Page 42 - Tribology in Machine Design

Basic principles of tribology 29

so that

is the probability that A/i is negative, i.e. the probability of asperity contact.
In the foregoing, his the mean value of the separation (see Fig. 2.11) and
i
o-* = (o-p 1 + (Tp 2) is the standard deviation.
The probability P(A/i^O) of asperital contact can be found from the
normalized contact parameter h, where A/T= h/a* is the number of standard
deviations from mean h. For this purpose, standard tables of normal
probability functions are used. The values of A/T represent the number of
standard deviations for specific probabilities of asperity contact, P(Ato^O).
They can be described mathematically in terms of the specific film thickness
or the lambda ratio, X, and the r.m.s. surface roughness R. Thus, from the
definition of the lambda ratio

where K! and R 2 are the r.m.s. roughnesses of surfaces 1 and 2, respectively.
If it is assumed that cr si % K t and cr S2« R 2, and that h, B d and B m are defined
as shown above, then

and finally

The general expression for the lambda ratio has the following form

If the contacting surfaces have the same surface roughness, then

Taking into account the above assumptions

If it is further assumed that R l=R 2=R and therefore pa sl =<r s2 =cr s, then