Page 106 - Tribology in Machine Design

92 Tribology in machine design

unrealistic model of the rough surface. When a more realistic surface model
is considered, the proportionality between load and real contact area can in
fact be obtained with an elastic mode of deformation.
It is well known that on real surfaces the asperities have different heights
indicated by a probability distribution of their peak heights. Therefore, the
simple surface model must be modified accordingly and the analysis of its
contact must now include a probability statement as to the number of the
asperities in contact. If the separation between the smooth surface and that
reference plane is d, then there will be a contact at any asperity whose height
was originally greater than d (Fig. 3.12). If (f)(z) is the probability density of
the asperity peak height distribution, then the probability that a particular
asperity has a height between z and z + dz above the reference plane will be
0(z)dz. Thus, the probability of contact for any asperity of height z is

Figure 3.12
If we consider a unit nominal area of the surface containing asperities, the
number of contacts n will be given by

Since the normal approach is (z — d) for any asperity and N (- and A f are
known from eqns (3.48) and (3.49), the total area of contact and the
expected load will be given by

and

It is convenient and usual to express these equations in terms of
standardized variables by putting h — d/a and s = z/a, o being the standard
deviation of the peak height distribution of the surface. Thus