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).