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Kinematics of Roughness in EHL
T h e Third Body Concept / D. Dowson et al. (Editors) 1996 Elsevier Science B.V.
50 1
Kinematics of Rougliness iu EHL G .E .M or ales-Espejel , J .A .C; r t w i wood * an (-1 .I .I,. M elgar O . “Insl,it,uto ‘I’ecnologico cle Monterrey, I T E S M , Mont,errey, Mexico DUniversity of Chrnhridge, Eng. Department, Chmhridge, 1 l . K .
III this paper, by considering only t h e central zone of t,he EHL cont>act,a n d by following t h e ideas of (;rrrnwood and Morales-Espe.jel 1994 [ti] that t,he viscosity is so high that. it disappears f r o m t h e eqiiations a n d t h e lubricant I)eronies i n effect just an elastic ‘t,hird hody’, a n d that, t h e roughness arriving at. t h e end of t,he inlet. produces a pumping effrct,, delivering a variable supply of oil which m u s t hr accommodated I)y t h e contact,, it? is shown t8tiat the pressure a n d film thickness variations in both; o w and two-sided roughness, can he described analytically simiilating t h e physical process. T h e results show t h e dancing around of t,he pressure ripples and t h e roughness during their passage through t h e contact, j u s t as in t h e experimental and numerical ohservations.
1
INTRODUCTION
Thr importance of t h e physical underst>andingof
the role of roughness in lubrication is clear ; it certainly appears t h a t scuffing and pitting a r c directly influenced by it. G r e a t effort hn3 been devoted to this topic in t h e recent years. Steady stat,e analysis (stationary roughness) have been carried out, by several authors to invrst igatr t h e deformation of waviness a n d real roughness when passing through an EIiL cont,act (e.g. Lee and Cheng 1973 [l], C h a n g rt al. 1989 [2], Venner et al. 1991 [3], Greenwood a n d Johnson 1992 [4], (ireenwood a n d Morales-Espqjel 1993 [ 5 ] ,e t c ) a11 of t h e m concluding t h a t t h e roughness cert.ainly did not pass through unchanged. ‘I’he physical understanding of the kinematic hcliaviour of surface features in E H L is j u s t beginning to emerge. lt has recently been observed experimentally ( K a n e t a et al 1992 [i’]) t h a t stirfaces witlh transversely orient>ated humps passiiig through a n EHL contact under rolling/sliding conditions produce variations in t h e film thickness, which m a y travel with velocities differing from those of t h e b u m p s which produce t,heni. Venner 1991 [8] has shown hy numerical simulation t h a t t h e pressure ripples generated in these cases, or with wavy surfaces, travel with t h e VPlocity of t h e rough surface a n d t h a t t h e film thick-
ness disturbances travel wit,h t h e average velocity of the Iiil)ricant,. Lubrecht a n d Veniier 1993 [9] have shown t h e (lancing around of pressures of a t w e s i d e d surface waviness in an EHL contact. using a sophistlicated multigrid model. Greenwood and Morales-Espqjel 1994 [ti], 1993 [lo] argued that, t h e transient. solution is m a d e of two separate parts: tjhe st,eady statmesolution (particular integral) and a complementary funct,ion tlur to the inlet, modulation of film a n d pressures. Their analysis is b a w d on t,he assrimpt,ion t h a t the contact geomet,ry in EHL can be represent,ed by that, of a n infinit,ely long c o n h c t with sinusoidal roughness, a n d a given nominal film thickness a n d mean pressure. T h e y also showed that, for typical EHL pressures viscosity effects are negligible, so t h e Reynolds equation c a n be linearized a n d solved analytically. T h e almost undeformed roughness at t h e inlet produces a pumping of the oil, giving a variable delivery much as in a standard gear-pump, which generates an excitation function of unknown a m p l i t u d e b u t with a frequency determined by t h e velocity of t h e rough surface. In tAhis paper t h e ideas of Greenwood a n d Morales-Espejel are applied to one-sided a n d twosided waviness, t o demonstrate how with a simple analytical model it is possible to show t h e dancing around of pressures a n d film thickness in time.
502 The results are compared with sophisticated numerical solutions and good agreement, is ohtained.
ANALYSIS
2
The one-dimensional Reynolds equation for compressihle Newtlonian fluids is:
i'
(Ph3
aX
")
12qaT
- a(Ph) - u8X
at
(1)
Following t h e arguments of Venner [8], for typical EHL contacts the term w 0 and therefore, the solution of equation ( I ) reduces t80 Ir x h(z - a t ) therefore, for one-sided roughness t,he film thickness variations seem to travel with the average velocity of the surfaces u and the pressure variations with the velocity of the rough surface. According to the arguments of Greenwood and Morales-Espejel [6] this is just a particular case of a more general situation, where the transient solution is made of the combination of two pari,s, some times one of bhem is dominat. For the general case of two-sided roughness, following Greenwood and Morales-Espejel one has:
2.1
Particular Integral
With non-dependence on time the Reynolds equation (1) hecoinrs dp
12111
_ -dz -(I112
p'h* - -)
Ph
where p' and h' represent the values of density and film thickness at pressure maximum. For sufficiently high mean pressures, this reduces to ph = p* h"
(3)
For a n infinitely long contact and sinusoidal undeformed roughness it can be shown that p* and h' may be approximated by the mean values p = p ( p o ) and h = h(p,,). The densiby ratio (Dowson and Higginson equation) can be linearized as
+
with C = (71 - P l ) / a , PI = P / ( l Q p o ) and = y/( 1 ypo). Therefore, the Reynolds equation (2) for high mean viscosities is reduced to
+
y1
-hh = -PP = 1 - C a ( p - P o )
(5)
For a moving two-sided sinusoidal roughness with velocity 141 for the lower surface and velocity u2 for the upper one, the film thickness ratio H = h / h is assumed to be
If = I
2r + Hasin[-(x x - ult)]+
Hbsin[-(x 2r
x
- u2t)]
According to (ireenwood and Johnson [4] sinusoidal film thickness variations produce nearly sinusoidal pressure variations, so 2r A P = ~ ( - pp o ) = Pasin[-(2
x
2r Pbsin[-(z
x
-uzf)]
- tilt)]+
(7)
by substituting equations (4) and (7) into the reduced Reynolds equation ( 5 ) it is possible to obtain the pressure amplit8udes
Having found t,he pressures, the corresponding elastic displacements v are found, from elastic theory and the assumption that the contact can be treated as an infinite wavy surface, then
V= vb
-h V
2r = Vasin[-(x
x
- uli)]+
2r sin[ -( x - uzt)]
x
with Va
= APa ,
vb
= APb
(10)
where A = 2X/(rE'ah) The film thirkneRs is the combination of the tindeformed roughness t and the elastic displace, is, measuring both roughness and ments t ~ that
503
'Table 1: Complete input, data for t,he examples.
Example One-sided roughness Venner and Luhrecht, [ 1 I] 'Two-sided roughness h b r e c h t and Venner [9]
h
x
pin
(;Pa-'
E' GPa
Ins-'
Pa s
m
inin
pn
inm
0.54
0.4
22
117
0.048
1.22
0.0127
0.184
0.12
0.058
2.0
(0.354)
22
220
0.97
0.04
0.014
0.5
0 25
0.125
Po
fY
displacement, a i ~positive outward from the cent.rr,line of the lubricant film,
Complementary Function
Equation (1) is not, linear, t)herefore its exact solirtion cannot be found by adding a particdar integral and a complementary function. However, one can extend the linearization used in the particular integral for this cwc. If again the viscosity is high, equation (1) is reduced to
The oil flow r = ( p h , ) / ( p h ) is approximated by
cmc
f711flJ'
of two-sided roughness the amplitudes are equr
neglecting higher order ternis. This can he writ,t,tw as
r=I
Once the amplitudes H, and Hb are k n o w n , t,he pressures can he calculated from eqirat,ions (8) and (7).
110
?I
rlw values of zmoI reprrsent the amplitiide of the sinusoidal rimghurss, for the I'lir value in parentheses ( ) represents an estimate from graphs
2.2
R
(;Pa
TE
+
21r + Z, sin[-(r x
TT
-~1t)]+
=
l + Z + ( A + ( ' ) [ A r ~ + ( A P n ~ + A P b ~ )(16) ] For the st,eady stmate solution, 2, = - ( A C')AP, and zb = - ( A t#hrreforeequation ( 16) reduces to
+
TIT
rE
+ r2T = ( A + (J)[APIT+ APzT]
= 1,
+ (I)hfb,
(17)
with PIT and rgT as the components for the surfaces 1 and 2 . Taking
it is easy to calculate the transient pressure ampli t.utles
A P n= ~ - , APwr = A', + C A', + C --TflT
-rbT
504
= 2X1,2/(7rmhEt)based on XI = Xu/irl with and A 2 = XU/uz. ‘rhus, the comp1et.e solution is
4.5
:-
4
3.5
E
4
32.5.
21.5 1 0.5
;
2
3
4 r
5
6
;
I
Figure 2 : Film thickness and pressures as a fiinct8ionof ttime, for S = 0.
2n sin[-(x A\+C Xi raT
4.5
- at)]-
4
0x 3.5
-E
2
3.
2.5 -
21.5
4.5
4
1
...............................
0.5 T
2.5
Figure 3: Film thickness and pressures as a function of time, for S = -1.
sin[-(r 27r
x
T
Figure I: Film thickness and pressures as a function of time, for S = 1. Note that for one-sided roughness, say 2 , = 0 these equations retain only two sinusoidal t8erins
- uzt)] , sin[-(2n xu2 : X
u
- uzt)]
for both, the coefficient of 1 is 112, and one can expect. periodic variation in 1 but, not4in E . Again as in [6] the amplitudes of r,T and P)T
cannot, he determined without analyzing the inlet. Here arbitrary values were set to reproduce the examples chosen from literature.
505
3
RESULTS
l’wo cases from literature have been chosen for sohitmion(one-sided and t,wo-sided roughness): t,he data are given in Table 1. The arbitrary amplit.iltles chosen for the complement,ary function are r c 3=~ 0.0, rbT = 0.06, for the one-sided roughness example and r , = ~ 0.025, T b T = 0.1, for t.he t,wositlrd roughness one. For conipa.rison, what, may I ) t x described as the ‘gear-pump’ values, f ( p = ( ’ ) z 4 , b rh are 0.55 for each surface. It. will he clear t,hat, however oseful t,he gearpump analogy may he in explaining 7uh.y there should he an ’excitation’, quant,it8at,ivelyit, is of lit,t.levalue. Of course, t,he pressures and viscosity arc’ relatively low at. the end of t,he inlet,, and t h e cont.rihution to t,he flow of the pressure-gradient t,t,rm need not he negligible: it woiild seem t,hat it. largely cancels the shear-flow t,erm, so that, t,lie vxcit,ation is much smaller t,han t,tie gear-pump ;Lnalogy would suggest,. ‘I‘he first set. of data (one-sided roughness) are taktw from a point rontact. example, however, tirre khey are used in a line contact, solut,ion.
3.1
One-sided Roughness
’l%ree rolling-sliding conditions are analyzed
= (S = I ) , u1 = 112 = ti (S = 0) and = 3712 (S = - I ) , where S is the slide t,o roll rai.io, S = (u2 - u l ) / V . Figures 1, 2 and 3 show the variation o f film t.hickness and pressures with time, for the three sli(le to roll ratios S = 1, S = 0 and 5’ = -1 at, a fixed point in z / b = 0.1875. In these figures t.hr%dimensionless time and pressures are tlefined as 7’ = ( C l ) / b and P = p / p o , in accord wit,h the rrf’erence. It is clear that, t,he wave1engt.h of / I and 1’ increases as the value of S, is reduced, which is (Jut, to the reduction in t.he rough surface velocity: i t must he emphasised that the wave1engt.h of the roughnr.w is the same in the t,hree cast’s. Venner and Luhrecht [ I l l show very similar resitlts for the cases S = 0 and S = -1, however, i t ] t,lwir plot for S = 1 trtiey report an increase of t,he nwan film thickness h., as the rough surface ent,ers i 1 i t . o the cont.act. It) seems that the vrlocit,y o f khe roiigh surface is so high that, the ‘ext,ra’ pttmped 111
711
1 3712
lubricant can no longer h e accommodated in the volume released by deforming tht. roughness. Figure 4 shows the variat,ion of pressures ant1 film thickness along 3: for different, times and S = I ( u 1 = $ 1 4 2 ) . It, displays t2he progression of the roughness in a complete cycle. Since t,he velocit,y of t.he upper surface ( ~ 2 is ) larger than the srnoot811surface velocity, t,he d e c t , of t.he excitation funct,ion is important8arid the deformed roughness clearly shows a. coinhination of waves, sirn ilarly t.he pressiires. Figure 5 displays the wavinrss result,s for t,he case of pure rolling, ,Y = 0 (11, = 112) i n different. t.imes of a complet,e cycle. For piire rolling, since XI = A 2 = X the coniplement8aryfunct,ion has t,he same wavelrngt,li as t.he particltlar integral and the deformed roughness keeps its siiiusoidal shape and t.he pressures t,oo. Figure 6 shows the results for S = -1 (u1 = 3u.2) i n a complete cycle. In t,his case the rough surface is irioving with the lower velocit,y, again the complementary function is modifying tlhe wavelength of t,he deformed roughness and pressures. I n t,he figure, the effect of t,he two waves is clear.
3.2
Two-sided Roughness
Here the analyzed rolling-sliding conditions are = 0, 112 = 211 (s = a), U,1 = 762 = 6 (s = 0 ) and 111 = 3112 (S= - I ) . Figures 7, 8 and 9 show the variations of pressures and film thickness along time, for a fixed value of z / b = 0.1875 and t,he t,tiree rolling-sliding cases. For tJhe first, case (S = 2) and this particular point,, the pressures remain lower or equal to t.he overall mean value p o at all times. For S = 0, the pressures are sinitsoidal ant1 since t,he waviness of the two surfaces rcniain i n phase, only the completnrnt,ary function prcwures show up. The last, case (S = - I ) is the only one solved by the reference giving very good agreerrient, wit,h Figure 9. Figure 10 displays pressures and shapes along z / b for a complete cycle i n t8ime when S = 2. Again t,he left, hand side shapes i n the graph represent, the original undeforrned roughness and its ph a.se. 211
506
1
2
O'
.0:2
.0:1
I
0
0.1
x Imnl
0.2
i
0.3
T = 0.0426
T = 0.0000 1
1 1.B 1.6 &
1.4
1.2 1 -0.8
5
I
= 0.6 0.4
O'
-0:2
-0:1
x lml
0:1
0:2
T = 0.1280
?' = 0.0853
21
1.8
O'
-of2
-011
1
0 x IW
0.1
T = 0.1706 Figiirr 4: Pressures and film thickn~ssas a function of
0.2
;P
1 0.3
for different times and S = 1.
013
507
2r--T--l
I
1
1.8 1.6
1.a
1.6
a 1.4
,1.4
1.2
1.2
1
1
-0.8
-0.8
-5
5
I
= 0.6
= 0.6
0.4
0.4
-0.2
-0.1
0 x IW
0.1
0.2
0.3
O
-0.2
T = 0.0000
-0.1
0 x [ml
0.1
0.2
0.3
T = 0.0640
2r--T---
2 1 " 1 " 1 I
1.8t
1.8
1.6 ,1.4
1.2 1
-5
-0.8 0.6
0.4
O'
-0:2
.0:1
I
0 x IW
0.1
T = 0.1280
0.2
o.2L-l---
I
O
0.3
-0.2
-0.1
0
0.1
0.2
x1 m
T = 0.1920
2r-l--l
1.8
-0.2
-0.1
0
X Iml
0.2
0.1
T = 0.2560 Figure 5: Pressures and film thickness as a function of
T
0.3
for different times and
S = 0.
3
508
1.6
1.6 ,1.4
.1.4
1.2
1.2
1
1
-0.8
-0.8 E
E. '0.6
2
I
'0.6
0.4 0.4
0.4 0.2
O'
0.2 -0:2
-0:1
I
0 x [ml
0.1
0.2
I
O'
0.3
-0:2
-0:1
1
x
0
0.1
0.2
T = 0.1280
T = 0.0000
1
2
1 .8t
O'
-0:2
.0:1
I
0
x lml
1
0.3
0.1
0.2
I
O'
0.3
-0:2
1
.0:1
0
x lmnl
0.1
0.2
T = 0.3840
T = 0.2560
-0:2
-0:l
x lml
17' = 0.5120
0:l
0:2
013
Figure 6: Pressures and film thickness as a funcbion of t for different. times and S = -1.
1
0.3
509
2.41
2
;
:
.
21.8
?!
i
1.5
1 a
0.5
I
O;
I 0.5
1
T
1.5
I
2
Figure 7: Film thickness a n d pressures as a fiinction of time, for S = 2 a n d two-sided rough ness.
T
Figure 9: Film thickness a n d pressures as a firnct,ion of t i m e , for S = - I a n d two-sided roughness. Figure 11 shows t h e results for t h e case of pure rolling S = 0, where the waviness of both surfaces remains in phase along t h e whole cycle. T h e pressures are only due to t h e complementary function and t,he shapes remain almost undeformed. All the waves keep t,tieir origiiial sinusoidal shapes. Figure 12 displays t,he res~rlt~s for t h e case when u1 = 3112 ( 5 ’ = -1)) since the two surfaces are moving with different, velocities, for 7 = 0, it IS easy t o observe t,hat t,he pressure variations, which are only due to the inlet excitation, are t h e addition of t>he contributions of each surface t o the complementary functioii.
2.2 2
8
$1.8
0.5
1
T
1.5
1
2
Figure 8: Film thickness and pressures as a fuiict.ion of time, for S = 0 and t,wo-sided roughness. At. t,he heginning of t h e cycle T = 0.0 only t h r complementary function pressures show up, since I I , = 0 it,s short wavelerigt,h is d u e to t8he velocity 112 a n d t h e shapes a p p e a r almost, undeformed. Along t h e cycle, t h e amplitude of t h e pressures increases a n d then decreases again, t h e waviness is deformed loosing its original sinrrsoidal shape.
4
CONCLUSIONS
Following Greenwood a n d Morales-Espejel [6], the transient micro-EHL solution consists of t,wo parts; a particular integral, which represents t h e st,eady s t a t e solution (with stationary roughness) moving wit,h the surface velocity, a n d a complementary function, produced by t,he inlet p u m p i n g effect from t
510
o'2r-- o'2r---r-0.15.
m
-
0 i? 0.1 -
Roughas
0.05. 1
0-
-0'07.5
.1
0.5
-0.5
1
-0.0
-5.5
1.5
T = 0.0000
-1
r r
0.5
0
-0.5
Xkl
1
5
= 0.0250
o'21--
1
0.15-
0
s!
0.1.
-
0.1.
1
I
0-
0-
-0.0
-8.5
-1
-0.5
0.5
0.5
1
1
I
7' = 0.0750
T = 0.0500
I
0-
5
T = 0.1000 Figure 10: Pressures and film thickness as a function of
3:
for different times and 5' = 2.
1.5
51 1
o'2-To'zm 0.15-
0.1 5 .
0
0
fi
E
0.1 -
0.1 -
0.05.
hitill R O U Q ~ S S
I
I
0-
0.
T = 0.0500
T = 0.0000
o'2--7--o ' 2 1 l 0.15-
0.15-
5!
4
E
0.1.
0.1 .
I
I
0.
0-
-0'07.5
-1
T = 0.1000
-0.5
0
x*
0.5
1
T = 0.1500
0.15 o2'/ 5!
a 0.1 -
I
0-
T = 0.2000 Figure 11: Pressures a n d film t,hirkness as a funct.ion of
I
for different, times a n d S = 0.
5 12
0.1 5 . 0 r
0 -0.0 -5.5
-1
0
-0.5
0.5
Yh
1
1.5
o'2r-- o'2r-t T = 0.0000
0.15
O.I5
I
I
L
-"05,5
-
0-
0-
-1
0.5
-0.5
-0'01.5
1
-1
-0.5
0.5
1
l' = 0.3000
T = 0.2000
0.15P
0.1 .
0.05.
MRoupms
I
0-
1.5
Kb
T = 0.4000
Figure 12: Pressures and film thickness as a function of
2
for different, times and S
= -1.
1.5
513 In this way the Reynolds equation is linearized and therefore the two separate solutions (for either the pressures or the film thickness) can he superposed. Both terms, the partlicular integral and t.he complementary function, can be found separat,ely by considering only an infinitely long heavily loaded contact, with known mean film t.hicknew h and mean pressure p,. Of course, since t h r inlet is not been considered, it is not possible 1.0 determine the amplitude of the excitation function and values have been arbitrarily chosen. The scheme is applied in tlwo examples from t,hv literature, with one and t,wo-sided waviness; t.hrrc. rolling-sliding conditions are studied. For t h c a first, example (one-sided rooghness), Figures 4,s and 6, it is clearly shown that, in the presenw of sliding, pressures and deformed shape are niade of a cornbinat,ion of several wavw and the shape h a s not entirely lost, its original wavelength, thv steady state roughness h a s not, disappeared and the amplitude of the complementary function is riot completely dominant, just. as predicted in G . E . Morales-Espejel [lo]. The results agree well witah Venner and Lubrecht [ l l ] . T h e second example (two-sided waviness), Figures 10, 1 I and 12 show well t8hedancing around of pressures and s h apes. When sliding is present, one can observe how the roughness is deformed to avoid coalition with the one on the other surface, and the pressures increase. For the pure rolling case, since the waviness remains in phase, only the complement,ary function in the pressures is seen. The result,^ agree well with Lribrecht, and Venner [9].
References Kwan Lee and H.S. Cheng. Effects of surface asperity on elastohydrodynamic lubrication, NASA report no. CR-2195,1973. L. Chang, C:. Cusano, T . F . Conry. Effects of lubricant rheology and kinemat,ic conditions on micro-elastohydrodynamic lubricattion, ASME J . of Trib. 111, 1989.
C.H. Venner, A.A. Lubrecht, W.E. ten
Napel. Numerical simulation of overrolling
of a surface featsure in a n EHL line contract8,
ASME .I. of Trih. 113,1991.
[4] .J.A. Greenwood and K.L. Johnson. T h e
behaviour of transverse roughness i n sliding elastohydrodynamic lubricated cont,acts. Wear, 153,page 107, 1992.
[5] J.A. Greenwood and G.E. Morales-Espqjel. The behaviour of real transverse roughness in a sliding EHL contact. Proc. 19th LeedsLyon Symp. on Trib. (1992), Elsevier Science, p p . 227-236, 1993.
[6] J.A. Greenwood and G . E . Morales-Espejel. The behaviour of transverse roughness in EHL contacts. Proc. Instn. Mech. Engrs. 208, Part .I, J . of Eng. Trih., pp. 121-132, 1994. [7] M. Kaneta, T. Sakai, and H. Nishikawa. Optical interferometric observations of the effects of a b u m p on point contact EHL, ASME, J . o f T r i b . 114,pp. 779-784, 1992. [8] C.H. Venner. Multilevel solution of the EHL line and point contact problems. Ph.D. t8hesis, University of Twente, Enschede, T h e Netherlands, ISBN 90-9003974-0,1991.
[Y] A.A. Luhrecht. and C.H. Venner. Aspects of
twmsided surface waviness in an EHL contact. Proc. 19th Leeds-Lyon Symp. on Trib. (1992), Elsevier Science, pp. 205-214, 1993.
[lo] G.E. Morales Espejel. Elastohydrodynamic lubrication of smooth and rough surfaces. Ph.D. thesis, Engineering Department, IJniversit,y of (:ambridge, 1993. [ll] C.H. Venner and A.A. Lubrecht. Numerical simulation of waviness in a circular EHL contact, under rolling / sliding. Proc 21st. Leeds-Lyon Sym. on Trib. (1994).
50 1
Kinematics of Rougliness iu EHL G .E .M or ales-Espejel , J .A .C; r t w i wood * an (-1 .I .I,. M elgar O . “Insl,it,uto ‘I’ecnologico cle Monterrey, I T E S M , Mont,errey, Mexico DUniversity of Chrnhridge, Eng. Department, Chmhridge, 1 l . K .
III this paper, by considering only t h e central zone of t,he EHL cont>act,a n d by following t h e ideas of (;rrrnwood and Morales-Espe.jel 1994 [ti] that t,he viscosity is so high that. it disappears f r o m t h e eqiiations a n d t h e lubricant I)eronies i n effect just an elastic ‘t,hird hody’, a n d that, t h e roughness arriving at. t h e end of t,he inlet. produces a pumping effrct,, delivering a variable supply of oil which m u s t hr accommodated I)y t h e contact,, it? is shown t8tiat the pressure a n d film thickness variations in both; o w and two-sided roughness, can he described analytically simiilating t h e physical process. T h e results show t h e dancing around of t,he pressure ripples and t h e roughness during their passage through t h e contact, j u s t as in t h e experimental and numerical ohservations.
1
INTRODUCTION
Thr importance of t h e physical underst>andingof
the role of roughness in lubrication is clear ; it certainly appears t h a t scuffing and pitting a r c directly influenced by it. G r e a t effort hn3 been devoted to this topic in t h e recent years. Steady stat,e analysis (stationary roughness) have been carried out, by several authors to invrst igatr t h e deformation of waviness a n d real roughness when passing through an EIiL cont,act (e.g. Lee and Cheng 1973 [l], C h a n g rt al. 1989 [2], Venner et al. 1991 [3], Greenwood a n d Johnson 1992 [4], (ireenwood a n d Morales-Espqjel 1993 [ 5 ] ,e t c ) a11 of t h e m concluding t h a t t h e roughness cert.ainly did not pass through unchanged. ‘I’he physical understanding of the kinematic hcliaviour of surface features in E H L is j u s t beginning to emerge. lt has recently been observed experimentally ( K a n e t a et al 1992 [i’]) t h a t stirfaces witlh transversely orient>ated humps passiiig through a n EHL contact under rolling/sliding conditions produce variations in t h e film thickness, which m a y travel with velocities differing from those of t h e b u m p s which produce t,heni. Venner 1991 [8] has shown hy numerical simulation t h a t t h e pressure ripples generated in these cases, or with wavy surfaces, travel with t h e VPlocity of t h e rough surface a n d t h a t t h e film thick-
ness disturbances travel wit,h t h e average velocity of the Iiil)ricant,. Lubrecht a n d Veniier 1993 [9] have shown t h e (lancing around of pressures of a t w e s i d e d surface waviness in an EHL contact. using a sophistlicated multigrid model. Greenwood and Morales-Espqjel 1994 [ti], 1993 [lo] argued that, t h e transient. solution is m a d e of two separate parts: tjhe st,eady statmesolution (particular integral) and a complementary funct,ion tlur to the inlet, modulation of film a n d pressures. Their analysis is b a w d on t,he assrimpt,ion t h a t the contact geomet,ry in EHL can be represent,ed by that, of a n infinit,ely long c o n h c t with sinusoidal roughness, a n d a given nominal film thickness a n d mean pressure. T h e y also showed that, for typical EHL pressures viscosity effects are negligible, so t h e Reynolds equation c a n be linearized a n d solved analytically. T h e almost undeformed roughness at t h e inlet produces a pumping of the oil, giving a variable delivery much as in a standard gear-pump, which generates an excitation function of unknown a m p l i t u d e b u t with a frequency determined by t h e velocity of t h e rough surface. In tAhis paper t h e ideas of Greenwood a n d Morales-Espejel are applied to one-sided a n d twosided waviness, t o demonstrate how with a simple analytical model it is possible to show t h e dancing around of pressures a n d film thickness in time.
502 The results are compared with sophisticated numerical solutions and good agreement, is ohtained.
ANALYSIS
2
The one-dimensional Reynolds equation for compressihle Newtlonian fluids is:
i'
(Ph3
aX
")
12qaT
- a(Ph) - u8X
at
(1)
Following t h e arguments of Venner [8], for typical EHL contacts the term w 0 and therefore, the solution of equation ( I ) reduces t80 Ir x h(z - a t ) therefore, for one-sided roughness t,he film thickness variations seem to travel with the average velocity of the surfaces u and the pressure variations with the velocity of the rough surface. According to the arguments of Greenwood and Morales-Espejel [6] this is just a particular case of a more general situation, where the transient solution is made of the combination of two pari,s, some times one of bhem is dominat. For the general case of two-sided roughness, following Greenwood and Morales-Espejel one has:
2.1
Particular Integral
With non-dependence on time the Reynolds equation (1) hecoinrs dp
12111
_ -dz -(I112
p'h* - -)
Ph
where p' and h' represent the values of density and film thickness at pressure maximum. For sufficiently high mean pressures, this reduces to ph = p* h"
(3)
For a n infinitely long contact and sinusoidal undeformed roughness it can be shown that p* and h' may be approximated by the mean values p = p ( p o ) and h = h(p,,). The densiby ratio (Dowson and Higginson equation) can be linearized as
+
with C = (71 - P l ) / a , PI = P / ( l Q p o ) and = y/( 1 ypo). Therefore, the Reynolds equation (2) for high mean viscosities is reduced to
+
y1
-hh = -PP = 1 - C a ( p - P o )
(5)
For a moving two-sided sinusoidal roughness with velocity 141 for the lower surface and velocity u2 for the upper one, the film thickness ratio H = h / h is assumed to be
If = I
2r + Hasin[-(x x - ult)]+
Hbsin[-(x 2r
x
- u2t)]
According to (ireenwood and Johnson [4] sinusoidal film thickness variations produce nearly sinusoidal pressure variations, so 2r A P = ~ ( - pp o ) = Pasin[-(2
x
2r Pbsin[-(z
x
-uzf)]
- tilt)]+
(7)
by substituting equations (4) and (7) into the reduced Reynolds equation ( 5 ) it is possible to obtain the pressure amplit8udes
Having found t,he pressures, the corresponding elastic displacements v are found, from elastic theory and the assumption that the contact can be treated as an infinite wavy surface, then
V= vb
-h V
2r = Vasin[-(x
x
- uli)]+
2r sin[ -( x - uzt)]
x
with Va
= APa ,
vb
= APb
(10)
where A = 2X/(rE'ah) The film thirkneRs is the combination of the tindeformed roughness t and the elastic displace, is, measuring both roughness and ments t ~ that
503
'Table 1: Complete input, data for t,he examples.
Example One-sided roughness Venner and Luhrecht, [ 1 I] 'Two-sided roughness h b r e c h t and Venner [9]
h
x
pin
(;Pa-'
E' GPa
Ins-'
Pa s
m
inin
pn
inm
0.54
0.4
22
117
0.048
1.22
0.0127
0.184
0.12
0.058
2.0
(0.354)
22
220
0.97
0.04
0.014
0.5
0 25
0.125
Po
fY
displacement, a i ~positive outward from the cent.rr,line of the lubricant film,
Complementary Function
Equation (1) is not, linear, t)herefore its exact solirtion cannot be found by adding a particdar integral and a complementary function. However, one can extend the linearization used in the particular integral for this cwc. If again the viscosity is high, equation (1) is reduced to
The oil flow r = ( p h , ) / ( p h ) is approximated by
cmc
f711flJ'
of two-sided roughness the amplitudes are equr
neglecting higher order ternis. This can he writ,t,tw as
r=I
Once the amplitudes H, and Hb are k n o w n , t,he pressures can he calculated from eqirat,ions (8) and (7).
110
?I
rlw values of zmoI reprrsent the amplitiide of the sinusoidal rimghurss, for the I'lir value in parentheses ( ) represents an estimate from graphs
2.2
R
(;Pa
TE
+
21r + Z, sin[-(r x
TT
-~1t)]+
=
l + Z + ( A + ( ' ) [ A r ~ + ( A P n ~ + A P b ~ )(16) ] For the st,eady stmate solution, 2, = - ( A C')AP, and zb = - ( A t#hrreforeequation ( 16) reduces to
+
TIT
rE
+ r2T = ( A + (J)[APIT+ APzT]
= 1,
+ (I)hfb,
(17)
with PIT and rgT as the components for the surfaces 1 and 2 . Taking
it is easy to calculate the transient pressure ampli t.utles
A P n= ~ - , APwr = A', + C A', + C --TflT
-rbT
504
= 2X1,2/(7rmhEt)based on XI = Xu/irl with and A 2 = XU/uz. ‘rhus, the comp1et.e solution is
4.5
:-
4
3.5
E
4
32.5.
21.5 1 0.5
;
2
3
4 r
5
6
;
I
Figure 2 : Film thickness and pressures as a fiinct8ionof ttime, for S = 0.
2n sin[-(x A\+C Xi raT
4.5
- at)]-
4
0x 3.5
-E
2
3.
2.5 -
21.5
4.5
4
1
...............................
0.5 T
2.5
Figure 3: Film thickness and pressures as a function of time, for S = -1.
sin[-(r 27r
x
T
Figure I: Film thickness and pressures as a function of time, for S = 1. Note that for one-sided roughness, say 2 , = 0 these equations retain only two sinusoidal t8erins
- uzt)] , sin[-(2n xu2 : X
u
- uzt)]
for both, the coefficient of 1 is 112, and one can expect. periodic variation in 1 but, not4in E . Again as in [6] the amplitudes of r,T and P)T
cannot, he determined without analyzing the inlet. Here arbitrary values were set to reproduce the examples chosen from literature.
505
3
RESULTS
l’wo cases from literature have been chosen for sohitmion(one-sided and t,wo-sided roughness): t,he data are given in Table 1. The arbitrary amplit.iltles chosen for the complement,ary function are r c 3=~ 0.0, rbT = 0.06, for the one-sided roughness example and r , = ~ 0.025, T b T = 0.1, for t.he t,wositlrd roughness one. For conipa.rison, what, may I ) t x described as the ‘gear-pump’ values, f ( p = ( ’ ) z 4 , b rh are 0.55 for each surface. It. will he clear t,hat, however oseful t,he gearpump analogy may he in explaining 7uh.y there should he an ’excitation’, quant,it8at,ivelyit, is of lit,t.levalue. Of course, t,he pressures and viscosity arc’ relatively low at. the end of t,he inlet,, and t h e cont.rihution to t,he flow of the pressure-gradient t,t,rm need not he negligible: it woiild seem t,hat it. largely cancels the shear-flow t,erm, so that, t,lie vxcit,ation is much smaller t,han t,tie gear-pump ;Lnalogy would suggest,. ‘I‘he first set. of data (one-sided roughness) are taktw from a point rontact. example, however, tirre khey are used in a line contact, solut,ion.
3.1
One-sided Roughness
’l%ree rolling-sliding conditions are analyzed
= (S = I ) , u1 = 112 = ti (S = 0) and = 3712 (S = - I ) , where S is the slide t,o roll rai.io, S = (u2 - u l ) / V . Figures 1, 2 and 3 show the variation o f film t.hickness and pressures with time, for the three sli(le to roll ratios S = 1, S = 0 and 5’ = -1 at, a fixed point in z / b = 0.1875. In these figures t.hr%dimensionless time and pressures are tlefined as 7’ = ( C l ) / b and P = p / p o , in accord wit,h the rrf’erence. It is clear that, t,he wave1engt.h of / I and 1’ increases as the value of S, is reduced, which is (Jut, to the reduction in t.he rough surface velocity: i t must he emphasised that the wave1engt.h of the roughnr.w is the same in the t,hree cast’s. Venner and Luhrecht [ I l l show very similar resitlts for the cases S = 0 and S = -1, however, i t ] t,lwir plot for S = 1 trtiey report an increase of t,he nwan film thickness h., as the rough surface ent,ers i 1 i t . o the cont.act. It) seems that the vrlocit,y o f khe roiigh surface is so high that, the ‘ext,ra’ pttmped 111
711
1 3712
lubricant can no longer h e accommodated in the volume released by deforming tht. roughness. Figure 4 shows the variat,ion of pressures ant1 film thickness along 3: for different, times and S = I ( u 1 = $ 1 4 2 ) . It, displays t2he progression of the roughness in a complete cycle. Since t,he velocit,y of t.he upper surface ( ~ 2 is ) larger than the srnoot811surface velocity, t,he d e c t , of t.he excitation funct,ion is important8arid the deformed roughness clearly shows a. coinhination of waves, sirn ilarly t.he pressiires. Figure 5 displays the wavinrss result,s for t,he case of pure rolling, ,Y = 0 (11, = 112) i n different. t.imes of a complet,e cycle. For piire rolling, since XI = A 2 = X the coniplement8aryfunct,ion has t,he same wavelrngt,li as t.he particltlar integral and the deformed roughness keeps its siiiusoidal shape and t.he pressures t,oo. Figure 6 shows the results for S = -1 (u1 = 3u.2) i n a complete cycle. In t,his case the rough surface is irioving with the lower velocit,y, again the complementary function is modifying tlhe wavelength of t,he deformed roughness and pressures. I n t,he figure, the effect of t,he two waves is clear.
3.2
Two-sided Roughness
Here the analyzed rolling-sliding conditions are = 0, 112 = 211 (s = a), U,1 = 762 = 6 (s = 0 ) and 111 = 3112 (S= - I ) . Figures 7, 8 and 9 show the variations of pressures and film thickness along time, for a fixed value of z / b = 0.1875 and t,he t,tiree rolling-sliding cases. For tJhe first, case (S = 2) and this particular point,, the pressures remain lower or equal to t.he overall mean value p o at all times. For S = 0, the pressures are sinitsoidal ant1 since t,he waviness of the two surfaces rcniain i n phase, only the completnrnt,ary function prcwures show up. The last, case (S = - I ) is the only one solved by the reference giving very good agreerrient, wit,h Figure 9. Figure 10 displays pressures and shapes along z / b for a complete cycle i n t8ime when S = 2. Again t,he left, hand side shapes i n the graph represent, the original undeforrned roughness and its ph a.se. 211
506
1
2
O'
.0:2
.0:1
I
0
0.1
x Imnl
0.2
i
0.3
T = 0.0426
T = 0.0000 1
1 1.B 1.6 &
1.4
1.2 1 -0.8
5
I
= 0.6 0.4
O'
-0:2
-0:1
x lml
0:1
0:2
T = 0.1280
?' = 0.0853
21
1.8
O'
-of2
-011
1
0 x IW
0.1
T = 0.1706 Figiirr 4: Pressures and film thickn~ssas a function of
0.2
;P
1 0.3
for different times and S = 1.
013
507
2r--T--l
I
1
1.8 1.6
1.a
1.6
a 1.4
,1.4
1.2
1.2
1
1
-0.8
-0.8
-5
5
I
= 0.6
= 0.6
0.4
0.4
-0.2
-0.1
0 x IW
0.1
0.2
0.3
O
-0.2
T = 0.0000
-0.1
0 x [ml
0.1
0.2
0.3
T = 0.0640
2r--T---
2 1 " 1 " 1 I
1.8t
1.8
1.6 ,1.4
1.2 1
-5
-0.8 0.6
0.4
O'
-0:2
.0:1
I
0 x IW
0.1
T = 0.1280
0.2
o.2L-l---
I
O
0.3
-0.2
-0.1
0
0.1
0.2
x1 m
T = 0.1920
2r-l--l
1.8
-0.2
-0.1
0
X Iml
0.2
0.1
T = 0.2560 Figure 5: Pressures and film thickness as a function of
T
0.3
for different times and
S = 0.
3
508
1.6
1.6 ,1.4
.1.4
1.2
1.2
1
1
-0.8
-0.8 E
E. '0.6
2
I
'0.6
0.4 0.4
0.4 0.2
O'
0.2 -0:2
-0:1
I
0 x [ml
0.1
0.2
I
O'
0.3
-0:2
-0:1
1
x
0
0.1
0.2
T = 0.1280
T = 0.0000
1
2
1 .8t
O'
-0:2
.0:1
I
0
x lml
1
0.3
0.1
0.2
I
O'
0.3
-0:2
1
.0:1
0
x lmnl
0.1
0.2
T = 0.3840
T = 0.2560
-0:2
-0:l
x lml
17' = 0.5120
0:l
0:2
013
Figure 6: Pressures and film thickness as a funcbion of t for different. times and S = -1.
1
0.3
509
2.41
2
;
:
.
21.8
?!
i
1.5
1 a
0.5
I
O;
I 0.5
1
T
1.5
I
2
Figure 7: Film thickness a n d pressures as a fiinction of time, for S = 2 a n d two-sided rough ness.
T
Figure 9: Film thickness a n d pressures as a firnct,ion of t i m e , for S = - I a n d two-sided roughness. Figure 11 shows t h e results for t h e case of pure rolling S = 0, where the waviness of both surfaces remains in phase along t h e whole cycle. T h e pressures are only due to t h e complementary function and t,he shapes remain almost undeformed. All the waves keep t,tieir origiiial sinusoidal shapes. Figure 12 displays t,he res~rlt~s for t h e case when u1 = 3112 ( 5 ’ = -1)) since the two surfaces are moving with different, velocities, for 7 = 0, it IS easy t o observe t,hat t,he pressure variations, which are only due to the inlet excitation, are t h e addition of t>he contributions of each surface t o the complementary functioii.
2.2 2
8
$1.8
0.5
1
T
1.5
1
2
Figure 8: Film thickness and pressures as a fuiict.ion of time, for S = 0 and t,wo-sided roughness. At. t,he heginning of t h e cycle T = 0.0 only t h r complementary function pressures show up, since I I , = 0 it,s short wavelerigt,h is d u e to t8he velocity 112 a n d t h e shapes a p p e a r almost, undeformed. Along t h e cycle, t h e amplitude of t h e pressures increases a n d then decreases again, t h e waviness is deformed loosing its original sinrrsoidal shape.
4
CONCLUSIONS
Following Greenwood a n d Morales-Espejel [6], the transient micro-EHL solution consists of t,wo parts; a particular integral, which represents t h e st,eady s t a t e solution (with stationary roughness) moving wit,h the surface velocity, a n d a complementary function, produced by t,he inlet p u m p i n g effect from t
510
o'2r-- o'2r---r-0.15.
m
-
0 i? 0.1 -
Roughas
0.05. 1
0-
-0'07.5
.1
0.5
-0.5
1
-0.0
-5.5
1.5
T = 0.0000
-1
r r
0.5
0
-0.5
Xkl
1
5
= 0.0250
o'21--
1
0.15-
0
s!
0.1.
-
0.1.
1
I
0-
0-
-0.0
-8.5
-1
-0.5
0.5
0.5
1
1
I
7' = 0.0750
T = 0.0500
I
0-
5
T = 0.1000 Figure 10: Pressures and film thickness as a function of
3:
for different times and 5' = 2.
1.5
51 1
o'2-To'zm 0.15-
0.1 5 .
0
0
fi
E
0.1 -
0.1 -
0.05.
hitill R O U Q ~ S S
I
I
0-
0.
T = 0.0500
T = 0.0000
o'2--7--o ' 2 1 l 0.15-
0.15-
5!
4
E
0.1.
0.1 .
I
I
0.
0-
-0'07.5
-1
T = 0.1000
-0.5
0
x*
0.5
1
T = 0.1500
0.15 o2'/ 5!
a 0.1 -
I
0-
T = 0.2000 Figure 11: Pressures a n d film t,hirkness as a funct.ion of
I
for different, times a n d S = 0.
5 12
0.1 5 . 0 r
0 -0.0 -5.5
-1
0
-0.5
0.5
Yh
1
1.5
o'2r-- o'2r-t T = 0.0000
0.15
O.I5
I
I
L
-"05,5
-
0-
0-
-1
0.5
-0.5
-0'01.5
1
-1
-0.5
0.5
1
l' = 0.3000
T = 0.2000
0.15P
0.1 .
0.05.
MRoupms
I
0-
1.5
Kb
T = 0.4000
Figure 12: Pressures and film thickness as a function of
2
for different, times and S
= -1.
1.5
513 In this way the Reynolds equation is linearized and therefore the two separate solutions (for either the pressures or the film thickness) can he superposed. Both terms, the partlicular integral and t.he complementary function, can be found separat,ely by considering only an infinitely long heavily loaded contact, with known mean film t.hicknew h and mean pressure p,. Of course, since t h r inlet is not been considered, it is not possible 1.0 determine the amplitude of the excitation function and values have been arbitrarily chosen. The scheme is applied in tlwo examples from t,hv literature, with one and t,wo-sided waviness; t.hrrc. rolling-sliding conditions are studied. For t h c a first, example (one-sided rooghness), Figures 4,s and 6, it is clearly shown that, in the presenw of sliding, pressures and deformed shape are niade of a cornbinat,ion of several wavw and the shape h a s not entirely lost, its original wavelength, thv steady state roughness h a s not, disappeared and the amplitude of the complementary function is riot completely dominant, just. as predicted in G . E . Morales-Espejel [lo]. The results agree well witah Venner and Lubrecht [ l l ] . T h e second example (two-sided waviness), Figures 10, 1 I and 12 show well t8hedancing around of pressures and s h apes. When sliding is present, one can observe how the roughness is deformed to avoid coalition with the one on the other surface, and the pressures increase. For the pure rolling case, since the waviness remains in phase, only the complement,ary function in the pressures is seen. The result,^ agree well with Lribrecht, and Venner [9].
References Kwan Lee and H.S. Cheng. Effects of surface asperity on elastohydrodynamic lubrication, NASA report no. CR-2195,1973. L. Chang, C:. Cusano, T . F . Conry. Effects of lubricant rheology and kinemat,ic conditions on micro-elastohydrodynamic lubricattion, ASME J . of Trib. 111, 1989.
C.H. Venner, A.A. Lubrecht, W.E. ten
Napel. Numerical simulation of overrolling
of a surface featsure in a n EHL line contract8,
ASME .I. of Trih. 113,1991.
[4] .J.A. Greenwood and K.L. Johnson. T h e
behaviour of transverse roughness i n sliding elastohydrodynamic lubricated cont,acts. Wear, 153,page 107, 1992.
[5] J.A. Greenwood and G.E. Morales-Espqjel. The behaviour of real transverse roughness in a sliding EHL contact. Proc. 19th LeedsLyon Symp. on Trib. (1992), Elsevier Science, p p . 227-236, 1993.
[6] J.A. Greenwood and G . E . Morales-Espejel. The behaviour of transverse roughness in EHL contacts. Proc. Instn. Mech. Engrs. 208, Part .I, J . of Eng. Trih., pp. 121-132, 1994. [7] M. Kaneta, T. Sakai, and H. Nishikawa. Optical interferometric observations of the effects of a b u m p on point contact EHL, ASME, J . o f T r i b . 114,pp. 779-784, 1992. [8] C.H. Venner. Multilevel solution of the EHL line and point contact problems. Ph.D. t8hesis, University of Twente, Enschede, T h e Netherlands, ISBN 90-9003974-0,1991.
[Y] A.A. Luhrecht. and C.H. Venner. Aspects of
twmsided surface waviness in an EHL contact. Proc. 19th Leeds-Lyon Symp. on Trib. (1992), Elsevier Science, pp. 205-214, 1993.
[lo] G.E. Morales Espejel. Elastohydrodynamic lubrication of smooth and rough surfaces. Ph.D. thesis, Engineering Department, IJniversit,y of (:ambridge, 1993. [ll] C.H. Venner and A.A. Lubrecht. Numerical simulation of waviness in a circular EHL contact, under rolling / sliding. Proc 21st. Leeds-Lyon Sym. on Trib. (1994).