Page 256 - Tribology in Machine Design

Friction, lubrication and wear in higher kinematic pairs 241

surprising that an increase in the viscosity with pressure is also a significant
factor in elastohydrodynamic lubrication. When sliding is a prevailing
motion in the contact, frictional heating causes a rise in the temperature in
the film which reduces the viscosity of the film. However, for reasons which
will be explained later, it is possible to separate the effects of pressure and
temperature.
Let us consider an isothermal film in which variation in the viscosity with
pressure is given by the equation

where /i 0 is the viscosity at ambient pressure and temperature and a is a
constant pressure coefficient of viscosity. This is a reasonable description of
the observed variation in the viscosity of most lubricants. Substituting this
relationship into the Reynolds eqn (6.18) gives

This modified Reynolds equation for the hydrodynamic pressure in the field
must be solved simultaneously with eqn (6.24) for the effect of elastic
deformation on the film shape. The solution to this problem can be
obtained numerically. There are a number of changes in the contact
behaviour introduced by the pressure-viscosity effect. Over an appreciable
fraction of the contact area the film is approximately parallel. This results
from eqn (6.26). When the exponent ap exceeds unity, the left-hand side
becomes small, hence h — h^ becomes small, i.e. hxh t = constant. The
corresponding pressure distribution is basically that of Hertz for dry
contact, but a sharp pressure peak occurs on the exit side, followed by a
rapid drop in pressure and thinning of the film where the viscosity falls back
to its ambient value /^ 0- The characteristic features of highly loaded
elastohydrodynamic contacts, that is a roughly parallel film with a
constriction at the exit and a pressure distribution which approximates to
Hertz but has a sharp peak near the exit, are now well established and
supported by experiments. It is sufficiently accurate to assume that the
minimum film thickness is about 75 per cent of the thickness in the parallel
section. The important practical problem is to decide under what
conditions it is permissible to neglect elastic deformation and/or variable
viscosity. Some guidance in this matter can be obtained by examining the
values of the two non-dimensional parameters, the viscosity parameter 0 V
and the elasticity parameter g e which are presented and discussed in
Chapter 2, Section 2.12.1. The mechanism of elastohydrodynamic lubri-
cation with a pressure dependent lubricant is now clear. The pressure
develops by hydrodynamic action in the entry region with a simultaneous
very large increase in the viscosity. The film thickness at the end of the
converging zone is limited by the necessity of maintaining a finite pressure.
This requirement virtually determines the film thickness in terms of the
speed, roller radii and the viscous properties of the lubricant. Increasing the
load increases the elastic deformation of the rollers with only a minor