Experimental Study of Central and Minimum Film Thickness
in Elastohydrodynamic Elliptic Contacts
I. Křupka, M. Hartl, R. Poliščuk and M. Liška
Faculty of Mechanical Engineering, Brno University of Technology,
Technická 2, 616 69 Brno, The Czech Republic
Colorimetric interferometry measurement technique has been used to explore the influence of ellipticity on
elastohydrodynamic (EHD) film thickness and shape. The results obtained for three curvature ratio have been
compared with the Hamrock and Dowson film thickness formulas and more recent solution by Nijenbanning,
Venner and Moes. It has been shown that for thinner films, the both speed and ellipticity has more dominant
influence on minimum film thickness than the Hamrock and Dowson formula predicted. This behaviour is in
a good agreement with those predicted by Nijenbanning, Venner and Moes computational analysis.
1. INTRODUCTION
Over the last few decades various solutions of
elastohydrodynamic (EHD) lubrication of elliptical
contacts have been published. They ranged from
analytical approximate solutions based on major
simplifications [Error! Reference source not
found.] to more informative numerical analysis
required efficient solvers [Error! Reference source
not found.–Error! Reference source not found.].
Currently four central film thickness formulas
derived by Archard and Cowking [Error!
Reference source not found.], Cheng [Error!
Reference source not found.], Hamrock and
Dowson [Error! Reference source not found.] and
Chittenden et al. [Error! Reference source not
found.] are available in the literature for elliptical
contacts. Because its validity is restricted to the
piezoviscous-elastic regime, Nijenbanning, Venner
and Moes [Error! Reference source not found.]
recently presented more general central film
thickness formula that incorporates asymptotic
behaviour.
Although the minimum film thickness is
the most important parameter for design purposes
there are only two formulas given by Hamrock and
Dowson [Error! Reference source not found.] and
Chittenden et al. [Error! Reference source not
found.] that enables its calculation for elliptical
contacts.
There have been done several attempts to
experimentally verify some of the above-mentioned
equations [Error! Reference source not found.],
[Error! Reference source not found.–Error!
Reference source not found.]. The most
remarkable and comprehensive is the study carried
out by Koye and Winer [Error! Reference source
not found.] that evaluated the Hamrock and
Dowson minimum film thickness formula for
ellipticity ratios ranging from 3.7 to 0.117.
All computational and experimental works are
limited to the relative thick films only. The aim of
this study is to get over this limitation and obtain
reliable information about the influence of ellipticity
on EHD film thickness and shape. Colorimetric
interferometry measurement technique has been
used to measure both central and minimum film
thicknesses down to a few nanometers for three
curvature ratio. Obtained results have enabled to test
the validity of both the Hamrock and Dowson film
thickness formulas and numerical analysis by
Nijenbanning, Venner and Moes.
2. EXPERIMENTAL APPARATUS AND
TECHNIQUE
EHD film thickness measurements were made
using a modified version of the automatic system for
the real-time evaluation of EHD film thickness and
shape developed at our laboratory. Because
the description of system design concept, hardware
and software has been given in detail elsewhere
[Error! Reference source not found.] it will not be
included here. The following brief system
description is given only for continuity and to
illustrate its major changes.
2.1. Experimental apparatus
Figure 1 shows a perspective view of
the measurement system configuration. This is
a conventional optical test rig equipped with
a microscope imaging system and a control unit.
Its principal parts are a steel spherical roller and
a glass disk with lower surface covered with a thin
semi-reflective chromium layer overlaid by a silicon
dioxide “spacer layer”, about 200 nm thick. A centre
together with a ball bearing is used to support the
spherical roller that is driven through a planet
gearbox by a servomotor. The glass disk is mounted
on a central shaft freely rotated in two precision ball
bearings ensuring nominally pure rolling conditions.
The disk is mounted on a pivoted lever arm through
that the load is applied to the contact.
Figure 1. Measurement system configuration.
Light originating at a xenon light source is
transmitted through light guide cable to
the episcopic microscope illuminator where is
reflected by the beam splitter toward the microscope
objective. It concentrates the light into the contact
where the beams are reflected at both
glass/chromium layer and lubricant/steel ball
interfaces. The reflected beams recombine giving
chromatic interference pattern whose colour
corresponds to a film thickness. The interferograms
are imaged into three CCD (charge-coupled device)
image sensors in a colour video camera and
the output signal is digitised by a frame gabber
board inside personal computer.
Control unit consists of a personal computer and
a series of data acquisition and control cards that
support bi-directional communication between
optical test rig and personal computer.
2.2. Measurement technique and
its implementation
The colorimetric interferometry measurement
technique [Error! Reference source not found.]
adapted to the thin films study [Error! Reference
source not found.] was used in this work. It is based
on the colorimetric analysis of chromatic Fizeau
interferograms produced by the concentrated contact
in an optical test rig.
The
frame-grabbed
interferograms
with
a resolution of 512 pixels × 512 lines are first
transformed from RGB to CIELAB colour space and
they are then converted to the film thickness map
using appropriate calibration and colour matching
algorithm. L*, a*, b* colour co-ordinates/film
thickness calibration is created from Newton rings
for flooded static contact formed between the steel
ball and the glass disc coated with chromium layer
only. All measurements are carried out at the same
disc position where the spacer layer thickness has
been previously determined.
All aspects of interferogram and experimental
data acquisition and optical test rig control are
provided by computer program that also performs
film thickness evaluation.
It is believed that the film thickness resolution of
colorimetric interferometry measurement technique
is about 1 nm. The lateral resolution of a microscope
imaging system used is 1.2 μm.
3. TEST DETAILS
3.1. Contact bodies
The contact bodies are in the spherical roller-onplate configuration. Their material properties are
summarised in Table 1.
Effective modulus (GPa)
124
Table 1
Material properties of contact bodies
Elastic modulus (GPa)
Poisson’s ratio
Rollers
Disc
212
0.300
81
0.208
Spherical rollers were machined from AISI
52100 steel, hardened to a Rockwell hardness of
Rc 60 and manually polished with diamond paste to
RMS surface roughness of 5 nm. The geometry of
spherical rollers before polishing is given in Table 2