Analysis of the oil distribution in starved elliptical E.H.D. contacts

Tribological Research and Design for Engineering Systems D. Dowson et al. (Editors) 9 2003 Elsevier B.V. All fights reserved.

685

ANALYSIS OF THE OIL DISTRIBUTION IN STARVED ELLIPTICAL E.H.D. CONTACTS B. Damiens a, P.M.E. Cann b, C.H. Venneff, A.A. Lubrecht a. Laboratoire de M6canique des Contacts, UMR CNRS 5514, INSA de Lyon, France b Department of Mechanical Engineering, Tribology Section, Imperial College, London SW7 2BX, UK. University of Twente, Tribology group, Enschede, The Netherlands

1. I N T R O D U C T I O N In a recent paper, the authors analyzed starved lubrication of elliptical contacts [1]. The aim was to predict quantitatively starved film decay as a function of contact operating conditions. In this respect the work was an extension of earlier analysis of circular contacts by Chevalier et al. [2]. Although the elliptical contact analysis allowed quantitative prediction of the starvation level, it failed to provide a single variable capable of describing starvation as a function of ellipticity. However, the elliptical analysis did show that the degree of starvation can vary substantially with position across the contact. This variation was not found for circular contacts and hence a simple transformation from circular to elliptical is not possible. The current paper therefore explores some of the issues raised in the elliptical contact analysis. The aims were: (a) to provide additional insight into the local character of starvation in elliptical contacts. (b) to quantify the amount of oil lost from the track in the inlet and outlet regions. (c) to analyze the local oil film redistribution in the inlet region The numerical calculations were performed using a Multi-Level algorithm [3]. NOMENCLATURE ellipse axis in rolling direction =

ellipse axis perpendicular to rolling direction

b - a/ec- (3wR/E')l/3(2t~g/Tc)l/3/t~

Figure 1. Starved elliptical contact.

D E'

domain X0 _< X <_ Xe,-Y0 _< Y < Y0 reduced modulus of elasticity 2 / m t - (I M2)/EI nu (] L'2)/E2 material parameter, G- ~E' film thickness central film thickness dimensionless film thickness -

G h hc H

-

-

-

H - h(2Rg)/(a21<) Hc

dimensionless HcII dimensionless thickness H0 dimensionless Hoit dimensionless Ho~t dimensionless

central film thickness fully flooded central film

rigid body displacement inlet oil film thickness outlet oil film thickness Ho~,tffdimensionless fully flooded outlet oil film thickness L dimensionless material parameter (Moes)

686

m M n p Ph P0 P R

L=G(2U)I/4 parameter of the elliptic integrals ]C and $ m = (1 - ~2)1/2 dimensionless load parameter (Moes) M = W(2U) -3/4 number of disk revolutions (overrollings) pressure maximum Hertzian pressure Ph = (3w)/(2~ab) constant (Roelands), p0 = 1.96 10 s dimensionless pressure, P = P/Ph reduced radius of curvature R = (R;

+

Rx

reduced radius of curvature in x direction

Ry

reduced radius of curvature in y direction

= (R;: +

5Hi~ inlet film thickness reduction

SHout outlet film thickness reduction dimensionless parameter /' 128 16~r55

] Aoil

dimensionless oil profile wave length with respect to b 70 viscosity at ambient pressure dimensionless viscosity, ~ - ~?/r/0 Vl, v2 poisson ratio of bodies 1 and 2 p density p0 density at atmospheric pressure fi dimensionless density, fi = p/po 0 filling rate, ratio of the oil film thickness and the gap height ellipticity ratio ~ - a/b

lC(m) Rx/Ry = ~2(K: - $ ) / ( $ - ~2K:) rc relative oil film thickness =

/

:)

) 1/3

=

f o / 2 1 / V / 1 - m2 sin2(r

$(m)

= fo/2 V / 1 - m 2 sin2(r162

s

= (e-

ro~,t

relative oil film thickness

7~c

central film thickness reduction,

2. S T A R V A T I O N

~o~t

total film thickness reduction, T~out - Hout/Hout:f f inlet length s c< r s i : dimensionless inlet length

The starved regime is characterised by a lack of lubricant in the inlet of the contact. An incomplete oil film profile is imposed in the inlet. Experimentally, this oil profile corresponds to the oil profile found in the outlet of the previous pass. From a numerical point of view, it constitutes a boundary condition and the meniscus position is treated as a free boundary problem. 0 represents the ratio of the oil layer thickness (Hoiz) and the total gap (H) between the surfaces (0 = Hoiz/H) see figure 2. In the inlet, the gap H between the surfaces decreases as X increases while the oil thickness Hoil remains constant. Hence the fractional film thickness 0 increases steadily until a step occurs (0 = 1). The lack of lubricant delays the pressure generation which starts at the point where the full film is formed (at the meniscus position). Figure 2 (obtained numerically with the model presented in appendix A), shows the variations of the different parameters: the pressure P, the fractional film 0, and the gap H between the surfaces along the X axis, in the plane Y = 0. The shape of the oil layer in the same plane is shown in figure 3.

=

s S s f] S::

fully flooded inlet length dimensionless fully flooded inlet length = s / / / a c< ( L / M ) 1/2 Um mean velocity in x, Um = (ul + u2)/2 U dimensionless speed parameter U = (~oum)/(E'R~) w external load W dimensionless load parameter W = w / ( E ' R 2) X, Y dimensionless coordinates X = x/a, Y = y/b Xo, Xe, Yo dimensionless domain boundaries z pressure viscosity index (Roelands) a pressure viscosity coefficient dimensionless parameter ~ = c~ph

~ -- L ( 3M~2tc 2 (I+R~/Ry)2 1652 )

687

1.4 1.2 1

",,..

0.8

/

"....

0.6

/

/

X\

\,

/ i ~ i/'....... >./ ........................................................ i.... '.,~......

0.4 0.2

~ . 0 -2.5 -2

-1.5

-1

-0.5

0

0.5

1

x Figure 2. Centreline profile Y = 0 of an elliptical contact t~ = 0.63, M = 300, L = 10 and re = 1.

As a first step the oil removal of a single pass is studied as a function of the amount of oil available in the inlet. In figure 4 the relative oil thickness re accounts for the fully flooded central film thickness and the relative compressibility, re ranges from 0 to oe. The relative central film thickness T~c accounts for the fully flooded film thickness and ranges from 0 to I. The function has two asymptotes. On the starved asymptote, the film thickness increases linearly with the oil layer T~e = re. On the fully flooded asymptote, the central film thickness T~ = 1 is independent of the oil layer thickness.

1.2

,

,,

,,,

/

....

,

-

.

,

,

/

Traditionally the central film thickness prediction is the main goal of numerical calculations. However, this work focuses on the oil film loss from inlet to outlet. Consequently, the ratio of the inlet oil layer thickness Hoit, the central film thickness He and the outlet oil layer Hour was studied, see figure 3. In this figure one observes a reduction of the oil layer thickness going from inlet to outlet. The difference between Hoil and Hour represents the oil removed from the track by a single contact pass. By varying the inlet oil layer Ho~t, the evolution of the contact starvation as a function of multiple passes can be modelled.

0.8

oe /

~

............. ..... , .................. ................. :::::::::::::=:::::::::: .....

/'/

0.6

O o ....

s"

0.4 0.2 0

1

0

0.5

,

1

,

1.5

,i

2

. . . . . . .

i

2.5

3

rc

Figure 4. 7~e = f(rc) for a circular contact with M = 10 and L = 20, 7 = 2.66.

Chevalier [2] studied the circular contact and showed that a good approximation to the function Tie = f(re) is given by equation (1).

0.35 ~1~

...........................

0.3 0.25

__

0.2 ,.~

0.15

9

0.1

re

7~c -- ~/i -Frc~

Hoil

~

Ho~ ,7--

0.05 0

-2.5

i

i

i

a

i

-2

-1.5

-1

-0.5

0

,

L

i

0.5

1

1.5

X Figure 3. Centreline oil profile h(x, y = O) for an elliptical contact ~ = 0.63, M = 300, L = 10 and rc=l.

(1)

In [i] Damiens et al. noticed the linear relation between the parameter '7 and a dimensionless length S or (L/M)I/2 characteristic of the inlet zone (see figure 5). The parameter 31 can be considered as a resistance to side-flow. With increasing ellipticity (decreasing t~), it becomes more difficult for the lubricant to flow around the contact and thus "7 increases. For an infinitely long line contact, no side-flow is possible and "7 will be infinite. This tendency is shown in figure

688

5 where 7 is plotted as a function of ( M / L ) 1/2 for different ellipticities.

16

,

~

i

,

,

~

,

,

o

14 12 10 8 6

0

i

i

i

!

i

i

i

i

2

4

6

8

10

12

14

16

IM/L

Figure 5. 3' as a function of v / M / L for different ellipticities: a = 0 . 1 4 , 0.22, 0.35, 0.63, 1.00, rc~_l.

3. E J E C T I O N

Previous studies on starvation [2] have focussed on the lubricant loss from the inlet region. This is a valid approximation for fully flooded and mildly starved conditions, but not for the severely starved cases. As the contact becomes increasingly starved, the pressure distribution approaches Hertz. Hence, the pressure gradients in inlet and outlet become comparable, see figure 6. As the film thickness distribution is similar in inlet and outlet, this means that the oil removal in inlet and outlet regions become comparable.

1.2

:::_-

/

0.4

In [4], Notary and Lubrecht studied the starvation behaviour of the rigid isoviscous elliptical contact. They gave a more accurate approximation of the starvation curve using a fraction of two polynomials. However, the improved precision is obtained at the cost of a dozen constants. It is unlikely that a simpler relation can be found for the piezoviscous-elastic (EHL) starvation curve. Hence we will stick with equation (1)in spite of its mediocre performance to approximate the oil ejection in all the starved regime. The rest of this paper will treat the points 2 and 3.

. . . . . . .

0.8 0.6

Using equation (1) recursively, 7 can be directly related to the film thickness reduction as a function of the number of passes allowing experimental validation [1]. For the circular contact, good agreement was found between the numerical and experimental results. However, for the elliptical contact the agreement was less satisfactory. A number of factors can explain this difference: 9 equation (1) is only an approximation 9 7 oil removal occurs in inlet and outlet 9 starvation can be non uniform across the contact

IN INLET AND OUTLET

0.2 0 -2

- 1.5

-1

-0.5

0

0.5

1

1.5

X Figure 6. Centreline pressure profile for 2 oil layer thicknesses rc = 1.0 and rc = 0.3 with s = 0.35, M = 100 and L = 5.

Figure 7 shows the relative amount of oil lost in the outlet region, compared to the overall oil lost. In this case the inlet covers the region from X = - c ~ to X = 0, whereas the outlet is defined by positive X values. On the central line (Y = 0), the inlet oil ejection is defined as: SHin = H o i l - fiHc. In the same way the outlet ejection is defined as: 5Ho~t = f i H c - Ho~t. Please note t h a t / 5 > 1, hence, 5Hour > 0 even if Hc < Hout. Since a pressure gradiant in the Y direction tends to empty the track, a positive amount of oil 5Ho~,t is removed from the centerline going from the contact center to the outlet meniscus. The percentage of the oil ejection taking place in the outlet can be defined as:

689

4.

%outlet ejection = 100 9

5Hour SHin -~- SHout

(2)

Figure 7 shows its variation as a function of

(M/L) 1/2 for an elliptical contact (n = 0.35) and two different starvation conditions. The parameter (M/L) 1/2 determines the ejection process only up to a first order. Furthermore, the values of the ejection are computed numerically, and contain a certain numerical error. This combination causes the variations observed in figure 7. Nevertheless, the global trend which is of interest in this section is clear and consistent. For large amounts of oil (r~ ~_ I) only 5% of the lubricant ejection occurs in the outlet. Whereas roughly 40% of the lubricant ejection occurs in the outlet for thin oil layers (rc ~- 0.3). This shows the increased importance of outlet ejection under heavily starved conditions. Asymptotically, for values of rc going to 0, the pressure distribution becomes symmetrical between inlet and outlet. As a result, the ejection in the inlet and outlet becomes comparable. A similar trend is observed in circular contacts. As a consequence, the oil ejection in the outlet can be neglected for moderately starved conditions r ~ I. However, for severely starved contacts r < 0.5, the ejection in the outlet is almost as important as the one occurring in the inlet, and hence cannot be neglected.

5O 45 9~

r~ -- 0.3

9

rc =

9

1.0

40 35

~

30

25 O 9

~

20 15 10

5

9 9 9

0

o0o

9

,,

2

4

6

a

I

8

10

9

12

i

14

( M / L ) 1/2 Figure 7. Proportion of the outlet ejection for the oil layers rc = 0.3 and rc = 1.0 for n = 0.35.

OIL

PROFILE

SMOOTHING

In practice, the oil layer in the Y direction is not constant. Due to, for example, manufacturing defects, the oil layer can be close to zero at certain points (see Chevalier et. al. [5]). In such a case, it is important to know if the lubricant available in the neighbourhood of the defect is mobile enough to replenish the local lubricant shortage. Hence, it is important to study the lubrication coupling between points placed at different Y positions across the width of a starved elliptical contact. This section uses a non smooth oil profile Hoiz(Y) described by equation (3) to feed the ino let of the contact. Aoit is the dimensionless wave length of the oil profile, it is the real wave length divided by b, the half width of the contact in the Y direction. Note that the mean oil film thickness is r, and that locally zero oil film thickness exists in the inlet. In order to describe the lubricant mobility, one focuses on the ratio of the waviness amplitudes between inlet and outlet.

Hoiz(Y) = r 9 1 + cos k ~oiZ

(3)

The oil redistribution that occurs in the contact takes place in the low pressure region where Poiseuille flow occurs. Like the oil ejection, the redistribution should be observed in the inlet and outlet. The coupling C between two points of w the profile decreases as their distance (bAoiZ) increases. The further the points are from each other, the more independent they will be, the less the starvation level in one point will influence the lubrication in the other. The coupling will increase when the inlet length increases because as the inlet length increases, the low pressure region (in which the redistribution is assumed to occur) increases. Hence, the governing parameter in the coupling phenomena should be the ratio of these two distance. If the inlet length s is assumed to be proportional to the fully flooded inlet length multiplied by the amount of oil r on the track the coupling parameter can be expressed as"

s ra(L/M) 1/2 ~ _ C = bAoiz bAoit _

=

r(L/M) 1/2 _ Aoit/n

(4)

690

Figure 8 shows the evolution of the ratio of the waviness amplitudes in the inlet and outlet as a function of the coupling C. The points were calculated with the model presented in appendix A for average oil layer thicknesses of r = 0.1, 0.3, 0.5, ellipticities of ~ = 0.1, 0.2, 0.5, 1.0, with the Moes numbers M = 10,30,100,300,1000 and L = 5,10,20, and oil profile wavelength o f /koil = 0.1,0.2,0.5.

to an ejection shared between two sites of similar importance, inlet and outlet, as starvation increases (r decreases). A coupling (stronger for the circular contact case) seems to flatten the "/out profiles, rendering the oil ejection more or less uniform across the contact. This lubrication coupling is an important parameter to understand local lubrication failure. A parameter C describing this coupling is found to depend on the contact operating conditions M and L, the contact ellipticity a, the starvation level r and the lubricant defect width Aoit.

1

0.8

~3 .~

APPENDIX

9

0.6

% 0.4

A. M O D E L E Q U A T I O N S

0.2

To describe the lubricant flux in the contact, the Elrod model [6,7] is used. Wijnant [8] studied the starved elliptical contact and derived the corresponding dimensionless Reynolds equation (6). Capillary forces were neglected in this analysis. The Elrod model divides the calculational domain into two sub-domains: In the first subdomain, the pressure P is zero and the lubricant film is incomplete. Two lubricant layers separated by an air layer stick to the rolling element and plane. The lubricant travels with the moving surfaces. The filling rate 0, which is the ratio of the thickness of the oil layers and the total gap between the surfaces, is smaller than 1. In the second sub-domain, the film is complete, the pressure is positive and a lubricant side flow is observed. in sub-domain 1, 0 < 0 < 1 and P = 0 and in sub-domain 2, 0 = 1 and P > 0. The behaviour can be condensed in equation (5) :

0

.

.

.

.

.

0.0l

,

,

,L

0.1

1

C Figure 8. Ratio of the amplitude in the inlet and outlet as a function of the coupling C

The fact that the calculated points are more or less lying on a single curve indicates that the parameter C is governing the coupling under starved conditions. The ability of the parameter C to predict the coupling could be improved if a better approximation of the inlet length was used. 5. C O N C L U S I O N S The starvation behaviour of circular and elliptical contacts has been studied and compared numerically. Increasing the ellipticity (decreasing ~) increases the resistance to side flow (7). The oil ejection mechanisms are modified when the starvation degree increases. As starvation increases (r decreases), the relative film thickness reduction increases (7 decreases up to a factor 2 in elliptical contacts). The oil ejection location moves from a single site in the inlet (in fully flooded and moderately starved contacts)

P(1-0)

=0

(5)

The oil profile is imposed in the inlet of the contact. The position of the boundary (meniscus) between the two sub-domains is found by solving simultaneously the system of equations composed by the load equation (8), the modified Reynolds equation (6), the complementarity equation (5)

691

REFERENCES

and the equation of elastic deformations (8). The problem is a free boundary problem. The dimensionless parameters for an elliptical contact are defined in the nomenclature. The resulting dimensionless Reynolds equation reads:

ox

ox

+

1. B. Damiens, C. H. Venner, P. M. E. Cann, and A. A. Lubrecht. Starved lubrication of elliptical e.h.d, contacts, accepted at the ASME, Journal of Tribology, 2003. 2. F. Chevalier, A. A. Lubrecht, P. M. E. Cann, F. Colin, and G. Dalmaz. Film thickness in starved ehl point contacts. ASME, Journal of Tribology, 120:126- 133, 1998. 3. C. H. Venner and A. A. Lubrecht. MultiLevel Methods in Lubrication. Elsevier, ISBN 0-444-50503-2, 2000. 4. M.P. Notary and A. A. Lubrecht. Starved lubrication of isoviscous rigid circular contacts.

oY

_O(~OH.___~)= 0 OX 7)" Xo < X < X~,-Yo < Y < Yo

(6)

With the boundary conditions"

P(Xo, Y) =P(Xe, Y) =P(X, -Yo) =P(X, ]!o) = 0 and O(Xo, Y) = Ho~t(Y)/H(Xo, Y).

presented at the 29th Leeds Lyon symposium on Tribology. 5. F. Chevalier, A. A. Lubrecht, P. M. E. Cann, and G. Dalmaz. Evolution of lubricant film defects in the starved regime. Proceedings of

The dimensionless gap between the surfaces is given by:

the 2~th Leeds-Lyon Symposium on Tribology, H(X, Y) = Ho + S X 2 + (1 - $ ) y 2 9 1 ff P(X', Y') dX'dY' +-~--~ JJz~ v/~2(X _ X,)2 + (Z _ y,)2

The dimensionless rigid body displacement H0 is coupled to the force balance equation, which, for an elliptical contact reads:

/ f z P(X, Y) d X d Y = 27r

T

(8)

The viscosity pressure relation proposed by Roelands [9] is used" ~/(P) = exp { aP---2-~ + (1 + PPh )Z)

(9)

with

(7)

pages 2 3 3 - 244, 1997. 6. H.G. Elrod. A cavitation algorithm. ASME, Journal of Lubrication Technology, 103:350 354, 1981. 7. H. G. Elrod and M. L. Adams. A computer program for cavitation and starvation problems. Proceedings of the 1st Leeds-Lyon Symposium on Tribology, pages 3 7 - 41, 1974. 8. H. Wijnant, Y. Contact Dynamics in the field of Elastohydrdynamic Lubrication. PhD thesis, University of Twente, Enschede, the Netherlands, ISBN : 90-36512239, 1998. 9. C . J . R . Roelands. Correlation aspects of the

viscosity-temperature-pressure relationship of lubricating oils. PhD thesis, Technical University of Delft, The Netherlands, 1966. 10. D. Dowson and G. R. Higginson. A numerical solution to the elastohydrodynamic problem.

Journal of Mechanical Engineering Science, 1 : 6 - 15, 1959. aPo = in 70 + 9.67

(10)

Z

The compressibility is taken into account using the density pressure relation proposed by Dowson and Higginson [10]"

fi(P) =

0.59 109 + 1.34Pph 0.59 109 + PPh

(11)