calculation of pressure distribution in ehd point contacts from experimentally determinated film...

Calculation of pressure distribution in EHD point contacts

from experimentally determinated film thickness

J. Molimarda,*, M. Querryc, P. Vergneb, I. Krupkac, M. Hartlc

aCentre SMS, ENSMSE, 158 Cours Fauriel, 42 027 Saint-Etienne, cedex 02, FrancebLMC/IET, UMR CNRS/INSA n85514, INSA bat. d’Alembert, 69621 Villeurbanne, Cedex, France

cFaculty of Mechanical Engineering, Brno University of Technology, 616 69 Brno, Czech Republic

Received 23 October 2002; received in revised form 30 June 2003; accepted 5 November 2003

Available online 2 November 2004

Abstract

In lubrication studies, film thickness is measured since 1963 using optical interferometry, but pressure distribution is also of great interest.

Unfortunately, this parameter is very difficult to measure. This paper presents a new method for pressure evaluation based on an inverse

elastic approach. This latter uses the film thickness field known precisely (G3 nm) by computerised treatment of interferograms. We present

here the whole process, from the image to the pressure field. A complete error analysis is conducted: the pressure field is known within 5%

and the spatial resolution is 3 grid steps, i.e. 6 mm.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Inverse analysis; EHL; Pressure distribution

1. Introduction

The knowledge of the pressure distribution over a contact

area in concentrated contacts has a great importance for

design of machine components such as various rolling

bearings, gears, cams and followers and traction drivers.

This is caused by the fact that the pressure fluctuation

influence the subsurface stresses and thereby the life of

rolling contacts. There are two main way to experimentally

determine the pressure distribution within elastohydrody-

namic (EHD) contacts: direct and indirect.

The former method is based on using the surface

microtransducers or on the detection of electromagnetic

radiation. The first pressure measurements were reported by

Kannel et al. [1] and subsequently by Cheng and Orcutt [2].

In both cases a small manganin resistance gauge was

deposited on the insulated surfaces of discs in disc machine

and variations in its resistance were measured to

determine the relatively low-resolution pressure profiles.

0301-679X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.triboint.2003.11.011

* Corresponding author. Tel.: C33 4 77 42 66 48, fax: C33 4 77 42 02

49.

E-mail address: [email protected] (J. Molimard).

This technique was further improved by Hamilton and

Moore [3] that reveal the difference between the predicted

and the observed form of the ‘Petrusevich pressure spike’.

Over the past decade, spectroscopic techniques have been

applied successfully for measurement of the pressure

distribution in lubricated contacts. Gardiner et al. [4,5]

used the frequency shift of lubricant’s Raman vibrational

modes to calculate pressure profiles for both static and

rolling EHD contacts. The similar approach has been

recently used by LaPlant at al. [6] to study spatial pressure

variation in high pressure EHD point contact. Lubricant

pressure study has also been carried out with Fourier

transform infrared (FTIR) adsorption spectroscopy [7],

however, with considerable lower spatial resolution.

Indirect method is a combination of experimental and

numerical approach in which measured film thickness

distribution is used as an input for pressure calculation

that is based on elastic deformation equation. This method

was used first by Paul and Cameron [8] that introduced high-

pressure microviscometer based on a glass ball in impact

with a glass target surface covered with a layer of the test

fluid. The shape of the contact was evaluated using optical

interferometry and the pressure distribution was obtained by

Tribology International 38 (2005) 391–401

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Notations

r density at pressure P

r0 density at atmospheric pressure

a contact radius

A specific refractivity

E* Young’s reduced modulus

(Ei, ni) elastic parameters of every solid

L wavelength in the Fourier approach

(m, n) co-ordinates in the Fourier space

n refractive index at pressure P

n0 refractive index at atmospheric pressure�Pðn;mÞ deformation in the Fourier space

p(x,y) pressure in the real space

R* equivalent radius of curvature�Uðn;mÞ out of plane displacement in the Fourier space

u(x,y) out of plane displacement in the real space

W total load applied in the contact

(x,y) co-ordinates in the real space

J. Molimard et al. / Tribology International 38 (2005) 391–401392

solving the elastic equation in polar coordinates. A similar

approach was also used by Wong et al.[9] and Larsson and

Lundberg [10] in their studies of squeeze film. More general

solution was published by Astrom and Venner [11] that

calculated the pressure distribution in a grease-lubricated

EHD contact from a film thickness map. It was created

employing image analysis [12] from chromatic interfero-

grams obtained from ball-on-plate test rig. The technique

has been further improved by Jolkin and Larsson [13].

The aim of this paper is to develop a Fast Fourier

Transform (FFT)-based technique for relatively simple and

fast determination of pressure distribution from measured

EHD lubricant film thickness. This approach is validated on

dry Hertzian contact and tested out using experimental

results for oil-lubricated EHD contact.

Fig. 1. Experimental apparatus.

2. Film thickness measurement

2.1. Experimental apparatus

The experimental apparatus is classical and have been

already described elsewhere [14]. It comprises three main

parts as Fig. 1 shows: a conventional optical tests rig, a

reflected light microscope with colour video camera and a

personal computer.

The film thickness was measured in a concentrated

contact formed between a transparent disk and a highly

polished steel ball of 25.4 mm in diameter. Depending on

the expected pressure range, the transparent disk could be

made of glass or sapphire. The composite roughness of such

a contact is 3 or 4 nm depending on the disk material. For

measurement purpose, the transparent disc is coated on its

underside with a thin semi-reflective chromium layer. If

needed, the chromium layer is overlaid by a silicon dioxide

‘spacer layer’, about 200 nm thick. Both surfaces are

independently driven by two motors with a slip about

0.1% maximally to produce nominally pure rolling

conditions.

The whole contact together with test fluid is enclosed in a

thermal insulated chamber that is heated with the help of

external heating circulator. Test fluid temperature is

monitored and maintained using a platinum temperature

probe.

The contact area is illuminated with a halogen light

source built in the episcopic microscope illuminator. The

chromatic interferograms produced by the contact are

captured by the colour video camera mounted on the

trinocular eyepiece tube and frame-grabbed by a personal

computer. The spatial density of captured images is about

0.9 mm.

2.2. Material and test conditions

The pressure calculation procedure is applied to an EHL

contact. The two elastic bodies are a sapphire disk and a

steel ball. The contact is lubricated with 5P4E lubricant and

the temperature within the contact is 50 8C. All the material

parameters are summarised in Table 1.

Table 1

Material parameters for EHL test

E* 144 GPa

R* 12.7 mm

Viscosity 170 mPa s

Piezoviscosity 28.4 GPaK1

J. Molimard et al. / Tribology International 38 (2005) 391–401 393

A constant linear speed of 0.3 m/s is applied on both

solids. The ball is pressed against the disk with a load of

16.56 N, thus, the maximum pressure according to Hertz

theory is 0.7 GPa. Central and minimum film thickness

according to Hamrock and Dowson formula should be 385

and 217 nm.

2.3. Measurement technique

Colorimetric Interferometry technique (CI) represents an

improvement of conventional chromatic interferometry in

which film thickness is obtained by colour matching

between interferogram and colour/film thickness

dependence obtained from Newton rings for a static contact.

Because this technique has been described previously, only

a block diagram of the film thickness evaluation process is

shown in Fig. 2. For further details, the reader is expected to

refer to authors’ previous publications [14,15].

First, the relation between colorimetric co-ordinates

(RGB or Lab) and thickness is established with a static

contact. Algorithms are different in the case Brno’s software

(Achiles) or Lyon’s one (Paris). In the first case, light

extrema in a monochromatic picture are used to evaluate the

thickness from interference equations and the shape is

interpolated between them. In the second case, Paris

Fig. 2. Film thickness evaluation usin

software is based on the hertzian shape. The centre and

contact radius are very important, but have been very

carefully managed. The precision on the centre is 0.2 pixel

and on the radius 0.1 pixel.

Second, for every unknown thickness, the colorimetric

co-ordinates (R,G B or L, a, b) are matched together to

recover the film thickness. In this step two, some differences

are obvious in the algorithm. Achiles minimizes the distance

between the (L, a, b) coordinates and a lookup table. But, a

distance is impossible to define using (R, G, B) colour

representation. So, the process used with Paris is based on

intensity matching for each colour plane. Last, a classical

refractive index vs pressure correction is introduced. This

correction is based on Lorentz–Lorenz and Dowson–

Higginson formula.

Because of differences between the two softwares, a

precise analysis on error sources is a tedious task. The

measurement quality will be summarized by the resolution

on the thickness measurement, evaluated to be G3 nm in

the 1–800 nm range. The reader should refer to the authors’

previous cross-comparison [16] or to [17] for further details.

3. Pressure calculation

3.1. Introduction

The mechanical analysis of an EHD contact is based

classically on the simultaneous resolution of Elasticity

(deformation of solids) and Reynolds (thin film fluid

mechanics) equations. The two sets of equations are related

together considering the following boundary conditions:

g colorimetric interferometry.


†
first, the shape of both deformed surfaces is supposed to
be the same as that of the film in the contact region,

defined as the non-null pressure zone;

†
second, the pressure applied by the fluid on the solids is
the same as that of the solids on the fluid.

In the considered problem, the film thickness is

experimentally known by DCI. As a consequence, the

geometry for Elasticity or Reynolds Equation is known.

Thus, it is possible to evaluate the pressure either by using a

fluid or elastic solid approach. Elasticity equations are

preferable because the material properties are very well

known. As a consequence, knowing the pressure from

Elasticity laws, it is possible to evaluate the lubricant

rheology using Reynolds Equation.

Astrom [11] developed such an approach using the

multigrid/multi-integration technique. We use here a novel

algorithm based on Fourier’s series [18,19]. This method is

simpler to implement than the multigrid/multi-integration

technique, for a very reasonable calculation time.

3.2. Solution technique

3.2.1. Principle

The basic principle of pressure calculation using Four-

ier’s series has been previously developed by Johnson [18]

and used for example by Stanley [19].

Johnson showed that if a normal pressure P(x,y)

generated on a semi-infinite smooth surface is a sine wave

Pðx; yÞ Z �Pðn;mÞcos 2pnx

L

� �cos 2pm

y

L

� �(1)

then, the out of plane displacement induced is also a sine

wave

Uðx; yÞ Z �Uðn;mÞcos 2pnx

L

� �cos 2pm

y

L

� �(2)

where (n/L) and (m/L) are the frequencies according to

the x and y directions. The relationship between amplitudes�Uðn;mÞ and �Pðn;mÞ is

�Pðn;mÞ Z �Uðn;mÞE�ðn2 Cm2Þ0:5

2(3)

In Eq. (3), E* is the reduced Young modulus, defined as

1

E�Z

ð1 Kn21Þ

E1

Cð1 Kn2

2Þ

E2

:

As an extension, any continuous pressure field P(x,y) can be

developed in Fourier Series as

Pðx; yÞ ZX

n

Xm

�Pðn;mÞcos 2pnx

L

� �cos 2pm

y

L

� �(4)

then, the out of plane displacement induced is

Uðx; yÞ ZX

n

Xm

�Uðn;mÞcos 2pnx

L

� �cos 2pm

y

L

� �(5)

In these last two expressions, �Uðn;mÞ and �Pðn;mÞ are the

Fourier coefficients related to U(x,y) and P(x,y), n and m are

the Fourier frequencies, and L is the length of the Fourier

transform.

This formulation appears to be very simple to implement,

compared with a Finite Element method or a multi-

grid/multi-integration algorithm. Furthermore, this method

is only based on surface considerations and the procedure is

not iterative, thus calculation time is strongly reduced.

Readers should consult Colin’s comparison between multi-

grid/multi-integration and FFT algorithms [20].

3.2.2. Evaluation of the out-of plane displacements

Eqs. (3)–(5) show that the pressure is completely and

exactly evaluated if the out-of plane displacement is known.

Therefore, film thickness in a lubricated contact arises from

the initial geometry of surfaces, from the elastic displace-

ment of the involved solids and from a translation due to the

hydrodynamic effect. So it is possible to deduct the surface

displacements from the film thicknesses found experimen-

tally within a constant. In this paper, only a smooth sphere

on plane contact is considered. Thus, instead of measuring

it, the unloaded surface is supposed to be parabolic.

Moreover, the curvature radii of the ball are the same in x

and y directions, described by a single parameter R*. Then,

the out of plane displacement is given by the expression

Uðx; yÞ Z hðx; yÞKx2

R�C

y2

R�

� �Cconstant (6)

Experimentally, h(x,y) is well known in the contact area and

near this contact area. But, for FFT calculation, the out-of-

plane displacement has to be known on a large distance, and

especially in places where no experimental data are

available.

In a first approximation, it is possible to consider that the

out-of-plane displacement outside the experimental area

have a Hertzian shape, characterised by its load W. Because

this function is a first approximation of the EHL displace-

ments, it is suitable if the experimental zone covers entirely

the contact area.

The displacement field u(x,y) is transformed in Fourier’s

space, then a first pressure field is calculated in Fourier’s

space using Eq. (3), and afterward in the classical Cartesian

space.

The unknown constant in the displacement expression

implies that the pressure field is known within a constant as

well as. Finally, this constant can be easily evaluated either

by considering a zero pressure on the boundaries of the FFT

domain, either by conservation of the load W. It is better to

use the first condition than the second because in this case,

the load is a result of the pressure calculation, and it is an

easy way to check the validity of pressure calculation.

Outside the contact area, and especially in the extrapo-

lated domain, the pressure should be null. This consider-

ation is used in an iterative procedure to reach a more

Fig. 3. Principle of the validation process.


precise extrapolated displacement field. The pressure is

calculated as explained before. In the extrapolated area, the

pressure is forced to zero. Then, it is possible to calculate a

displacement field from this modified pressure distribution

inverting Eq. (3). Now, the experimental displacement is put

in this latter displacement field, and pressure is calculated

again. The process stops if the residual pressure outside the

experimental area is negligible.

3.3. Validation

3.3.1. Method

The resolution of the pressure evaluation is related to

some physical or geometrical parameters. Eq. (3) shows that

the pressure error is directly proportional to the reduced

Young’s modulus. The other parameters (centre position,

Fig. 4. Cross section of the pressure field obtained

FFT domain, load W, curvature radius R*, noise level on

film thickness) have a more complex influence.

A complete evaluation has been performed in order to

characterise the resolution and the spatial resolution of the

pressure calculation method. A pure numerical simulation is

conducted to avoid experimental hazard. Two elastic bodies

are pressed together in static contact conditions. First, Hertz

theory is applied to calculate the reference pressure and the

deformed surface on a grid of 100 per 100 steps. Then,

another pressure field is evaluated from the latter with the

FFT approach for the given contact conditions. Finally, the

result is compared to the reference pressure field.

As to study the influence of the uncertainty on any

parameter, its value is changed within the uncertainty range

before FFT calculation. A schematic view of this process is

given Fig. 3.

Particularly difficult conditions are defined in all the

following cases: we have simulated numerically a static

sapphire-on-disk contact (E*Z144 GPa, R*Z12.7 mm). A

load of 5.9 N is applied, leading to an Hertz pressure of

0.53 GPa. The deformed shape is known as a grid of 100 per

100 with a step of 2.2 mm. In this case, the contact area, the

out-of-plane displacement and the load are very small, thus

very noise-sensitive. But, these contact conditions seem

experimentally realistic.

One should remark that because of the chosen contact, it

is obvious that the iterative calculation of the extrapolated

displacement field is a non-sense. Consequently, this

capability has not been used in the validation.

A cross section in the middle of the reference pressure

field for the given contact conditions is shown Fig. 4. The

FFT approach is conducted with the Hertz deformed shape,

as described before. In this first illustration, no noise is

added. The mean difference between these two pressure

fields is 0.5% of the maximum pressure. This clearly shows

with Hertz theory or FFT-based approach.


the capability of this FFT calculation to recover a pressure

for a given displacement.

3.3.2. Centre position

In order to evaluate the out-of-plane displacement,

Eq. (6) shows that it is necessary to superimpose a deformed

state (the film thickness) to an undeformed one. In practice,

this means that the centres of the deformed and undeformed

fields must be merged together. On the experimental

thickness map, a classical image analysis algorithm is

used for the centre evaluation. We can consider that the

centre coordinates are calculated with an error less than 0.5

grid step. The influence of such uncertainty has been

evaluated using the method described before.

The given conditions have been used, but deformation

and pressure fields are calculated with the FFT-based

approach with a shift of 0.1, 0.5, 1, and 2 grid steps on the

centre co-ordinates in x and y directions. Then, pressure

across the contact is compared to the reference pressure

given by Hertz Theory. Results plotted Fig. 5 shows that

errors grow with the distance from the centre of the contact,

but they remain weak. For the 0.5 shift, the mean error in the

contact area is 1.3%.

3.3.3. FFT domain

The use of a FFT-based algorithm induces a periodicity.

Everything happens as if an infinity of contacts occurs

between the surfaces. So, the transformation domain must

be large enough to prevent interactions between these

contacts. But of course, if the transformation domain is too

large, the calculation time becomes very long.

In order to optimise the FFT domain, the same

calculation was performed with FFT domains of 256, 512,

Fig. 5. Influence of a

1024 and 2048 grid steps. Contact conditions are the same

as before. Thus, the contact radius is 73 mm, which is 33.8

grid steps. The FFT domain can be represented as 7.6, 15.1,

30.3 and 60.6 times the contact radius.

Fig. 6 shows that the relative error on the maximum

pressure is under 1% if the FFT length is 20 times the

contact radius. For most cases, this means that the FFT size

should be 1024 grid steps. With these conditions, the

calculation time remains very reasonable: less than 1 min

with a common personal computer.

3.3.4. Load W

As mentioned before, the load has to be used to extend

artificially the FFT domain. This parameter is measured on

the experimental test-rig within 0.16 N. As it is an absolute

error, for a characterisation purpose, the load case should be

in the lower part of the experimental range. The former

conditions (E*Z144 GPa, R*Z12.7 mm, WZ5.9 N) cor-

respond to a sapphire on steel contact with a Hertz pressure

of 0.5 GPa. Thus, this is a realistic low load case.

We have used the deformed field calculated by Hertz

theory in these conditions, as mentioned in ‘Method’

section. Then the pressure field has been calculated using

FFT approach with supposed loads between 5.7 and 6.1 N.

The pressure variation is evaluated lower than 0.3% in the

contact area. This section shows that the function outside the

experimental area has a few influences on the results.

3.3.5. Curvature radius

The equivalent curvature radius appears in Eq. (6). For a

ball-on-disk apparatus, curvature is only related to the ball.

Measurements on a UBM profilometer lead to a curvature

radius known as 12.7G0.1 mm. If this uncertainty is

centre position.

Fig. 6. Influence of FFT length on the pressure evaluation.


applied to the pressure calculation, it has been explained

before, the pressure along a contact diameter is known

within G0.5%.

3.3.6. Sensitivity to an experimental noise

The experimental film thickness field is supposed to be

evaluated within 3 nm. This section analyses the error on the

pressure field induced by this noise. Two different types of

Fig. 7. Effect of a punctual no

noise are introduced in the simulated deformed shape: a

‘punctual’ noise or a random noise. In each case, the noise

level is 3 nm. The first one is used to characterize the

spreading of a displacement discontinuity on the pressure

field. In so far, it is a way to evaluate the spatial resolution.

This punctual noise is 1 by 1 grid step large and its height is

equal to the noise level (3 nm) in order to keep realistic

values. The second one evaluates the smallest change in

ise on the pressure field.

Fig. 8. Effect of a random noise on the pressure field.


the pressure that can be seen as a physical variation, in other

terms, the resolution.

First, Fig. 7 shows the effect of the punctual noise on the

pressure estimation across the contact. The error generated

is important, but also very localised. Thus, this example

gives an idea on the spatial resolution of the inverse pressure

estimation: if the acceptable precision range is 1%, the

spatial resolution for this example is 3 grid steps. In fact, this

value depends on the gap in the discontinuity, and should be

adjusted in case of a rough contact for example.

Second, Fig. 8 presents different pressure profiles across

the contact in the case of a random noise representative of

the one encountered in film thickness measurements (3 nm).

The pressure field is very noisy: the mean noise level is

about 10% of the mean pressure. But, if a 3 per 3 spatial

filter is used on this pressure map, the values are in quite a

good agreement with Hertz values, and the mean error is

within 2%. This is in good agreement with the discussion on

spatial resolution developed before.

The chosen conditions (high reduced Young’s modulus,

low load) are particularly tough: another example simulat-

ing a glass-on-steel contact (E*Z64 GPa, R*Z12.7 mm,

WZ19.6 N) gave an error level within 5% of the mean

pressure without filtering.

Fig. 9. Iterative procedure for pressure and film thickness evaluation.

4. Applications: pressure in an EHL contact

4.1. Refractive index correction procedure

For the pressure calculation, this example is a bit more

complicated: the refractive index changes with the density,

as described by Lorentz–Lorenz laws.

1

r

n2 K1

n2 C2Z A (7)

where A is the specific refractivity, r and n the density and

the refractive index for any thermo-mechanical conditions

(temperature or pressure) within the fluid domain.

Experimentally, calibration curves are established with a

lubricant at atmospheric pressure. Of course, in the EHL

contact, the lubricant flow induces a very high pressure, and

consequently a change in the lubricant density. The most

popular way to describe this change is Dowson–Higginson

Fig. 10. Evolution of pressure during the iterative procedure.


relationship

r

r0

Z 1 C0:6P

1 C1:7P(8)

where r0 is the density at atmospheric pressure and r the

density for the pressure P.

The effect of this change on the film thickness is

noticeable and a correction has to be applied. Eqs. (7) and

(8) are sufficient for the evaluation of the refractive index

Fig. 11. Film thickness across the cont

under pressure. But, the pressure has to be known.

Classically, it is given by Hertz theory. The FFT procedure

could lead to a more precise pressure distribution, but the

film thickness is erroneous. Thus, it is necessary to use an

iterative procedure, as described by Marklund [21].

The procedure used is summarised Fig. 9: at first, a film

thickness without any correction is evaluated. The corre-

sponding pressure map is calculated with Hertz theory. Then,

a first pressure correction can be applied. The corrected film

act after the iterative procedure.

Fig. 12. Final film thickness map.


thickness leads to a pressure map calculated with FFT

procedure. Now, this pressure distribution can be used with

the initial film thickness for the calculation of the corrected

film thickness. The process ends when the variations

between iteration j and iteration jC1 on thickness or

pressure are within the noise level.

4.2. Results

Fig. 10 presents the pressure variations during the

iterative procedure. Differences between the Hertzian

shape and the final one are obvious: at contact inlet, the

two pressure distributions are close together; after the

middle of the contact, experimental pressure remains high

and increases with the well known Petrusevich peak. At the

outlet, the experimental pressure strongly falls down before

the Hertzian distribution. During the iterative procedure,

Fig. 13. Final pre

the pressure distribution becomes smoother and the

Petrusevich peak is lower when convergence is achieved

than after the first iteration.

The film thickness across the contact has been presented

Fig. 11 after the iterative procedure. Fig. 11 shows also the

difference between the final film thickness and an estimation

of the film thickness obtained with a correction of refractive

index based on a Hertz pressure. The latter appears to be

enough for central film thickness evaluation, but fails near

the contact boundaries. This implies that the minimum film

thickness near the outlet is under-evaluated of 6% with a

non-iterative approach.

Final thickness and pressure maps can also be presented

as 2D plots, as shows Figs. 12 and 13. The comparison

between these two maps shows a difference in the

pertubated area. Considering thickness film, this area is

very close to Hertz contact, whilst the non-null pressure

ssure map.


zone is smaller. In such a case, the definition of a contact

area becomes unprecize.

5. Conclusion

This work has presented a new method for pressure

estimation from an experimental thickness map.

This method, based on Fast Fourier Transform is very

simple to implement. Practically, the most important

problem is that the out-of-plane displacement must be

extended outside the experimental area. An Hertz shape is a

good approximation of the deformed surfaces for the two

applications. This latter can be refined using a simple

iterative procedure if needed. Calculations can be conducted

on a simple office computer and results are given in less than

1 min.

Using the validation developed Section 3.2, it is possible

to evaluate the resolution of the inverse pressure evaluation

method. The main part of the resolution loss is related to the

experimental uncertainties on film thickness, even if the

height is known within 3 nm.

The example developed in Section 3.2 is a difficult

situation for every parameter. But, for a FFT domain of 1024

points, using a 3 per 3 spatial filter, the resolution is 5.1% of

the maximum pressure and the spatial resolution is 6.5 mm

per 6.5 mm, to be compared with a contact radius of 73 mm.

The application developed indicates some practical

interest for such a calculation. This example deals with

classical EHL conditions. An iterative procedure has been

built in order to evaluate more precisely the refractive index,

the pressure and the thickness in the contact area. This

procedure shows important differences in the film thickness

map (6%) if the pressure field used is calculated from FFT

method instead of the classical Hertz profile. Thus, for thick

films, it is necessary to use such an approach for a precise

evaluation of the film thickness map. As to the pressure

profile itself, its shape shows clearly the classical Petruse-

vich peak at the contact outlet. A comparison between a

complete EHL solutions and semi-experimental pressure

distributions will be held in the near future.

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calculation of pressure distribution in ehd point contacts from experimentally determinated film...

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