Elastohydrodynamic Solutions for Point Contact

It will be recognized that the elastohydrodynamic problem as presented in Section 3.5.1 is formulated in terms of plane strain and collinear contact. Obviously the simple Reynolds equation is not applicable to spherical geometry or crossed-axis contact. Geometrically both of these cases fall into the category of point contact. In actuality, if elastic deformation is involved, the contact region is a circle or an ellipse. 

Treatments of the elastohydrodynamic problem for such cases have been published by Archard and Cowking [20], by Cheng [21] and by Hamrock and Dowson [22]. Of these, the work of Hamrock and Dowson is the most comprehensive. The general Reynolds equation is written as 

Uj being the surface velocity of direction, U2 the surface velocity 

The coordinates X and 2 define the orientation of the plane perpendicular to the thickness of the lubricant film. Transformations may be employed to accommodate the geometry of the boundary surfaces or to throw the treatment into non-dimensional form. 

In solving the elastohydrodynamic problem for point contact, the Reynolds equation is coupled with the expressions for the elastic deformation of the bounding surfaces and for the influence of pressure and temperature on the viscosity of the lubricant, as in the solution for line contact. However, a single traverse across the contact zone does not suffice as the integration path in the case of point contact, where the contact area is elliptical or circular instead of rectangular. This brings into play the ellipticity parameter, which is simply 

---