Figure 12-23 plots the largest peak value of
the shear stress
3 occurring at any value of
x
across
the patch zone, and so is a composite plot of the peak shear stress value in
each
z
plane. For 0 <
< 0.5 the peak value remains within 60–80%
of its largest value over the first a
of depth and is still 58–70%
of its peak value at z / a = 2.0. As the coefficient of friction is increased
to 0.5 or greater, the normalized maximum shear-stress value becomes equal to and is constant across the contact-patch
surface.
The
limited variation of
max
over
small
z
depths may explain why some pitting failures
appear to start at the surface and some below it. With a relatively
uniform-magnitude maximum shear stress over the entire near-surface region, any
inclusion in that region of the material creates a stress concentration and
serves as a crack initiation point. The fact that the peak value of the maximum
shear stress occurs at slightly different transverse locations at different
depths within the contact zone is irrelevant, since an inclusion at any
particular depth will pass through that location once per revolution and be
exposed to the peak stress value.
FIGURE 12-22
Principal stresses below surface at
x
/
a
= 0.3 for cylinders in
combined rolling and sliding with
= 0.33
FIGURE 12-23
Peak values of maximum
shear stress at all values of
x
/
a
for
cylinders in combined rolling and sliding with 0
≤ ì ≤
0.5
EXAMPLE 12-4
Stresses
in Combined Rolling and Sliding of Cylinders.
Problem
:
A radial
track cam and cylindrical roller follower have a combination of rolling and
sliding. Find the maximum tensile, compressive, and shear stresses in the cam
and roller.
Given:
The roller
radius is 1.25 and the minimum radius of curvature of the cam is 2.5 in. The
cam is 0.875 in thick and the roller is 1-in-long. The force is 500
lb,
normal to the contact plane.
Assumptions:
The roller axis is exactly parallel to the cam track surface. Both
materials are steel. The coefficient of friction is 0.33.
Solution:
1
The contact-patch geometry is found in the same way as was done in Example
12-3. Find the material constants from equation 12.9a (p. 355).
The
geometry constant is found from equation 12.15a (p. 360)
and
the patch half-width from equation 12.15b (p. 360).
where
a
is the half-width of the contact patch. The
rectangular contact-patch area is then
2
The average and maximum contact pressure can now be found from equations 12.14b
and
The
tangential pressure is found from equation 12.22
f
(p.
370):
3
With
= 0.33, the principal stresses in the contact
zone will be maximal on the surface (z
= 0) at x =
0.3a from the centerline as shown in Figures 12-20
(p. 364) and 12-22 (p. 369). The applied stress components are found from
equation 12.23a (p. 370) for the normal force and equation 12.23b (p. 370) for
the tangential force.
4
Equations 12.24a and b (p. 371) can now be solved for the total applied
stresses along the
x
,
y,
and
z
axes.
5
Since the roller is short, we expect a plane stress condition to exist. The
stress in the third dimension is then:
6
Unlike the pure rolling case, these stresses are not principal because of the
applied shear stress. The principal stresses are found from equations 12.5 (p.
348) using a cubic root finding solution.
The
maximum shear stress is found from the principal stresses using equation 12.6
(p. 349).
7
The principal stresses are maximum at the surface as seen in Figures 12-20 and
12-22.
Copyright 2004, Industrial
Press, Inc., New York, NY