**
12.13 CYLINDRICAL CONTACT
**

Cylindrical
contact is common in machinery. A non-crowned cylindrical roller follower
running against a face cam is one example. Roller bearings are another
application. The mating cylinders can be both convex, one convex and one
concave (cylinder-in-trough), or, in the limit, a cylinder-on-plane. In all
such contacts there is the possibility of sliding as well as rolling at the
interface. The presence of tangential sliding forces has a significant effect
on the stresses compared to pure rolling. We will first consider the case of
two cylinders in pure rolling and later introduce a sliding component.

Photoelastic analysis of
contact stresses under a cam-follower (Source: V. S. Mahkijani, Study of
Contact Stresses as Developed on a Radial Cam Using Photoelastic Model and
Finite Element Analysis. M.S. Thesis, Worcester Polytechnic Institute, 1984)
[15]

**
Contact Pressure and Contact Patch in Parallel
Cylindrical Contact
**

When
two cylinders roll together, their contact patch will be rectangular as shown
in Figure 12-11
*
b
*
(p. 353). The pressure distribution will be a
semi-elliptical prism of halfwidth
*
a
*
. The contact zone will look as shown in Figure
12-13 (p. 355). The contact pressure is a maximum
*
pmax
*
at
the center and zero at the edges as shown in Figure 12-14 (p. 357). The applied
load
*
F
*
on the contact patch is equal to the volume of
the half-prism:

where
*
F
*
is the total applied load and
*
L
*
is
the length of contact along the cylinder axis. This can be solved for the
maximum pressure:

*
*

The
average pressure is the applied force divided by the contact-patch area:

*
*

Substituting
equation 12.14c in 12.14b gives

We
now define a cylindrical geometry constant that depends on the radii
*
R
*
1
and
*
R
*
2 of the two cylinders, (Note that it is the
same as equation 12.9b (p. 356) for spheres.)

*
*

To
account for the case of a cylinder-on-plane,
*
R
*
2 becomes
infinite, making 1/
*
R
*
2 zero. For a cylinder-in-trough,
*
R
*
2
becomes negative. Otherwise
*
R
*
2 is finite and positive, as is
*
R
*
1.
The contact-patch half-width
*
a
*
is then found from

*
*

where
*
m
*
1 and
*
m
*
2 are material constants as defined in equation
12.9a (p. 355).

The
pressure distribution within the semi-elliptical prism is

*
*

which
is an ellipse, as shown in Figure 12-11 (p. 353).