leeds LS2 9JT. U.K. Kansai University, Faculty of Engineering, 3-35 Yamate-cho 3 chome, Suita-shi, Osaka 564, Japan.
J.E.L.
DR.
T.J.
Smith(*)
University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, N2L 3G1, Canada.
DR.
R.W.
Snidle,
University College, Department of Mechanical Engineering, Newport Road, Cardiff. CF2 1TA. U.K.
PROF. T. Someya,
The University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, 7-3-1 Hongo, Runkyo-ku, Tokyo, 113 Japan.
DR.
Imperial College of Science and Technology, Department of Mechanical Engineering, Tribology Section, Exhibition Road, London SW7 2BX, U.K.
DR.
R. Stanway,
A.K.
Tieu.
The University of Liverpool, Department of Mechanical Engineering, P.O. Box 147, Liverpool. L64 3RX, Merseyside, U.K.
The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K. University of Wollongong, Department of Mechanical Engineering, P.O. Box 1144, Wollongong, N.S.W. 2500, Australia.
MR.
F. Vergne
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
DR.
Ph. Vergne(*)
Institut National des Sciences Appliquees, Laboratoire de Mecanique des Contacts, Batiment 113, 20 Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
University of Durham, Department of Engineering and Applied Science, South Road, Durham. DH1 3LE. U.K.
DR.
H.A. Spikes
DR.
The University of Tokyo, Department of Mechanical Engineering, Hongo, Bunkyo-ku, Tokyo 113, Japan.
PROF. S. Wads(*)
Waseda University, School of Science and Engineering, Tokyo, Japan.
DR.
M.H. Walton,
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN. U.K.
DR.
S H. Wang(*)
Shanghai University of Technology, 149 Yanchang Road, Shanghai, China.
DR.
Y. Wang(*)
Tsinghua University, Research Centre of Tribology, Beijing, China.
DR.
C. Wate rhouse(* )
Imperial College of Science and Technology, Department of Mechanical Engineering, Tribology Section, Exhibition Road, London SW7 2BX, U.K.
DR.
S.
.
Wen(*)
Tsinghua university, Research Centre of Tribology, Beijlng, China.
687
TITLE NAME -DR.
D.F. Wilcock(*)
Tribolock Inc. 1949 Hexam Road, Schenectady, N.Y. 12309, U.S.A.
MR.
J. Worden(*)
University of Wollongong, Department of Mechanical Engineering, P.O. Box 1144, Wollongong, N.S.W. 2500, Australia.
DR.
Y.W. Wu(*)
China Institute of Mining and Technology, Beijing, China.
MR.
H. Xu,
The University of Leeds, Institute of Tribology Department o f Mechanical Engineering, Leeds LS2 9JT. U.K. Wuhan Research Institute of Material Protection, Wuhan, China.
DR.
S.M. Yan(*)
China Institute o f Mining and Technology, Bei jing, China.
PROF. H.H. Zhang(*)
Shanghai University of Technology, 149 Yanchang Road, Shanghai, China.
PROF. Z. Zhang,
Shanghai University of Technology, Research Institute of Bearings, 149 Yanchang Road, Shanghai, China.
MR.
The University o f Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT, U.K.
G. Zhu,
NOT PRESERT AT TEE SpwOSIUn
This Page Intentionally Left Blank
689
Fluid Film Lubrication - Osborne Reynolds Centenary, edited by D.Dowson, C.M. Taylor,M. Godet and D. Berthe Elsevier Science PublishersB.V., Amsterdam,1987-Printed in The Netherlands
13th LEEDS-LYON SpWoSIUn ON TRIBOLOGY FLUID FILM LUBRICATION
- OSBORNE WYNOLDS
CEmNARY
8th-12th SEPTEPIBBR 1986 LIST OF DELEGATES TITLE
NAUR
AFFILIATION/ADDRESS
TITLE NAUE --
AFFILIATION/ADDRESS
DR.
C.I. Adderley
Rolls-Royce & Associates Ltd., P.O. Box 31, Raynesway, Derby, U.K.
PROF. H. Blok
4 Dr. H. Colijnlaan, Flat 19, 2283 XM Rijswijk,The Netherlands.
MR.
S.D. Advani
Michell Bearings plc Scotwood Road, Newcastle upon Tyne. NE15 6LL. U.K.
DR.
The University of Edinburgh, Department of Mechanical Engineering, The King's Buildings, Edinburgh, EH9 3JL, U.K.
PROF. K.J. Aho
MR.
Tampere University of Technology, P.O. Box 527 SF-33101, Tampere, Finland.
A.F. Alliston-Greiner, University Engineering Department, Trumpington Street, Cambridge CB2 1PZ
MR
H. Anderson
U.S. Navy, David Taylor Naval Ship Res. Div. Center, Anapolis, Maryland, 20084, U.S.A.
DR.
A. Artiles
Mechanical Technology Inc. 968 Albany-Shaker Road, Latham, N.Y. 12110, U.S.A.
MR.
A.D. Ball
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
J. Banks
Davy McKee (Sheffield) Ltd., General Engineering Division, Prince of Wales Road, Sheffield S9 4EX. U.K.
PROF. F.T. Barwell
26, Woodland Way, Brighton, BN1 8BA.
PROF. D. Berthe
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
DR.
DR.
P. Bezot
A.G. Blahey
Universite de Nice, Laboratoire Physique Matiere Condensee, (UA 1901, Parc Valrose, 06034, Nice, France. Esso Petroleum Canada, P.O. Box 3022, Sarnia, Ontario, Canada N7T 7M1
D.A. Boffey
PROF. J.F. Booker,
Cornell University, Department of Mechanical 6 Aerospace Engineering, Upson Hall, Ithaca, New York 14853, U.S.A.
PROF. R. Bosma,
Twente University of Technology, Department of Mechanical Engineering, (Tribology), Postbox 217, Enschede, The Netherlands.
DR.
B. Bou-Said
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
PROF. D. Bradley,
The University of Leeds, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
It.
F. Bremer
Nederlandse Philips Bedrijven B.V. CFTfSAQ 2100, 5600 MD Eindhoven, The Netherlands.
MR.
K.R. Brockwell
National Research Council of Canada, 3650 Wesbrook Mall, Vancouver, British Columbia, V6S 2L2, Canada.
PROF. R.H. Buckholz,
Columbia University, Department of Mechanical Engineering, New York 10027, U.S.A.
DR.
Associacion de Investigacion, Centro Tutelado Goblerno Vasco Tekniker, C/.Isasi, d n , Eibar (Guipuzcoa), Spain
R. Bueno,
690 AFFILIATION/ADDRESS
TITLE NAME -MISS 'R.M. Burke
MR.
P. Caisso,
PROF. A. Cameron
MR.
J. J. Chapman
DR.
T.H.C.
Childs
DR.
R.J. Chittenden
Elsevier Science Publishers, Science Tech. Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands.
AFPILZATION~ADDRESS
TITLE NA)(E --
Institut National des Sciences Appliques, Laboratoire de Mecanique.des Contacts , Batiment 113, 20, Avenue Albert Einstein, 69621 Villetirbanne , Lyon, France.
PROF. G. Dalmaz &
Societe Europeenne de Propulsion, B.P. 802, 27207 Vernon, Cedex, France. DR.
M.P. Dare-Edwards
Shell Research Ltd., Thornton Research Centre, P.O. Rox 1, Chester CHI 3SW, U.K.
MR.
J.E. Davies
CSIR, NMERI, P.O. Box 395, Pretoria 0001, South Africa.
MR.
N. Dekker
LIPS BV., Postbox 6 , 5150 BB Drunen, The Netherlands.
DR.
R. Delbourgo
The Scientific Counsellor, Ambassade de France, Silver City House, 62 Brompton Road, London SW3 1BW.
DR.
J.A. Dominy,
Rolls-Royce plc, P.0. Box 31, Derby, DE2 8BJ.
Cameron-Plint Ltd., 39 Maids Causeway, Cambridge. CB5 8DE.U .K. Royal Military College of Science, Land Systems Group, Shrivenham, Swindon. SN6 8LA. U.K. University of Bradford, Department of Mechanical Engineering, Bradford, W. Yorks, BD7 lDP, U.K. The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K.
PROF. D. Dowson PROF. B.N. Cole
The University of Leeds, Department of Mechanical Engineering, Leeds LS2 9JT, U.K.
PROF. H. Christensen
The University of Trondheim, Department of Machine Design, N-7034, Trondheim-NTH, Norway.
SIR
D.G. Christopherson, 43 Lensfield Road, Cambridge, U.K.
MR.
D.M. Clarke
DR.
DR.
A.J. Croft
P.E. Dale,
Newcastle upon Tyne Polytechnic, School of Mathematics and Stats. Ellison Place, Newcastle upon Tyne NE1 8ST, U.K. Coventry (Lanchester) Polytechnic, Department of Mathematics, Priory Street, Coventry, CVl 5FB, U.K. Ontario Hydro Research, Mechanical Testing & Development, 800 Kipling Avenue, Toronto, Ontartio, M8Z 5S4, Canada.
MR.
R. Driver
PROF. J.L. Duda
The University of Leeds, Institute of Tribology, Dept. o f Mechanical Eng. Leeds LS2 9JT. U.K. British Timken, Main Road, Duston, Northampton NN5 6UL. U.K. Pennsylvania State Univ. Department o f Chemical Engineering, 160 Fenske Laboratory, University Park, PA 16802, U.S.A. PSA - Etudes et Recherches, 18 rue'des Fauvelles, 92250 - La Garenne, Colombes, France.
ING.
J. Dupin
MR.
E. Edeline,
Societe Europeenne de Propulsion, B .P. 802, 27207 Vernon, France.
PROF. H.G. Elrod,
14 Cromwell Court, Old Saybrook, CT 06475, U.S.A.
DR.
Renseelaer Polytechnic Institute, Mechanical Engineering Department, Troy, N.Y. 12181, U.S.A.
C.M.McC
Ettles
691
TITLE NAME --
TITLE NAME -DR.
P.A. Evans
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
F.M.H.
El-Minia University, Department of Mechanical Engineering, Egypt
DR.
C.G.
Ezzat
Floyd
Michael Neale 6 Associations, 43 Downing Street, Farnham, Surrey, GU9 7PH. U.K.
PROF. J. Frene
University of Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteur Pineau, 86022 Poitiers, Cedex, France.
PROF. T. Fujii
Kansai University, Faculty of Engineering, 3-35 Yamate-cho 3 chome, Suita-shi, Osaka 564, Japan.
IR
.
MR.
J.G. Fijnvandraat Philips Research Laboratory, P r o f . Holstln. 2, 5600 JA, Eindhoven, The Netherlands. W.W. Gardner
Waukesha Bearings Corporation, Box 798, Waukesha, WI 53187, U.S.A.
DR.
P.H. Gaskell
The University of Leeds, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
DR.
D.T. Gethin
University College Swansea, Singleton Park, Swansea, SA2 8PP
PROF M. Godet
MR.
D.W.F.
Goslin,
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
ASSOC.Qiangai Gu, PROF.
MR.
J.W. Hadley,
Shell Research Limited, Thornton Research Centre, P.O. BOX No. 1, Chester. CH1 3SH.
MR.
W.E. Hale,
University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, N2L 3G1, Canada
MR.
R. Hall,
The University of Leeds, Department of Mathematics, Leeds LS2 gJT, U.K.
PROF. B.J.
Hamrock,
The Ohio State University, Department of Mechanical Engineering, 206 West 18th Avenue, Columbus, Ohio 43210-1107, U.S.A.
MR.
R.T. Harding
The University of Leeds, Institute of Tribology, Dept. of Mech. Eng. Leeds LS2 gJT, U.K.
DR.
C.J. Harries
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT, U.K.
DR.
H. Hashimoto
Tokai University, Department of Mechanical 6 Production Engrg. Faculty of Engineering, 11 17 Kitakaname, Hiratsuka-shi, Kanagawaken, Zip Code 259-12, Japan.
MR.
S.S. Hassan,
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
PROF. K. Hayashi,
Osaka Industrial University, 83 Inari-Onmaecho Fukakusa, Fushimi-ku, Kyoto, Japan.
.
YARD Ltd., Charing Cross
Tower, Glasgow, G2 4PP. U.K.
AFFILIATION~ADDRESS
Dr
.
Chairman o f Mechanical Engineering Department of Wuhan Institute o f Technology, Mafangshan, Wuhan, China. MR.
G.J.J.
van Heijningen, TH Delft, Department of Mechanical Engineering, Laboratory for Machine Elements 6 Tribology, P.O. Box 5039, 2600 GA Delft, The Netherlands.
J.I. Hellstrom,
Nynas Industri AB, P.O. Box 194, S-14901, Nynashamn, Sweden.
692
TITLE NAME --
AFFILIATION/ADDRESS
Nederlandse Philips B.V. Laboratorium Wy-p-30, Postbus 218, 5600 MD, Eindhoven, The Netherlands.
Lulea Technical University, S-95187 Lulea. Sweden.
PROF. Bo. 0. Jacobson
The University of Birmingham, Department of Mechanical Engrg. South West Campus, P.O. Box 363, Birmingham. B15 2TT. U.K.
SKF Engineering & Research Centre B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands.
MR.
H. Heshmat,
Mechanical Technology Inc., 968 Albany Shaker Road, Latham, New York 12110, U.S.A.
DR.
Erik B. Hoglund
MR.
C.J. Hooke.
N.A. Hopkinson
PROF. Y. Hori,
DR.
DR.
MR.
L.G.
Houpert
K. Ikeuchi
G.E. Innes
Royal Aircraft Establishment, Naval Engineering Department,Pyestock, Farnborough, Hants. U.K.
Technical University of Denmark, Building 403, Department of Machine Elements, DK 2800 Lyngby, Denmark.
MR.
B. Jobbins
SKF Engineering b Research Centre, B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands.
The University of Leeds,Institute of Tribology, Dept. of Mechanical Eng. Leeds. LS2 9JT. U.K.
DR.
Hans L. Johannesson, Chalmers University of Technology, Division of Machine Elements, S-412, 96 Goteborg, Sweden.
MR.
J.H. Johnson
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN, U.K.
MR.
D.A. Jones,
The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K.
MRS.
Elisabet Kassfeldt Lulea Technical University, S-95187, Lulea, Sweden.
DR.
G. Karami,
Shiraz University, Department of Mechanical Engineering, School of Engineering, Shiraz, Iran.
MR.
J. Kinnunen
Tampere university of Technology, P.O. Box 527, SF-33101 Tampere, Finland.
Kyoto University, Department of Mechanical Engineering, Faculty of Engineering, Kyoto University, Kyoto 606, Japan University of Toronto, Department of Mechanical Engrg. 5 King's College Road, Toronto, Ontario, Canada. M5S 1A4.
E. Ioannides
SKF Engineering & Research Centre, B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands.
Ives-Smith
PROF. J. Jakobsen
Lloyd's Register of Shipping, 71 Fenchurch Street, London EC3M 4BS. U.K.
The University of Leeds, Institute of Tribology, Dept. of Mech. Eng. Leeds. LS2 9JT. U.K.
DR.
A.J.D.
R.W. Jakeman
Z.M. Jin,
Morgan Construction, 15, Belmont Street, Worcester, N.A. 01605, USA.
MR.
J.A.H.M.
MR.
C.L. Innis,
Y. Ishibashi
MR.
University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, Hongo, Bunkyo-ku, Tokyo 113, Japan.
DR.
MR.
AFFILIATION/ADDRESS Jacobs
MR.
DR.
TITLE NAME --
Nippon Steel Corporation, 11th Floor, Bucklersbury House, 3 Queen Victoria Street, London EC4N 8EL. B.L. Technology Ltd., Department of Mathematics, Coventry (Lanchester) Polytechnic, Priory Street, Coventry. CV1 5FB. U.K.
693
TITLE HAEIE -MR.
MR.
C.C. Kweh
D.J. Lacy,
DR.-ING Otto R. Lang
AFPILIATIOIJ/ADDRESS University College, Department of Mechanical Engineering & Energy Studies, Newport Road, Cardiff. CF2 1TA.
TITLE NAME --
AFPILIATION/ADDRESS
MR.
A.A. Lubrecht,
Twente University of Technology, Postbus 217, 7500 AE Enschede, The Netherlands.
MR.
J.R. McDonald
Superintendent, Quality Engineering Test Establishment, Department of national Defence, Ottawa, Canada
Ricardo Consulting Engineers plc, Bridge Works, Shoreham-by-Sea , West Sussex BN4 5FG. U.K.
DIPL-ING. N. Madzia,
Daimler-Benz AG, Abt. EZBM, Postfach 60 02 02, D 7000, Stuttgart 60, West Germany.
STEYR Walzlager, Abt. ETT. Postfach 120, 4400 Steyr, Austria.
DR.
Industrial llnit of Tribology, Department of Mechanical Engineering, The University of Leeds , Leeds LS2 9JT.
11
C.N. March,
MR.
A.K-C Lau
The University of Leeds, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
J. Lauzirika
Asociacion de Investigacion, Centro Tutelado Goblerno Vasco, Tekniker, C/.Isasi,s/n, Eibar (Guipuzcoa), Spain.
MR.
F.A. Martin
MR.
H.J. van Leeuwen
Eindhoven University of Technology, Department of Mechanical Engineering, WH 04-102, NL-5600, MB Eindhoven, The Netherlands.
The Glacier Metal Co. Ltd. , 368 Ealing Road, Alperton, Wembley, Middlesex HA0 IHD, U.K.
DR.
J.B. Medley,
Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhoven, The Netherlands.
University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, N2L 3G1, Canada
DR.
J.O. Medwell,
University College Swansea, Department of Mechanical Engineering, Singleton Park, Swansea SA2 8PP
MR.
A.O. Mian,
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering,Leeds LS2 9JT. U.K.
MR.
H. Middlebrook
National Centre of Tribology, U.K.A.E.A., Risley, Warrington, Cheshire, WA3 6AT, U.K.
MR.
A. Mikula,
Kingsbury Inc. 10385 Drummond Road, Phila, PA 19154, U.S.A.
MR.
Norbert Mittwollen Technische Universitat Braunschweig,Inst. f . Maschinenelemente U. Fordertechnik, Langer Kamp 19B, D-3300 Braunschweig, West Germany.
DR.
A. Leeuwestein
MR.
A.J. Leopard
Glacier Metal Co. Ltd., Ealing Road, Alperton, Wembley, Middlesex HA0 1HD. U.K.
MR.
P.M. Leslie
Rolls-Royce plc, Leavesden, Watford, WD2 7B2, Herts. U.K.
PROF. Hans J. Leutheusser, University of Toronto, Department of Mechanical Engineering, Toronto, Ontario M5S 1A4, Canada. DR.
P.J. Lidgitt,
Ministry of Defence (PE), DGME, ME 233, Block B, Room 90, Foxhill, Bath, Avon. BA1 5AB. U.K.
DR.
S. Lingard
University of Hong Kong, Department of Mechanical Engineering, Pokfulam Road, Hong Kong.
MR.
P.M. Lo
The Glacier Metal Co. Ltd., 368 Ealing Road, Alperton, Wembley, Middlesex HA0 lHD, U.K.
PROF. H. Marsh,
The University of Durham, Department of Engineering, South Laboratories, South Road, Durham DHl 3LE.
694
TITLE N M --
AFFILIATION/ADDRESS
MR.
F.R. Mobbs
The University of Leeds, Department of Mechanical Engineering, Leeds IS2 9JT. U.K.
MR.
C. Mondet,
Renault Vehicules Industriels, Renault VI, Der Avenue No. 1, 69800 St. Priest, France.
Lt.Cdr.J.E.
Moorman RN
Kyoto University, Department of Mechanical Engineering, Kyoto 606, Japan.
MR.
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN, U.K.
DR.
J.E. Mottershead, University of Liverpool, Department of Mechanical Engineering, Liverpool, L69 3BX.
PROF. T. Murakami,
N. Ohtsuki
Kyushu University, Department of Mechanical Engineering, Faculty of Engineering, 6-10-1 Hakozaki, Higashi-ku, 812 Japan.
MR.
T.M. Osman
The University of Leeds, Department of Mechanical Engineering, Leeds. LS2 9JT. U.K.
DR.
M.S. Ozogan,
Humberside College of Higher Education, School of Engineering, George Street, Hull, U.K.
GKN Vandervell Ltd., Norden Road, Maidenhead SL6 4RG. U.K.
DR.
D.W. Parkins
Cranfield Institute of Technology, College of Manufacturing, Building 50, Cranfield, Bedford, MK43 OAL. U.K.
MR.
J.P. Pedron,
Universite de Poitiers, Laboratoire de Mecanique des Solides, 40 Avenue du Recteur Pineau, 86022 Poitiers, Cedex, France.
RENAULT Automobile, Direction des Laboratoires Automobile, 8a 10 av Emile Zola, 92 WG Boulogne, Billancourt, France.
DR.
M.R. Philpott
Technical University of Denmark, Department of Machine Elements, Building 403, DK 2800 Lyngby, Denmark.
I.B.M. Almaden Research Center, Department K331802, 650 Harry Road, San Jose, CA 95120, U.S.A.
DR.
J.F. Pierre,
CERMO, 21 Rue Pinel, 75013 Paris, France.
MR.
N. Pinfield,
Nutterworth Scientific Ltd., P.O. Box, Westbury House, Bury Street, Guildford, Surrey GU2 5BH. U.K.
DR.
E.G. Pink,
GEC Research Limited, Whetstone, Leicester, LE8 3LH.
MR.
Rolls-Royce plc, P.O. Box 31, Derby, DE2 8BJ. U.K.
A. Nitanai
MR.
D.D.
R. Nicholson,
MR.
Kansai University, Faculty of Engineering, 3-35 Yamate-cho 3, Suita, Osaka 564, Japan.
MR.
Kyushu University, Department of Mechanical Engineering, Faculty of Engineering, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812, Japan.
S. Natsumeda,
G.T. Nielsen,
M. Ogata
Columbia University, Department of Mechanical Engineering, S.W. Mudd Bldg. Room 236, New York 10027, U.S.A.
MR.
DR.
MR.
PROF. Coda H.T. Pan,
The University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113 Japan.
PROF. D. Nicolas
AFFILIATION/ALIDRESS
R.N.E.C., Manadon, Plymouth, Devon. U.K.
PROF. A. Mori,
P.G. Morton,
TITLE NAME --
Parker
Tamagawa University, Faculty of Engineering, 6-1-1 Tamagawa gakuen Machida shi, Tokyo, Japan.
695
PROF. T.F.J.
MR.
MR.
DR.
MR.
MR.
TITLE NAME --
AFFILIATION/ADDRESS
TITLE NAME -Quinn
T.G. Rajaswamy,
K.T. Ramesh,
G.S. Ritchie,
J.A. Ritchie,
C. Rodwell
DR.
J.E.L.
MR.
T.E. Simmons,
Senior Engineer, Tribology Laboratory, Corp. R. & D., Bharat Heavy Electricah Ltd., Vikas Nagar, Hyderabad, 500593, India.
Morgan Construction, 15, Belmont Street, Worcester, NA 01605, U.S.A.
DR.
R.W. Snidle,
Brown University, Box D, Division of Engineering, Providence, RI 02912, U.S.A.
University College, Department of Mechanical Engineering, Newport Road, Cardiff. CF2 1TA. U.K.
PROF. T. Someya,
The University of Tokyo, Department of Mechanical Engineering, Faculty of Engineering, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113 Japan.
ASSOC.Ertao Song, PROF.
The University of Leeds, Department of Applied Mathematical Studies, Leeds. LS2 9JT. U.K.
Vice President, Wuhan Institute of Technology, Mafangshan, Wuhan, China.
PROF. A. Soom,
GEC Large Machines Ltd., Special Contracts Div. Rushton Avenue, Thornbury, Bradford, BD3 8JZ.
University of Buffalo, Department of Mechanical h Aerospace Engineering, Buffalo, N.Y. 14260; U.S.A.
DR.
H.A. Spikes
Imperial College of Science and Technology, Department of Mechanical Engineering, Tribology Section, Exhibition Road, London SW7 ZBX, U.K.
DR.
M. Stanojevic
GKN Vandervell Ltd., Norden Road, Maidenhead, Berkshire SL6 4BG. U.K.
DR.
R. Stanway,
The University of Liverpool, Department of Mechanical Engineering, P.O. Rox 147, Liverpool. L64 3BX, Merseyside, U.K.
Ms
S. Taheri
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
M.G. Talks,
British Coal, H.Q. Technical Department, Ashby Road, Stanhope, Bretby, Burton on Trent, DE15 OQD. U.K.
United States International University, School of Engineering Applied Sciences, The Avenue, Bushey, Hert WD2 2LN. U.K.
GEC Research Ltd., Engineering Research Centre, Whetstone, Leicester, LE8 3LH, U.K.
Purdue University, School of Mechanical Engineering, Purdue University, West Lafayette, In 47906, U.S.A.
DR.
M.D. Savage,
The University of Leeds, Department of Mathematics, Leeds LS2 9JT, U.K.
MR.
J. Schwarzenbach
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
MR.
J. Seabra
N.P. Sheldrake
Simmons
&
PROF. S . Sadeghi,
MR.
AFFILIATION/ADDRESS
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France. The University of Leeds, Institute of Tribology Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
University of Durham, Department of Engineering and Applied Science, South Road, Durham. DH1 3LE. U.K.
696
TITLE NAME --
AFFILIATION/ADDRESS
TITLE N M E --
PROF. M. Tanaka
The University of Tokyo, Department of Mechanical Engineering, Hongo, Bunkyo-ku, Tokyo 113, Japan.
MR.
H. Xu,
The University of Leeds, Institute of Tribology Department of Mechanical Engineering, Leeds LS2 9JT. U.K.
DR.
The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K.
MR.
M. Yanagimoto,
NSK Technical Centre Europe, Harkortstr. 15 4030 Ratingen, West Germany.
DR.
Yatao Zhang
Chalmers University of Technology, Division of Machine Elements, S-412, 96 Goteborg, Sweden.
C.M. Taylor
Ir.
Care1 Thijsse,
Nederlandse Philips Bedrijven B.V. C.F.T./SAQ 2113 P.O. Box 5600 MD Eindhoven, The Netherlands.
DR.
A.K. Tieu,
University of Wollongong, Department of Mechanical Engineering, P.O. Box 1144, Wollongong, N.S.W. 2500, Australia.
DR.
J. Tripp,
SKF Engineering 6 Research Centre, B.V., Postbus 2350, 3430 DT Nieuwegein, The Netherlands.
Dr.1r.M. Vermeulen
University of Gent, Laboratory of Machines & Machine Construction, St. Pietersnieuwstraat, 41, 9000 Gent, Belgium.
MR.
F. Vergne
Institut National des Sciences Appliques, Laboratoire de Mecanique des Contacts, Batiment 113, 20, Avenue Albert Einstein, 69621 Villeurbanne, Lyon, France.
DR.
N.C. Wallbridge
The University of Leeds, Institute of Tribology, Dept. of Mechanical Eng. Leeds LS2 9JT. U.K.
DR.
M.H. Walton,
GEC Research Ltd., Engineering Research Centre, Mechanical Laboratory, Lichfield Road, Stafford ST17 4LN, U.K.
MR.
M.N.
Shell Research Limited, Thornton Research Centre, P.O. Box 1, Chester, CH1 3SH, U.K.
Webster
PROF. W.O. Winer
Georgia Institute of Technology, School of Mechanical Engineering, Atlanta, Georgia 30332, U.S.A.
PROF. Z. Zhang,
Shanghai University of Technology, Research Institute of Bearings, 149 Yanchang Road, Shanghai, China.
MR.
The University of Leeds, Institute of Tribology, Department of Mechanical Engineering, Leeds LS2 9JT, U.K.
G. Zhu,