Thin film, time dependent, micro-EHL solutions with real surface roughness

Lubrication at the Frontier / D. Dowson et al. (Editors) © 1999 Elsevier Science B.V. All rights reserved.

163

Thin film, time dependent, micro-EHL solutions with real surface roughness. C D Elcoate t , H P Evans, T G Hughes and R.W. Snide. School of Engineering, University of Wales Cardiff, Cardiff, CF2 1XH This paper gives results of a time dependent analysis of the elastohydrodynamic (EHL) contact between a smooth disk and a rough one taken from a scutfing experiment. The rough steel surface is finished by transverse grinding and a line contact EHL model is used which can be expected to give a good approximation to conditions on the contact centre-line. A non-Newtonian fluid model is adopted and the EHL equations are solved using a coupled method. Results are obtained for situations where the micro contact film thicknesses are considerably smaller than the typical roughness dimensions. It is found that within the Hertz contact area the local shape of the deformed surfaces is controlled by the direction of relative sliding. Detailed roughness features only influence the hydrodynamic pressures (a) when the contact is in pure rolling, and (b) in the case of sliding when the micro contact film thickness is very thin compared to the roughness. 1. I N T R O D U C T I O N

The mechanism responsible for the protection of the surfaces of heavily loaded rolling/sliding contacts such as those between gear teeth is elastohydrodynamic lubrication (EHL). However, an important feature of gear tooth contacts is that the surfaces produced by present day manufacturing methods have roughness features that are of the same order or even significantly greater than the predicted thickness of the EHL film. Consequently they operate under conditions described as "partial", "mixed" or "micro" EHL. In available theoretical solutions of both the dry contact and micro-EHL problems the presence of significant roughness leads to a severe tippling in the contact pressure distribution with maximum values far in excess of the Hertzian values expected when the surfaces are perfectly smooth. A practical problem which is directly associated with these effects in hardened steel gears is that of "micropitting". This is pitting (rolling contact fatigue) on the scale of the roughness, as opposed to classical pitting which is on the scale of the nominal Hertzian contact. With increased use of hardened gears micropitting has become of widespread concern. The work reported in this paper forms part of an ongoing study of the lubrication of gear tooth t now at Frazer Nash Consultancy Ltd, Bristol.

surfaces with the specific aim of developing a fundamental understanding of the physical mechanisms of micropitting. Factors which affect the formation or otherwise of a lubricant film in such contacts include average and relative surface speeds, load, temperature, lubricant properties and component surface finish. Film-forming conditions in heavily loaded, high temperature gearboxes used in aerospace practice (for example) are often particularly severe because of the need to operate with relatively thin oils. A surface roughness R a value of 0.4 p.m is typical for such gears which must be considered in the context of typical predicted film thickness values in the range 0.05 - 0.5 p.m. If micro-EHL theory is to be of any use in helping to understand problems such as micropitting it must therefore be capable of modelling contacts having ~ ratios of 0.1 or less. Numerical modelling of rough surface EHL contacts has been undertaken by numerous investigators in recent years. Initial efforts involved stationary idealised roughness features Goglia et al. (1984), Lubrecht et al. (1988), Chang et al. (1989) and Kweh et al. (1989). Real roughness features were introduced by Venner and ten Napel (1992) who used a roughness profile with an R~ of 0.04 p.m, and by Kweh et al. (1992) who

164

used a profile with an R a of 0.3 txm that was of the same order as the film thickness. Transient effects have been studied by Chang and Webster (1991), Venner and Lubrecht (1994) and Greenwood and Morales-Espejel (1994) using sinusoidal features or waveforms. Chang Webster and Jackson (1993) and Chang and Zhao (1995) have drawn attention to the difference between Newtonian and non-Newtonian lubricant models in transient micro-EHL. Real roughness was included in a transient line contact by Ai and Cheng (1994). More recently transient point contact analyses have been presented by Venner and Lubrecht (1996) using sinusoidal features and Xu and Sadhegi (1996) using measured roughness data. Where real roughness data has been incorporated in transient solutions it is generally of a relatively low amplitude compared to the film thickness in the numerical solution. This is indicative of the numerical difficulties that occur when the roughness features are large compared to the residual film thickness. The work reported in this paper follows on from an investigation into the benefits of coupling the elasticity and flow equations Elcoate et al (1998). This technique was described by Okamura (1982) who used a fully coupled Newton-Raphson model. This was subsequently extended by Houpert and Hamrock (1986) to enable analysis of heavily loaded, smooth line contacts assuming Newtonian lubricant behaviour. The advantages of coupling the elastic and hydrodynamic equations have not been exploited in the past because of the resulting 'full matrix' problem and the computing resources required to solve it. These difficulties have been largely overcome by the authors (Evans and Hughes, Elcoate et al (a)), so that time dependent models using this technique are now realistic as demonstrated in this paper.

2. NOTATION b

El, E2 E'

Hertzian semi-dimension. Young's moduli of the two surfaces. 2 / E ' = ( 1 - v 2 ) / E l + ( 1 - v2)/E2 .

h hmin

film thickness. minimum film thickness. heffeetiveeffective film thickness. p pressure. R radius of relative curvature. l / R = 1/R 1 + I / R 2. R l, R2 radius of curvature of the surfaces. Ra Roughness average of surface. u elastic deflection. O mean entraining speed = (Ul + U2) / 2. U l, U 2 surface speeds relative to contact. load per unit length. W! co-ordinate in direction of rolling. x pressure viscosity coefficient. o~ parameters in density relationship. )', K heffective/Ra

rl rio p Po 1;0 V 1, V 2

viscosity viscosity at zero pressure density. density at zero pressure. Eyring shear stress. Poisson's ratios of the two surfaces. slide roll ratio = 2(U1- U2)/(U l + U2).

Other symbols are defined in the text. 3. FORMULATION In conventional involute gears the teeth are finished in a direction transverse to that of rolling/sliding and contact is nominally along a line. The results presented in this paper have therefore been obtained with a one dimensional EHL line contact model. Both surfaces are treated as semi-infinite elastic solids so that the displacement normal to the surface is given by:

ux/=- ~E---4I7 p(s) 1

r

ds

--o0

where r is the co-ordinate of the point (usually 2b downstream of the contac0, relative to which the displacement is established. The film thickness is given by: h(x) = q)(x, t) - ~ 4 f°° p(s) I l n[x - : ds + C ~E' -~o r

(1)

where qo(x,0 incorporates both the relative curvature of the two surfaces and the measured

165

roughness of the run-in, ground surface. The constant C is selected to obtain the required load. Isothermal conditions are assumed and the viscosity and density of the lubricant are taken to depend on pressure as follows: 11= rioe l+yp P=Po~ l+Kp The non-Newtonian flow equation adopted is the second order, one dimensional, modified Reynolds equation developed by Conry et al. (1987) to which the appropriate squeeze film term is added to enable time dependence to be considered. ~ph)

t9 (lah 3 o~pS/ - U/9(0h)

(2)

where: S = 3(X cosh X - sinh E) I

E3

1+

1~2(U2 _ U1)2

)-,2

x2h2

sinh 2 5".

and: E= h c3P 2x 0 tgx The representative shear stress, x0, beyond which level the fluid response is non-Newtonian is taken as constant. An implicit formulation for the time dependent term in equation (2) is adopted and each time step

consists of a small number of iterative cycles (typically 4) within which linearised numerical representations of equations (1) and (2) are solved in a coupled scheme as described by Elcoate et al

Co). A finite element method using the weak formulation of the Galerkin weighted residual approach has been adopted to solve equation (1). The domain is discretized using one dimensional quadratic elements and the numerical integration of the resultant equations is carded out on a local elemental level by 3 point Gauss quadrature. 4. ROUGHNESS MODEL

The roughness data used for qo(x,t) is taken from a test disc using a surface profilometer. The ground surface analysed was that from a well run in but unscuffed experimental disc. The disc measured was one used by Patching (1994) in a two disc scuffing rig. The disc was manufactured by transverse grinding to simulate the finish found on gear teeth in aerospace auxiliary gearboxes. The initial distribution of surface heights at manufacture was essentially Gaussian. After running the surface was measured in a circumferential direction along the centre of the running track with heights taken at a spacing of 3.5 ~tm. The resultant profile is shown in Figure 1, where the rounding off of asperity tips caused by the running-in process is apparent. The profile has an R a value of 0.32 ~tm with maximum peak to valley dimensions of

3.0 2.5 '~::L 2.0

"i~

1.5

1.0 0.5 0.0 0

250

500

750

1000

1250

Traverse/Itm

Figure 1. Section of surface profile used in contact (metal above curve) showing extent of running-in.

1500

166

specifies the parameters that remain unchanged in the analysis. The load adopted results in a smooth surface Hertzian contact having b = 0.358 mm and a maximum Hertzian contact stress of 1.07 GPa.

approximately 2 p.m. The resolution adopted for the profilometer gives approximately 200 height measurements over the Hertz contact area. A cubic spline interpolation is used to provide surface heights at intermediate points as required at each timestep in the time dependent solution. In this way the surface is always represented by the same collection of piecewise cubic splines. This approach has been found useful in eliminating spurious squeeze film effects that can be introduced in interpolating between measured height values. The computing nodes are fixed relative to the point of contact so that there is no time dependence of the basic curvature terms of the two surfaces.

Table 1 Conditions assumed for the real rou~:hness analyses R 19.05 mm xo 5.0 MPa w' 600 kN/m rio 0.0048 Pas E 227.3 GPa cz 11.1 GPa "1 The behaviour of the rough surface in this EHL contact can most easily be appreciated by constructing an animated sequence of film and pressure distributions as the time steps progress. The striking feature that becomes apparent in viewing this animation is that the film thickness on an individual micro contact is formed at the inlet to the Hertz area and the deformed shape of the asperity then progresses through the contact in a broadly unaltered way. The pressure distribution associated with the deformed shape keeps pace with the micro contact as it moves. This is in contrast to earlier studies e.g. Venner and Lubrecht (1994)

5. REAL ROUGHNESS RESULTS Results were obtained for a sequence of eleven slide-roll ratios ~ = ± 2, ± 1.2, ± 0.6, ± 0.3, ± 0.15 and 0.0. The convention adopted is that positive values have the rough surface moving faster and case ~ = -2 has a stationary rough surface for which the problem is not time dependent. A range of three entraining velocities was used to simulate different film thickness conditions. Table 1 2.0

10.0

1.5

7.5

1.0

5.0

j

0.5

ft

'%

2.5

0.0

0.0

-2.5

-0.5 -1.5

-1

-0.5

0

0.5

1

1.5

x/b Figure 2. Pressure, film thickness and undeformed roughness profile (thin curve) shown at time step 5000.

167

2.0

10.0

1.5

7.5

//// V

S oo

--~

~"-

J

oo

-0.5

-2.5 -1.5

-1

-0.5

0

0.5

1

1.5

x/b Figure 3. Pressure, film thickness and undeformed roughness profile (thin curve) shown at time step 5500. where film shape disturbances were seen to move through the Hertz contact area at the entrainment velocity.

and film thickness at two such positions for the case of ~ = - 0 . 3 (timestep 5000 and 5500). The entrainment velocity of 25 m/s results in minimum film thicknesses of the order 0.12 ~tm as the different surface roughness features pass through the contact area. This compares with a Dowson and Higginson smooth surface value of 0.43 ~tm.

Figures have been obtained with the rough surface profile in a number of fixed positions which are offset by 0.5b. Figures 2 and 3 show the pressure 2.0

10.0

1.5

I I

7.5

J t~

C,~ 1.0

5.0

g~

0.5

V

"1/

p

r

2.5

0.0

0.0 -1.5

-1

-0.5

0

0.5

1

1.5

Figure 4. Pressure (heavy curve) and film thickness (thin curve) for section of profile at five timesteps during traverse of contact area with { = -0.6. Undeformed roughness profile for section also shown at top left.

168

3 .-¢ 2

0

0.1

0.2

0.3

0.4

0.5

Figure 5. Superimposed film thickness at five positions during traverse of contact area with g = -0.6. entry to the Hertz contact, at three positions within the contact and at the exit to the contact. (The second and third positions correspond to the timesteps illustrated in Figures 2 and 3 for ~ = -0.3) The deformed shape of the micro contact can be seen to be essentially the same within the Hertz contact as it progresses from entry to exit. This is emphasised in Figure 5 where the five timesteps have been aligned with each other and drawn on a larger scale.

Between the two timesteps illustrated the surface features have moved through 0.5 b and can clearly be recognised not only in the undeformed geometry shown offset at the bottom of the figure, but also in the deformed geometry of the contact. A particular length of surface defined by 100 node points and its associated pressure distribution is shown in Figure 4 at five positions during its transit of the contact area with ~= -0.6. It is shown at the 2.0

1.5

10.0

F_ I

7.5

1.0

5.0

0.5

2.5

,.=

g~

0.0

0.0 -1.5

-1

-0.5

0

0.5

1

1.5

x/b Figure 6. Pressure (heavy curve) and film thickness (thin curve) for section of profile at five timesteps during traverse of contact area with ~ = 0.6. Undeformed roughness profile for section also shown at top left.

169

3 :::L 2

0

0.1

0.2

0.3

0.4

0.5

x/b Figure 7. Superimposed film thickness at five positions during traverse of contact area with ~ = 0.6. adjusting to the sense of the relative motion of the surfaces. In Figure 5, the smooth surface is moving faster so that the inlet to the micro contacts is on the left, and the micro contacts align so that they form a converging film in the direction of entrainment relative to the micro contact. When the direction of sliding is reversed then so is the slope tendency of the micro contacts. In Figure 7 the rough surface is moving faster so that relative to

Figures 6 and 7 show the corresponding results when the slide roll ratio has the opposite sign, i.e. = 0.6. Again the deformed shape of the micro contact can be seen to be essentially the same throughout the contact with the pressure form adjusting to produce this shape at the various locations. A comparison of figures 5 and 7 shows that the deformed shape within the contact is 1.5

4

1.0

~

2 0.5 1

0.0

0

-1.5

-1

-0.5

0

0.5

1

x/b Figure 8. Pressure and film thickness at the same profile position with ~ = -1.2, -0.6 and -0.3.

1.5

170 4

1.5

3 1.0

b

i

0.5

0.0 .5

-1

-0.5

0

0.5

1

1.5

x/b Figure 9. Pressure and film thickness at the same profile position with ~ = 1.2, 0.6 and 0.3. the asperity entrainment is from fight to left, i.e. in the opposite sense to the entrainment of the whole contact. Once a micro contact is within the Hertz contact area the deflected shape adjusts so that a converging film is again formed in the direction of entrainment relative to the micro contact. This observation supports the basic assumption made by Evans and Snidle (1996) in proposing a scuffing failure model based on side leakage from deep valley features. The effect of magnitude of sliding speed is also interesting. When cases are compared at the same position of roughness there is considerable similarity between those having the same sign of ~, with the case ~ = 0 emerging as a clearly very special case. Figure 8 compares cases with = -1.2, -0.6 and -0.3 for a particular roughness location. The deflected shape of the micro contacts can be seen to be similar with differences in micro contact film thicknesses in the Hertzian region. At the inlet the smallest film on a micro contact occurs for the case with the highest sliding speed, but the position is reversed for micro contacts in the exit where the smallest film thickness is associated with the lowest sliding speed. The differences in pressure between the three cases follow a

corresponding pattern as can be expected from elasticity considerations. The same conditions but with the rough surface running faster, i.e. ~ = 1.2, 0.6 and 0.3, are shown in Figure 9. In these cases there is less variation between micro contact film thicknesses. Those in the inlet now have smaller films when the sliding velocity is smallest, and this pattern is generally the case for all the micro contacts. Again the differences between the pressure curves correspond with lowest pressure associated with lowest film. For positive slide roll ratio the squeeze film effect will be largest at the highest value of ~, whereas when g is negative the squeeze film effect will be largest at the smallest absolute value of ~. This effect explains the different dependence of micro contact film thickness at the inlet to the Hertzian region.

When the sliding speed is identically zero the result is a special case as illustrated in Figure 10. The fluid in this regime behaves very differently as the square root term in the non-Newtonian factor S becomes identically unity. Consequently the effective viscosity is much higher than in all the other cases considered (where there is sufficient

171 1.5

!

1.0

2

El =L

0.5

/ 0

0.0 .5

-1

-0.5

0

1.5

0.5

x/b

Figure 10. Pressure and film thickness with ~ = O. 2.5

12.5

2.0

10.0

1.5

7.5

Et

5.0 gl,

0.5

2.5

II

0.0

0.0

-o.5

~

1

I

-1.5

-1

-0.5

v

1

1

0

0.5

-

1

-2.5

1.5

x/b Figure 11. Pressure, film thickness and undeformed roughness profile (thin curve) shown at time step 5500 for case with U=5 m/s, ~ =-0.3.

172

sliding to make the t e r m

YI(U2- Ul)

result of the corresponding dry contact analysis (note that in comparison with Figure 11 the pressure scale has been reduced by a factor of two). The maximum pressure in dry contact is about 3.5 GPa and there is a significant increase in the number of individual micro contacts. The subsurface stress field arising from the pressure loadings illustrated in figures 11 and 12 will be very different, as will the number and intensity of the pressure loading cycles experienced as the surfaces move past each other.

significant

"t;oh

compared to confined in through the dictated by features first

unity). As a result the liquid remains the surface features as they pass contact and the pressure response is squeeze film effects as the asperity come into the contact area.

The same pattern is seen to emerge when results are examined at conditions with lower entrainment velocities of 10 m/s and 5 m/s. The minimum film thicknesses on the micro contacts for these cases are of the order 0.04 ~tm and 0.02 }xm respectively to be compared with Dowson and Higginson values of 0.23 txm and 0.14 }xm. At these very thin film values there is less variation possible over the contact area. The waviness in the run-in profile is more completely removed and there is, inevitably, a tad beating e il g tendency to higher pressures on the load i cell p il g asperities. This effect can be seen in comparing latt,~ t ue Figure 3 with Figure 11 where the latter figure nt s ~ ff corresponds to the lowest entrainment speed of IveIt inr this :t is 5 m/s with ~ - -0.3. However , even extremely thin film situation the hydrodynamic Mr¢, I ~ c co pressure remains significant betweent tttt the Imicro ied onr the contacts and the load is not carded ~ 1thee individual micro contacts. Figure 122 sla shows 4.0

6. CONCLUSIONS The conclusions that can be drawn at this stage of the investigation based on the analysis of lubricated contacts with a single run-in ground surface are as follows. Film thickness is controlled by micro contact shape rather than by roughness detail. , Inside the Hertzian contact area micro contacts have films that are related to the relative (sliding) velocity of the surfaces. . Pure rolling is a special case where the lubricant pressure is strongly influenced by the roughness detail. • t,At very v,." low ~, values (0.0625 for the case c, ~si ker~, the detailed shape of the asperity lands considered) • 12.5

/ ij ,l I/ ltl/lla/I,

-

3.0

2.0

10.0

!

7.5

.11 ,!,!11-111111.11'1tlt I1 I lllll

III

1.0

..= 5.0

h

_

0.o

.

_1 L~

2.5

,L.

-1.0 -1.5

-1

0.0 -0.5

0

0.5

1

x/b Figure 12. Pressure (upper curve) and film thickness distribution in dry contact.

1.5

173

begins to influence the hydrodynamic pressure distribution. The results also draw attention to the need to consider the actual shape of rough surfaces in any lubrication contact analysis and not the 'as manufactured' ones. To understand the processes at work in the failure of such ground surfaces it is necessary to consider the nmning-in process and how it is related to the loading and kinematic history of the contact. 7. A C K N O W L E D G E M E N T

The work reported in this paper has been supported by EPSRC Grant GR/L90996. REFERENCES

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