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Surface Roughness in Elastohydrodynamically Lubricated Contacts
Thinning Films and Tribological Interfaces / D. Dowson et al. (Editors) © 2000 Elsevier Science B.V. All rights reserved.
25
Surface Roughness in Elastohydrodynamically Lubricated Contacts C.H. Venner a, M. K a n e t a b, and A.A. Lubrecht c a University of Twente, Faculty of Mechanical Engineering, Enschede, The Netherlands. bKyushu Institute of Technology, Department of Mechanical Engineering, Kitakyushu, Japan. CLaboratoire de M@canique des Contacts, UMR CNRS 5514, INSA de Lyon, France. The nominal film thickness at which EHL contacts have to operate reliably has decreased to a level where the effects of the surface microgeometry on the film thickness can not be neglected. To predict the effects of the microgeometry on the film thickness one should be able to predict its deformation as a function of the operating conditions. Recent work on the effects of waviness in EHL contacts has shown that the amplitude reduction of harmonic patterns in line and circular contacts under pure rolling could be described as a function of a single nondimensional parameter. In the present paper this work is extended to elliptical contacts. The result is a simple formula that predicts the amplitude reduction of waviness in an EHL contact under pure rolling as a function of its wavelength and the operating conditions of the contact. This formula can be used as a simple tool to obtain a first estimate of the deformed roughness profile inside the contact for any microgeometry. Subsequently, attention is directed to experimental validation of the observed behaviour. Although further experiments are needed the first results are very encouraging.
1. I n t r o d u c t i o n
Nowadays EHL contacts have to operate reliably under conditions where the film thickness is much smaller than could have been imagined when the classical film thickness equations were developed [1,2]. As a result the microgeometry of the surface will have a significant effect on the operation of the contact and to predict performance, a detailed understanding of the effects of microgeometry on film formation is as important as ever before. Fortunately the available tools to investigate these effects have improved considerably over the years. For example, experimental techniques have advanced to a level where roughness effects on film thickness can be measured accurately inside the contact [3-7]. Also theoretically impressive progress has been made in the past decade. The introduction of fast and stable algorithms for numerical simulations has led to the situation where the transient non-smooth EHL point contact problem can be simulated in great detail, e.g. see [8-11]. Also the underlying mechanisms governing the solution of the transient EHL problem have become well understood [12,13].
The challenge researchers and engineers face now is to use this experimental and theoretical capability not just to illustrate effects for single cases, but to derive engineering tools, i.e. simple prediction formulas for practical use, that describe the effects of surface roughness on the lubricant film. Obviously due to the deformation of the roughness profile the well known h/a ratio with h taken as the nominal film thickness and a as the standard deviation of the undeformed roughness profile is of little use to quantify the effect of a given roughness profile on the film thickness. A first step towards an improved parameter could be a simple tool to predict the deformed roughness profile inside the contact. To obtain such a tool by studying the rough surface problem directly, e.g. by means of numerical simulations using rough surfaces as input, is rather cumbersome. For example, the enormous amount of d a t a generated (different surfaces, load conditions etc) would make it hard to extract general tendencies. The authors have therefore followed an alternative route. Instead of studying the rough surface problem, the deformation of harmonic patterns (waviness)
26
has been determined in detail. First the line contact problem was studied [14]. It was shown that the reduced relative amplitude under pure rolling could be predicted accurately with a simple function of a single non-dimensional parameter. Theoretical support for this behaviour was provided by Hooke [15]. Next it was shown that the amplitude reduction of harmonic patterns in a circular contact followed the same behaviour [16-18]. These results show that there appears to be a unifying mechanism governing the amplitude reduction of harmonic patterns in EHL contacts. In the present paper this subject is investigated further by studying the amplitude reduction of harmonic patterns in elliptical contacts. It is confirmed that indeed a unifying mechanism exists. As a result the amplitude reduction of a harmonic pattern in an EHL contact under pure rolling can be predicted quite accurately with a single formula. This formula can serve as a simple tool to predict the deformed geometry inside the contact for any surface microgeometry by applying it to each of the Fourier components of the microgeometry followed by backtransform to obtain the deformed profile. Subsequently, attention is directed to experimental validation. Using a ball on disk machine film thickness measurements have been performed for the case of an artificially produced shallow waviness pattern on the ball. The measured film thickness profiles are compared with those obtained by numerical simulations.
2. N o m e n c l a t u r e Hertzian contact length a = ( 3 I R / E ' ) I / Z ( 2 ~ E / r ) 1/3 A Dimensionless amplitude b Hertzian contact width b=a/~ c Hertzian approach c = (~I(2R))(~cI~) D ratio reduced radii of curvature D = R~/R~ I:) Dimensionless elastic deformation E Modulus of elasticity
E'
Reduced modulus of elasticity 21E' - (1 v2)lE1 + (1 v2)lE2 Elliptic integral (second kind) E = ~ ( 1 - ~2) External nominal load Dimensionless external force Dimensionless materials parameter G=aE' Film thickness Dimensionless film thickness H =h/c Dimensionless time step Dimensionless mesh size in X, Y Elliptic integral (first kind) : ~ ( 1 - ~2) Dimensionless material parameter L : G(2U) 1/4 Dimensionless load parameter M = W ( 2 U ) -3/4 Pressure Maximum Hertzian pressure Ph = (3 f ) / ( 2 r a b ) Dimensionless pressure P = P/Ph Reduced radius of curvature I l R = IlR~ + l l R y Reduced radius of curvature in x lln~ = llR~l + l/n~2 Reduced radius of curvature in y 1 / R v = 1/Rljl + 1/Rij2 Radius of curvature surface 1,2 in x Radius of curvature surface 1,2 in y Slip parameter S=ull~ Dimensionless shape factor s = (z- ~c)/(~c~c) Time Dimensionless time T = tu,/(2a) Surface velocity Sum velocity u, = (Ul + u2) Average velocity fi = (ul + u2)/2 Dimensionless speed parameter 2U = (17ou.)l(E'R=) Dimensionless load parameter W = F/(E'R 2) Coordinate in direction of rolling -
E
/ F G h H hT
hx,hy
M P Ph P R RX
Rv R z 1 ~Rx2
R~ I , R,2 S
t T
a
f~
Us
U W X~ X !
-
27 X, X'
Dimensionless coordinate X = x/a, X' = x'/a Coordinate perpendicular to x Dimensionless coordinate Y = y/b, Y ' = y'/b Viscosity index (Roelands) Pressure viscosity coefficient Dimensionless parameter
y, y' Y, Y' z a c~
5 ~ - O~ph
5 A
Mutual approach Dimensionless mutual approach A =~lc Coefficient in Reynolds' equation
e
-(~H3)/(~)
Ellipticity ratio n, = a/b Poisson ratio Dimensionless speed parameter = 6(rlousa)/(c2ph) Wavelength in x direction Wavelength in y direction Viscosity Dimensionless viscosity
u A Ax A~ 7/ ~/
Dimensionless natural frequency a 2 = (4fa2)(mu2c) a~ - ( 8 f R E ) ( m u ~ H , ( ) Density Dimensionless density - p/po slip parameter
p fi E
subscripts a,b i,d F 0 1,2
3.1. Equations In this section dimensionless equations are given describing the EHL point contact under fully flooded conditions. The dimensionless equations have been obtained using the Hertzian dry contact parameters (see Nomenclature) and the lubricant viscosity and density at ambient pressure. This gives the following dimensionless variables: X-x/a H = h/c P - - P/Ph P -- PlPo
Y =y/b A = 5/c T = tu~/(2a) (7 = ~7/Yo
The resulting dimensionless Reynolds equation is:
inlet, outlet undeformed, deformed force constant, e.g. at ambient pressure surface 1, 2
O~- "-- L 1/3(Ir2t¢(1_/~2) 2 )1/3 ~" (~-~) 16(E-~2 JK)2) : ! 132)~//r:)1/3 (167r~E_~2K) 5 1/3 ~4(1--~2)K0 ) (K-E) ~2 (E-~ ~ ~)
16(E-a
0 (0P
OX
c-ff-~ O(pH) OX
O(#H) cOT
=0
(1)
where
-- (Ul -- U2)/U
Relations between parameters
")'
In the first part of this paper the deformation of specific harmonic surface patterns is determined. From these results a formula is derived that accurately predicts the reduced amplitude as a function of the wavelength and operating conditions.
0 = ~/~o
gtn
D
3. Theory
/5H3 e = rlA
with
~ = 6us~?oa c2ph
and
a b The boundary conditions are P ( X a , Y) = P(Xb, Y) = P ( X , Ya) = P ( X , Yb) = O. In addition the cavitation condition should hold: P ( X , Y, T) > O. The lubricant is assumed to be compressible according to the Dowson and Higginson equation [1] and the viscosity is assumed to depend on the pressure according to the Roelands equation [19]. Introducing the dimensionless variables as defined above in the film thickness equation gives
28
H(X, Y,T) = -A(T) + SX 2 + (1 - S)Y 2 + T~(X, Y, T) +/)(X, Y)
(2)
with D(X,Y) denoting the dimensionless elastic deformation: /)(X, Y) = ~
1 //
P(X',Y',T)dX'dY'
~,¢2(X _ X,)2 + ( y _ y,)2
(3)
A(T) represents the dimensionless "mutual approach" of two remote points in the solids. By remote it is meant that the elastic deformation at these points is negligible. The other terms in Equation (2) describe the undeformed shape of the gap between the surfaces, where the first part describes the smooth shape, and T~(X, Y, T) is used to indicate surface features or roughness. S = S(~) is a shape factor only depending on the geometry of the undeformed solids, see Nomenclature. The value of A in Equation (2) is determined by the equation of motion: 1 dA
+ 3 / P(X, Y, T ) d X d Y = F(T)
(4)
where F(T) describes the oscillations relative to the nominal load and ~,~ is the dimensionless natural frequency. If the acceleration term is neglected Equation (4) reduces to the usual condition of force balance:
2~ P(X, Y, T ) d X d Y = -5-
(5)
Different sets of dimensionless parameters are in use to characterise EHL contacts. Here the Moes dimensionless parameters will be used. They are defined as:
. f ( E t R x ) 3/4 M = E, R2
~?ous
~?ous L = oLEI E IRx
(6)
(7)
and
D = Rx
R~
(8)
The relation between M, L, D and the parameters ~, A and ~ is given in the nomenclature. The aim of the theoretical investigation is to determine the amplitude reduction of harmonic surface patterns of the form: T~(X, Y, T) = ,4 cos
2~r (A=ia) cos
(
2~r ()q,/b)
with f( = X, + ST with S = Ul/fi, and
.,4 = Ai 10 -10(max(0' (;/o))2) x-x
(10)
This particular shape was chosen to avoid discontinuous derivatives, and to start the calculation with a smooth surface geometry. The deformed amplitude of the pattern as a function of the operating conditions will be determined according to:
2Ad = m a x U ( 0 , 0, T) - min U(0, 0, T) T
T
(11)
This definition is suited for all cases except for purely longitudinal waviness Ax = c~. For that case it needs to be generalised as is explained in [16]. 3.2. N u m e r i c a l S o l u t i o n The equations were discretised on a uniform grid with second order accuracy in space and time. For the discretization of the wedge and squeeze term a second order narrow upstream scheme was used referred to as NU2, see [16] and [20]. The discrete equations per time step were solved using multilevel techniques to accelerate convergence of the relaxation process. The elastic deformation integrals were computed using Multilevel Multi-Integration. For the basic techniques involved the reader is referred to [21], [22] and for recent developments to [23]. An overview of the techniques is also presented in [24]. The calculations were performed using a domain - 2 . 5 < X < 1.5 and - 2 < Y < 2 using a grid with 257 x 257 points. The time step was chosen equal to the spatial mesh size, i.e. with A T -- A x = A y = 0.015625. As we are interested in small amplitude waviness the amplitude of the undeformed wave was taken as 20 % of the central film thickness obtained when assuming perfectly smooth surfaces.
29
The simulation was started with the waviness outside the domain at X8 - -2.5. Subsequently the pattern moves into the contact with the velocity ul of the wavy surface. Monitoring of the central film thickness to determine Ad is started at T - T8 where/"8 denotes the time after which the solution has become periodic, i.e. when all running in effects have disappeared. To obtain a time independent value for Ad it is important that monitoring only starts for T > T~. As a general rule T~ can be taken as the time at which the slowest of the induced or real wave reaches the end of the contact. Subsequently the monitoring time should be taken sufficiently long to ensure that a real maximum and minimum have occured. As we assume pure rolling in the present paper it is sufficient to have a monitoring time that is (a multiple of) Ax/a. 3.3. W a v i n e s s a m p l i t u d e r e d u c t i o n In [16] results were presented for a circular tact and pure rolling (S = 1 , ~ - 0). In paper it was shown that the relative reduced plitude of a harmonic pattern was accurately dicted by the formula:
Ad
conthis ampre-
1
A-"~. = 1 + 0.15](r)V2 + 0.015(f(r)~72) 2
(12)
V o( (.~/a)5~1"5L-2
where { e l - -l f(r) 1 -
of the central film thickness computed for the smooth surface situation. Figure 1 shows the relative deformed amplitude Ad/Ai as a function of f ( r ) V 2 defined by (13) and (14). The dashed lines in the figure represent the predictions of equation (12). The figure shows that for each value of D a curve similar to the one for the circular contact is obtained, however, with decreasing D it shifts to the right. This shift suggests that V2 is only a special case of a more general parameter and that when represented as a function of this general parameter the curves would all fall onto a single curve. In [16] it is noted that the powers of M and L in the parameter V2 were chosen such that, for the circular contact, it reflects the same dependence on the Hertzian pressure and rolling speed as the parameter previously found for the line contact problem in [14]. This was the first evidence that there could be a unifying mechanism controlling the deformation of waviness in an EHL contact. Further evidence was given in [18] where it was shown that amplitude reduction results for line and circular contacts exhibit the same behaviour and are a function of a single parameter which is given by"
if r > 1 otherwise
(13)
with r = Az/Au, and
V2 - ( A / a ) ( M 1 / 2 / L 1/2)
(14)
with )~ = min()~x, )~u) In this paper it is investigated if and how this formula can be extended to elliptical contacts. Computations were carried out for load conditions 50 < M _ 2000, L 12.05, and 0.04 <_ D _ 1, and pure rolling. These cases have been selected to ensure that the contact is always in the piezoviscous elastic regime. Three different patterns were considered, i.e. transverse (Ay = c~), isotropic (Ax - A~), and longitudinal (Ax = oo). As mentioned in the previous section in all cases the amplitude of the waviness was taken as 20 %
(15)
The physical meaning of this parameter is that it represents the ratio of the wavelength in the entrainment direction to the length of the inlet pressure sweep. Based on these results one may thus expect the same mechanism to govern the elliptical contact. For a circular contact V2 as defined by (14) satisfies (15). However, for an elliptical contact it does not. The shift of the curves with changing D is thus explained by the fact that for a given D at the same value of M the Hertzian contact pressure is different. To obtain a parameter for the elliptical contact that satisfies (15), M in V2 should be replaced by 7 M giving:
V - (A/a)L-1/2(~/M) 1/2
-
-
~l/2v 2
(16)
where 7 = 16(E-
~2JK)2)
(17)
30
with 7 - 1 for ~ = 1 ( D = I ) . As can be verified using the relations given in the nomenclature: V_
/ ~rv2~__~3 ()~/a)C~3/2L_2
0.8
(18)
Figure 2 presents the calculated results as a function of f ( r ) V . The figure shows that all results now indeed collapse onto a single curve which is accurately described by equation (12) if V2 is replaced by V = ')'1/2~72 or simply by V according to (18). In this latter case it can also be used for the line contact problem. Strictly for a line contact one should replace the factor x / ( 2 r 3 ) / 3 in Equation (18) by (2r) 3/4, however, from a practical point of view the difference is small. 3.4. S u r f a c e D e f o r m a t i o n The formula predicting the deformation of sinusoidal waves described in the previous section reduces the need to perform full scale numerical simulations to predict the effects of surface roughness or single features on the film thickness. For those cases where the amplitude of the roughness or feature is relatively small compared to the film thickness the formula presented provides two simple and practical ways to analyse the behaviour. In engineering surfaces there is generally one component that dominates the roughness spectrum. When the behaviour of this component is analysed, it is possible to make qualitative predictions concerning the behaviour of the complete roughness. A second possibility is to use Fourier decomposition to decompose the roughness profile into harmonic components. Then the amplitude reduction of each component is calculated and finally the deformed profile is obtained by adding up all the deformed components. Subsequently the effects of the roughness or feature on the film thickness can be estimated using the deformed profile. The time consuming full scale numerical simulations can then be saved for those cases where they are really needed, i.e. when the amplitudes are large or when accurate results are required.
""°#o ~ : . ~ x D=0.35 + "..~ll~i~x D=0.12 @ D=0.04 x
"¢.~) x
0.6 Ai
0.4 0.2
1
........ I 1
..,o
........ , 10
......
,
D'--'I
o
D=0.35 + =
D=0.12
~
_
(9
D=0.04 x
"~~)x
0.4
_
~+ex ~+e %+
-
0 0.1
........ I 1
• .J.'.
0.8 -
'"6"1+,'~
........ i 10
' U'"l
-
, , ,~,,.j.. 100
........
"'.. o St+@^ "".o + + ~ x
_4~ 0.6 Ai
-
, ,'q v,,,.J_ 100
](~)v:
R,
0.6
0.2
%.~
~ , + e
.
Ai
_
~,+ex
O.8 A_~
-
"~++~x
-
0 0.1
_
D=I o D=0.35+ "o8 .~-@~ D=0.12@ "'o ' ~ ~ = 0 , 0 4 x k o / ~
I
_
0.4
0
.......
0.1
I
1
. . . . . . . .
I
10
/(~)v~
,
," : t , , . d
.
100
Figure 1. Relative deformed amplitude as a function of .f(r)V2 and D for transverse waviness
(top), isotropic waviness (center) and longitudinal waviness (bottom) in an elliptical contact. The dashed line represents the predictions of (12).
31
1 _y-. . . . . . .
, .......
0.8
D=0.35 D=0.12 D=0.04
0.6
4.
i D = I '' " .... o '
+ @ x
A_~ Ai
0.4
%
0.2. . . . . . . .
0 0.1
,
. . . . . . . .
1
,
,'<>,~
9.,.,j
_ _
10
100
f(r)V
D=I D=0.35 D=0.12
0.8 _ A_~ Ai
0.6
_
o + G
D=0.04
_
x
_
0.4 0.2
-
0 0.1
-
........
J
........
t
1
,'0,~ ¢,,,. a _ _
10
100
f(r)V
'
!
'w
t
i
l!
|
!
!
. 1 . , . |
,
.
!
if|
0.8 A_.,L Ai
0.6
'" D'---i D=0.35 D-0.12 D=0.04
|
|fill]
o 4(9 x
0.4 0.2 0 0.1
l
1
l
10
.
100
/(r)V Figure 2. Relative deformed amplitude as a function ] ( r ) V and D for transverse waviness (top),
isotropic waviness (center) and longitudinal waviness (bottom) in an elliptical contact. The dashed line represents the predictions of (12) (with V2 replaced by V).
Experimental
Validation
The theoretical results presented in the previous section were to some extent validated by Guangteng et al. [7]. The deformation of a single feature in an EHL contact was studied using optical interferometry. The paper describes how the measured amplitude increases with increasing speed. Increasing speed means decreasing V and results in reduced elastic deformation of the ridge and thus in an increase in amplitude. However, with a further increase in the speed a flattening was observed, which is not predicted theoretically. Unfortunately, only results only for a single feature are presented. Moreover its amplitude was about 9 times larger than the film thickness, whereas the theory described previously is based on the assumption that the amplitude of the feature is small compared to the film thickness. Therefore a separate validation was undertaken. Some first results are presented below. 4.1.
Setup
The duochromatic optical interferometry technique [25] was used to measure the shape and thickness of the lubricant film in an EHL circular contact. The experimental equipment used in this investigation is a ball and disc apparatus as schematically described in Figure (3). The duochromatic interference fringe pattern, obtained by a xenon light source with a flash duration of 20#s through red and green filters, was recorded with a high-speed VCR (200 frames per second) and a 35-mm camera attached to a microscope. The lubricant used is a mineral bright stock and was held at room temperature. Results will be presented for pure rolling (E = 0) with a rolling velocity of 0.024, 0.048 and 0.096 m/s. The load is kept constant at 39.2 N giving a maximum Hertzian pressure of 0.54 GPa, and a Hertzian contact radius of 0.185 mm. The parameter values and the values of various sets of nondimensional parameters for these contact conditions are summarized in Table 1 and Table 2. On the ball a shallow waviness pattern was sputtered. A stylus trace of the pattern is given in Figure 4. The pattern is almost harmonic.
32
stylus-trace 9.8
HIGH-SPEED] /
VCR
VIDEO~ CAMERA ST O OPE
~
35ram CAMERA
~
GAGE
RECORDER I
I
COUNTERI
I
i
ENCOOER PULLEY
Figure 3. Schematic diagram of the experimental
apparatus. R E'
y0 a z
12.52 10 -3 1.171011 39.2 0.024 1.22 2.25 10 -8 0.484
Ira]
[Pa] 0.048
0.096
IN] [m/s]
[Pas] [Pa -1]
Table 1 Parameters and their values in the experiments.
9.3
130.77 6.61
79.90 7.88
47.51 9.36
A
12.23 1.65 10 -2
3.19 10 -2
6.39 10 -2
W G U
2.086 10 -6 2633 2.0 10 -11
11]1(!~ | m E -:~[lll(! ~ll
M
IV
[1.42
]1.02
0.72
Table 2 Va/ues of severa/groups of dimensionless param-
eters.
I
I
0 0.1 [mm]
I
,
0.2
0.3
Figure 4. Stylus trace of the surface pattern sputtered on the ball.
A two parameter least square fit against a sinusoidal function gives a wavelength of 60 #m and an amplitude of 0.14#m. Taking different parts of the trace the wavelength varies little. However, the amplitude varies between 0.13 and 0.16 #m, depending on the part of the trace taken in the least square fit. Table 2 also gives the values of the V parameter obtained assuming A = 60#m. In the following section the measured film thickness will be presented and compared with the results obtained with full numerical simulations using input from the profile. It is also verified to what extend the change of the amplitude of the film thickness oscillations in the center of the contact with speed matches the predicted behaviour. 4.2.
L
I
9.4
! j
I,
I
-0.1
I
h[,m] 9.5
9.2 -0.3
t
I
-0.2
I
9.6
ENCODER i~I',4--
I
9.7
XENONFILTER~ MICROSCOPE LAMP
PULLEY
I
Results
Figure 5 shows the interferometry picture of the film thickness in the contact at the time the waviness is fully inside the contact for fi = 0.024 m/s. Figure 6 shows the profile at the line y = 0 as a function of z for a similar situation. The figures show that the film thickness on a large scale exhibits the usual shape, i.e. a nearly fiat film thickness in the center of the contact and a horse shoe shaped constriction at the end. "Superimposed" on this mean profile is an oscillation due to the waviness. For this case the average film thickness computed from the measured profile is about 0.28 # m and the amplitude of oscillation about 0.15 #m.
33
-1.9 -2.00
-O.W X
" -2.50
Figure 5. Interferometry picture of measured film thickness, fi - 0.024 m / s , E - O. 1.5
0.5
I
I
I
I
- 0.024
I
I
u
u
0.9
"1'00" k ' ~ 0
'
-0.3
l
I
1
I
I
-0.2
-0.1
0
0.1
0.2
0.3
Figure 6. Measured film thickness h [#m] at the line y - 0 as a function o f x [mm] for - 0.024 m / s and E = O.
This behaviour can be easily explained. In the central region of the contact, the flow is dominated by the shear flow (Couette term). Hence, any feature on the surface in this region will move through this region at the average speed of the surfaces. For pure rolling this implies that any feature, once in the high viscosity region, will move through the contact maintaining its shape. This means that the wavelength of the oscillations will be the wavelength of the original profile. Only the amplitude of the oscillations can be smaller due to the reduction occuring when entering the high viscosity region.
-2.00
~~~-1.9
-0.91(
" -2,~
Figure 7. Snapshot of c o m p u t e d P (top) and H (bottom) as a function of X and Y , for fi = 0.024 m / s and E - O.
For the same conditions a numerical simulation was carried out. The profile shown in Figure 4 was scaled to obtain a pattern in dimensionless variables. Subsequently, a discrete function T~(X, T) was obtained by a single wave from the profile and repeating it periodically. Figure 7 shows the dimensionless pressure P and the dimensionless film thickness H as a function of X and Y at an instant the waviness is completely in the contact. Figure 8 shows the calculated film thickness h in # m at the centerline of the contact. The average film thickness in
34
the central region of the contact obtained from the computed profile is 0.25 ttm which is slightly larger than the value of 0.24 # m obtained for the central film thickness when assuming perfectly smooth surfaces. The amplitude of the oscillation in the theoretical result is about 0.13 #m. Comparing the Figures 6 and 8 and the values given above it is concluded that the predicted and measured film thickness agree quite well. The difference in the average level and in the amplitude of the film thickness is roughly 10%. For ~2 = 0.024 m / s one finds that V = 1.423. Figure 2 predicts that for a harmonic transverse waviness at this value of V the deformed amplitude should be close to the original amplitude. Using Equation (12) one obtains A d / A i = 0.80. This value is a bit too small because the formula for small V underestimates the computed results. However, it is sufficiently close to unity to indicate that in practice the waviness will tend to move through the contact virtually unchanged, as was indeed observed in the experimental results. Figure 9 shows the measured centerline film profile for the cases fi = 0.048 and 0.096 m / s . Due to the increased speed one can expect the nominal film thickness to increase. For fi = 0.048 m / s the average film thickness in the central region is 0.42 # m and for fi = 0.096 m / s it is 0.70 #m. However, compared to the increase in nominal film thickness the amplitude of the film thickness oscillations induced by the waviness changes little. In both cases it roughly equals the amplitude of the undeformed profile. Based on the values of V for these cases this is according to the expectations. A larger value of the velocity implies a smaller value of V. Hence, if the amplitude reduction for fi = 0.024 m / s is small, for the higher velocities it will only be smaller. Figure 10 shows the computed results for the cases ~ = 0.048 m / s and 0.096 m / s . The values of the average film thickness in the center of the contact obtained from the computed results are 0.40 and 0.62 # m respectively. Comparing these values with the values obtained from the measured film thickness and comparing the Figures 9 and 10 shows that also for the higher velocities the agreement between measured and computed results is quite good.
1.5
1 h 0.5
0 -0.3
i -0.2
, -0.1
i I , 0 0.1 0.2 0.3 x Figure 8. Calculated film thickness h [#m] at the line y - 0 as a function o f x [mm] for - 0.024 m / s and E - O.
'
'
1.5
',
•
-o'04s ! fi -- 0:096"" i " "
1I 0.5
0 I
-0.3
1
I
I
I
I
-0.2
-0.1
0
0.1
0.2
0.3
Figure 9. Measured film thickness h [l~m] at the line y - 0 as a function of x [ram] for fi - 0.048 and 0.096 m / s and E = O. 1.5
\!:
'
'
' u - = 0 ' 0'4 8.
--"/
0.5
0
-0.3
!
I
I
I
I
-0.2
-0.1
0
0.1
0.2
x
0.3
Figure 10. Calculated film thickness h [lzm] at the line y = 0 as a function of x [mm] fi = 0.048 and 0.096 m / s and E = O.
35
Finally, the formula presented in Section 3.3 predicts for these cases a relative deformed amplitude of 0.86 and 0.90. From a practical point of view these values indicate a waviness moving through the contact virtually maintaining its amplitude, as was indeed found. Summarizing, it is concluded that the film thickness profiles predicted using full numerical simulations show good agreement with the measured film thickness profiles. The experimental results confirm the theoretically predicted tendency that high frequency waviness moves through the contact virtually unchanged in amplitude, and that once this is the case, an increase of the speed will hardly change the picture as could be expected based on the fact that it reduces the value of the V parameter. 5. C o n c l u s i o n The deformation of harmonic waviness in an EHL contact under pure rolling appears to be governed by a single unifying mechanism. The amplitude reduction in line and circular contacts could be described by a single equation. The effect of pattern orientation and anisotropy could be incorporated in the same relation. Finally, as was demonstrated here the influence of the contact ellipticity can be included in the equation too, such that the circular contact and the line contact become specific cases of a general equation. The relative reduced amplitude of general harmonic waviness in a general EHL contact can be predicted by:
Ad
1.
(19)
2.
where
-
REFERENCES
1
A--~. = 1 + 0.15](r)V + 0.015(](r)V) 2
f(r) =
can be used as an engineering tool to obtain a first estimate of the deformation of any microgeometry under given load conditions by decomposing the microgeometry into its harmonic components. The deformed amplitude of each component can then be calculated with the formula. Finally, the deformed profile is obtained by adding up all the deformed components. Using optical interferometry experiments were carried out to validate the predictions of the theory. For the cases considered a good agreement was found between experimental and theoretical results, both in terms of amplitude reduction as well as in the details of the film thickness profiles. However, more experiments should be performed focussing on larger wavelengths and smaller speeds, to examine the complete predicted curve. Finally, the results presented in this paper were restricted to pure rolling. For the deformation of the microgeometry pure rolling may be seen as a worst case, i.e. the deformation will be smallest. If sliding occurs the fluid will force itself past surface features leading to an increased deformation. However, to accurately predict this latter case the model presented in Section 3.1 needs to be extended including non-Newtonian lubricant behaviour. These aspects are the subject of ongoing research.
/ el-~ 1
I,
if r > 1 otherwise
(20) 3.
with r = )~x/)~ and
V = ~/~~-(A/a)~3/2L-2
(21)
with A = min(Ax, A~) This formula expresses that, in terms of the dimensionless "wavelength" V, high frequency patterns deform little whereas low frequency components tend to flatten completely. The equation
4.
5.
Dowson, D., and Higginson, G.R., 1966, "Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication," Pergamon Press, Oxford, Great Britain. Hamrock, B.J., and Dowson, D., 1978, "Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III- Fully Flooded Results," ASME JOT, 99, pp. 264-276. Kaneta, M., and Cameron, A., 1980, "Effects of Asperities in Elastohydrodynamic Lubrication," ASME JOT, 102, 374-379. Kaneta, M., Sakai, T., and Nishikawa, H., 1992, "Optical Interferometric Observations of the Effects of a Bump on Point Contact EHL," ASME JOT, 114, pp. 779-784. , 1996, Cann, P.M., Spikes, H.A., Hutchinson, J.,,"The development of a Spacer Layer Imaging Method (SLIM) for Mapping Elastohy-
36
10.
11.
12.
13.
14.
15.
drodynamic Contacts," STLE Tribology Transactions, V 39, 4, pp 915-921. K a n e t a , J., K a n a d a , T., and Nishikawa, H., 1997, "Optical Interferometric Observations of the Effects of a Moving Dent on Point Contact EHL," Proc. 23rd Leeds-Lyon Symposium on Tribology, Ed. D. Dowson et al., Elsevier Tribology Series 32, pp. 69-79. G u a n g t e n g , G., Cann, P.M., Spikes, H.A., and Olver, A., 1999, "Mapping of Surface Features in the Thin Film Lubrication Regime," Proc. 1998 Leeds Lyon Tribology Conference, Elsevier Tribology Series 36, Ed. D. Dowson et al., pp. 175-183. Venner, C.H., and Lubrecht, A.A., 1994, "Numerical Simulation of a Transverse Ridge in a Circular EHL Contact, under rolling/sliding", ASME JOT, 116, pp. 751-761. Venner, C.H., and Lubrecht, A.A., 1994, "Numerical Simulation of Waviness in a Circular EHL Contact, under rolling/sliding", Proc. 20nd Leeds-Lyon Symposium on Tribology, Ed. D. Dowson et al., Elsevier Tribology Series 30, pp. 259-272. Ehret, P., Dowson, D., and Taylor, C.M., 1998, On lubricant transport conditions in elastohydrodynamic conjunctions, Proc. Roy. Soc. A 454, p 763-787. Jiang, X, Hua, D.Y., Cheng, H.S., Ai, X., Lee, S.C., 1999, "A mixed Elastohydrodynamic Lubrication Model With Asperity Contact," ASME Journal of Tribology, V 121, pp 481491. Morales Espejel, G.E., 1993, "Elastohydrodynamic Lubrication of Smooth and Rough Surfaces," PhD. Thesis, University of Cambridge, Department of Engineering. G r e e n w o o d , J.A., and Morales Espejel, G.E., 1994, "The Behaviour of Transverse Roughness in EHL Contacts," Proc. IMechE, 208, pp. 121-132. Venner, C.H., Couhier, F., Lubrecht, A.A., and G r e e n w o o d , J., 1997 "Amplitude Reduction of Waviness in Transient EHL Line Contacts," Proc. 1996 Leeds Lyon Tribology Conference, Elsevier Tribology Series 32, Ed. Dowson et al., pp. 103-112. Hooke, C.J., 1997, "Surface Roughness Modification in Elastohydrodynamic Line Contacts operating in the elastic Piezoviscous Regime,", Proc. Inst. Mech. Engrs Part J., V 212 J2, pp. 145-162.
16. Venner, C.H., Lubrecht, A.A., 1999, "Amplitude Reduction of Anisotropic Harmonic Surface Patterns in EHL Circular Contacts under Pure Rolling, " Proc. 1998 Leeds-Lyon Tribology Conference, Elseviers Tribology Series, 36, Ed. D. Dowson et al., pp 151-162. 17. Hooke, C.J., 1999, Surface Roughness modification in EHL line contacts- the effects of roughness wavelength, orientation and operating condition, Proc. 1998 Leeds-Lyon Tribology Conference, EIseviers Tribology Series, 36, Ed. D. Dowson et al., pp 193-202. 18. Hooke, C.J., a n d Venner, C.H., 1999, "Surface Roughness Attenuation in Line and Point Contacts," Proc. ImechE, part J. Journal of Engineering Tribology, in press. 19. Roelands, C.J.A., 1966, "Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils" Ph.D. Thesis, Technical University Delft, Delft, The Netherlands, (V.R.B., Groningen, The Netherlands). 20. Venner, C.H., Morales Espejel, G.E.,, 1999, "Amplitude Reduction of Small Amplitude Waviness in Transient EHL Line Contacts," Proc. ImechE, part J. Journal of Engineering Tribology, in press. 21. Lubrecht, A.A., 1987, "Numerical Solution of the EHL Line and Point Contact Problem Using Multigrid Techniques," Ph.D. Thesis, University of Twente, Enschede, The Netherlands, ISBN 909001583-3. 22. Venner, C.H., 1991, "Multilevel Solution of the EHL Line and Point Contact Problems," Ph.D. Thesis, University of Twente, Enschede, The Netherlands. ISBN 90-9003974-0. 23. W i j n a n t , Y.H., (1998), "Contact Dynamics in the field of Elastohydrodynamic Lubrication," Ph.D. Thesis, University of Twente, Enschede, The Netherlands. ISBN 90-36512239. 24. Venner, C.H., a n d Lubrecht, A.A.,, (1999), "Multigrid Techniques: A fast and etficient method for the numerical simulation of EHL point contact problems," Proc. ImechE, part J. Journal of Engineering 7kibology, in press. 25. Foord, C.A., W e d e v e n , L.D., Westlake, F.J. a n d C a m e r o n , A., (1969-70), "Optical elastohydrodynamics," Proc. Instn. Mech. Engrs., 184 P t 1 No 28, 487-505.
25
Surface Roughness in Elastohydrodynamically Lubricated Contacts C.H. Venner a, M. K a n e t a b, and A.A. Lubrecht c a University of Twente, Faculty of Mechanical Engineering, Enschede, The Netherlands. bKyushu Institute of Technology, Department of Mechanical Engineering, Kitakyushu, Japan. CLaboratoire de M@canique des Contacts, UMR CNRS 5514, INSA de Lyon, France. The nominal film thickness at which EHL contacts have to operate reliably has decreased to a level where the effects of the surface microgeometry on the film thickness can not be neglected. To predict the effects of the microgeometry on the film thickness one should be able to predict its deformation as a function of the operating conditions. Recent work on the effects of waviness in EHL contacts has shown that the amplitude reduction of harmonic patterns in line and circular contacts under pure rolling could be described as a function of a single nondimensional parameter. In the present paper this work is extended to elliptical contacts. The result is a simple formula that predicts the amplitude reduction of waviness in an EHL contact under pure rolling as a function of its wavelength and the operating conditions of the contact. This formula can be used as a simple tool to obtain a first estimate of the deformed roughness profile inside the contact for any microgeometry. Subsequently, attention is directed to experimental validation of the observed behaviour. Although further experiments are needed the first results are very encouraging.
1. I n t r o d u c t i o n
Nowadays EHL contacts have to operate reliably under conditions where the film thickness is much smaller than could have been imagined when the classical film thickness equations were developed [1,2]. As a result the microgeometry of the surface will have a significant effect on the operation of the contact and to predict performance, a detailed understanding of the effects of microgeometry on film formation is as important as ever before. Fortunately the available tools to investigate these effects have improved considerably over the years. For example, experimental techniques have advanced to a level where roughness effects on film thickness can be measured accurately inside the contact [3-7]. Also theoretically impressive progress has been made in the past decade. The introduction of fast and stable algorithms for numerical simulations has led to the situation where the transient non-smooth EHL point contact problem can be simulated in great detail, e.g. see [8-11]. Also the underlying mechanisms governing the solution of the transient EHL problem have become well understood [12,13].
The challenge researchers and engineers face now is to use this experimental and theoretical capability not just to illustrate effects for single cases, but to derive engineering tools, i.e. simple prediction formulas for practical use, that describe the effects of surface roughness on the lubricant film. Obviously due to the deformation of the roughness profile the well known h/a ratio with h taken as the nominal film thickness and a as the standard deviation of the undeformed roughness profile is of little use to quantify the effect of a given roughness profile on the film thickness. A first step towards an improved parameter could be a simple tool to predict the deformed roughness profile inside the contact. To obtain such a tool by studying the rough surface problem directly, e.g. by means of numerical simulations using rough surfaces as input, is rather cumbersome. For example, the enormous amount of d a t a generated (different surfaces, load conditions etc) would make it hard to extract general tendencies. The authors have therefore followed an alternative route. Instead of studying the rough surface problem, the deformation of harmonic patterns (waviness)
26
has been determined in detail. First the line contact problem was studied [14]. It was shown that the reduced relative amplitude under pure rolling could be predicted accurately with a simple function of a single non-dimensional parameter. Theoretical support for this behaviour was provided by Hooke [15]. Next it was shown that the amplitude reduction of harmonic patterns in a circular contact followed the same behaviour [16-18]. These results show that there appears to be a unifying mechanism governing the amplitude reduction of harmonic patterns in EHL contacts. In the present paper this subject is investigated further by studying the amplitude reduction of harmonic patterns in elliptical contacts. It is confirmed that indeed a unifying mechanism exists. As a result the amplitude reduction of a harmonic pattern in an EHL contact under pure rolling can be predicted quite accurately with a single formula. This formula can serve as a simple tool to predict the deformed geometry inside the contact for any surface microgeometry by applying it to each of the Fourier components of the microgeometry followed by backtransform to obtain the deformed profile. Subsequently, attention is directed to experimental validation. Using a ball on disk machine film thickness measurements have been performed for the case of an artificially produced shallow waviness pattern on the ball. The measured film thickness profiles are compared with those obtained by numerical simulations.
2. N o m e n c l a t u r e Hertzian contact length a = ( 3 I R / E ' ) I / Z ( 2 ~ E / r ) 1/3 A Dimensionless amplitude b Hertzian contact width b=a/~ c Hertzian approach c = (~I(2R))(~cI~) D ratio reduced radii of curvature D = R~/R~ I:) Dimensionless elastic deformation E Modulus of elasticity
E'
Reduced modulus of elasticity 21E' - (1 v2)lE1 + (1 v2)lE2 Elliptic integral (second kind) E = ~ ( 1 - ~2) External nominal load Dimensionless external force Dimensionless materials parameter G=aE' Film thickness Dimensionless film thickness H =h/c Dimensionless time step Dimensionless mesh size in X, Y Elliptic integral (first kind) : ~ ( 1 - ~2) Dimensionless material parameter L : G(2U) 1/4 Dimensionless load parameter M = W ( 2 U ) -3/4 Pressure Maximum Hertzian pressure Ph = (3 f ) / ( 2 r a b ) Dimensionless pressure P = P/Ph Reduced radius of curvature I l R = IlR~ + l l R y Reduced radius of curvature in x lln~ = llR~l + l/n~2 Reduced radius of curvature in y 1 / R v = 1/Rljl + 1/Rij2 Radius of curvature surface 1,2 in x Radius of curvature surface 1,2 in y Slip parameter S=ull~ Dimensionless shape factor s = (z- ~c)/(~c~c) Time Dimensionless time T = tu,/(2a) Surface velocity Sum velocity u, = (Ul + u2) Average velocity fi = (ul + u2)/2 Dimensionless speed parameter 2U = (17ou.)l(E'R=) Dimensionless load parameter W = F/(E'R 2) Coordinate in direction of rolling -
E
/ F G h H hT
hx,hy
M P Ph P R RX
Rv R z 1 ~Rx2
R~ I , R,2 S
t T
a
f~
Us
U W X~ X !
-
27 X, X'
Dimensionless coordinate X = x/a, X' = x'/a Coordinate perpendicular to x Dimensionless coordinate Y = y/b, Y ' = y'/b Viscosity index (Roelands) Pressure viscosity coefficient Dimensionless parameter
y, y' Y, Y' z a c~
5 ~ - O~ph
5 A
Mutual approach Dimensionless mutual approach A =~lc Coefficient in Reynolds' equation
e
-(~H3)/(~)
Ellipticity ratio n, = a/b Poisson ratio Dimensionless speed parameter = 6(rlousa)/(c2ph) Wavelength in x direction Wavelength in y direction Viscosity Dimensionless viscosity
u A Ax A~ 7/ ~/
Dimensionless natural frequency a 2 = (4fa2)(mu2c) a~ - ( 8 f R E ) ( m u ~ H , ( ) Density Dimensionless density - p/po slip parameter
p fi E
subscripts a,b i,d F 0 1,2
3.1. Equations In this section dimensionless equations are given describing the EHL point contact under fully flooded conditions. The dimensionless equations have been obtained using the Hertzian dry contact parameters (see Nomenclature) and the lubricant viscosity and density at ambient pressure. This gives the following dimensionless variables: X-x/a H = h/c P - - P/Ph P -- PlPo
Y =y/b A = 5/c T = tu~/(2a) (7 = ~7/Yo
The resulting dimensionless Reynolds equation is:
inlet, outlet undeformed, deformed force constant, e.g. at ambient pressure surface 1, 2
O~- "-- L 1/3(Ir2t¢(1_/~2) 2 )1/3 ~" (~-~) 16(E-~2 JK)2) : ! 132)~//r:)1/3 (167r~E_~2K) 5 1/3 ~4(1--~2)K0 ) (K-E) ~2 (E-~ ~ ~)
16(E-a
0 (0P
OX
c-ff-~ O(pH) OX
O(#H) cOT
=0
(1)
where
-- (Ul -- U2)/U
Relations between parameters
")'
In the first part of this paper the deformation of specific harmonic surface patterns is determined. From these results a formula is derived that accurately predicts the reduced amplitude as a function of the wavelength and operating conditions.
0 = ~/~o
gtn
D
3. Theory
/5H3 e = rlA
with
~ = 6us~?oa c2ph
and
a b The boundary conditions are P ( X a , Y) = P(Xb, Y) = P ( X , Ya) = P ( X , Yb) = O. In addition the cavitation condition should hold: P ( X , Y, T) > O. The lubricant is assumed to be compressible according to the Dowson and Higginson equation [1] and the viscosity is assumed to depend on the pressure according to the Roelands equation [19]. Introducing the dimensionless variables as defined above in the film thickness equation gives
28
H(X, Y,T) = -A(T) + SX 2 + (1 - S)Y 2 + T~(X, Y, T) +/)(X, Y)
(2)
with D(X,Y) denoting the dimensionless elastic deformation: /)(X, Y) = ~
1 //
P(X',Y',T)dX'dY'
~,¢2(X _ X,)2 + ( y _ y,)2
(3)
A(T) represents the dimensionless "mutual approach" of two remote points in the solids. By remote it is meant that the elastic deformation at these points is negligible. The other terms in Equation (2) describe the undeformed shape of the gap between the surfaces, where the first part describes the smooth shape, and T~(X, Y, T) is used to indicate surface features or roughness. S = S(~) is a shape factor only depending on the geometry of the undeformed solids, see Nomenclature. The value of A in Equation (2) is determined by the equation of motion: 1 dA
+ 3 / P(X, Y, T ) d X d Y = F(T)
(4)
where F(T) describes the oscillations relative to the nominal load and ~,~ is the dimensionless natural frequency. If the acceleration term is neglected Equation (4) reduces to the usual condition of force balance:
2~ P(X, Y, T ) d X d Y = -5-
(5)
Different sets of dimensionless parameters are in use to characterise EHL contacts. Here the Moes dimensionless parameters will be used. They are defined as:
. f ( E t R x ) 3/4 M = E, R2
~?ous
~?ous L = oLEI E IRx
(6)
(7)
and
D = Rx
R~
(8)
The relation between M, L, D and the parameters ~, A and ~ is given in the nomenclature. The aim of the theoretical investigation is to determine the amplitude reduction of harmonic surface patterns of the form: T~(X, Y, T) = ,4 cos
2~r (A=ia) cos
(
2~r ()q,/b)
with f( = X, + ST with S = Ul/fi, and
.,4 = Ai 10 -10(max(0' (;/o))2) x-x
(10)
This particular shape was chosen to avoid discontinuous derivatives, and to start the calculation with a smooth surface geometry. The deformed amplitude of the pattern as a function of the operating conditions will be determined according to:
2Ad = m a x U ( 0 , 0, T) - min U(0, 0, T) T
T
(11)
This definition is suited for all cases except for purely longitudinal waviness Ax = c~. For that case it needs to be generalised as is explained in [16]. 3.2. N u m e r i c a l S o l u t i o n The equations were discretised on a uniform grid with second order accuracy in space and time. For the discretization of the wedge and squeeze term a second order narrow upstream scheme was used referred to as NU2, see [16] and [20]. The discrete equations per time step were solved using multilevel techniques to accelerate convergence of the relaxation process. The elastic deformation integrals were computed using Multilevel Multi-Integration. For the basic techniques involved the reader is referred to [21], [22] and for recent developments to [23]. An overview of the techniques is also presented in [24]. The calculations were performed using a domain - 2 . 5 < X < 1.5 and - 2 < Y < 2 using a grid with 257 x 257 points. The time step was chosen equal to the spatial mesh size, i.e. with A T -- A x = A y = 0.015625. As we are interested in small amplitude waviness the amplitude of the undeformed wave was taken as 20 % of the central film thickness obtained when assuming perfectly smooth surfaces.
29
The simulation was started with the waviness outside the domain at X8 - -2.5. Subsequently the pattern moves into the contact with the velocity ul of the wavy surface. Monitoring of the central film thickness to determine Ad is started at T - T8 where/"8 denotes the time after which the solution has become periodic, i.e. when all running in effects have disappeared. To obtain a time independent value for Ad it is important that monitoring only starts for T > T~. As a general rule T~ can be taken as the time at which the slowest of the induced or real wave reaches the end of the contact. Subsequently the monitoring time should be taken sufficiently long to ensure that a real maximum and minimum have occured. As we assume pure rolling in the present paper it is sufficient to have a monitoring time that is (a multiple of) Ax/a. 3.3. W a v i n e s s a m p l i t u d e r e d u c t i o n In [16] results were presented for a circular tact and pure rolling (S = 1 , ~ - 0). In paper it was shown that the relative reduced plitude of a harmonic pattern was accurately dicted by the formula:
Ad
conthis ampre-
1
A-"~. = 1 + 0.15](r)V2 + 0.015(f(r)~72) 2
(12)
V o( (.~/a)5~1"5L-2
where { e l - -l f(r) 1 -
of the central film thickness computed for the smooth surface situation. Figure 1 shows the relative deformed amplitude Ad/Ai as a function of f ( r ) V 2 defined by (13) and (14). The dashed lines in the figure represent the predictions of equation (12). The figure shows that for each value of D a curve similar to the one for the circular contact is obtained, however, with decreasing D it shifts to the right. This shift suggests that V2 is only a special case of a more general parameter and that when represented as a function of this general parameter the curves would all fall onto a single curve. In [16] it is noted that the powers of M and L in the parameter V2 were chosen such that, for the circular contact, it reflects the same dependence on the Hertzian pressure and rolling speed as the parameter previously found for the line contact problem in [14]. This was the first evidence that there could be a unifying mechanism controlling the deformation of waviness in an EHL contact. Further evidence was given in [18] where it was shown that amplitude reduction results for line and circular contacts exhibit the same behaviour and are a function of a single parameter which is given by"
if r > 1 otherwise
(13)
with r = Az/Au, and
V2 - ( A / a ) ( M 1 / 2 / L 1/2)
(14)
with )~ = min()~x, )~u) In this paper it is investigated if and how this formula can be extended to elliptical contacts. Computations were carried out for load conditions 50 < M _ 2000, L 12.05, and 0.04 <_ D _ 1, and pure rolling. These cases have been selected to ensure that the contact is always in the piezoviscous elastic regime. Three different patterns were considered, i.e. transverse (Ay = c~), isotropic (Ax - A~), and longitudinal (Ax = oo). As mentioned in the previous section in all cases the amplitude of the waviness was taken as 20 %
(15)
The physical meaning of this parameter is that it represents the ratio of the wavelength in the entrainment direction to the length of the inlet pressure sweep. Based on these results one may thus expect the same mechanism to govern the elliptical contact. For a circular contact V2 as defined by (14) satisfies (15). However, for an elliptical contact it does not. The shift of the curves with changing D is thus explained by the fact that for a given D at the same value of M the Hertzian contact pressure is different. To obtain a parameter for the elliptical contact that satisfies (15), M in V2 should be replaced by 7 M giving:
V - (A/a)L-1/2(~/M) 1/2
-
-
~l/2v 2
(16)
where 7 = 16(E-
~2JK)2)
(17)
30
with 7 - 1 for ~ = 1 ( D = I ) . As can be verified using the relations given in the nomenclature: V_
/ ~rv2~__~3 ()~/a)C~3/2L_2
0.8
(18)
Figure 2 presents the calculated results as a function of f ( r ) V . The figure shows that all results now indeed collapse onto a single curve which is accurately described by equation (12) if V2 is replaced by V = ')'1/2~72 or simply by V according to (18). In this latter case it can also be used for the line contact problem. Strictly for a line contact one should replace the factor x / ( 2 r 3 ) / 3 in Equation (18) by (2r) 3/4, however, from a practical point of view the difference is small. 3.4. S u r f a c e D e f o r m a t i o n The formula predicting the deformation of sinusoidal waves described in the previous section reduces the need to perform full scale numerical simulations to predict the effects of surface roughness or single features on the film thickness. For those cases where the amplitude of the roughness or feature is relatively small compared to the film thickness the formula presented provides two simple and practical ways to analyse the behaviour. In engineering surfaces there is generally one component that dominates the roughness spectrum. When the behaviour of this component is analysed, it is possible to make qualitative predictions concerning the behaviour of the complete roughness. A second possibility is to use Fourier decomposition to decompose the roughness profile into harmonic components. Then the amplitude reduction of each component is calculated and finally the deformed profile is obtained by adding up all the deformed components. Subsequently the effects of the roughness or feature on the film thickness can be estimated using the deformed profile. The time consuming full scale numerical simulations can then be saved for those cases where they are really needed, i.e. when the amplitudes are large or when accurate results are required.
""°#o ~ : . ~ x D=0.35 + "..~ll~i~x D=0.12 @ D=0.04 x
"¢.~) x
0.6 Ai
0.4 0.2
1
........ I 1
..,o
........ , 10
......
,
D'--'I
o
D=0.35 + =
D=0.12
~
_
(9
D=0.04 x
"~~)x
0.4
_
~+ex ~+e %+
-
0 0.1
........ I 1
• .J.'.
0.8 -
'"6"1+,'~
........ i 10
' U'"l
-
, , ,~,,.j.. 100
........
"'.. o St+@^ "".o + + ~ x
_4~ 0.6 Ai
-
, ,'q v,,,.J_ 100
](~)v:
R,
0.6
0.2
%.~
~ , + e
.
Ai
_
~,+ex
O.8 A_~
-
"~++~x
-
0 0.1
_
D=I o D=0.35+ "o8 .~-@~ D=0.12@ "'o ' ~ ~ = 0 , 0 4 x k o / ~
I
_
0.4
0
.......
0.1
I
1
. . . . . . . .
I
10
/(~)v~
,
," : t , , . d
.
100
Figure 1. Relative deformed amplitude as a function of .f(r)V2 and D for transverse waviness
(top), isotropic waviness (center) and longitudinal waviness (bottom) in an elliptical contact. The dashed line represents the predictions of (12).
31
1 _y-. . . . . . .
, .......
0.8
D=0.35 D=0.12 D=0.04
0.6
4.
i D = I '' " .... o '
+ @ x
A_~ Ai
0.4
%
0.2. . . . . . . .
0 0.1
,
. . . . . . . .
1
,
,'<>,~
9.,.,j
_ _
10
100
f(r)V
D=I D=0.35 D=0.12
0.8 _ A_~ Ai
0.6
_
o + G
D=0.04
_
x
_
0.4 0.2
-
0 0.1
-
........
J
........
t
1
,'0,~ ¢,,,. a _ _
10
100
f(r)V
'
!
'w
t
i
l!
|
!
!
. 1 . , . |
,
.
!
if|
0.8 A_.,L Ai
0.6
'" D'---i D=0.35 D-0.12 D=0.04
|
|fill]
o 4(9 x
0.4 0.2 0 0.1
l
1
l
10
.
100
/(r)V Figure 2. Relative deformed amplitude as a function ] ( r ) V and D for transverse waviness (top),
isotropic waviness (center) and longitudinal waviness (bottom) in an elliptical contact. The dashed line represents the predictions of (12) (with V2 replaced by V).
Experimental
Validation
The theoretical results presented in the previous section were to some extent validated by Guangteng et al. [7]. The deformation of a single feature in an EHL contact was studied using optical interferometry. The paper describes how the measured amplitude increases with increasing speed. Increasing speed means decreasing V and results in reduced elastic deformation of the ridge and thus in an increase in amplitude. However, with a further increase in the speed a flattening was observed, which is not predicted theoretically. Unfortunately, only results only for a single feature are presented. Moreover its amplitude was about 9 times larger than the film thickness, whereas the theory described previously is based on the assumption that the amplitude of the feature is small compared to the film thickness. Therefore a separate validation was undertaken. Some first results are presented below. 4.1.
Setup
The duochromatic optical interferometry technique [25] was used to measure the shape and thickness of the lubricant film in an EHL circular contact. The experimental equipment used in this investigation is a ball and disc apparatus as schematically described in Figure (3). The duochromatic interference fringe pattern, obtained by a xenon light source with a flash duration of 20#s through red and green filters, was recorded with a high-speed VCR (200 frames per second) and a 35-mm camera attached to a microscope. The lubricant used is a mineral bright stock and was held at room temperature. Results will be presented for pure rolling (E = 0) with a rolling velocity of 0.024, 0.048 and 0.096 m/s. The load is kept constant at 39.2 N giving a maximum Hertzian pressure of 0.54 GPa, and a Hertzian contact radius of 0.185 mm. The parameter values and the values of various sets of nondimensional parameters for these contact conditions are summarized in Table 1 and Table 2. On the ball a shallow waviness pattern was sputtered. A stylus trace of the pattern is given in Figure 4. The pattern is almost harmonic.
32
stylus-trace 9.8
HIGH-SPEED] /
VCR
VIDEO~ CAMERA ST O OPE
~
35ram CAMERA
~
GAGE
RECORDER I
I
COUNTERI
I
i
ENCOOER PULLEY
Figure 3. Schematic diagram of the experimental
apparatus. R E'
y0 a z
12.52 10 -3 1.171011 39.2 0.024 1.22 2.25 10 -8 0.484
Ira]
[Pa] 0.048
0.096
IN] [m/s]
[Pas] [Pa -1]
Table 1 Parameters and their values in the experiments.
9.3
130.77 6.61
79.90 7.88
47.51 9.36
A
12.23 1.65 10 -2
3.19 10 -2
6.39 10 -2
W G U
2.086 10 -6 2633 2.0 10 -11
11]1(!~ | m E -:~[lll(! ~ll
M
IV
[1.42
]1.02
0.72
Table 2 Va/ues of severa/groups of dimensionless param-
eters.
I
I
0 0.1 [mm]
I
,
0.2
0.3
Figure 4. Stylus trace of the surface pattern sputtered on the ball.
A two parameter least square fit against a sinusoidal function gives a wavelength of 60 #m and an amplitude of 0.14#m. Taking different parts of the trace the wavelength varies little. However, the amplitude varies between 0.13 and 0.16 #m, depending on the part of the trace taken in the least square fit. Table 2 also gives the values of the V parameter obtained assuming A = 60#m. In the following section the measured film thickness will be presented and compared with the results obtained with full numerical simulations using input from the profile. It is also verified to what extend the change of the amplitude of the film thickness oscillations in the center of the contact with speed matches the predicted behaviour. 4.2.
L
I
9.4
! j
I,
I
-0.1
I
h[,m] 9.5
9.2 -0.3
t
I
-0.2
I
9.6
ENCODER i~I',4--
I
9.7
XENONFILTER~ MICROSCOPE LAMP
PULLEY
I
Results
Figure 5 shows the interferometry picture of the film thickness in the contact at the time the waviness is fully inside the contact for fi = 0.024 m/s. Figure 6 shows the profile at the line y = 0 as a function of z for a similar situation. The figures show that the film thickness on a large scale exhibits the usual shape, i.e. a nearly fiat film thickness in the center of the contact and a horse shoe shaped constriction at the end. "Superimposed" on this mean profile is an oscillation due to the waviness. For this case the average film thickness computed from the measured profile is about 0.28 # m and the amplitude of oscillation about 0.15 #m.
33
-1.9 -2.00
-O.W X
" -2.50
Figure 5. Interferometry picture of measured film thickness, fi - 0.024 m / s , E - O. 1.5
0.5
I
I
I
I
- 0.024
I
I
u
u
0.9
"1'00" k ' ~ 0
'
-0.3
l
I
1
I
I
-0.2
-0.1
0
0.1
0.2
0.3
Figure 6. Measured film thickness h [#m] at the line y - 0 as a function o f x [mm] for - 0.024 m / s and E = O.
This behaviour can be easily explained. In the central region of the contact, the flow is dominated by the shear flow (Couette term). Hence, any feature on the surface in this region will move through this region at the average speed of the surfaces. For pure rolling this implies that any feature, once in the high viscosity region, will move through the contact maintaining its shape. This means that the wavelength of the oscillations will be the wavelength of the original profile. Only the amplitude of the oscillations can be smaller due to the reduction occuring when entering the high viscosity region.
-2.00
~~~-1.9
-0.91(
" -2,~
Figure 7. Snapshot of c o m p u t e d P (top) and H (bottom) as a function of X and Y , for fi = 0.024 m / s and E - O.
For the same conditions a numerical simulation was carried out. The profile shown in Figure 4 was scaled to obtain a pattern in dimensionless variables. Subsequently, a discrete function T~(X, T) was obtained by a single wave from the profile and repeating it periodically. Figure 7 shows the dimensionless pressure P and the dimensionless film thickness H as a function of X and Y at an instant the waviness is completely in the contact. Figure 8 shows the calculated film thickness h in # m at the centerline of the contact. The average film thickness in
34
the central region of the contact obtained from the computed profile is 0.25 ttm which is slightly larger than the value of 0.24 # m obtained for the central film thickness when assuming perfectly smooth surfaces. The amplitude of the oscillation in the theoretical result is about 0.13 #m. Comparing the Figures 6 and 8 and the values given above it is concluded that the predicted and measured film thickness agree quite well. The difference in the average level and in the amplitude of the film thickness is roughly 10%. For ~2 = 0.024 m / s one finds that V = 1.423. Figure 2 predicts that for a harmonic transverse waviness at this value of V the deformed amplitude should be close to the original amplitude. Using Equation (12) one obtains A d / A i = 0.80. This value is a bit too small because the formula for small V underestimates the computed results. However, it is sufficiently close to unity to indicate that in practice the waviness will tend to move through the contact virtually unchanged, as was indeed observed in the experimental results. Figure 9 shows the measured centerline film profile for the cases fi = 0.048 and 0.096 m / s . Due to the increased speed one can expect the nominal film thickness to increase. For fi = 0.048 m / s the average film thickness in the central region is 0.42 # m and for fi = 0.096 m / s it is 0.70 #m. However, compared to the increase in nominal film thickness the amplitude of the film thickness oscillations induced by the waviness changes little. In both cases it roughly equals the amplitude of the undeformed profile. Based on the values of V for these cases this is according to the expectations. A larger value of the velocity implies a smaller value of V. Hence, if the amplitude reduction for fi = 0.024 m / s is small, for the higher velocities it will only be smaller. Figure 10 shows the computed results for the cases ~ = 0.048 m / s and 0.096 m / s . The values of the average film thickness in the center of the contact obtained from the computed results are 0.40 and 0.62 # m respectively. Comparing these values with the values obtained from the measured film thickness and comparing the Figures 9 and 10 shows that also for the higher velocities the agreement between measured and computed results is quite good.
1.5
1 h 0.5
0 -0.3
i -0.2
, -0.1
i I , 0 0.1 0.2 0.3 x Figure 8. Calculated film thickness h [#m] at the line y - 0 as a function o f x [mm] for - 0.024 m / s and E - O.
'
'
1.5
',
•
-o'04s ! fi -- 0:096"" i " "
1I 0.5
0 I
-0.3
1
I
I
I
I
-0.2
-0.1
0
0.1
0.2
0.3
Figure 9. Measured film thickness h [l~m] at the line y - 0 as a function of x [ram] for fi - 0.048 and 0.096 m / s and E = O. 1.5
\!:
'
'
' u - = 0 ' 0'4 8.
--"/
0.5
0
-0.3
!
I
I
I
I
-0.2
-0.1
0
0.1
0.2
x
0.3
Figure 10. Calculated film thickness h [lzm] at the line y = 0 as a function of x [mm] fi = 0.048 and 0.096 m / s and E = O.
35
Finally, the formula presented in Section 3.3 predicts for these cases a relative deformed amplitude of 0.86 and 0.90. From a practical point of view these values indicate a waviness moving through the contact virtually maintaining its amplitude, as was indeed found. Summarizing, it is concluded that the film thickness profiles predicted using full numerical simulations show good agreement with the measured film thickness profiles. The experimental results confirm the theoretically predicted tendency that high frequency waviness moves through the contact virtually unchanged in amplitude, and that once this is the case, an increase of the speed will hardly change the picture as could be expected based on the fact that it reduces the value of the V parameter. 5. C o n c l u s i o n The deformation of harmonic waviness in an EHL contact under pure rolling appears to be governed by a single unifying mechanism. The amplitude reduction in line and circular contacts could be described by a single equation. The effect of pattern orientation and anisotropy could be incorporated in the same relation. Finally, as was demonstrated here the influence of the contact ellipticity can be included in the equation too, such that the circular contact and the line contact become specific cases of a general equation. The relative reduced amplitude of general harmonic waviness in a general EHL contact can be predicted by:
Ad
1.
(19)
2.
where
-
REFERENCES
1
A--~. = 1 + 0.15](r)V + 0.015(](r)V) 2
f(r) =
can be used as an engineering tool to obtain a first estimate of the deformation of any microgeometry under given load conditions by decomposing the microgeometry into its harmonic components. The deformed amplitude of each component can then be calculated with the formula. Finally, the deformed profile is obtained by adding up all the deformed components. Using optical interferometry experiments were carried out to validate the predictions of the theory. For the cases considered a good agreement was found between experimental and theoretical results, both in terms of amplitude reduction as well as in the details of the film thickness profiles. However, more experiments should be performed focussing on larger wavelengths and smaller speeds, to examine the complete predicted curve. Finally, the results presented in this paper were restricted to pure rolling. For the deformation of the microgeometry pure rolling may be seen as a worst case, i.e. the deformation will be smallest. If sliding occurs the fluid will force itself past surface features leading to an increased deformation. However, to accurately predict this latter case the model presented in Section 3.1 needs to be extended including non-Newtonian lubricant behaviour. These aspects are the subject of ongoing research.
/ el-~ 1
I,
if r > 1 otherwise
(20) 3.
with r = )~x/)~ and
V = ~/~~-(A/a)~3/2L-2
(21)
with A = min(Ax, A~) This formula expresses that, in terms of the dimensionless "wavelength" V, high frequency patterns deform little whereas low frequency components tend to flatten completely. The equation
4.
5.
Dowson, D., and Higginson, G.R., 1966, "Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication," Pergamon Press, Oxford, Great Britain. Hamrock, B.J., and Dowson, D., 1978, "Isothermal Elastohydrodynamic Lubrication of Point Contacts, Part III- Fully Flooded Results," ASME JOT, 99, pp. 264-276. Kaneta, M., and Cameron, A., 1980, "Effects of Asperities in Elastohydrodynamic Lubrication," ASME JOT, 102, 374-379. Kaneta, M., Sakai, T., and Nishikawa, H., 1992, "Optical Interferometric Observations of the Effects of a Bump on Point Contact EHL," ASME JOT, 114, pp. 779-784. , 1996, Cann, P.M., Spikes, H.A., Hutchinson, J.,,"The development of a Spacer Layer Imaging Method (SLIM) for Mapping Elastohy-
36
10.
11.
12.
13.
14.
15.
drodynamic Contacts," STLE Tribology Transactions, V 39, 4, pp 915-921. K a n e t a , J., K a n a d a , T., and Nishikawa, H., 1997, "Optical Interferometric Observations of the Effects of a Moving Dent on Point Contact EHL," Proc. 23rd Leeds-Lyon Symposium on Tribology, Ed. D. Dowson et al., Elsevier Tribology Series 32, pp. 69-79. G u a n g t e n g , G., Cann, P.M., Spikes, H.A., and Olver, A., 1999, "Mapping of Surface Features in the Thin Film Lubrication Regime," Proc. 1998 Leeds Lyon Tribology Conference, Elsevier Tribology Series 36, Ed. D. Dowson et al., pp. 175-183. Venner, C.H., and Lubrecht, A.A., 1994, "Numerical Simulation of a Transverse Ridge in a Circular EHL Contact, under rolling/sliding", ASME JOT, 116, pp. 751-761. Venner, C.H., and Lubrecht, A.A., 1994, "Numerical Simulation of Waviness in a Circular EHL Contact, under rolling/sliding", Proc. 20nd Leeds-Lyon Symposium on Tribology, Ed. D. Dowson et al., Elsevier Tribology Series 30, pp. 259-272. Ehret, P., Dowson, D., and Taylor, C.M., 1998, On lubricant transport conditions in elastohydrodynamic conjunctions, Proc. Roy. Soc. A 454, p 763-787. Jiang, X, Hua, D.Y., Cheng, H.S., Ai, X., Lee, S.C., 1999, "A mixed Elastohydrodynamic Lubrication Model With Asperity Contact," ASME Journal of Tribology, V 121, pp 481491. Morales Espejel, G.E., 1993, "Elastohydrodynamic Lubrication of Smooth and Rough Surfaces," PhD. Thesis, University of Cambridge, Department of Engineering. G r e e n w o o d , J.A., and Morales Espejel, G.E., 1994, "The Behaviour of Transverse Roughness in EHL Contacts," Proc. IMechE, 208, pp. 121-132. Venner, C.H., Couhier, F., Lubrecht, A.A., and G r e e n w o o d , J., 1997 "Amplitude Reduction of Waviness in Transient EHL Line Contacts," Proc. 1996 Leeds Lyon Tribology Conference, Elsevier Tribology Series 32, Ed. Dowson et al., pp. 103-112. Hooke, C.J., 1997, "Surface Roughness Modification in Elastohydrodynamic Line Contacts operating in the elastic Piezoviscous Regime,", Proc. Inst. Mech. Engrs Part J., V 212 J2, pp. 145-162.
16. Venner, C.H., Lubrecht, A.A., 1999, "Amplitude Reduction of Anisotropic Harmonic Surface Patterns in EHL Circular Contacts under Pure Rolling, " Proc. 1998 Leeds-Lyon Tribology Conference, Elseviers Tribology Series, 36, Ed. D. Dowson et al., pp 151-162. 17. Hooke, C.J., 1999, Surface Roughness modification in EHL line contacts- the effects of roughness wavelength, orientation and operating condition, Proc. 1998 Leeds-Lyon Tribology Conference, EIseviers Tribology Series, 36, Ed. D. Dowson et al., pp 193-202. 18. Hooke, C.J., a n d Venner, C.H., 1999, "Surface Roughness Attenuation in Line and Point Contacts," Proc. ImechE, part J. Journal of Engineering Tribology, in press. 19. Roelands, C.J.A., 1966, "Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils" Ph.D. Thesis, Technical University Delft, Delft, The Netherlands, (V.R.B., Groningen, The Netherlands). 20. Venner, C.H., Morales Espejel, G.E.,, 1999, "Amplitude Reduction of Small Amplitude Waviness in Transient EHL Line Contacts," Proc. ImechE, part J. Journal of Engineering Tribology, in press. 21. Lubrecht, A.A., 1987, "Numerical Solution of the EHL Line and Point Contact Problem Using Multigrid Techniques," Ph.D. Thesis, University of Twente, Enschede, The Netherlands, ISBN 909001583-3. 22. Venner, C.H., 1991, "Multilevel Solution of the EHL Line and Point Contact Problems," Ph.D. Thesis, University of Twente, Enschede, The Netherlands. ISBN 90-9003974-0. 23. W i j n a n t , Y.H., (1998), "Contact Dynamics in the field of Elastohydrodynamic Lubrication," Ph.D. Thesis, University of Twente, Enschede, The Netherlands. ISBN 90-36512239. 24. Venner, C.H., a n d Lubrecht, A.A.,, (1999), "Multigrid Techniques: A fast and etficient method for the numerical simulation of EHL point contact problems," Proc. ImechE, part J. Journal of Engineering 7kibology, in press. 25. Foord, C.A., W e d e v e n , L.D., Westlake, F.J. a n d C a m e r o n , A., (1969-70), "Optical elastohydrodynamics," Proc. Instn. Mech. Engrs., 184 P t 1 No 28, 487-505.