Leaf Coppin
113
Investigation into the Traction
Coefficient in Elastohydrodynamic
Lubrication
Y.S. Wang Harbin Institute of Technology, Harbin, China and Henan University of Science and Technology, Luoyang, China B.Y. Yang Henan University of Science and Technology, Luoyang, China L.Q. Wang Harbin Institute of Technology, Harbin, China
Abstract
The elastohydrodynamic traction coefficients of two Chinese aviation lubricating oils
were investigated for various loads, rolling velocities, and lubricant inlet temperatures
iising a self-made test rig. Traction coefficient versus slide-to-roll ratio curves were gen-
erated. The concept of critical load varying with the lubricant temperature is proposed.
This paper presents a new empirical formula for the dynamic pevforniance design of
high-speed rolling bearings, that relates traction coefficient with normal load, rolling ve-
locity, and lubricant inlet temperature. The coefficients of the formula may be computed
by regression analysis of the experimental data. Two example calculations are presented.
The predicted results from the formula agree well with experimental obserziations.
Keywords
rolling bearings, elastohydrodynamic lubrication, traction coefficient, empirical
formula, rolling velocity, lubricant temperature
INTRODUCTION
When a lubricated rolling bearing is operating at high speed an elastohydro-
dynamic (EHD) film is developed and excessive slip between rolling elements
shears this oil film to generate a traction force. The traction force between the
lubricant and the rolling element interfaces can cause balls and rollers to accel-
erate, decelerate, skid, or skew. Thus cage instabilities and the life of a rolling
bearing are associated with the traction behaviour of the lubricant. Since the
1960s, many researchers14 have presented various rheological models to calcu-
late the traction force. Unfortunately, the limitations of rheological models and
Tribotest journal 11-2, December 2004. (11) 113 lSSN 1354-4063 $35.00 (2630/1204)
Wang, Yung, and Wang 114
the lack of data for rheological parameters along with the complexity of numer-
ical iteration techniques have restricted the practical application of those models
for traction prediction in engineering. Therefore, the authors’ current aim is to
model the traction behaviour with an empirical formula which can be con-
veniently and quickly used to compute the traction coefficient in simulating the
dynamic performance of rolling bearings. Based on a large number of traction
tests on two Chinese aviation lubricants, a new empirical formula for the trac-
tion coefficient is proposed. The method of calculating the traction coefficient
used in this paper can be adapted to other lubricants working under the same operating conditions. This will not only be of significance in the dynamic behav-
iour design of rolling bearings, but will also help lay the foundations of further
research on traction theory.
PREVIOUS METHOD TO CALCULATE TRACTION COEFFICIENT
The method that has been used in many studies5-’ to calculate the traction co-
efficient needs four basic equations that can be summarised as:
1 dT Gdt 7 = -- + F(T, T*, q)
where 7 is the shear rate, T is the shear stress, G and q are the shear modulus
and the viscosity of the lubricant, T* is the reference stress or the limiting shear
stress, p, c, and k are thermal characteristics of the lubricants, u, T, p, and p are
the sliding velocity, absolute temperature, pressure, and lubricant density, x, y,
and z are the coordinates along and perpendicular to the rolling velocity and
across the lubricant film, respectively, and -a is the abscissa of the inlet of the
Hertzian zone.
Eq. (l), which was presented by Johnson and Tevaarwerk2 and by Bair and
Wir~er,~ has been the more widely accepted rheological model that describes the
lubricant behaviour as a non-linear viscous flow superimposed on a linear
elastic strain. Eq. (2) is an energy equation including the convection and heat
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lnuestigation into the traction coeflicient in elastohydrodynamic lubrication 115
Figure 1 Construction of the test rig"
Disc soecimen Motor I
Ball specimen
Motor II
dissipation. Eq. (3) is the momentum equation. Boundary conditions on temper-
ature are calculated by Eq. (4)? Given the thickness of the lubricant film and
rolling velocities, the above equations are solved simultaneously to obtain tem-
peratures, velocities, and shear stress distributions over the contact. The traction
force can be obtained by integrating the shear stress distribution. The traction
coefficient is the ratio of the traction force to the applied normal load between
the interacting surfaces.
Eqs. (1)-(4), which may have to be calculated a great number of times be-
fore convergence solutions can be obtained, are quite computationally complex
even after several simplifying assumptions are made about the operating cnvi-
ronment and film properties.@ The complexity of the traction model makes the
computer program rather impractical to use. Therefore, the implementation of
this method for practical design tools is somewhat difficult, particularly in the
case of dynamic performance simulation of rolling bearings. This situation has
led to the simple method presented in this paper, in which an algebraic equation
is used to predict the traction coefficient. The constitutive coefficients of the
equation can be obtained by regression analysis of the available experimental
data.
TEST RIG
The traction experiment was carried out using an improved ball-on-disc test rig
designed and built by Yang et al.," as shown in Figure 1. The details of the test
Tribotest journal 11-2, December 2004. (11) 115 lSSN 1354-4063 $35.00
