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SMRE
Homework 1 Answers
Failure Models; Qualitative Analysis
1
A type of safety valve has been shown to exhibit constant failure rate with respect to all
failure modes. A field-study has produced a value of MTTF for the valve of 2345 days.
Assume that the valve operates in a continuous manner and that the failure modes are
independent of each other.
a.- Determine the valve failure rate.
Answer Since the failure rate is constant, the failure probability is exponential and
λ=
1
1
=
= 0.000426
MTTF
2345
in failures per day.
b.- Determine the probability that the valve will survive for 100 days without failing.
Answer For exponentially distributed failures, the reliability (survival) function, giving
the probability of survival to time t is given as
Z
R(t) = P r(T > t) =
∞
t
Z
f (t)dt =
t
∞
λ exp(−λt)dt = exp(λt)
and since t = 100 then R(100) = exp(−0.000426 × 100) = 0.958.
2
Use the relationships summarized on Table 2.1 (p. 20) in your text and the Maple software,
to derive all the entries contained in Table 2.2 (p. 64).
1
Answer For a (partial) answer, see Maple files Hw1-P2A.pdf and Hw1-P2A.mw.
3
Failure times of 22 Ball Bearings (in cycles) have been collected. The values are as follows:
Time of Failure
17880000
28920000
33000000
41520000
42120000
45600000
48480000
51840000
51960000
54120000
55560000
67800000
68640000
68880000
84120000
93120000
98640000
105120000
105840000
127920000
128040000
173400000
Bearing
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Use the MINITAB software to answer the following questions:
a) Compute descriptive statistics for the data,
b) Construct the histogram,
c) Find good-fitting distributions to the data,
d) Select the best fitting distribution(s) and determine the distribution parameters and
confidence intervals.
2
e) Use the parameter values of the best fitting distribution and the Maple software to
determine analytical expressions for the failure probability distribution function, the survival
probability function, the failure rate (hazard) function, the MTTF and the MRL for the
bearings.
Answer See MINITAB files Hw1-P3A.pdf and Hw1-P3A.MPJ and Maple files Hw1P3A.mw and Hw1-P3A2.pdf.
4
United Airlines Flight 232 crashed on July, 19th, 1989 in Sioux City IA after experiencing
an unprecedent catastrophic failure. Research the accident to answer the following:
a.- Construct an Event Tree with the initiating event being the specific failure that
ultimately led to the crash (e.g. Fig. 3.23, p. 110, text).
b.- Construct a Fault Tree that could have helped predict the accident (e.g. Fig. 3.19,
p. 102, text).
c.- Construct a FMECA worksheet (e.g. Fig. 3.13, p. 89, text) that could have helped
prevent the accident.
Answer The accident was finally blamed on the presence of a metallurgical defect called
”hard alpha” in the Ti-alloy engine disk. The undetected hard inclusion nucleated a crack
that broke the disk. High velocity metal pieces broke through the containment and ultimately
severed key hydraulic lines making flying the airplane next to impossible. Please examine
answer provided by Mas Hongoh and Marc Shaw in their portfolios for details.
3