Page 52 - Shigley's Mechanical Engineering Design

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Introduction to Mechanical Engineering Design 27
1–10 When one knows the true values x 1 and x 2 and has approximations X 1 and X 2 at hand, one can
see where errors may arise. By viewing error as something to be added to an approximation to
attain a true value, it follows that the error e i , is related to X i , and x i as x i = X i + e i
(a) Show that the error in a sum X 1 + X 2 is

(x 1 + x 2 ) − (X 1 + X 2 ) = e 1 + e 2
(b) Show that the error in a difference X 1 − X 2 is
(x 1 − x 2 ) − (X 1 − X 2 ) = e 1 − e 2
(c) Show that the error in a product X 1 X 2 is

e 1 e 2
x 1 x 2 − X 1 X 2 = X 1 X 2 +
X 1 X 2
(d) Show that in a quotient X 1 /X 2 the error is

x 1 X 1 X 1 e 1 e 2
− = −
x 2 X 2 X 2 X 1 X 2
√ √
1–11 Use the true values x 1 = 7 and x 2 = 8
(a) Demonstrate the correctness of the error equation from Prob. 1–10 for addition if three cor-
rect digits are used for X 1 and X 2 .
(b) Demonstrate the correctness of the error equation for addition using three-digit significant
numbers for X 1 and X 2 .
1–12 A solid circular rod of diameter d undergoes a bending moment M = 1000 lbf in inducing a
3
stress σ = 16M (πd ). Using a material strength of 25 kpsi and a design factor of 2.5, deter-
mine the minimum diameter of the rod. Using Table A–17 select a preferred fractional diameter
and determine the resulting factor of safety.
1–13 A mechanical system comprises three subsystems in series with reliabilities of 98, 96, and
94 percent. What is the overall reliability of the system?
1–14 Three blocks A, B, and C and a grooved block D have dimensions a, b, c, and d as follows:
a = 1.500 ± 0.001 in b = 2.000 ± 0.003 in
c = 3.000 ± 0.004 in d = 6.520 ± 0.010 in

d
w a b c
Problem 1–14
A B C
D

(a) Determine the mean gap ¯w and its tolerance.
(b) Determine the mean size of d that will assure that w ≥ 0.010 in.

1–15 The volume of a rectangular parallelepiped is given by V = xyz. If
x = a ± a, y = b ± b, z = c ± c, show that
V a b c
= + +
V ¯ ¯ a b ¯ ¯ c