Elasto-hydrodynamic lubrication of centrally pivoted thrust bearings

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Elasto-hydrodynamic lubrication of centrally pivoted thrust bearings

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1995 J. Phys. D: Appl. Phys. 28 2371 (http://iopscience.iop.org/0022-3727/28/11/023) View the table of contents for this issue, or go to the journal homepage for more

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J. Phys. D: Appl. Phys. 28 (1995) 2371-2377. Printed in the UK

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Elasto-hydrodynamic lubrication of centrally pivoted thrust bearings

J A Greenwood and J J Wu University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK Received 20 April 1995 Abstract.’ Textbooks on lubrication show that a centrally pivoted hydrodynamic plane slider bearing can carry no load, and imply lhat in practical bearings the pivot must be offset. In practice offset pivots are the exception. This is attributed in the technical literature to a complex combination of thermal and elastic distortions. We show that operation is in principle possible purely as the result of the hydrodynamic pressures bending the pad to a desirable shape. Calculations for a wide pad suggest that successful operation requires only that a parameter p/4/(E/hmin) lie within the very wide range 20-330.

1. Introduction

Textbooks on hydrodynamic lubrication all give the analysis of a wide, plane, inclined pad; and note that optimal behaviour can be obtained for different running speeds by arranging that the pad is pivoted at a certain distance downstream of its centre, at the location of the centre of pressure for the tilt corresponding to the greatest load capacity. They may comment that the centre of pressure can only be at the centre of the pad when the tilt is zero, and that the load capacity is then also zero; and the more practical books may remark, apparently without noticing that this invalidates the entire analysis, that in practice centrally pivoted pads work well ‘and are often used in practice because of the convenience of being able to operate the bearing in either direction’. The Tribology Handbook (Neale 1973) has no doubts: it gives design rules for centrally pivoted pads and considers no others (and, contrasting with its treatment of journal bearings, gives no results of theoretical analyses). Neal and Soliman (1992) deduce from their experimental measurements that, although an offset pivot ‘provides. benefits which .. . may be of crucial importance ..., in many cases the penalties incurred in the use of centrally pivoted pads would be of little practical significance’. How does the centrally pivoted pad manage to ignore Michell’s solution of Reynolds’ equation (Michell 1905, 1950). and succeed in carrying a load? There is no doubt about our laboratory demonstration (air) thrust bearing: the ‘plane’ pads are not in fact planar but (very slightly) curved; and Raimondi and Boyd (1955) made the fundamental discovery that a very slight pad curvature completely alters the picture: the load capacity of a wide, centrally pivoted pad of given, constant, curvature can approach that of the optimally pivoted plane pad. Our calculations agree with their graphs, and show that an acceptable load capacity (i.e. one which is greater than half the optimum) is obtained 0022-3727/95/112371+07$19.50

0 1995 IOP Publishing Ltd

when the ratio of the ‘crown’ of the pad to the minimum film thickness lies in the range 0.035-2.5. This is a comfortably large range and the crowning needed is almost undetectable. The optimum ( p = fih$J(qul) = 0.16005) is when this ratio is 0.38: a crown of perhaps 4 fim. (Subsequently Raimondi (1961) showed that similar results apply for a square pad; recent calculations (Greenwood 1993) show that exceeds the value for the optimally pivoted plane square pad ( P = 0.0707) for 0.2 < S/ h ~ 0.12) are achieved for B , between 20 and 330. Figure 2 shows the variation of H,, = h,"/ho. The two parts of the curve correspond to h k n at the end of the pad ( B , c 100) and hhn at an internal point ( B , > 100): it appears that the maximum load capacity occurs just ufer h,, leaves the end of the pad. Figure 3(a) shows the film thickness for B, = 1 0 0 note that the shape is closer to two straight sections meeting at an angle than to a constant curvature; the corresponding pressures (figure 3(b)) are more or less parabolic, though the sides are almost straight. (The pressure has inflections at h = ISh* (X = -0.31) and at h = hmin (X = +0.40).) We have been unable to obtain solutions above B, = 1000 because of numerical instability; figure 4 shows the results for B , = 800. The shape of the pad is much as before, but the tilt is slightly greater, so that the film thickness now increases at the exit. The pressures now appear almost bell-shaped, the inflections having moved much nearer the centre of the pad there is a region of very slightly negative pressures near the end, which is clearly insufficient to give rise to cavitation, but suggesting that, for larger values of B , this would have to be allowed fort.

d2u El= m(x) dxZ

+

where EZ is the bending rigidity per unit width of the pad and m is the bending moment per unit width, related to the pressure p ( x ) by

with m = m' = 0 at x = rtf/2. It is convenient to take U = U' = 0 at the centre x = 0 and to introduce the pad inclination separately, so the film thickness is h = ho - p x U@). (3)

+

We introduce non-dimensional variables x = Xf. h = Hho, U = Vho, p = 9,5 and m = M(,512) (where overbarsdenote average values). Then (4)

which is solved separately (by standard numerical methods) for X c 0 and X > 0, using the boundary conditions M = M' = 0 at X = -I2 and at X = +I' 2 ' it can be shown that if we only consider pressure distributions whose centre of pressure is at the pivot X = 0 (which is the condition for moment equilibrium of the pad), then M will be continuous at X = 0. The displacements are given by -d 2=V

dXZ

BM

where B = p14/(EZh0), with V = V' = 0 at X = 0. Reynolds' equation becomes d9 =_6 7 ~ 1H - H* dX

phi

N3

H = 1 -AX+ V

*+:

where H* is chosen so that 9 = 0 at X = this again is solved by standard numerical methods. Then ).(E p l j h o ) must be found so that the centre of pressure is at X = 0 a Newton-Raphson iteration works very effectively. The results of (6) are scaled so that the mean value of q is unity in accordance with the definition of 9 (this incidentally gives the value of P = ,%i/(qul)), and substituted into (4) to repeat the cycle. The iteration may be started with a parabolic pressure distribution for B c 200 or with triangular pressures for B > 200; or more conveniently, in a series of solutions, with the pressures from the previous solution. (This proved to be the only way of obtaining solutions for B > 450.) 2374

3.1. Comparison with a constant curvature pad

The pad shape is no longer completely defined by the crown as i t was in Raimondi's work on pads of given, constant. curvature; but the ratio of the crown to the minimum film thickness again provides a useful measure of the shape. The values are very similar to the constant curvature case: the optimum is when 6/h,i, = 0.432, and P, exceeds 0.08 for S / h d n = 0.03G2.16. 3.2. Friction At this point we emphasize again that real bearings, except at low speeds, produce heat which reduces the viscosity of the oil. The effect on the friction (and so on the heat generation) is likely to be far more significant than the effect f Qualimtiivel$ it is cleac what happens for larger values of E,: the ntio of inlet film thickness LO minimum film thickness steadily increases. and lhe region of appreciable pressures steadily contract%so thal, to carry the same load, the minimum film thickness must also decrease.

Electro-hydrodynamic lubrication of bearings

I

I

0

0.5

X/I

'i-

I

-0.5

I

I

0

0.5

X/ {

Figure 4. Film shape and pressure distribution for B,,,= 808 (pressures just negative at rear). on the film thickness; the results which follow will be of very limited applicability. Friction is normally specified by the non-dimensional group F, = fh,in/(qul), where f is the friction force per unit width, and we shall follow this usage. However, clearly FO = f h o / ( q u l ) would be a better variable (i.e. one using the film thickness at the pivot) since its range is less and its behaviour simpler: FO rises gently from 1 to a maximum of 1.028 at B, = 750 before falling slightly to 1.025 at E , = 1000, the limit of our calculations. Because the variation of H,i, is relatively complicated, so is that of F,, as shown in figure 5; but in the useful range of B, the friction is consistently some 10% greater than that for the optimum plane pad (Fm= 0.754). The lubricant flow under the pad fuh' may be characterized by h*/h,jn. The (simpler) variation of h*/ho is included in figure 2. 4. Discussion

This appears to be the first attempt to produce a general theory of thrust bearings. It seems likely that the results of the present analysis can be applied qualitatively to a

pad of finite width; that there will again be a wide range of acceptable values of B, = p l ' / ( E I h ~ n ) perhaps , with the optimum value still around 100, and a corresponding maximum load capacity p h k , / ( q u l ) , probably similar in value to the 0.0707 found for the optimally pivoted plane square padt. It is not easy to make a comparison with the results of Sternlicht eta/ (1961a, b) because their variations in p are accompanied by consequential variations in inlet and outlet temperatures and hence by variations in the mean viscosity. As a result, even when ph$/(qul) possesses a maximum, this does not imply that h ~ has " a maximum. We note a conflict between the results found by varying the pad thickness in (1961a) (using the less accurate calculation " found, of pad deformation), for wbich a maximum h ~ was " increased with and those in (1961b), for which h ~ steadily increasing pad thickness. It seems clear that for large, fast-moving pads the viscosity variation is a major effect, and that we cannot estimate h,i, without performing a complete heat-flow analysis to find the mean viscosify. It is less clear that the viscosity distribution need be known. Thermo-elastic t Martin (1970) suggests a value of 0.0621 for pads which are slightly crowned in both radial and circumferential directions 2375

J A Greenwood and J J Wu

1.05

I

I

I

I

B,

-

Figure 5. The variation of friction with pad stiffness for a plane pad with optimum pivot ( K = 1 . 3 ) Fm=0.75. pad distortion might well be an important effect; but this depends critically on the temperature distribution in the pad. We find the postulate used by Castelli and Malanoski, that the back of the pad remains at the oil inlet temperature, startling (though highly conducive to pad bending!): we should expect a rather restricted oil flow round the back of the pad. Ettles (1987) pointed out that his regression formula (see the introduction) is equivalent to assuming an effective speed there of 0 . 0 1 5 ~ ;and that heat transfer coefficients are somewhat higher with point pivots than with line pivots, which impede the flow more. Certainly there is no simple picture possible with large pads. For smaller pads, Neal and Soliman (1992) reported tests on a four-pad thrust bearing in which they varied the pivot location with no other changes. The pads were sector-shaped, 38 mm long and 1 1 mm thick, and carried a mean pressure of 3.1 MN mm2. With a central pivot, a minimum film thickness of IO /m (at 4000 rpm) was measured, giving a value of B,,, = 28, which for a wide pad would give P,,, = 0.13 instead of the optimum value Pm = 0.16. The results that they give suggest that the curvature of the pad increased considerably as the pivot was progressively offset, which is understandable as a simple geometric effect if the cause is elastic bending by the hydrodynamic pressures. At the same time the pad temperature decreased, showing that thermal distortion was not the major cause.

4.1. Pad thickness We know of no discussion of the thickness requirements of a Michell pad other than a calculation given by Michell himself (Michell 1950) of the thickness needed to avoid elastic deflections which might invalidate the theory (Michell believed that the pad should remain planar). His recommendation is that the thickness of the pad should be at least 0.4 times the length: 'a somewhat thinner pad may

be used, but d / l can hardly be allowed in any case to be less than 0.30'. This is for an offset pivot: presumably for a central pivot (and so a shorter 'beam') the corresponding figures would be 0.32 and 0.25. The Engineering Sciences Data Unit, like the textbooks, offers no guidance on the design of centrally pivoted pads (though recognizing that they are 'often' used): for optimally pivoted ones the data sheet (ESDU 1983) recommends (without comment) thicknesses of 0.34.35 times the length. If the results of the present analysis can indeed be applied to pads of finite width, then we should make For a steel pad ( E = 200 GN m-') of uniform thickness d ( I = d 3 / 1 2 )this gives

d1 % (0.56-1.4)

which emphasizes how tolerant this mechanism is. Choosing as our preferred value B,,, = 80 (on the grounds that thermal distortion is to be added) gives d / l = 0.9 x 10-4(pI/h4n)''3.

For the 200 mm pads studied by Sternlicht et al and by Castelli, with p = 5 MN m-' and h4" = IO pm, this gives d / l = 0.42: the pads used had d / l = 0.28;but for smaller pads and pressures ( I = 50 mm, = 4 MN n r 2 ) we find d/Z = 0.25, close to current practice. 5. Conclusion

The operation of centrally pivoted Michell pads can be explained as the direct effect of the hydrodynamic pressures bending the pad to a desirable shape. Success requires that a parameter B , = p14/(EZhmin)should lie within the range 20-330, assuming that the results for a wide pad can be taken as a guide to the behaviour of a square pad.

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Electro-hydrodynamic lubrication of hearings

References Castelli V and Malanoski S B 1969 Method of solution of lubrication problems with temperature and elasticity effects: application to sector tilting-pad bearings J. Lub. Tech. 101 634-40 Engineering Sciences Data Unit 1983 Item 83004, Calculation methods for steadily loaded, off-set pivot. tilting pad thrust bearings Ettles C M M 1980 Size effects in tilting pad thrust bearings Wear 59 23 1-45 -1987 Three-dimensional computation of thrust bearings Proc. 13th Leeds-Lyoii SyinposiiNn on Tribology: Fluid Film Lubrication Greenwood J A 1993 Rectangular plane sliders, Cambridge University Engineering Department Report C-mech TR52 Martin F A 1970 Tilting pad thrust bearings: rapid design aids Pmc. Ins!. Mech Eng. 184 120-38 Michell A G M 1905 The lubrication of plane surfaces Z Math. Phys. 52 123-37 Lubrication (London: Blackie) -1950

Neal P B and Soliman M A M 1992 Influence of pivot location on the performance of tilting-pad thrust bearings lmlilule of Mechnnical Engineering Seminar on Plain Bearings Energy EIficiency and Design Neale M J (ed)1973 Tribology Handbook (London: Butterworths) Raimondi A A 1961 Inlhence of IongitudinaI and transverse profile on the load capacity of pivoted pad bearings ASLE Trans 3 265-76 Raimondi A A and Boyd J 1955 Influence of surface profile on the load capacity of thrust bearings with centrally pivoted pads ASME Tram. 77 321-30 Robinson C L and Cameron A 1975 Studies in hydrodynamic thrust bearings Phil. Trans. R. Soc. A 278 351-84 Sternlicht B. Carter G K and Arwas E B 1961a Adiabatic analysis of elastic, centrally pivoted, sector, thrust-bearing pads J. Appl. Meck 28 179-87 Stemlicht B. Reid J C and Arwas E B 1961b Performance of elastic, centrally pivoted, sector thrust bearing pads J. Basic Eng. 83 169-78

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