Page 51 - Tribology in Machine Design

38 Tribology in machine design

model of contact between two real surfaces, R e is the relative radius of
curvature of the contacting surfaces, E is the effective elastic modulus, N is
the asperity density, r is the average radius of curvature at the peak of
asperities, cr* is the standard deviation of the peaks and d e is the equivalent
separation between the mean height of the peaks and the flat smooth
surface. The ratio of lubricant pressure to total pressure is given by

where A is the specific film thickness defined previously, h is the mean
thickness of the film between two actual rough surfaces and h 0 is the film
thickness with smooth surfaces.
It should be remembered however that eqn (2.80) is only applicable for
values of the lambda ratio very near to unity. For rougher surfaces, a more
advanced theory is clearly required. The fraction of the total pressure, p c,
carried by the asperities is a function of dja* and the fraction carried
hydrodynamically by the lubricant film is a function of h 0/h. To combine
these two results the relationship between d e and h is required. The
separation d e in the single rough surface model is related to the actual
separation of the two rough surfaces by

where <r s is the standard deviation of the surface height. The separation of
the surface is related to the separation of the peaks by

for surfaces of comparable roughness, and for <7*%0.7<7 S. Combining these
relationships, we find that

Because the space between the two contacting surfaces should accom-
modate the quantity of lubricant delivered by the entry region to the
contacting surfaces it is thus possible to relate the mean film thickness, h, to
the mean separation between the surfaces, s. Using the condition of
continuity the mean height of the gap between two rough surfaces, h, can be
calculated from

where Fi(s/a s) is the statistical function in the Greenwood-Williamson
model of contact between nominally flat rough surfaces.
It is possible, therefore, to plot both the asperity pressure and the film
pressure with a datum of (h/a s). The point of intersection between the
appropriate curves of asperity pressure and film pressure determines the
division of total load between the contacting asperities and the lubricating
film. The analytical solution requires a value of h/a s to be found by iteration,
for which