Page 44 - Shigley's Mechanical Engineering Design

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Introduction to Mechanical Engineering Design 19
goal. Thus, if the objective reliability is to be 99.4 percent, as above, what combination
of materials, processing, and dimensions is needed to meet this goal? If a mechanical
system fails when any one component fails, the system is said to be a series system. If
the reliability of component i is R in a series system of n components, then the relia-
i
bility of the system is given by
n

(1–5)
R = R i
i=1
For example, consider a shaft with two bearings having reliabilities of 95 percent and
98 percent. From Eq. (1–5), the overall reliability of the shaft system is then
= 0.95 (0.98) = 0.93
R = R 1 R 2
or 93 percent.
Analyses that lead to an assessment of reliability address uncertainties, or their
estimates, in parameters that describe the situation. Stochastic variables such as stress,
strength, load, or size are described in terms of their means, standard deviations, and
distributions. If bearing balls are produced by a manufacturing process in which a
diameter distribution is created, we can say upon choosing a ball that there is uncertainty
as to size. If we wish to consider weight or moment of inertia in rolling, this size uncer-
tainty can be considered to be propagated to our knowledge of weight or inertia. There
are ways of estimating the statistical parameters describing weight and inertia from
those describing size and density. These methods are variously called propagation of
error, propagation of uncertainty, or propagation of dispersion. These methods are
integral parts of analysis or synthesis tasks when probability of failure is involved.
It is important to note that good statistical data and estimates are essential to per-
form an acceptable reliability analysis. This requires a good deal of testing and valida-
tion of the data. In many cases, this is not practical and a deterministic approach to the
design must be undertaken.

1–13 Dimensions and Tolerances
The following terms are used generally in dimensioning:
• Nominal size. The size we use in speaking of an element. For example, we may spec-
1
1
ify a 1 -in pipe or a -in bolt. Either the theoretical size or the actual measured size
2 2
1
may be quite different. The theoretical size of a 1 -in pipe is 1.900 in for the outside
2
1
diameter. And the diameter of the -in bolt, say, may actually measure 0.492 in.
2
• Limits. The stated maximum and minimum dimensions.
• Tolerance. The difference between the two limits.
• Bilateral tolerance. The variation in both directions from the basic dimension. That
is, the basic size is between the two limits, for example, 1.005 ± 0.002 in. The two
parts of the tolerance need not be equal.
• Unilateral tolerance. The basic dimension is taken as one of the limits, and variation
is permitted in only one direction, for example,
1.005 +0.004 in
−0.000
• Clearance. A general term that refers to the mating of cylindrical parts such as a bolt
and a hole. The word clearance is used only when the internal member is smaller than
the external member. The diametral clearance is the measured difference in the two
diameters. The radial clearance is the difference in the two radii.