The determination of the pressure-viscosity coefficient of a lubricant

Transcription

1 The determination of the pressure-viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements (II) - high L values Harry van Leeuwen a, b * a Eindhoven University of Technology, Eindhoven, Netherlands b Shell Global Solutions, PAE Laboratory, Hamburg, Germany Abstract The pressure-viscosity coefficient of a traction fluid is determined by fitting calculation results on accurate film thickness measurements, obtained at different speeds, loads, and temperatures. Through experiments, covering a range of 5.6 < M < 000,. < L < 7.5, film thickness values are calculated using a numerical method and approximation formulas from twelve models. It is concluded that, to assess the pressure-viscosity coefficient of the fluid, the Chittenden et al. approximation formula applied to circular contacts is the best choice, having an inaccuracy in between (-5%, +%). This expression has been used far outsides the regime of the numerical data where it was based upon. Keywords EHL, experiment, elastohydrodynamic lubrication, pressure-viscosity coefficient, central film thickness measurement, film thickness approximation formula, rolling contact, circular contact, point contact * addresses : H.J.v.Leeuwen@tue.nl, Harry.vanLeeuwen@shell.com - -

2 . INTRODUCTION In a first paper [] a method to determine the pressure-viscosity coefficient α from film thickness experiments was explored, by fitting it on the measurement results. This pressure-viscosity coefficient is of crucial importance in the elastohydrodynamic lubrication (EHL) of heavily loaded contacts. The results were satisfying for the test lubricant, HVI60, a mineral oil, and in particular if the Chittenden et al. [] approximation was used. It may be argued that the conditions were favourable, or that they were close to the assumptions laying the basis of the formulas employed. It was also found that the older and simpler formulas gave a better estimate than the newer more complex ones, which are claimed to have a better fit on a larger domain. All tests were carried out at L < 7, the lubricant or so-called Moes [] parameter, containing, see also equation (.). It was suggested that the latter models would perform better than the former if L were appreciably larger. Viscometers have been in use for many years and are most appropriate to determine the pressure-viscosity coefficient of lubricants, see Bair [4]. Only a few institutes have a high-pressure viscometer, while many have film thickness measurement devices. Hence it is sensible that film thickness measurements are used to assess. Recently, a few authors [5, 6] used an approach which resembles the method in [] for several fluids having a high. The well known Hamrock and Dowson [7] formula was employed. But is its use justified, and how close are its predictions to the proper values? The pressure-viscosity coefficient was not known. And is the Hamrock and Dowson approximation a good choice, while a dozen more of these formulas exist? The present work does not intend to explore the boundaries of the assumptions in classical EHL work, but investigates the tenability of the film models when the conditions in the experiment are further beyond the regime where the better part of the numerical data, forming the basis of these models, were obtained. In other words, physical experiments where the lubricant parameter L is high, or the load parameter M is very high or very low, with a fluid which exhibits Newtonian behaviour, were performed and served as a basis for the method. The four nondimensional film thickness groups, describing the EHL problem of an elliptical EHL contact, are []: Er Re e u 0 h R F E R M r e E R u r e 0 u L E r 0 Er R e Rt R e 4 4 film thickness parameter Meldahl or load parameter lubricant parameter radii of curvature ratio - - (.)

3 where R e is the radius of curvature in entrainment direction, and R t in transverse direction (normal to entrainment). In most cases, also in this study, circular contacts are studied. Hence =, reducing the set in (.) to nondimensional groups. The other symbols in (.) are explained in the nomenclature. Only the formulas suggested by Moes and co-authors ([], [8], [9], and [0]) are based on a wide (L, M) regime, and it might be expected that these models may prove more appropriate.. METHOD AND EQUIPMENT.. Analysis and methodology The method used in this study is described in detail by van Leeuwen []. It utilises an accurate film thickness measurement technique [], and central film thickness formulas from the literature. The pressure-viscosity coefficient is the only unknown, and is adapted so that the root mean square error in the predicted film thickness values (obtained from the formula) with respect to their measured counterparts is at a minimum. The applicability has been proven for a mineral lubricant with a low value for, see []. Rather than the central film thickness, the minimum film thickness has always been of more interest to tribologists, see Chaomleffel et al. []. However, most approximation formulas have been derived for the central film thickness. The behaviour of the minimum film thickness in elliptical contacts is even more complex than for the central film thickness. The location of the minimum film thickness may occur at the centre line, or at the side lobes, see Moes [], Venner [8] and Hooke []. Chaomleffel et al. [] use the central film thickness formula by Nijenbanning et al. [9] as a basis, and apply a correction table for certain values of L and M to attain at a value for the minimum film thickness. Although Moes [] states that the correction factor h min /h c varies between 0.65 and 0.8 for his 4 asymptotes, thereby suggesting 0.75 as a good average, Venner [8] shows that the factor h min /h c varies between 0. and 0.77 for <M<000, 0<L< 5, and Chaomleffel et al. [] state that in the very thin film domain with low L and high M numbers the value might be even lower than 0.9. This was corroborated by the numerical calculations in this study: this factor can be lower than 0.0 (for L =.85, M = 5000). Also, as [] and [8] show, the relation between this factor and M and L is a complex one. Therefore, it is preferable to use the central film thickness in assessing the value of the pressure-viscosity coefficient. Full multigrid numerical calculations will be performed, see Venner and Lubrecht [0], for many of the experimental conditions. They are subject to assumptions from classical EHL theory, as isothermal flow, Roelands [4] viscosity-pressure behaviour, and density-pressure behaviour as in Dowson and Higginson [5]. All film thickness approximation formulas are essentially based upon assumptions like these or more restrictive ones, see []. In addition, all of these formulas postulate fully-flooded conditions. - -

4 It is reasoned that if numerical simulation results for film thickness are close to the experimental ones, the latter do comply with theoretical assumptions, and the approximation formulas may be applied. If a perfect approximation formula, which would perfectly predict the numerical results, would exist, this would be close to the experimental values. Conversely, if this perfect formula would be employed to predict the value of the pressure-viscosity coefficient, it must be close to the value which a high-pressure viscometer would determine. Which of the approximations is closest to this ideal approximation, is decided in section 4. In this study the reciprocal asymptotic isoviscous pressure * is used, since this value has shown a good correspondence with film thickness (see Bair [6], [7]). It is defined as * 0 dp (.) p i, as 0 p.. Equipment A PCS Instruments EHL Ultra Thin Film Measurement System [] is run in central film thickness measurement mode under pure rolling. The ball is supported by three angular contact ball bearings and driven by traction in the contact. The radius at the glass disc was fixed at r = m. At this radius the ball is supposed to roll without spin [8]. Other test rig data are provided in Table. Measurements were performed at 4, 0, and 50 N (corresponding to a Hertzian pressure of about 0., 0.5, and 0.7 GPa, respectively). The speed range was 0.0. m/s. The temperatures were 40, 60 and 80 degrees C. The measurement procedure is as follows. Ball and glass disc are first cleaned in toluene in an ultrasonic bath during at least 0 minutes, next with ethanol, dry in air, and finally mechanically cleaned using lens tissue. The first measurement was done after thermal equilibrium was established, which takes at least 0 minutes. The measurement was started at a medium rolling speed, e.g. 0. m/s, next down to the minimum speed, then up to the maximum, and at last back to the first speed. For each measurement result five readings were averaged and if the standard normalized deviation was smaller than, accepted. A higher L value than in [] can be attained by increasing, or E r, which is most effective, and further 0, u, and decreasing R e. The speed is limited to.7 m/s, while beyond this speed the films are too thick to be measured accurately. As E r and R e are fixed in the device, the two parameters which allow a higher L value are and 0. A fluid having a high and viscosity was found in Santotrac S70. The viscosity data are listed in Table, which also provides the test data

5 .. Film thickness formulas Twelve film thickness models were used. The appendix provides a concise overview of all the formulas employed in this study. One model was added to the used in [], which stems from Evans and Snidle [9]. This one is of the interpolation type, fitted on numerical results, and discussed in more detail in the appendix. All other formulas are extensively discussed in []. Many are for elliptical contacts, and will be used for circular contacts here. Long elliptical contacts have by definition their major ellipse axes in entrainment direction. The approximation formulas reported here are: From the Hamrock family: () Hamrock and Dowson [7] for circular and short elliptical contacts () Hamrock et al. [0] for circular and short elliptical contacts () Chittenden et al. [] for circular and long elliptical contacts (and arbitrary entrainment direction) Other film models: (4) Archard and Cowking [] for circular hard contacts (5) Evans and Snidle [9], for circular contacts (6) Hooke [] for arbitrary elliptical contacts (7) Sutcliffe [] for long elliptical contacts (8) Greenwood [] for circular contacts (note that this model was not suggested by Greenwood) From the Moes family: (9) Venner [8] for circular contacts (0) Nijenbanning, Venner and Moes [9] for circular and short elliptical contacts () Venner and Lubrecht [0] for circular contacts, and () Moes [] for arbitrary elliptical contacts Figure illustrates the large differences in the values of some of these formulas for L = 0. The proper choice of a good approximation formula is important, as roughly seen nondimensional film thickness H L //4 so h 4/. Alone a systematic error of 0% in the prediction of film thickness may result in an error of up to 0% in the prediction of the pressure-viscosity coefficient. Only formulas from (4), (6), and (7) are based on the assumption of an incompressible fluid. Greenwood [] and Nijenbanning et al. [9] discuss compressibility, which induces at least one extra nondimensional group. All other models, including [9], employ the same Dowson-Higginson pressure-viscosity relationship for their numerical results, including the parameters therein. No attempt has been done to develop a proper nondimensional group

6 .4. Numerical calculations For numerical calculations, specific software written for circular EHL contacts by Venner [4] has been used. It is a multilevel finite difference code, where the general EHL assumptions as outlined in. have been used. This program employs the Roelands parameters R, p R and z, see [], where z is found through equation (5b) in []. The pressure-density relationship is according to Dowson and Higginson, see equation (6) in [], which reference provides the value used for p R too. The viscosity and pressureviscosity coefficient from Table were used in the calculations.. RESULTS The Chittenden et al. [] approximation formula is based on numerical simulations of EHL elliptical contacts, and those of Hamrock and Dowson [7]. Only a few of them are circular contacts (most are elliptical with = 6.6). Figure shows the experimental conditions represented in the (M, L) domain. Contour lines of nondimensional film thickness H have been constructed using the formulas from Moes []. Also lines of constant nondimensional film thickness H, according to Chittenden et al. [], have been depicted. The conditions of the experiments described in this report are also depicted, omitted are a few data at M > It is seen that these circumstances are well outsides the regime of the numerical data. Also, the maximum L value is about 7.5, exceeding the value in [] by a factor of.5. Figure shows the results for central film thickness measurements at 40, 60 and 80 0 C, respectively, and values obtained by multigrid calculations [4]. In general, close agreement is observed between the measured and calculated data. This implies that the model used in the calculations can be used to describe the behaviour of S70 within the range of the measurements. In addition, it is likely that the experimental data may be analyzed by fitting approximation formulas, to assess the value of the pressure-viscosity coefficient. The deviation in the theoretical film thickness from the experimental results, by using the formulas pointed out in., using the known value for the pressure-viscosity coefficient, can be up to +0% or -0%. This is illustrated by Figure 4, for S70 at 0 N and 40 0 C. When the estimated value for, predicted by a model formula, is used, this narrows to -5% +0%, see Figure 5. At lower entrainment speed values two film thickness values were evaluated, by using the first measurement results (with speed going down) and the later ones (with speed going up), see section.. Table shows the results of the analysis for the 6 series of experiments. Table a contains the results of the optimisation of the pressure-viscosity coefficients for the 6 measurement series and models, and the target values

7 Table b demonstrates the averaged deviations from the known values per measurement series, the overall averaged relative deviations, as well as the average absolute deviation and the standard deviation in the relative deviation. The first overall value discloses a systematic deviation, while the last two illustrate scatter. Significant underestimation is found with the approximation formulas from Sutcliffe [], Greenwood [], and all of the Moes family [], [8], [9], and [0] formulas, which all predict 0% too low pressure-viscosity values. The Moes family results have low scatter. Severe overestimation is a result of using Archard and Cowking s [] formula. An acceptable average estimate is found by employing the expressions from Hooke [] and Evans and Snidle [9], but their scatter is too large to be acceptable. Not much difference exists between Hamrock and Dowson s [7] and Hamrock et al. s [0] results, which have a consistent underestimation of by about 0% average. The Chittenden et al. [] based predictions have the smallest deviations. Table c shows the standard deviation in the film thickness prediction for Santotrac S70. Values better than *0-9 m (or for the average 6*0-9 m) are in bold italic. The best film thickness values are obtained with Moes family [], [8], [9], and [0] expressions. Sutcliffe s expression [] also yields a good curve fit for the film thickness. It is noted that these expressions severely underestimate the pressure-viscosity coefficient. Table d contains the correlation between calculated and measured film thickness values for Santotrac S70. All values better than are in bold italic characters. It is seen that the Sutcliffe [], next the Moes family [], [8], [9], and [0] and next the Hamrock and Dowson family [], [7], and [0], have the highest correlation. 4. DISCUSSION The results in section show that the mutual differences between the models are large. When the criteria from [], where a mineral oil was studied, are applied: a. the deviation in the result is less than % (which is of the order of up to *0-9 Pa - ); b. the standard deviation for the film thickness prediction is better than *0-9 m; c. the correlation is better than none of these models meet all requirements. All predictions which show a consistent underestimation, use a formula which yields an overestimation of the film thickness over the range of the experiments. Among them are the formulas from Sutcliffe [], Greenwood [], and all Moes film equations [], [8], [9], and [0]. The behaviour of Evans and Snidle s [9] formula differs from what the authors state, which is due to the use in a much wider range than it was originally made for. Archard and Cowking s [] model is greatly overestimating. Hooke s formula [] yields deviations at both ends, with a trend towards overestimation, and has a high standard deviation - 7 -

8 and low correlation. The well known Hamrock and Dowson [7] formula, which is very often applied, is inferior to the Chittenden et al. []. Lubrecht et al. [5] showed that Hamrock and Dowson s numerical results and the resulting curve fit are very good, but outsides the range of the numerical experiments this curve fit performs less well than the formula provided by Chittenden et al. []. From many perspectives, the last one shows the best prediction, deviation, and correlation over the range of the experiments. Their results are closest to the measured values, in a range of (-.5, 7.5%) for thicker films than 0 nm. It must be noted that Chittenden et. al s [] central film thickness formula applied to noncircular contacts is prone to gross errors up to a factor of, even if the entrainment is along a minor or major ellipse axis, see Sharif et al. [6]. For heavily loaded circular contacts, with central film thickness values of 0 nm and onwards, they found that the deviation is in between -% and + 9%, and these conditions prevail in the current measurements. Just as in [], the Moes family of formulas shows the best values for standard deviation and correlation. However, the predicted film thicknesses are too high, as was also commented by Chaomleffel et al. []. Table 4 indicates the error in the film thickness compared to the measured one, using the viscosity data of Table. Table 4 also lists averages of all percentage errors in the calculated film thicknesses with respect to the measurements, and the averaged values of the absolute deviations. The largest contributions are obtained at extremely thin films, of the order of less then 0 nm (at 0.0 m/s, 50 N and 80 0 C means L =. and M = 000). If films below 8 nm are omitted, the last row but one in Table 4 results, and if all films thinner than 0 nm are dropped, the last row results. It turns out that film thicknesses acquired with the Moes equations [], [8], [9], and [0] are.5 % too high, see also Figure. Most of the approximation formulas, except the Moes family, have been fitted on numerical results in the piezoviscous elastic (VE) regime, see also the Appendix. Fig. shows that indeed the measurements were performed far beyond the area where Chittenden et al. [] and Hamrock and Dowson [7] did their calculations. However, at high M values the slope in these curve fits is close to the Moes [] approximation. Hence, within the limits of L < 0 and M < 000, the behaviour of the extrapolated Chittenden et al. contours for high M values will be not too far off the numerical results, which is supported by Figure 5. Although not perfect, the Chittenden et al. [] formula for circular contacts may therefore be used with care, and some confidence, beyond the limits of the original numerical data. The differences are to be expected at low M values, as is witnessed by the breaking up of the contour lines for H and H

9 There is a considerable influence of load on the prediction of the pressure-viscosity coefficient, as can be seen from comparison between the data obtained for 40 0 C and 4, 0 and 50 N. Numerical analysis results, based on the known pressure-viscosity coefficient, are less prone to such a variation. This means that the load dependency in none of the film thickness approximations is perfect. In the optimisation procedure the square error in the film thickness is used. An optimisation through logarithmic film thickness values will much better acknowledge the influence of very thin films. This was performed on almost all experiments, but did not result in any noticeable reduction of the errors in the predictions. If only Moes [] VE asymptote is employed, the results hardly become any better. Probably the values of some or all constants in his asymptotic formulas have been chosen too low. The high correlation, low standard deviation and good experiences of Chaomleffel et al. [] with the Moes approximation suggest that they have the potential to be very good, but for now it must be concluded that these Moes formulas cannot be recommended as they are. Several traction fluids are known for their inlet shear thinning, even under pure rolling conditions [], [7], and [8]. Also, thermal effects may reduce film thickness at higher speeds []. However, the numerical calculations are so close to the experimental results that inlet shear thinning is considered not to be important and the assumptions from. applicable. An evaluation of the thermal correction factor C T, as provided by Hamrock et al. [0] for pure rolling: Hz * 0.4. L Er CT * L * u L T K f yields a minimum rolling velocity of about m/s to reach a reduction by 5%. This would imply that at speeds beyond 0.5 m/s the film thickness drops due to shear heating. If Figure is reproduced on a linear scale, showing more details at higher speeds, see Figure 6, indeed the measured film thickness is a little smaller than the calculated one for speeds beyond 0.7 m/s. But this may also be due to the measurement method, which has a (small) systematic error when the film thickness is measured with a fringe order higher than (which corresponds to more than about 80 nm). From all results, the decrease in measured film thickness compared to the calculated one at high rolling speeds was found to be largest at 0N and 60 0 C. Table 5 also shows the results for a data analysis where all measurements at speeds higher than 0.56 m/s are excluded. The changes in inaccuracy are of minor importance, and the general trend is the same as seen before. The standard deviation improves, 7 models have less than nm now. The correlation of all predictions lessens slightly. (4.) - 9 -

10 This implies that in order to assess the value of the pressure-viscosity coefficient, the measurement regime is not very critical, as long as the experiments are carried out in the VE domain. In this study, 0 to 0 film thickness measurements in a speed range of 0.0 m/s suffice. A considerable improvement is only to be expected from a new approximation formula, which should be much closer to the numerical simulations than any existing one. These considerations lead to the conclusion that the literature does not provide an almost perfect film thickness approximation formula for circular EHL contacts. The most reasonable approximation is the Chittenden et al. [] expression, when applied to circular contacts. 5. CONCLUSIONS AND RECOMMENDATIONS () Film thickness in a circular EHL contact was measured using a traction oil, in a wide range of conditions (<M<000,.<L<7.5). Twelve approximation formulas for the central film thickness in EHL circular contacts have been compared in the assessment of the pressure-viscosity coefficient of a traction fluid (S70). () The numerical calculation results were close to the experimental values, suggesting that the fluid is behaving in a Newtonian manner in the range of the measurements. () In the measurement range of this study the formula from Chittenden et al. [], if applied to circular contacts, appeared to have the lowest inaccuracy, in between 4.7% and + %. The Chittenden et al. formula for central film thickness can be recommended for estimating the value of the pressureviscosity coefficient of a lubricant through an interferometric device with reasonable accuracy, just as in []. (4) The validity of the Chittenden et al. [] expression transcends the conditions of the numerical experiments where it was originally based upon. (5) The Moes [], [8], [9], and [0] formulas consistently yield too small values for the pressure-viscosity coefficient and predict too high values of film thickness. Their small standard deviation and high correlation suggest that an improved central film thickness equation should be possible. (6) All film thickness formulas investigated have deficiencies, leaving room for improvement

11 Acknowledgements The author wishes to acknowledge Kees Venner (Twente University, Enschede, Netherlands) for discussions and substantial help. Also Brian Papke (Shell Global Solutions US, Houston, USA), and Glyn Roper (Shell Global Solutions UK, Thornton, UK) are acknowledged for suggestions and support. This work is a follow up of earlier work, carried out under the Marie Curie Host Fellowship for the Transfer of Knowledge Programme of the European Commission []. - -

12 REFERENCES [] van Leeuwen, H. The determination of the pressure viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements, Proc. IMechE, Part J: J. Engineering Tribology, 009, (J8), 4-6 [] Chittenden, R.J., Dowson, D., Dunn, J.F and Taylor, C.M. A Theoretical analysis of the isothermal elastohydrodynamic lubrication of concentrated contacts, part I: Direction of lubricant entrainment coincident with the major axis of the Hertzian contact ellipse, Proc. R. Soc. Lond., 985, A 97(8), [] Moes, H. Lubrication and Beyond, Lecture Notes, Code 55, Twente University Press, Enschede, Netherlands, 000, 66 p. [4] Bair, S. On the concentrated contact as a viscometer, Proc. IMechE, Part J: J. Engineering Tribology, 000, 4 (J6), 55-5 [5] Pensado, A.S., Comuñas, M.J.P., and Fernándes, J. The Pressure Viscosity Coefficient of Several Ionic Liquids, Tribology Letters, 008, (), 07 8 [6] Bantchev, G., and Biresaw, G. Elastohydrodynamic study of vegetable oil polyalphaolefin blends, Lubr. Sc., 008, 0(4), 8 97 [7] Hamrock, B.J and Dowson, D. Isothermal Elastohydrodynamic Lubrication of Point Contacts. Part III Fully Flooded Results, Trans. ASME Series F, J. Lubr. Technol., 977, 99(), [8] Venner, C.H. Multilevel solution of the EHL line and point contact problems, Ph.D. Dissertation, Twente University, Netherlands, 99, 8 pp. [9] Nijenbanning, G., Venner, C.H and Moes, H. Film thickness in elastohydrodynamically lubricated elliptic contacts, Wear, 994, 76(), 7-9 [0] Venner, C.H. and Lubrecht, A.A., Multilevel methods in lubrication, 000, 79 pp. (Elsevier, Amsterdam, Tribology Series No. 7) [] Johnston, G.J., Wayte, R and Spikes, H. A. "The Measurement and Study of Very Thin Lubricant Films in Concentrated Contacts," Tribol. Trans., 99, 4(), [] Chaomleffel, J.-P., Dalmaz, G., and Vergne, P. Experimental results and analytical film thickness predictions in EHD rolling point contacts, Tribol. Int., 007, 40(0-, Special Issue), [] Hooke, C.J., Minimum film thickness in lubricated point contacts operating in the elastic piezoviscous regime, Proc. IMechE, Part C: J. Mech. Engineering Sc., 988, 0 (C), 7 84 [4] Roelands, C.J.A. Correlational aspects of the viscosity-temperature-pressure relationship of lubricating oils, Ph.D. Thesis, Delft University, V.R.B., Groningen, 966, 495 pp. [5] Dowson, D., and Higginson, G.R. Elasto-hydrodynamic lubrication, nd (SI) edition, 977, 5 pp. (Pergamon, Oxford) [6] Bair, S. An Experimental Verification of the Significance of the Reciprocal Asymptotic Isoviscous Pressure for EHD Lubricants, Tribol. Trans., 99, 6(), 5-6 [7] Bair, S. High Pressure Rheology for Quantitative Elastohydrodynamics, 007, 40 pp. (Elsevier, Amsterdam, Tribology and Interface Engineering Series No. 54) [8] Baker, R., Personal communication with PCS Instruments, 00 [9] Evans, H.P., and Snidle, R.W. The isothermal Elastohydrodynamic lubrication of spheres, Trans. ASME Series F, J. Lubr. Technol., 98, 0(4), [0] Hamrock, B.J., Schmid, S.R. and Jacobson, B. Fundamentals of Fluid Film Lubrication, nd edition, 004, 699 pp. (Dekker, Basel) [] Archard, J.F. and Cowking, E.W. Elastohydrodynamic lubrication at point contacts, Proc. Instn. Mech. Engrs., , 80(Part B) In: Proceedings of the Symposium on Elastohydrodynamic Lubrication, Leeds, September 965, paper. [] Sutcliffe, M.P.F. Central film thickness in elliptical contacts, Proc. IMechE, Part C: J. Mech. Engineering Sci., 989, 0,

13 [] Greenwood, J.A. Film thicknesses in circular elastohydrodynamic contacts, Proc. IMechE, Part C: J. Mech. Engineering Sci., 988, 0(C) 7 [4] Venner, C.H., Software Code ehld0 version.00, Twente University, Netherlands, 008 [5] Lubrecht, A.A., Venner, C.H., and Colin, F. Film thickness calculation in elasto-hydrodynamic lubricated fine and elliptical contacts: the Dowson, Higginson, Hamrock contribution, Proc. IMechE, Part J: J. Engineering Tribology, 009, (J, Special Issue), 5-55 [6] Sharif, K.J., Barragan de Ling, F.dM., Martin, M.J., Alanou, M.P., Evans, H.P., and Snidle, R.W. Film thickness predictions for elastohydrodynamic elliptical contacts over a wide range of radius ratios with consideration of side starvation, Proc. IMechE, Part J: J. Engineering Tribology, 000, 4 (J), 6-78 [7] Bair, S., Qureshi, F., and Kotzalas, M. The low-shear-stress rheology of a traction fluid and the influence on film thickness, Proc. IMechE, Part J: J. Engineering Tribology, 004, 8 (J), [8] Bair, S. Shear thinning correction for rolling/sliding elastohydrodynamic film thickness, Proc. IMechE, Part J: J. Engineering Tribology, 005, 9 (J),

14 NOMENCLATURE a Hz semi width of the Hertzian contact ellipse (generally in entrainment x direction) m b Hz semi width of the Hertzian contact ellipse (generally normal to entrainment y direction) m c parameter in Evans and Snidle s viscosity-pressure equation (A5b) m /N C T thermal correction factor, see equation (4.) - E r equivalent modulus of elasticity N/m F normal load N h film thickness m h c central film thickness (film thickness at x = y = 0) m Ĥ nondimensional number for film thickness in an elliptical contact, after Moes [], see (.) - k ellipticity ratio, k = b Hz /a Hz - K f thermal conductivity of the fluid N/s 0 C L nondimensional number for the pressure-viscosity coefficient after Moes [], see (.) - L* nondimensional viscosity-temperature parameter, see equation (A5b) - M nondimensional number for the load, after Moes [], see (.) - p fluid film pressure N/m p i,as asymptotic isoviscous fluid film pressure N/m R e effective radius of curvature in the direction of entrainment m R t effective radius of curvature perpendicular to the direction of entrainment m T temperature 0 C u mean entrainment or rolling velocity in x direction, u = ½ (u + u ) m/s x coordinate in the direction of entrainment m y coordinate in the direction normal to entrainment and along the thin film m Greek symbols: pressure-viscosity coefficient m /N * reciprocal asymptotic isoviscous pressure (= /p i,as ) m /N γ shear velocity in the fluid s - dynamic viscosity of the fluid Ns/m 0 dynamic viscosity of the fluid at ambient pressure, p = 0 Ns/m nondimensional parameter in Evans and Snidle s viscosity-pressure equation (A5b) - Hz Hertzian (maximum dry contact) pressure N/m effective radii of curvature ratio (crowning ratio), = R t /R e, see (.) - Indices: c central value (at the centre, at x = y = 0) e in the direction of entrainment IE isoviscous-elastic IR isoviscous-rigid max maximum value min minimum value t in transversal direction VE piezoviscous-elastic VR piezoviscous-rigid 0 under ambient conditions - 4 -

15 Appendix: Nondimensional central EHL film thickness formulas Below follows a survey of all central film thickness formulas which were employed in this study. Most are taken from [], which also provides nondimensional minimum film thickness formulas, where possible. They are all written in the Ĥ, M, L groups notation. Note that all formulas are for the piezoviscous elastic (VE) sub domain, except those from [], [8], [9], and [0]. Some equations employ, the ratio of effective radii of curvature (crowning ratio), while in others k, the ratio between the contact ellipse semi axes or ellipticity ratio, is preferred. Hamrock et al. [0] provide a simple approximation, credited to Hamrock and Brewe, between the two: k for (A0) This relationship can also be used for <, by changing the indices, but mind using the proper value in the film thickness equation, which may have k < (or <, so called long contacts, because they are long in the direction of entrainment). Also note that for circular contacts k = =. A Hamrock and Dowson s [7] formula The original Hamrock and Dowson [7] central film thickness formula is transformed into the Ĥ, M, L groups using Table 7 from []. As the film thickness formulas based on the original groups H, U, W, G contain one redundant group, always a remaining term will be found. If in this case the remaining group U is replaced by.78, which means that U does not vary much around.69*0 -, the original Hamrock and Dowson equation can be written as:. 0.7k e L M VE,c (A) A Hamrock et al. [0] formula The later Hamrock et al. [0] formula is based on the same data as equation (A), but Hamrock et al. have chosen exponents for their nondimensional numbers h, g E, g V which yield:.5 0.7k e L M VE,c (A) This explains why the results of employing Hamrock and Dowson s or Hamrock et al. s formula are almost identical

16 A Chittenden et al. s [] formula Chittenden et al. [] used the Hamrock and Dowson [7] data, added data for lower L numbers (see Fig. ), and also considered entrainment in an arbitrary direction. For other conditions than circular the results may be less good than Hamrock and Dowson s, see [6]. ˆ H VE,c e L M (A) A4 Archard and Cowking s [] central film thickness formula VE, c.58 L M (A4a) where represents the Archard and Cowking side-leakage factor: (A4b) A5 Evans and Snidle s [9] film thickness formulas Evans and Snidle s [9] formulas for minimum and central circular contact EHL film thickness have not been reviewed in [] and will therefore briefly explained here. The fluid is compressible, using a Dowson and Higginson [5] pressure-density relationship. A two parameter equation has been favoured to describe the pressure-viscosity behaviour: 0 c. p (A5a) resulting in a reciprocal asymptotic isoviscous pressure * for c (A5b) The nondimensional central film thickness approximation for circular contacts in Moes numbers is: ˆ H VE,c ( ). 7 L M (A5c) - 6 -

17 For completeness, the minimum film thickness approximation for circular EHL contacts is provided: ˆ H VE, min ( ). 90 L M (A5d) The authors came to their result by fitting the numerical data over 9 results. They do not claim any accuracy, but remark that their formulas yield 8 7% lower central film thickness and 0 % lower minimum film thickness in comparison to Hamrock and Dowson s [7] formulas, when these are applied to the numerical cases which they studied. In the study reported now it was found that for 0<M<55, 8<L<4 Evans and Snidle s [9] central film thickness results were % lower than the experimental results, and 8 9% lower than those of Hamrock and Dowson [7]. However, deviations outside this (M.L) range were significant, and in an opposite direction, as is seen in the results and discussion. A6 The Hooke [] central film thickness formula The Hooke [] central film thickness at the contact centre expressed in the Moes numbers reads: ˆ 0.44 H 4 VE, c L M (A6) A7 The Sutcliffe [] central film thickness formula k n. n L4 M ˆ 5 4, H VE c ' k a (A7a) The Sutcliffe [] formula expressed in the Moes numbers reads: for 0.4 k and 0 n, where c a k n 0.446a 5 5 a F a a R e Er Re 0.97 ',067 c L L M M (A7b) - 7 -

18 A8 The Greenwood [] central film thickness formula Note that Greenwood [] did not provide a central film thickness curve fit. Based on his results it was argued in [] that his results can be approximated by c 0.94 L (A8) A9 The Venner [8] central film thickness formula The Venner [8] formula for the central film thickness in EHL circular contacts, attributed to Moes, is based on three asymptotes. The asymptotes read in this case IR asymptote: IE asymptote: VE asymptote: ˆ H IR,c 47. M IE,c VE,c.96 M.70 M 9 9 L 4 (A9a) and these asymptotes merge into a combined approximation for the central film thickness in EHL circular contact: c where r exp ˆ r ˆ r H t H s r H ˆ VE s 0 exp M 6 L 8 t exp 0.9 M 6 L 6 IE IR s s (A9b) A0 The Nijenbanning et al. [9] central film thickness formula Moes distinguishes four asymptotes in Nijenbanning et al. [9]: IR asymptote: VR asymptote: IE asymptote: VE asymptote: IR,c VR,c IE,c VE,c M ln ln 0.6 L M M 5 L 4 (A0a) - 8 -

19 The central film thickness in EHL elliptical contacts now is: c where sˆ IR s ˆ s 8 ˆ IE exp. IE IR VR VE sˆ (A0b) A The Venner and Lubrecht [0] central film thickness formula Venner and Lubrecht [0] provide an approximation formula for the central film thickness in EHL circular contacts. They define 4 asymptotes: IR asymptote: VR asymptote: IE asymptote: VE asymptote: ˆ IR,c 4.4 M H VR,c IE,c VE,c 0.9 L.96 M.5M 9 L 4 (Aa) which are combined in c IR s sˆ s 4 8 ˆ IE 0. 8 VR 8 8 ˆ VE where (Ab) sˆ IE exp. IR It is seen that the factors in equation (Aa) can be obtained from (A0a) by equating = there, except the IE asymptote, where Venner and Lubrecht [0] favoured the corresponding expression in equation (A9a) from [8]. A The Moes [] central film thickness formula A comparison between the approximations in Moes [] and Nijenbanning et al. [9] reveals that they are identical, apart from a minor dissimilarity in a shift factor. The Moes [] expressions will therefore not be repeated here

20 This shift factor is to compensate for side leakage effects at long contacts ( << ). The shift factor in (A0b) amounts (.8) , whilst this factor is in Moes []. Therefore, the differences in results from the Nijenbanning et al. equation (A0b), and the one from Moes [] will be minimal, if notable at all. Therefore [] must be seen as an account of the choices made in [9]. Nijenbanning et al. [9] explicitly argue that their curve fit is valid for circular and short elliptical contacts, based on results obtained at =,.5, and 5, without details on the inaccuracy. Moes [] explicitly states that the same expressions hold for long, circular and elliptical contacts, and that the inaccuracy is better than 0% if 0.40, 0 L 5, and 5 -/ M 000. Moes [] also discusses the derivation of the asymptotes in detail and comments on the final form of the film thickness equation

21 Table captions Table : Ball and disc data Table : Properties of lubricants and test conditions Table a: Overview of estimates for the pressure-viscosity coefficient of Santotrac S70 (in Pa - ) Table b: Deviation from the target value of the estimates for the pressure-viscosity coefficient of Santotrac S70 (in %). Best values in bold italic. Table c: Standard deviation in the film thickness prediction for Santotrac S70 (in m). Best values in bold italic. Table d: Correlation in calculated and measured film thickness values for Santotrac S70 (-).Best values in bold italic. Table 4: Average error in film thickness measurements per series, using correct pressure-viscosity coefficient for Santotrac S70 (in %). Extremes in bold italic, values for a reduced number of experiments in italic, see text. Table 5: Estimates, deviations and correlations for Santotrac S70 at 0N and 60 0 C if measurements at rolling speeds > 0.56 m/s are excluded. Best values in bold italic. Figure captions Figure : Nondimensional film thickness diagram for a circular EHL contact, H = H(M) for L = 0. Legend to approximations: Moes; Venner and Lubrecht; Chittenden et al.; Archard and Cowking; Evans and Snidle; Hooke; Sutcliffe Figure : Contour map of nondimensional film thickness H vs. nondimensional load M and lubricant L parameter for a circular contact. Legend: contour lines for H = (based on Moes []); contour lines for H = 0 (based on Chittenden et al. []) ; VR/VE transition; Chittenden et al. [] and Hamrock and Dowson [7] data; S70 at 4N and 40 0 C; S70 at 0N and 80 0 C; S70 at 0N and 60 0 C; + S70 at 0N and 40 0 C; S70 at 50N and 80 0 C; S70 at 50N and 40 0 C Figure : Central film thickness behaviour for S70 at 0 N (log scales). Legend: measurements at 40 0 C, at 60 0 C, and at 80 0 C.; calculations Figure 4: Deviations of the predicted from the measured value of the central film thickness, for the known value of the pressure-viscosity coefficient at 0 N and 40 0 C. Legend: Sutcliffe; Greenwood; Moes; Nijenbanning et al.; Hamrock and Dowson (977); - Venner (99), Venner and Lubrecht (000); Hamrock et al. (004); numerical solution ; Chittenden et al.; Evans and Snidle; Hooke; Archard & Cowking Figure 5: Deviations of the predicted from the measured value of the central film thickness, for the predicted value of the pressure-viscosity coefficient at 0 N and 40 0 C. Legend: Evans and Snidle; Greenwood; Hamrock and Dowson (977); - Venner (99); Chittenden et al.; Hamrock et al. (004); Sutcliffe; Moes; Nijenbanning et al.; Venner and Lubrecht (000); numerical solution ; Archard and Cowking; Hooke Figure 6: Central film thickness behaviour for S70 at 0 N (linear scales). Legend: measurements at 40 0 C, at 60 0 C, and at 80 0 C.; calculations - -

22 Table : Ball and disc data Steel ball Glass disc diameter 9.05*0 - m 0. m running track radius m E modulus.07*0 Pa 7.5*0 0 Pa Poisson coefficient RMS surface roughness.5 nm 5 nm Table : Properties of Santotrac S70 and test conditions Santotrac S70 temperature ( 0 C) dynamic viscosity 0 (Pa.s) 97.65*0 -.4*0-5.5*0 - pressure-viscosity coefficient * (GPa - ) Test conditions range of load parameter values M range of lubricant parameter values L load (N) 0 4, 0, 50 0 maximum contact pressure (GPa) rolling speed minimum range (m/s) maximum,55,55,7 - -

23 Table a: Overview of estimates for the pressure-viscosity coefficient of Santotrac S70 (in Pa - ) Overview of estimates for the pressure-viscosity coefficient of Santotrac S70 (in Pa - ) temp 0 C load N target Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 40 4 N,68E-08,54E-08,96E-08,9E-08,4E-08 4,44E-08,9E-08,76E-08,0E-08,80E-08,74E-08,8E-08,74E N,68E-08 4,59E-08,6E-08,6E-08 4,08E-08 4,E-08,9E-08,7E-08,5E-08,E-08,98E-08,04E-08,98E N,68E-08 4,E-08,49E-08,4E-08,9E-08 4,09E-08,6E-08,49E-08,85E-08,85E-08,7E-08,77E-08,7E N,00E-08,8E-08,77E-08,77E-08,06E-08,0E-08,8E-08,E-08,5E-08,4E-08,E-08,6E-08,E N,50E-08,E-08,9E-08,9E-08,9E-08,E-08,86E-08,7E-08,E-08,9E-08,84E-08,9E-08,85E N,50E-08,47E-08,7E-08,7E-08,5E-08,99E-08,0E-08,78E-08,98E-08,96E-08,88E-08,9E-08,89E-08 Table b: Deviation from the target value of the estimates for the pressure-viscosity coefficient of Santotrac S70 (in %). Best values in bold italic. Deviation from the target value of the estimates for the pressure-viscosity coefficient of Santotrac S70 (in %) temp 0 C load N target Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 40 4N,68E-08 -,9% -9,7% -0,6% -4,7% 0,7% -0,5% -4,9% -,% -4,0% -5,7% -,% -5,5% 40 0 N,68E-08 4,8% -,7% -,9% 0,7% 7,5% 6,4% -6,% -9,0% -5,4% -9,0% -7,4% -9,0% N,68E-08 5,0% -5,% -7,4% 6,%,% -,6% -,4% -,6% -,6% -6,0% -4,6% -5,8% 60 0 N,00E-08 7,4% -7,8% -7,8%,9% 0,4% 9,% -9,7% -5,7% -9,6% -,7% -,4% -,6% 80 0 N,50E-08,% -,% -,% -4,6% -5,5% 4,5% -,7% -5,8% -,6% -6,% -,5% -6,0% N,50E-08 8,9% -9,% -9,% 0,4% -0,5% 0,8% -8,9% -0,9% -,7% -4,7% -,7% -4,5% average deviation,6% -9,% -9,9% 0,0%,% 4,8% -9,0% -6,% -,% -4,0% -,% -,9% average absolute deviation,9% 9,% 9,9% 6,4% 4,%,% 9,0% 6,%,% 4,0%,%,9% standard deviation of deviation 5,% 6,% 6,% 8,9% 7,% 4,5%,9% 5,0%,%,8%,5%,7% - -

24 Table c: Standard deviation in the film thickness prediction for Santotrac S70 (in m). Best values in bold italic. Standard deviation in the film thickness prediction for Santotrac S70 (in m) temp 0 C load N Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 40 4N,9E-08,70E-08,9E-08,9E-08,E-08,50E-08 8,89E-09,65E-08,5E-08,56E-08,50E-08,56E N,69E-08 7,64E-09 9,97E-09,00E-08,08E-08,95E-08 8,6E-09,66E-08 8,67E-09 7,98E-09 8,E-09 7,96E N,48E-08,5E-09 4,46E-09 4,4E-09,0E-08,64E-08 6,48E-09 4,6E-09 4,78E-09 4,54E-09 4,57E-09 4,5E N 6,94E-09,87E-09 5,9E-09,0E-09 5,65E-09 7,6E-09,4E-09,97E-09,85E-09,78E-09,96E-09,78E N 5,09E-09,0E-09,99E-09,99E-09 5,7E-09 5,44E-09,96E-09,9E-09,9E-09,79E-09,8E-09,79E N 6,44E-09,67E-09,04E-09,0E-09,7E-09 7,0E-09,8E-09,58E-09,60E-09,7E-09,75E-09,70E N,E-09,8E-09,8E-09,84E-09 6,7E-09,76E-09,4E-09,45E-09,80E-09,4E-09,48E-09,4E-09 av stdrd deviation,8e-08 5,7E-09 6,8E-09 6,49E-09 7,54E-09,50E-08 5,06E-09 7,E-09 6,59E-09 5,55E-09 5,5E-09 5,55E-09 Table d: Correlation in calculated and measured film thickness values for Santotrac S70 (-).Best values in bold italic. Correlation in calculated and measured film thickness values for Santotrac S70 (-) temp 0 C load N Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 40 4N 0,9875 0, ,9967 0,9966 0, , ,9985 0,9955 0,999 0, ,9960 0, N 0,985 0,9984 0,9976 0,997 0,9967 0,9805 0,9985 0,9979 0, ,998 0,998 0, N 0,9959 0, ,9996 0,9997 0,9964 0,999 0, ,999 0,9997 0,9994 0,9994 0, N 0, ,9994 0,998 0,9996 0, , , ,9997 0,9995 0,9999 0,9999 0, N 0,998 0, ,9988 0,9980 0,997 0,9957 0,9980 0, ,9987 0, , , N 0, ,9974 0, , , , ,9987 0,997 0,998 0, , ,9986 av correlation 0, ,9980 0, ,9976 0, , ,9985 0, , ,9989 0,9980 0,

25 Table 4: Average error in film thickness measurements per series, using correct pressure-viscosity coefficient for Santotrac S70 (in %). Lowest values in bold italic; values for a reduced number of experiments in italic, see text. Average error in film thickness measurements per series using correct for Santotrac S70 (in %) temp 0 C load N Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 40 4N -,% 8,8% 7,5%,8% -,4% -0,4% 9,4% 7,8%,0% 5,5%,8% 5,4% 40 0 N -9,9% 4,%,% 0,0%,6% -,0% 0,5% 5,%,4% 5,% 4,% 5,% N -8,% 5,0% 4,4% -,5% 6,6% -9,5%,5%,% 5,% 8,% 7,5% 8,% 60 0 N -,0% 9,9% 7,9%,4% 6,5% -5,%,7% 8,5% 6,5% 8,4% 7,% 8,4% 60 0 N -5,6%,7% 8,9%,4%,% -8,5%,4% 0,7% 7,5% 9,% 7,8% 9,0% 80 0 N -6,0%,5% 0,9% 6,% 6,% -8,9% 4,% 9,5% 0,0% 0,9% 9,5% 0,9% N -9,9% 5,6%,8% 6,5% 46,5% -,% 4,8% 40,7%,4%,%,%,% N -,6% 9,4% 7,%,% 6,% -4,7% 0,0%,6% 5,8% 7,% 6,% 7,% N -,7% 5,%,5% 7,6% 9,% -5,7% 6,9% 8,7%,%,8%,7%,8% av dev av abs dev -9,9%,% 9,4% 4,% 6,4% -,% 4,% 0,8% 8,% 0,% 8,9% 0,% 0,%,% 9,5% 6,7% 9,%,6% 4,% 0,8% 8,% 0,% 8,9% 0,% Table 5: Estimates, deviations and correlations for Santotrac S70 at 0N and 60 0 C if measurements at rolling speeds > 0.56 m/s are excluded Estimates of Santotrac S70 at 0N, 60 0 C and rolling speeds maximum 0.56 m/s temp 0 C load N target Archard & Cowking Hamrock & Dowson Hamrock et al. Chittenden et al. Evans & Snidle Hooke Sutcliffe Greenwood Venner Nijenbanning et al. Venner & Lubrecht Moes 60 0 N,00E-08 4,08E-08,8E-08,87E-08,8E-08,77E-08,5E-08,E-08,56E-08,49E-08,9E-08,45E-08,9E-08 deviation,00e-08 5,8% -5,8% -4,4% 6,% -7,9% 7,4% -6,4% -4,9% -7,% -0,5% -8,4% -0,4% standard deviation correlation 5,09E-09,0E-09,99E-09,99E-09 5,7E-09 5,44E-09,96E-09,9E-09,9E-09,79E-09,8E-09,79E-09 0,9944 0, ,9979 0,9979 0,9909 0,9948 0,9979 0,9968 0,9978 0, ,9976 0,

26 film thickness H load parameter M Figure : Nondimensional film thickness diagram for a circular contact, H = H(M) for L = 0. Legend to approximations: Moes; Venner and Lubrecht; Chittenden et al.; Archard and Cowking; Evans and Snidle; Hooke; Sutcliffe 00 lubricant parameter L H' = 0 H' = 7 H = 0 H = 5 H = 4 0 H' = 5 H = H' = H = H' = H' = load parameter M Figure : Contour map of nondimensional film thickness H vs. nondimensional load M and lubricant L parameter for a circular contact. Legend: contour lines for H = (based on Moes []); contour lines for H = 0 (based on Chittenden et al. []) ; VR/VE transition; Chittenden et al. [] and Hamrock and Dowson [7] data; S70 at 4N and 40 0 C; S70 at 0N and 80 0 C; S70 at 0N and 60 0 C; + S70 at 0N and 40 0 C; S70 at 50N and 80 0 C; S70 at 50N and 40 0 C - 6 -

27 central film thickness (nm) deviation from experimental result (-) 0,0 0,0,00 0,00 rolling speed (m/s) Figure : Central film thickness behaviour for S70 at 0 N (log scales). Legend: measurements at 40 0 C, at 60 0 C, and at 80 0 C.; calculations 0,0% 0,0% 0,0% 0,0% -0,0% -0,0% -0,0% 0,00 0,00,000 0,000 entrainment speed u (m/s) Figure 4: Deviations of the predicted from the measured value of the central film thickness, for the known value of the pressure-viscosity coefficient at 0 N and 40 0 C. Legend: Sutcliffe; Greenwood; Moes; Nijenbanning et al.; Hamrock and Dowson (977); - Venner (99), Venner and Lubrecht (000); Hamrock et al. (004); numerical solution ; Chittenden et al.; Evans and Snidle; Hooke; Archard & Cowking - 7 -