Page 81 - Tribology in Machine Design
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68 Tribology in machine design
The approach between two relatively distant and strain-free points, such
as Qi and Q 2, consists not only of the surface effect z l + z 2, but also of the
approach of Qi and Q 2 relative to M t and M 2, respectively, which are the
deformations w { and w 2 due to the, as yet, undetermined pressure over the
contact area. The total approach or deflection <5, with substitution from eqn
(3.1) and (3.2), is
For symmetry, the area of contact must be bounded by a circle, say of radius
a, and Fig. 3.3 is a special case of Fig. 3.1. A trial will show that eqn (3.3) will
be satisfied by a hemispherical pressure distribution over the circular area.
Thus the peak pressure at centre 0 is proportional to the radius a, or
Po = ca. Then, the scale for plotting pressure is c=p 0/a. To find Wj and w 2 at
M in eqn (3.3), an integration, pds, must first be made along a chord GH,
2
2
2
which has the half-length GN = (a —r sin </>)*. The pressure varies as a
semicircle along this chord, and the integral equals the pressure scale c
Figure 3.3 times the area A under the semicircle, or
By a rotation of line GH about Mfrom </>=Oto cf) = n/2 (half of the contact
circle), the shaded area of Fig. 3.3, is covered. Doubling the integral
completes the integration in eqn (3.3), namely
Now the approach 6 of centres Q± and Q 2, is independent of the particular
points M and radius r, chosen in the representation by which eqn (3.4) was
2
obtained. To make the equation independent of r, the two r terms must be
2
equal, whence it follows that the two constant terms are equal. The r terms,
equated and solved for a, yield the radius of the contact area
