sliding ehl line contacts

Transient Processes in Tribology G. Dalmaz et al. (Editors) 9 2004 Elsevier B.V. All fights reserved

243

The effect of two-sided roughness in rolling/sliding ehl line contacts Andreas Almqvist and Roland Larsson Luleh University of Technology, Division of Machine Elements, SE-971 87 Lule~, Sweden ABSTRACT In most theoretical studies carried out to date on the effect of surface roughness in elastohydrodynamic lubrication (EHL) one surface is considered smooth and one as being rough. In real tribological contacts however, both surfaces normally have similar roughness heights. When modelling a rolling contact it is possible to simply sum the roughness of the two contact surfaces but in a sliding EHL contact, a continuously changing effective surface roughness occurs. The aim of this work was to investigate the influence of elementary surface features such as dents and ridges on the film thickness and pressure. This was done numerically using transient non-Newtonian simulations of an EHL line contact using a coupled smoother combined with a multilevel technique. Four different "overtaking" phenomena were investigated; ridge-ridge, dent-ridge, ridge-dent, and dent-dent. It was shown that the minimum film-thickness produced by a ridge is further reduced in a dent-ridge overtaking event. The squeeze effect seen in the ridge-ridge case resulted in large deformations and filmthickness heights comparable to the corresponding smooth case just before the overtaking event occurred. These local effects arising from simulating two-sided roughness were compared to simulations using a traditional "onesided rough surface contacting a perfectly smooth surface." Keywords elastohydrodynamic lubrication, transient effects, coupled method, non-Newtonian, surface roughness NOTATION amni

matrix coefficients

b

Hertzian half width

E"

effective modulus of elasticity, 2/E'=(1-vlZ)/E1+(1-vlZ)/EI

S

non-Newtonian parameter surface roughness profile i

t

time

At

time increment

T

dimensionless time

h

film thickness

AT

dimensionless time increment

hmin

minimum film thickness

Ul

velocity of upper surface

hoo

integration constant

ll2

velocity of lower surface

H

dimensionless film thickness

Us

sum velocity, us =/,/i

Hmi,

dimensionless minimum film thickness

Hoo

dimensionless integration constant

W

externally applied load

I

integration interval

X

spatial coordinate

P

pressure

X

dimensionless spatial coordinate

Ph

maximum Hertzian pressure

AX

dimensionless spatial mesh size

PO

coefficient (Roelands equation)

Xic

central position of dent/ridge i

P

dimensionless pressure

viscosity index (Roelands equation)

R

reduced radius of curvature 1/R = 1/RI+1/R2

pressure viscosity index (Roelands equation)

slide-to-roll ratio, s = 2.(ul-u2)/(Ul+U2)

coefficient in Reynolds equation

+ U2

mean velocity, u - u~/2

244 ,Z

dimensionless speed parameter

17

viscosity

r/o

viscosity

r/

dimensionless viscosity

v/

Poisson ratio of surface material 1

89

Poisson ratio of surface material 2

P

density

p0

density at ambient pressure

/9

dimensionless density

r0

representative or Eyring shear stress

/'0

dimensionless representative or Eyring shear

at ambient pressure

stress measure of dent caused minimum film thickness

1 INTRODUCTION A machined surface is never perfectly smooth. The lubrication of such surfaces in elastohydrodynamic lubrication (EHL) contacts is normally a highly transient process. Two different types of transient effects occur in an EHL line contact subjected to a rolling/sliding motion. In the first case, pure rolling, the film thickness will change transiently at each position along the contact as a result of the shifted roughness profile. It is possible to simplify the theoretical model of the pure rolling case by assuming all roughness to be located at one surface while letting the other surface be smooth. The roughness profile used in this model is simply the sum of roughness profile heights of both surfaces. In the second case, a combined rolling and sliding motion, the roughness profiles of each surface shift relative to each other resulting in asperity "overtaking" phenomena occurring within the contact Most simulations of EHL with rough surfaces assume the rolling case, i.e. that the roughness is located on only one of the surfaces [1-4]. The reason for this is that it becomes much more difficult to define a simple model if the roughness and relative movement of both surfaces has to be considered. Clearly, a realistic analysis of a contact experiencing a sliding motion is not possible if the second transient effect is not considered. Changes in the film thickness in cases with roughness on only one surface are very much controlled by the local inlet conditions. One example is the distortion

phenomena where the passage of an asperity causes a distortion in the film-thickness which travels through the contact at the speed of the fluid and not the speed of the asperity through the contact. However, in the case where both surfaces are assumed to be rough and have a relative motion, local effects may occur inside the EHL conjunction that cause film breakdown and/or extreme pressure peaks. Such effects would be due to the constantly changing sum of the surface roughness when distortions from asperity overtaking phenomena inside the contact area are considered. This would in turn cause changing volumes between the asperities which will influence film pressure and also beating capacity. A number of investigations taking into account roughness on both surfaces have been presented, for example, [5-7]. However no systematic study of the two-sided roughness problem (such has been done by Lubrect and Venner [3] for the one-sided roughness case) has been attempted. This paper presents our first attempt to make such a systematic study of two-sided roughness. In order to describe the pressure and film-thickness between sliding rough surfaces in a highly pressurized hydrodynamically lubricated conjunction, the numerical solver has to be able to handle transient effects. Moreover, if the simulation aims to model the function of a real application, the solver needs to include a non-Newtonian description of the fluid, internal cavitation, thermal effects, plasticity, etc. There are still much to be done in this area, but the character of many of these effects makes it possible to model them in simplified way whilst still obtaining valuable information. The most commonly used numerical model of an EHL line contact is based on Reynolds equation coupled with an equation for the film-thickness, empirical laws for viscosity and density, and a force-balance equation. See, for example, Venner and Lubrecht [4] for a detailed view on multi-level techniques for the numerical solution of the equations. The transient squeeze effect that occurs when a dent and ridge overtake each other, almost causing film breakdown, is the main focus of this work. This event was chosen since it would intuitively not be considered harmful. However, simulations show that a film breakdown can occur as a result of such an overtaking event. Five different sliding-to-rolling ratios, a test with a doubled load and a test with doubled ridge amplitude were carried out. It was shown that, for all amplitude cases, there exists a wavelength for which the dent ridge overtaking produces a minimum film thickness.

245

r/(x, t ) - r/o exp

2 THEORY

This work focuses on one of the four different overtaking phenomena that may occur in an EHL line contact. Pressure build-up and film formation related to these phenomena could have been studied without taking into account non-Newtonian and thermal effects in order to simplify the analysis. However, Non-Newtonian effects are included according to Conry et al. [8]).

1+ p

(4)

Po where

a, po and z are mutually dependent: Z

Po - --(In(r/o)+ 9.67). 6g

(5)

p ( x , t ) d x - w(t). P(x, t) >_O .

Equations

2.1

- 1+

The line contact problem to be solved comprises the time dependent non-Newtonian Reynolds equation (1), the film thickness equation (2), Roelands and Dowson and Higginson equations for the viscosity (3) and density (4). At all times, the force balance condition (5) determines the integration constant hoo(t). Also, the cavitation condition (6) is used to ensure that all negative pressures obtained during the solution process are removed.

(6)

The surface dents/ridges were modeled by equation (7),

g/i(x,t)-2"lO-7"li

_loIx-x4(t)]z 10 ~'lO-'mi)

X

(7)

xc!,,t1

COS

-

1 0 -4 9 m

Ox

-o

(1) where the dent~ump central position is given by: C

where oh 8-

with x s as the initial position. To simplify varying the amplitudes and wavelengths of the surface dents/ridges, two help parameters where introduced; li and mi. A variation of li corresponds to a variation in the amplitude of the dent/ridge whilst a variation of mi corresponds to a variation in the wavelength of the dent/ridge.

12r/

S - 3(Ecosh(E)- sinh(E)) E3

1+ r/(u,-u2) X roh sinl~(Y~

I/

12

hop

E

--

~

S

X i ( t ) - X + uit

The line contact equations (1) - (6) were transformed into dimensionless form by the set of dimensionless variables:

m

2r 0 ax 2

x + 8(x,,)+ h(x,,)- ho(,)+-2e (x,,)+ (x,,)

(2)

H - hR/b 2

X-

P - P / Ph

p - p l Po ,

r - tu~/(26)

~ - q/qo

where

where the elastic deformation is given by 6 ( x , t ) - - ~ E4S ~ l n l x -

P(X,t)-Po

Ph =

s[p(s,t)ds.

0.59- 10 -9 q- 1.34p(x, t) 0.59" 10 -9 -[- p ( x , t )

(3)

2w b - / 8 w R rob' ~ :rE

x/b

246

2.2 Numerics

In the Reynolds approach used in this work, the Reynolds equation and the equation for the film thickness are coupled so that pressure and film thickness are solved simultaneously (Evans et al. [5], Holmes [9]). Multilevel techniques (Venner, Venner and Lubrecht [2-4]) have been used to accelerate the convergence of the Reynolds approaches based on serial solution methods, i.e., the pressure is the only unknown parameter. Here the multilevel technique is extended to accelerate the solution process of the coupled approach. The full multilevel algorithm is based on a specially developed iterative relaxation process. This relaxation process is a generalized form of the Jacobi method that addresses the coupling between the Reynolds and the film thickness equations. A block matrix system of the discrete forms of the two equations can be composed and hence the smoother will be referred to as the "Block-Jacobi" method. If the ordinary Jacobi method is applied, only the main diagonal of the determining block-matrix would be used to give pressure and film thickness increments. This kind of iterative method would not converge. In order to account for the coupling between pressure and film thickness, the Block-Jacobi method makes use of the main diagonals in each of the four blocks (see Appendix). The Block-Jacobi method is stable but has the disadvantage of slow convergence. Convergence can, however, be accelerated by using the applied multilevel technique. When the Block-Jabobi method is applied to solve the 2n-by-2n (n is the number of spatial nodes) block matrix system, it is reduced to a 2-by-2 matrix system

(8):

al2ilIPi]-[s 1,

[ alli a21i a22i Hi

(8)

gi

with coefficients given in Appendix. The nature of the EHL line contact problem introduces some difficulties that have to be dealt with. The convergence of the force-balance equation is very important in order to obtain sufficient accuracy of H 0 within each time step. This can be seen from the pressure-part solution of the coupled system, viz. 19. =

a22is -- al2ig i allia22i - al2ia21i

(9)

where the nominator contains the time dependence of the Reynolds equation. The nominator of (9)

expresses, numerically, the time derivative of the filmthickness and thus the time derivative of hoo(t), i.e. a small At will increase the influence of an error in hoo(t). The physical explanation of this behaviour is the squeeze mechanism represented by (:3/c3t in (1). This is important not only when using a coupled solution method, a serial solution method will be affected in the same way. In the EHL conjunction outlet the cavitation condition (6) is imposed. The solution of the Reynolds equation, which admits a negative pressure, is not valid here and the resulting non-Newtonian parameter S will therefore drop to zero. Correcting of S by forcing S = 1 when OP/aX = 0 leads to an effective viscosity r / / S which equals the ambient viscosity r/0. This gives an almost smooth extension of S in the region of zero pressure. The importance of such an extension relates to the Block-Jacobi relaxation of (7), since (6) implies P - 0 in the cavitation zone and the whole domain is considered at all iterations. In the solution process this, correction successively moves the starting point of the cavitation zone to a certain point. 2.3 Error estimation

In order to validate the results it is important to estimate the errors. The strategy used in this work was to check the maximum error in the whole domain (both spatially and temporally). In the case of a line contact, the defined error functions are twodimensional. The time error measure is defined by (10) since all errors measured were obtained by halving the step size.

~'time(At)-max[f~ i,k [ i,2k(-~l-tYi) i,k (At I. That is, the solution profiles

~

i,k

(10)

(pressure/film-

thickness) are considered as a function of the step size ( At ) in time. The indices i and k represents the spatial and the time coordinate respectively. However, this error will of course include the error in the spatial domain, e.g., the finer the spatial step size ( A x ) the smaller the

error~time(At).

The same principle was

adopted to define the spatial errors by considering the solution profiles as functions of the spatial step size.

e'x(AX)-max]~ 2i,k(--2)-(1)(Ax). i,k i,k l

(11)

This treatment of numerical errors is further discussed in [10]. The spatial error includes the time errors in the same way as the time error includes the spatial

247 error. In addition, both the time and the spatial error measures include a convergence error since the solution processes is not numerically exact. The convergence error is the maximum residual value of the Reynolds equation at each time step. This measure of the convergence error is sufficient since the residual value and thus the error in the film thickness equation are small in comparison. It was found that a relationship existed between the step size in time and the spatial step size. That is, for the running conditions in this work, u sAt/2 has to be less than or equal to the step size Ax ( A T <_ AX ). Applying this condition yields the appropriate behaviour of the time error (10), i.e.,

Etime(At/2)/Cam e ( A t ) ~ 1/2.

However, in the case of a slowly moving asperity

(12asperity <-~0.25/t ),

contacting a smooth surface, the

transient effects are well resolved even with values of u~ At / 2 larger than Ax. In this work the maximum time error in the film thickness was less than 5% of the corresponding smooth minimum film thickness. This maximum time error was found in the simulations with s = 1/3 and with the largest dent amplitude to dent wavelength ratio. All the results shown were simulated with at least four different time steps and in some cases, spatial discretisation was checked by a set of three different spatial mesh sizes. It was found that the time error in all cases was larger than the spatial error even when using the smallest number of nodes at the finest level, e.g., 257. Therefore, all simulations performed have a resolution of 257 nodes since the computational time in this case is decreased by roughly a factor of four compared to a resolution of 1025 nodes.

3 RESULTS AND DISCUSSION The focus of this section is on the film thickness thinning effect caused by a dent-ridge overtaking event. The other three possible cases of surface features overtaking each other have also been briefly studied and are described below. In the case of ridge-ridge overtaking the pressure spike produced is by far the largest, when comparing surface features of the same amplitude and wavelength, and this type of overtaking also produce the thinnest film thickness. In the beginning of a dentdent overtaking event, a pressure decrease due to the diverging surfaces occurs and at the end of the event the contact geometry causes a rapidly convergent gap that produce a large pressure spike. The pressure spike

produced in this case is of comparable magnitude to the pressure spike in the ridge-ridge event with ridges of ha/f the amplitude. The case of the ridge-dent overtaking event was found to have less extreme pressure spikes and minimum film thicknesses. However, it is not possible to investigate such a case without a two-sided surface roughness treatment. Thus, it is possible that the transient effects may be more significant in a contact with roughness on both surfaces. The case studied in more detail, a dent-ridge overtaking event, is most interesting because of the potential film breakdown which may occur as an elastically deformed asperity is allowed to recover to its original height. Because of the lubricant film thinning effect in a dent-ridge overtaking event, several different parameter studies were performed. Five different slide-to-roll ratios (s) were initially selected, starting from 5/3, decreasing to 1/3 in 1/3 steps. Sets of simulations with fixed dent wavelength and varying dent amplitude were carried out, keeping the ridge amplitude and wavelength constant. Also, for the case of s - 5/3, both dent amplitude and wavelength were varied giving a total of 42 combinations. A set of simulations with doubled load and varying the dent amplitude only, were also carried out. In addition, again for the case of s = 5/3, the dent amplitude was kept constant and simulations with doubled ridge amplitude varying only the dent wavelength were carried out. Figure 1 shows intermediate time steps of the dent-ridge overtaking event in the case of dent parameter lg = 1 and mi = -5 and s = 4/3. It is clear that the dent causes a thinning effect when overtaking the ridge. This is due to the pressure release, experienced by the already compressed ridge, under the time of the overtaking causing a thinning of the film thickness to occur. To compare the different sliding-to-rolling ratios, a measure of the minimum film thickness caused by the dent was defined, see equation (12). Simulations of the ridge passage only for each of the sliding-torolling ratios tested were used as a reference for the dent-ridge simulations. The exact position of the overtaking was set as -0.3253-b, in order to have a suitable "window" for all sliding-to-rolling ratios tested. It is important to remember that this measure assumes a large value when the actual film thickness is small. ~(l, m) :

-lmin~

(X,Z)- n

ref(X, T 1

min(H,,m (X:o))

(12)

248 1.5

-1

-

/0.15--~--____~

,

,

,

"0.16..-.._

, _ . ~ 0.~ 5 - ~ - - - - - - - - - - - - ~

. ~ ' ~ ~ ' o . 2 ~

1 -2

~qr

~ 0 . 2

~

-

O5

-3

%

1,5

-1.5

-1

-0.5

0

0.5

1

-4

"~

0.4

1

o.5

5

~2

-1.5

-1

-0.5

0

05

2

1

,,,

15

O.3~~~.~bi

3

4 rni

5

6

Figure 2. Contour map of ~., y-axis corresponds to dent amplitude, x-axis to dent-wavelength, lr~dge= 1 and mridge= 1 and s=5/3

"~ 1

0.5

0 -2

-,s

-,

-0.s

0

02

-1.5

-1

-0.5

0

~ o~

0.5

Figure 1. Dimensionless pressure (-), film thickness (-) and reference film thickness (- .) at four different time steps showing the dent caused thinning effect for s=4/3.

The set of simulations where both dent amplitude (li) and dent wavelength (mi) were varied are shown as a contour map of the measure~:, in Figure 2. Dent -li and -mi were varied to give a total of forty-two combinations in the ranges -1...-6 and 1...7 respectively. Ridge parameters were held constant, l~ =land mF1. The figure shows that the larger the amplitude of the dent, the larger were the effects of the dent and thus the thinner the film thickness became. As far as the dent wavelength is concerned, an optimal and worst case wavelength in the range of wavelengths simulated were found. This is interesting since in many tribological applications dents are thought of as lubricant depots and these simulations show that there is an optimal dent wavelength to maintain the greatest film thickness possible; at least when considering an EHL line contact.

When comparing the different sliding-to-rolling ratios, the dent amplitude (li) was varied as in Figure 2 but here the dent wavelength (mi) was held constant, i.e., m,---1. Ridge parameters were also kept constant using ,the same values as previously. As Figure 3 shows, there is a clear relationship to the sliding-to-rolling ratio and that the effects increase with increasing s. The effects caused by the dent are of both greater magnitude and increase more rapidly with dent li with a higher sliding-to-rolling ratio. The overall maximum of ~: is approximately 0.4, i.e., the dent causes a film thickness decrease that corresponds to 40% of the smooth surface minimum film thickness value. 1

0.45 -e-

0.4

s=5/3 s=4/3 --A-- s = 1 - ~ - s = 2/3 -~- s=1/3

035 0.3 ~0.25 0.2 o15 0.1

0.0'._,

i

i

-2

t

i

-3

I

i,

i

-4

i

i

i

-5

Figure 3. ~: as a function of the slide-to-roll ratio S. x-axis corresponds to dent amplitude.

In the case of s = 5/3, dent wavelength (mi) was varied both with and without variation of ridge wavelength (mi). The results are shown in Figure 4, where the dent amplitude (li) was kept constant at -2

i

-6

249

and the dent wavelength (mi) varied in the range 1...6 together with ridge amplitude li = 1 and 2. These results show that the measure ~: assumes both a local maximum and a local minimum. That is, a dent wavelength (mi) somewhere between 1 and 3 seems to be the worst as far as its effects on film thickness are concerned and somewhere between 4 and 6 seems to be the optimum wavelength. Figure 4 also shows that the dent causes a greater effect in the case of the doubled ridge amplitude (li). However, doubled ridge amplitude (l~) does not increase the effect more than a few percent in the parameter range chosen but the difference seem to increase with increasing wavelength. Together with the results shown in Figure 2, this indicates a tendency towards film breakdown if one or a combination of the parameters, ridge amplitude (l~), dent wavelength (mi) and dent amplitude (li) are combined in an unfortunate way. In real world situations, such combinations will occur frequently.

02'[i I --B- /ridge = -1 /ridge -2

0.26

0.24

0.22

0.2

0.5

9- B - -

W = 10kN w = 20kN

I

I

0.45

0, 0.35 ~ 0.a 0.25 0.2 0.15 o.!

i

i

i

-2

i -3

i

i,

i

i

-4

i

1

-5

Figure 5. ~ as a function of load in the case of s = 5/3. x-axis corresponds to dent amplitude.

Table 1. Fixed parameters Parameter

Value

Dimension

R1

0.01

m

R2

2

m

E1 E:

206 xl 0 9

Pa

206 xl 0 9

Pa

v/

0.3

1/2

0.3

r/0

0.14

Pas

6g

2.10x10 -8

1/Pa

Po

1.98x108

Pa

z

0.6

m,

Figure 4. ~ as a function of bump amplitude in the case of s = 5/3. x-axis corresponds to dent wavelength.

In order to investigate the effect of contact loading, another set of simulations with s = 5/3 and with the same set of dent and ridge parameters as those behind the results shown in Figure 3 were carried out. The results, shown in Figure 5, indicate that doubling the load gives rise to larger dent caused effects then the reference simulations. It can also be seen that the minimum film thickness corresponds to approximately 45% of the smooth film thickness and that the slope of is greater in the case of the doubled load. In all these simulations, several significant input parameters where held constant, some of which are shown in Table 1.

It is not possible to draw general conclusions from the investigation presented here, but it is possible to see a trend towards film thinning when an asperity is free to recover its shape in the valleys or dents between asperities on the opposing surface. Clearly, at a certain combination of lubricant and roughness parameters film breakdown will probably occur. Such a breakdown will be due to low pressure or even intemal cavitation in the valleys or dents rather than to very high contact pressures as might intuitively be thought to be the cause of a film breakdown. In the cases presented here, no actual film breakdown was observed. However, in a line contact of finite length, film thinning will be greater due to side leakage from the valleys which will lead to a less efficient squeeze effect allowing asperities to grow more rapidly and to a greater extent. This will

i -6

250 eventually lead to film breakdown. Further investigations will therefore focus on intemal cavitation phenomena and side leakage effects. It is, of course, also necessary to introduce more realistic surface roughness features. The smaller the wavelengths become, the smaller the amount of sliding required to cause this type of local film thinning.

4. C O N C L U S I O N S - Certain effects in a rolling/sliding EHL contact cannot be studied without using a two-sided roughness treatment. A dent-dent overtaking event gives rise to a pressure spike equivalent to a ridge-ridge overtaking event where the ridges have half the amplitude of the dent. Bearing capacity is therefore not significantly affected by this type of overtaking event. -

A ridge-ridge overtaking event produces the thinnest films and largest pressure spikes compared to the other three possible overtaking phenomena for a given asperity amplitude. However, film breakdown could not be simulated in this work because of the model used. -

- Local pressure effects resulting from a ridge-dent overtaking event are of comparable magnitude to the dent-dent case. Because of the involvement of a ridge, the film becomes slight thinner causing this case to have a somewhat greater negatively effect on bearing capacity. In a sliding contact, a dent overtaking a ridge can cause film breakdown. This could occur with too large a ridge, a too wide or deep dent, too high a load, or a combination of these parameters. -

- There is a dent wavelength in a dent-ridge overtaking event that optimizes film thickness. As the sliding-to-rolling ratio increases, the film thickness decreases in a dent-ridge overtaking event.

Asperities in EHL Point Contacts", Proc. 29 th LeedsLyon Symposium on Tribology. (ISBN, pp etc.) [2] Venner. C. H., 1991, "Multilevel Solution of the EHL Line and Point Contact Problems", PhD Thesis,. University of Ywente, Enschede, The Netherlands. [3] Venner. C. H., 1994, "Transient Analysis of Surface Features in an EHL Line Contact in the Case of Sliding", ASME Journal of Tribology, Vol 116, pp. 186-193. [4] Venner. C. H. and Lubrecht. A.A., 2000, "Multilevel Methods in Lubrication", Tribologi Series, 37, Elsevier. [5] Elcoate. C. D., Evans. H. P., Hughes T. G. and Snidle R. W., 2001, "Transient Elastohydrodynamic Analysis of Rough Surfaces Using a Novel Coupled Differential Deflection Method", Proc. Inst. Mech. Eng., Vol 215, Part J, pp. 319-337. [6] Yao. J., Hughes. T. G., Evans. H. P. and Snidle. R.W. "Elastohydrodynamic Response of Transverse Ground Gear Teeth". Proc. 28 th Leeds-Lyon Symposium on Tribology. (ISBN, pp etc.) [7] Chang. L. 1995, "A Deterministic Model for Line Contact Partial Elastohydrodynamic Lubrication", Tribology International, Vol. 28, pp. 75-84. [8] Conry. Y. F., Wang. S, Cusano. C., "A ReynoldsEyring Equation for Elastohydrodynamic Lubrication in Line Contacts", Transactions of ASME, Vol. 109, 1987, pp 648-658. [9] Holmes. J. A. M., 2002, "Transient Analysis of the Point Contact Elastohydrodynamic Lubrication Problem using Coupled Solution Methods", PhD Thesis, Cardiff University. [10] Almqvist. A., Almqvist. T., Larsson. R., 2003, "A Comparison Between Computational Fluid Dynamic Approaches and Reynolds Approaches for Simulating Transient EHL Line Contacts", Accepted for publication in Tribology International.

-

- A ridge with larger amplitude will cause only slightly thinner films. - The contact load influences the film thinning effect. Load related effects are somewhat larger than those for increasing ridge amplitude.

A

P

P

E

N

D

I

X

The dimensionless discrete form of the timedependent Reynolds equation (4) can be written as:

(18) REFERENCES [1] Ehret. P., Dowson. D. and Taylor. C. M., 1996, "Time-Dependent Solutions with Waviness and

2AX -

A_@(/Sk ,,k-l) = 0 , i H i k - P -k-i i ni

251 where

,

gk

1 : i+-2

~176+ ~'i__.l ~

Ei

2

/,

--

(H:)

p~

m k ]]i

Z~

"

(19)

~ is the non-Newtonian factor according to Conry et al [8]. For the film-thickness dimensionless form is used:

the

following

that the main diagonal is zero. The system (24) does not address the coupling between pressure and filmthickness. Thus, a Jacobi relaxation of the blockmatrix system (23) will not converge. However, the system (24) can be modified to address this coupling, i.e.,

diag(All )Pnkew+ diag(Al2 )Hnkew F k-1 - A,,Polkd -- A12Hoktd

g ~ _ g o k ~ X~_2 ~---1 ~ Ki, _P[ + (20)

-o, where K/j=

X ~ - X j +--7-

lnX i-Xj+

-1 -

(21)

k : diag(A2 1 )p knew + diag(A22)H,,ew Gk

9

k

*

(25)

k

- A21Pota - A22 Hota

The Block-Jacobi relaxation method is based on solving the block-matrix system (23) for the "new" pressure and film-thickness profiles. It is therefore straightforward to rewrite the system to indices form, i.e. to the two-by-two matrix system (12), viz.

[ali o,IE'I a2~~ a22 i

H~

g~

with coefficients: Equations (20) and (21) can be written in operator form if the k-1 term in (21) is moved to the r.h.s of the equation and the terms not depending on the pressure and/or the film-thickness moved to the r.h.s, of equation (21), viz.

all'

=

a12i

-"

IL~(P,H)-F*-'

a2'i= 2AX2l n ( ~ )

(22)

1

--

[L2(P,H)-G*

a22 i -" 1

If the coefficients (19) and the density are linearized by means of the previously determined values of P and H obtained from any relaxation process, then it would be possible to re-write the operator system (22) as a block-matrix system, i.e.:

and

fi= - -2 A ~X

[ All

Al21IPnkew ] - I F k - I

A21 A jLH ew

LG ,

.

9 ( A 11) P k * k * k diag new -- F k-1 -- AllPola -- A12H~ [diag(A22 ) n n e w = G - A21Pot * kd - AzzHoi * kd 9

k

ck )

i-12 +

i+l

2( 3~+2AX AT1)~

~,

)IAX'~2 CIr 4

Ir

ir

--k Hi-2 k ) -p i k - l g k l -- Pi-2 "

A @ - --k-1 Pi

l

(23)

If the Jacobi method is applied to solve the above block-matrix system then the updated, "new" pressure and film-thickness would be obtained from the following equations: I

(ck

(z~X') 2

(24)

k

where diag(Atm } is a diagonal matrix with the main diagonal of Arm. Azm are the same as Arm except

g i = H ko-t

x?

1 ~ Kijp~ -~l~ -~z,k j~i

g~ -1 9