Fluid Film Lubrication, Osborne Reynolds Centenary: Proceedings (Leeds-Lyon Symposium on Tribology Proceedings)

FLUID FILM LUBRICATION OSBORNE REYNOLDS C ENT ENARY TRIBOLOGY SERIES 11 FLUID FILM LUBRICATION OSBORNE REYNOLDS CENT...

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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

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 2 5 P.O. Box 2 1 1, 1000 AE Amsterdam, The Netherlands

Distriburors for rhe Unired Stares and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, NY 10017, U S A .

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,

199-206,443-450,629-642

Printed in The Netherlands

copyright was not transferred

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


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


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

>
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 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.


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


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


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 (

\

+ 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.

-


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/(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

0.49

-0.067

W

-0.073

ER 2

.

-

0.68 1.118

R

where

W

ER

R

-=hm

0.53

1.194

E

2

-1

= El

E2

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.

-

-

-

3.2

Traction

J o h n s o n and T e v a a r w e r k ( 2 ) h a v e shown t h a t

209 under i s o t h e r m a l elastohydrodynamic conditions, the shape of t h e t r a c t i o n c u r v e s i n r o l l i n g - s l i d i n g c o n t a c t s c a n b e d e s c r i b e d by t h e n o n - l i n e a r Maxwell r h e o l o g i c a l m o d e l b a s e d on t h e E y r i n g f l u i d t h e o r y . T h i s m o d e l , i n two-dimentional s h e a r i s w r i t t e n : U d s

jZ

7

T

7

--x+xO sin h &

i, =

G

dx

T~

p

7

U

drz

rZ

ro

7

7e

p

= --+-G dz

with U

sin h

0

& T

=- u1+u2 2

ix=

u 1-u 2+nz h

*

I

Y,

-

-

h

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

.

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

-

-

-

-

-

-

w -w2-m

4

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

g -5 .

8

.

,

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
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.


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,.


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


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
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

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0

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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:

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(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

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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.


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


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

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.62

600 NEUTRAI

24 40

--

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60

R 620 15

750 PALE

26.2 40.3 60 80

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---

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20.6 40.3

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.24 .336

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60

40

1300 PALE

5P4E

60 80

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--

.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


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.


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

. 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

300

1.2;;;

2 50

1 .o;

>

C P)

$200

0.8

m

$150

0.6

.ri

2 & C

0.42 U

0.2

50

a,

E

0

3

5

10

15 20 Time m i n

25

30

>

Fig. 13 Changes in electric resistance and volume of conductive rubber with time in immersion swelling test present simulator test within 20 min. In contrast, the swelling of oil S-1 of the lowest viscosity brings about a considerable change in electric resistance, but its contribution to measured value on degree of separation is at most half. So, fluid film formation with S-1 is considered to be superior to that with oil S-10. This fluid film formation is probably promoted by the exudation of lubricant from swollen rubber within the contact area or in the inlet zone by similar mechanism proposed by McCutchen (16) and Mansour et a1 (17). The adsorbed film penetrated through the elastomer (18) also may facilitate fluid film formation.

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


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

.

-

I-

Fig. 1 S c h e n a t i c diagram Taylor-Couette apparatus vortices.

of a showing

typical Taylor

T a y l o r v o r t i c e s are t o r o i d a l i n form and a secondary laminar flow constitute For superimposed on the basic Couette flow. many y e a r s it h a s b e e n s u g g e s t e d t h a t t h e sequence o f flows o c c u r r i n g as the Taylor number is increased is Couette flow, Taylor v o r t e x flow, wavy Taylor v o r t e x flow w i t h t r a v e l l i r q azimuthal waves s u p e r i m p s e d on t h e

3r4).

396

End

End P l a t e

Plate

X XX

+ O

r

b X b

I

3

t

x

o

x

Y O

I

x o

ox

ll

+

Y

x o

1

0"

t

X

S X

0

t

Y

X

x

D

a

0

+

x o

%

o

x o

x

X

Z

- 0

b 0

t 9

U

d

X

Y

X

'0

+

Y;

%

O

a

o

t

x

o

Y

0

+ +

00

+ +

+

8

0

+

+

0

o

u;

+

0

+.j+

t

+

t

O

+ +

+ +

++

+ A a

Y; +

O

o

A 0

A'

- __------

-----

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

0

a

0

0

d

A

A

--.

Annulus Mid length Fig.3

As f o r Fig. 2.

'I!/Tc= 0.1, 4 cells values of stream function := -0.008233,

v2

= -0.004116,

V , = 4.000620, +4 = 0.000032,

5' = 0.000016. O.C. = o u t e r c y l i n d e r , I.C. = i n n e r cylinder

.

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.


403

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.

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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


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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 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


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


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

=

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

.,,, I

c

b

a

, I ,,

I

,, I

e

c

c

-.*

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

\

\

\

\

\

\

, \

\ \ \

a

\

\

\

b

\

... .

\

C

..

\

\

\

d

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


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


d


€


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


E

fluidity (reciprocal viscosity), m/N-s

P

fluid density, kg/m3

T


@

dissipation function, J/m3-s

9


+ + 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 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.


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


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

-

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

-

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

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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

-

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.

-

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/:

-

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

-

"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/

/3/

/4/

/5/

/6/

/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 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


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


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 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


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


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


z


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.


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


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.


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


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/alo0 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

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