Page 353 - Shigley's Mechanical Engineering Design

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328 Mechanical Engineering Design
To determine the surface fatigue strength of mating materials, Buckingham designed
a simple machine for testing a pair of contacting rolling surfaces in connection with his
investigation of the wear of gear teeth. Buckingham and, later, Talbourdet gathered large
numbers of data from many tests so that considerable design information is now
available. To make the results useful for designers, Buckingham defined a load-stress
factor, also called a wear factor, which is derived from the Hertz equations. Equations
(3–73) and (3–74), p. 124, for contacting cylinders are found to be

2 2
2F 1 − ν /E 1 + 1 − ν /E 2
2
1
b = (6–59)
πl (1/d 1 ) + (1/d 2 )
2F
p max = (6–60)
πbl
where b = half width of rectangular contact area
F = contact force
l = length of cylinders
ν = Poisson’s ratio
E = modulus of elasticity
d = cylinder diameter
It is more convenient to use the cylinder radius, so let 2r = d. If we then designate
the length of the cylinders as w (for width of gear, bearing, cam, etc.) instead of l and
remove the square root sign, Eq. (6–59) becomes

2 2
4F 1 − ν /E 1 + 1 − ν /E 2
2 1 2
b = (6–61)
πw 1/r 1 + 1/r 2
We can define a surface endurance strength S C using
2F
p max = (6–62)
πbw
as
2F
S C = (6–63)
πbw
which may also be called contact strength, the contact fatigue strength, or the Hertzian
endurance strength. The strength is the contacting pressure which, after a specified
number of cycles, will cause failure of the surface. Such failures are often called wear
because they occur over a very long time. They should not be confused with abrasive
2
wear, however. By squaring Eq. (6–63), substituting b from Eq. (6–61), and rearrang-
ing, we obtain
F 1 1 2 1 − ν 1 2 1 − ν 2 2
+ = πS C + = K 1 (6–64)
w r 1 r 2 E 1 E 2

The left expression consists of parameters a designer may seek to control independently.
The central expression consists of material properties that come with the material and
condition specification. The third expression is the parameter K 1 , Buckingham’s load-
stress factor, determined by a test fixture with values F, w, r 1 , r 2 and the number of