Application of Probability to Mechanical Design
49
Eq. (2.16)
~2 1 ~2
= --z x +~z~
Zz ]A2y
r-y
or
¯ 2,~2 2~2
~2 ~y-x + Px z}
(2.22)
Z z -
#4y
The derivation Eqs. (2.12)-(2.16) is for what is termed uncorrelated
variables. This means all the variables in the equation are independent
of each other and a single variable can be changed without changing the
value of the rest. This may be seen when examining Eq. (2.12). The case
of E, Youngs modulus, from a tension test is not a result of uncorrelated
variables where the
E = -
(2.23)
stress in the sample is divided by the strain. Hence stress or strain can not be
varied independently of each other. The Eq. (2.23) consists of a measure
the force applied and the elongation because of it making Young’s Modulus
Eq. (2.23) a correlated variable. The mean and standard deviations are discussed by Haugen [2.18] and Miscke [2.42] for both correlated and
uncorrelated variables.
It should also be noted that many of the terms like "E" and "a" are
quoted as uncorrelated variables. The relationship for coefficient of variation is developed which is the Gaussian standard deviation divided by
its mean and multiplied by 100 for
Cv ~
=- × 100 a percentage.
(2.24)
which gives a percentage variation which becomes a constant number for
various materials. Some of the values quoted in the literature and
[2.18,2.19,2.44,2.53] are shown in Table 2.1. Haugen [2.18] performed an
extensive study of many design parameters.
EXAMPLE 2.9. When parts are placed in an assembly the overall
average assembly dimension and its variation are important. The problem
of the stack up variation in parts can be examined using Eq. (2.18) and
the stack up of z = x + y. If the coefficient of variation for x and y are
C~x
~x + 0.01 Cvy-ZY-+ 0.01
/z x
2.576
py
2.576
The dimensions vary 1°/0 about the means 4-2.576 ~ approximately for 99%
49
Eq. (2.16)
~2 1 ~2
= --z x +~z~
Zz ]A2y
r-y
or
¯ 2,~2 2~2
~2 ~y-x + Px z}
(2.22)
Z z -
#4y
The derivation Eqs. (2.12)-(2.16) is for what is termed uncorrelated
variables. This means all the variables in the equation are independent
of each other and a single variable can be changed without changing the
value of the rest. This may be seen when examining Eq. (2.12). The case
of E, Youngs modulus, from a tension test is not a result of uncorrelated
variables where the
E = -
(2.23)
stress in the sample is divided by the strain. Hence stress or strain can not be
varied independently of each other. The Eq. (2.23) consists of a measure
the force applied and the elongation because of it making Young’s Modulus
Eq. (2.23) a correlated variable. The mean and standard deviations are discussed by Haugen [2.18] and Miscke [2.42] for both correlated and
uncorrelated variables.
It should also be noted that many of the terms like "E" and "a" are
quoted as uncorrelated variables. The relationship for coefficient of variation is developed which is the Gaussian standard deviation divided by
its mean and multiplied by 100 for
Cv ~
=- × 100 a percentage.
(2.24)
which gives a percentage variation which becomes a constant number for
various materials. Some of the values quoted in the literature and
[2.18,2.19,2.44,2.53] are shown in Table 2.1. Haugen [2.18] performed an
extensive study of many design parameters.
EXAMPLE 2.9. When parts are placed in an assembly the overall
average assembly dimension and its variation are important. The problem
of the stack up variation in parts can be examined using Eq. (2.18) and
the stack up of z = x + y. If the coefficient of variation for x and y are
C~x
~x + 0.01 Cvy-ZY-+ 0.01
/z x
2.576
py
2.576
The dimensions vary 1°/0 about the means 4-2.576 ~ approximately for 99%