The behaviour of transverse roughness in sliding ... - Science Direct

107

Wear, I53 (1992) 107-117

The behaviour of transverse roughness in sliding elastohydrodynamically lubricated contacts J. A. Greenwood Department WV

and K. L. Johnson

of Engineering, Universi& of Cambridge, Tnrmpington Street, Cambridge, CB2 1PZ

(Received August 21, 1991)

Abstract Recent calculations have shown that in a sliding elastohydrodynamically lubricated contact, any initial roughness largely disappears and is replaced by (often large) pressure variations. This paper gives an elementary analysis of the process for transverse roughness which provides the conditions under which it occurs, and allows the magnitude of the pressure ripples to be estimated. If the fluid is non-Newtonian, the behaviour will be very different.

1. induction Recent advances in ~mputational technique have enabled a number of authors [l-5] to study the behaviour of single transverse ridges or dents, or transverse sinusoidal ripples, on the stationary surface in a sliding elastohydrodynamic contact. Indeed, solutions have been obtained for the more difficult problem of longitudinal waviness [2-53, and the much more difficult transient problem of moving roughness [l, 3, 41, but these do not concern us here. In this paper we consider the extent to which the quantitative behaviour found by computation can be predicted from simple consideration of an infinite sinusoidal surface, and so separate the effect of roughness from that of the bulk elastohydrodynamically lubrication (EIIL) contact. The full numerical solutions show that if we start with a regular sinusoidal surface roughness, transverse to the entraining direction, then in simple sliding with the rough surface stationary, the waviness virtually disappears, to be replaced by sinusoidal pressure ripples (see Fig. 1). This occurs both when the ripples are across a line contact [l, 33, an elliptical contact [2] or a circular contact [3]. A single ridge also disappears, and is replaced by a pressure ripple of much the same size as if the ridge were part of a wavy surface [l, 31. In contrast, Lubrecht et al. [5] show rather limited flattening of sinusoidal roughness across a circular contact; we shall see below why this should be. It is possible that the basic result, the almost perfect flattening, should have been predicted theoretically, perhaps by consideration of the relative stiffness of film and elastic solid (see ref. 6); here we simply accept it as our starting point. It follows, as Kweh et al. [2] point out, that the amplitude of the pressure ripples is what we should expect from elastic contact theory; we shall develop this remark to obtain a criterion for asperity flattening.

0 1992- Elsevier Sequoia. All rights reserved

108

43 iZ

E

0.1 i_

1 L-

0.2 n Y

-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

x/a

Fig. 1. Longitudinal centre line pressure distributions and film thickness profiles for an elliptical contact (from ref. 2): (a) 140 “C, A=025 pm, (b) SO “C, A=025 km, (c) 50 “C, A =0.25 pm, (d) 140 “C, A=O. The undeformed roughness is shown superimposed upon the film profile for 140 “C.

2. Analysis: jn~m~~ssible

Newtonian lubricant

We consider the roughness to be sinusoidal ripples, perpendicular to the direction, so that lubrication will be governed by the one-dimensional equation, and we enquire whether the pressure needed to flatten these ripples can indeed be generated. With a finite contact, the pressures can only numerically, but if the contact area contains many ripples, then we may use elastic solution for a half-space covered by ripples. It is well known [7] that deformation of the surface of a half-space due to a pressure p=p+p1

entraining Reynolds’ elastically be found the simple the elastic

cos 2Trxlh

is v=

*11-d -

-

T

E

cos %x/A

(lb)

However, the pressure acts on both bodies forming the contact, and a pressure which eliminates the roughness on one body produces (if the two are the same material) a (negative) roughness of the same amplitude on the other. We are interested not in flattening one surface but in producing a film which is parallel, or almost so; we need therefore the combined deformation, which is

PPl cos 2nxlh v= rrE’ where

2/E'= (1 - v~~)IE~I- (1 - Y%%

(2)

109

Thus, an initial sinusoidal roughness of amplitude A can only give a parallel film if there are pressure ripples of amplitude pt. = ?rE‘AD superimposed on the smooth surface EHL pressure distribution. Now, the one-dimensional Reynolds’ equation for a lubricant whose viscosity obeys the Barus law q= q. exp(cup) can be written

h-h* h

_

-

h2

dp z

exp(-of4

(3)

6770~1

To give sinusoidal

h-h* h

=-

-

nz@

3%&h

pressure

of amplitude

pl, the film shape must be

exp( - q?) exp( - apl cos k-x/A) sin Zmclh

Since we are assuming

l-h/h*=yexp(-cpl

variations

an almost parallel

cos 0) sin 8 .

film, i.e. I (h -h*)/h I < 1 we have (5)

where

p==(nplh*2/3~O~Ih) exp( -q5) and 8=2nxlA Figure 2 shows these film thickness variations for a range of values of cyp, (scaled for convenience t’o keep the ma&mum amplitude constant). For ql 1)

rt ~ exp(stpl -*I (@Ppl + a)‘” Thus, the greatest

deviation p1 hr2

T

from a parallel

film is k&*, with

1 (7)

mBx^ 3Je rlminUlh (al)1 +iy

where qmin is the viscosity at the pressure minima (0=~). The analysis will be valid provided (1) the pressure ripples are small enough for the pressure to remain positive everywhere @, 2 the film thickness is almost constant. solutions: there is no trace of a Reynolds what they refer to as a ‘microconstriction’ that is, where Pmin is small. The appearance

exp( - CYP,in)when h = 0.2 mm; hence for This is confirmed by the various numerical ripple, except that Goglia et al. [l.] show at the edges of their hertzian contact is the same as our Reynolds ripple.

For a compressible lubricant the term (h -h *) in the Reynolds equation is replaced by h-h *p*Ip where h * and p* are the values at the pressure maximum, Accordingly,

(8)

~-h*=(~-h*)~+h*(~*/~-1~ where (h -h *lo is the previous result for an incompressible to take the density variation to be P -=PO

It-n,

(9)

l-+&J

with y= 2.266 X lo-’ have -p* P

fluid. It is usual in EHL

1p

m* N-r,

I-cos

(Y-P)Pl 1+

P@- +Pl)

p= 1.683 x 10e9 m2 N-l.

1+

*/oT+tpx

8 cos e)

Then

for, p=_Is+p~

cos 0 we

(10)

Figure 3 shows an example of the film thickness variation for a compressible fluid, with the amplitude of the Reynolds term arbitrarily taken to be one-fifth of the compressibility term. It is clear that because of the local nature of the Reynolds ripple, and the slow variation of the compressibility term, it is possible and convenient to treat the two terms separately. It is easy to show that for plausible values of p and p1 we can put eos e= - 1 in the denominator in eqn. (10) without serious error (or even drop p1 from the

111

f III Reynolds 0

ripple

Compressrbllity

X Film shape

Fig. 3. Combined film thickness due to ‘Reynolds’ variation and compressibility, q, = 10, kI = 0.2&. denominator completely!) whereupon compressibility leads to a sinusoidal film thickness variation, in phase with the original roughness, with amplitude k2h*, where &S!Pl kz= (1+&411ax)(~ +

W) %&&I

The original idea that the film becomes parallel must now be modified; the amplitude k&* is left after a change of shape of amplitude ~J@/+I&‘, so the original amplitude must be

=pt

(Y--P)h* $#+ (l+ aDmax)(l + [

Y?Lin)

1

(12)

This determines p1 (by iteration, starting with pmDx=pmi,,=fi). We note that for mineral oil and steel at 1 GN me2 the ratio of the two terms is oil compliance

44:

elastic compliance so that the elastic term will normally be dominant and the roughness will disappear. However, for roughness of very short wavelength this does not hold and the roughness will persist. 2.2. Comparison with complete numerical elastohydrodynamic lubrication solutions The most comprehensive information on the magnitude of the pressure ripples is given by Venner [3], who has a graph showing the increased maximum pressure Ap as a function of wavelength. Table 1 gives the values read from his graph, together with our values from eqn. (12). Agreement is excellent. The slightly smaller effect of pressurep at longerwavelengths that we find compared with Venner may be an error in reading his graph, or a real defect of our model due to our assumption of an infinite number of pressure ripples.

112 TABLE

1

Comparison

of amplitudes

of pressure

ripples with Venner

h (mm)

0.125

0.25

0.375

0.5

0.625

0.75

Venner Ap G-1) & @=3)

1.41

0.67 0.70

0.44 0.47

0.32 0.35

0.27 0.30

0.23 0.26

Present theory Pl @=I) PI @=3)

1.406

0.675 0.707

0.459 0.472

0.347 0.354

0.279 0.284

0.233 0.236

A = 0.5 pm, E’ = 226, all pressures

are in G.

TABLE 2 Comparison

with ref. 2

Case c, 50 “C Case b, 80 “C

1.34 0.70

Values in brackets

0.379 (0.33) 0.408 (0.36)

are estimates

r\=0.2 mm, A=O.25 pm, E’=227

0.00014 0.00035

0.0371 (0.04) 0.0210 (0.01-0.02)

from graphs given by Kweh et al [2]. GN m-*,p=l GN m-‘, ur=SO m s-l.

TABLE 3 Comparison

with Lubrecht

et al., ref. 5

h* (pm)

PI (GN m-‘)

0.229

0.482

A=0.05 mm, A=O.l m-‘.

pm, E’-200

0.853 GN m-‘,p-0.5

0.102

k&* (pm)

M*

0.195

0.023

GN m-‘, a=20

m2 GN-‘,

(pm)

q0uI=0.0833

N

We find, in agreement with Venner, that for h =0.125 mm at 1 GPa the amplitude of the pressure variation exceedsa so that the solution must be rejected: the pressure would have to be negative, and cavitation will occur. The flattening found by Venner is too complete to permit a numerical comparison; all that can be said is that our theory and his calculations agree that the residual variation of film thickness is small. Kweh et al. f2] give a case where the variation is just measurable; a comparison is given in Table 2. It should be noted that Kweh et al. consider transverse ripples crossing a circular hertzian contact, so our assumption of an infinite wavy surface is drastic. It is, however, surprising that the pressures for the finite contact should be the smaller ones. A final comparison may be made with Lubrecht et al. [53, who studied the effect of transverse ridges (among other cases) crossing a circular EHL contact. Unlike the other papers cited, they found large film thickness variations across the contact, accompanied by pressures falling to near zero in the valleys. Our calculations for their case are given in Table 3.

113

The combination of moderate #a and short wavelength gives a Reynolds ripple larger than the original waviness. Clearly, our analysis is inapplicable, but nonetheless it correctly indicates that flattening does not occur, and predicts the amplitude of the pressure variations with reasonable accuracy.

3. Discussion

Most of the authors quoted seem not to have been concerned with the aspects of wavy-surface EHL which interest us - the degree of smoothing, or even the magnitude of the pressure ripples - except in so far as they may lead to surface fatigue. Their primary interest is in the effect of waviness on the minimum film thickness and, in particular, whether this occurs at the rear of the contact as for smooth surfaces. They appear to have answered this question, but in an unhelpful way: both the average film thickness and the minimum film thickness may be either increased or decreased by the waviness - depending entirely on the phase of the waviness with respect to the end of the inlet. Apparent effects of wavelength seem to be due to the consequential, but inadvertent, phase shift, and are of no real interest, To us, the important consequence is that in heavily loaded contacts the waviness disappears (as does apparently a random roughness), but is replaced by large pressure ripples. (It is demonstrated by Goglia et al. [l], and Venner [3], and tacitly assumed in our theory, that the location of the waviness has no effect on the magnitude of the ripples.) The magnitude p1 of the pressure ripples is given by eqn. (12)

pl=A

2h z [

(Y-PP* + (1+f@4p,,)(l+ypmi,)

1 -l

and it appears that this will be valid, and the residual waviness (given by eqn. (11)) will be small, provided the minimum pressure is positive, say for example, pl