RECENT ASSIGNMENT

1
ENGG961 Tieling and Richard _ Autumn 2015
Subject: ENGG961: Systems Reliability Engineering
Assignment 2: Reliability Data Modelling
Purpose:
To complete failure data modelling by applying the knowledge learnt.
Learning objectives covered:
1. Understand and apply reliability concepts and terminology.
2. Understand and apply the basic mathematics involved in reliability engineering.
3. Understand and make use of the relationships amongst the different reliability functions.
4. Collect and analyse reliability data (times to failure and times to repair) using empirical and
parametric methods (exponential, Weibull, normal and lognormal are in syllabus); collect and
analyse failure times of repairable systems to determine the intensity function (power law model).
Tasks
1. Generator is a critical component in power drive-train system in a wind turbine. The generators must
pass the required reliability testing before they can be applied to wind turbines. Three suppliers are
providing generators for wind turbine manufacturers. Extensive reliability testing has resulted in the
determination of the failure distribution for each vendor’s generator, see below:
Vendor Failure Distribution
Wind Power Generator Weibull distribution with scale parameter ? = 6,250 operating days and
shape parameter ? = 1.25.
GE Generator Lognormal distribution with median time tmed = 4,000 operating days and
s = 0.80 (s is shape parameter, see Page 81 of the prescribed textbook)
Siemens Generator F(t) = 1 ? a
2
/(a+t) 2 where 0 ? t (measured in operating days, a = 6,570)
Compare each vendor’s product by finding (total 20 marks)
1. R(10 years) (1 year =365 days) (2 marks)
2. The MTTF and median time to life (2 marks)
3. The mode of each distribution model by plotting pdf (5 marks)
4. The 95-percent design life (2 marks)
5. The reliability for the next 5 years if it has survived the first 10 years (4 marks)
6. Plotting the hazard function (3 marks)
7. Whether the hazard function is DFR, CFR, or IFR (2 marks)
2. Fifty automobiles using a new type of motor oil were monitored over a period of time to determine
when the oil needed replacing due to the level of contaminants. These times were recorded in tens of
miles. Several units were censored from the study as a result of vehicle losses. Motor-oil failures are
believed to follow a Weibull distribution.
2
ENGG961 Tieling and Richard _ Autumn 2015
1770 2034 2876 3200+ 2390 5751 553+ 1450 2319 682
2220 2207+ 654+ 1855 1393 480 1526 4006 3069 2100
1230 5050+ 2019 2622 3675 1714 810+ 2146 1819+ 1793
1187 2300 2859 2038 2180 2330 2110+ 2550 1980 890
1500 2750 2450 1110 1220+ 1250 4000 3150 850 3200
+ means censored data
Answer the following questions (total 24 marks)
1. Derive the maximum likelihood estimates and determine a replacement interval in miles based
upon a 95-percent design life. Compare this to the mean time to failure (MTTF) and median time
to failure. (8 marks)
2. Use Least Squares estimate to find the model parameters and give R2
value of the distribution
model obtained for the given data set. (8 marks)
3. Plot the data points and the fitted models obtained using maximum likelihood estimates and the
Least Square estimate on the Weibull Probability Paper (WPP). (8 marks)
3. A company manufactures various household products. Of concern to the company is its relatively low
production rate on its powdered detergent production line because of the limited availability of the line
itself. The line fails frequently generating considerable downtime. The line has two primary failure modes:
Mode A reflects operation failures such as jams, breaks, spills, and overflows on the line and Mode B
represents mechanical and electrical failures of motors, glue guns, rollers, belts, etc. Over the last 65 line
start-ups, the following times in hours until the line shut down were recorded:
Time
mode
0.1
A
0.3
A
0.3
A
0.4
A
0.5
A
0.6
A
0.8
A
1.1
A
1.1
A
1.2
A
1.5
A
1.8
A
1.9
A
Time
mode
2.1
A
2.3
A
2.8
A
4.0
A
5.7
A
8.7
A
9.4
A
10.0
A
10.1
A
11.7
A
13.7
A
15.1
B
15.3
A
Time
mode
19.1
A
19.3
A
19.6
B
21.3
A
23.2
B
24.9
A
25.2
A
32.7
A
34.4
B
47.3
A
59.9
B
64.8
A
65.5
B
Time
mode
73.1
B
86.8
B
93.0
B
99.0
B
103.1
B
115.5
B
118.3
B
118.8
B
122.7
B
125.0
B
134.4
B
135.4
B
147.6
B
Time
mode
149.1
B
160.4
B
161.5
B
163.4
B
165.4
B
180.5
B
181.1
B
192.6
B
196.2
B
199.6
B
204.1
B
205.2
B
211.4
B
Answer the following questions (total 50 marks)
1) From among the exponential, Weibull, normal and lognormal distributions find the best fit for
each failure mode based upon the Least Squares R
2
value. (10 marks)
2) From among the exponential, Weibull, normal and lognormal distributions find the best fit for
Mode B failure based upon Chi-Square Goodness-of-Fit test. (10 marks)
3) Find a best fit model from among Weibull, gamma, normal and lognormal distributions by
comparison of AICc and BIC values calculated for each of the four distribution models selected
for Mode A and Mode B failure. (10 marks)
4) Use the parameter estimates by the Least Squares approach to compute the reliability that the line
will operate for one hour (1) without a Mode A failure, (2) without a Mode B failure, and (3)
without either. (5 marks)
3
ENGG961 Tieling and Richard _ Autumn 2015
5) Referring to the results obtained in 3), can you find a better fit model for the test data including
Mode A and Mode B failure? If not, explain why? (10 marks)
6) What conclusion can be reached concerning the operational failures? (5 marks)
4. A reliability engineer is asked to answer the following questions. Can you help answer them? (total 6
marks)
1) If two distributions have the same MTTF, can it be concluded that the reliabilities given by the
two distributions are of the same? Give a short discussion. (3 marks)
2) One maintenance engineer argues that after periodical maintenance service, the system reliability
is improved because of maintenance actions applied and removal of degraded or possibly faulty
parts from the system, therefore, the system reliability is plotted as follows:
ti (i = 1, 2, …, ) is the time when the ith maintenance service is performed.
Do you agree with the point as argued? If not, give your opinion with a short discussion. (3 marks)
Learning Guides
In order to complete the assignment tasks, you need to read Chapter 4, Chapter 12 and Chapter 15 given
that you have learnt Chapter 1 to Chapter 3; and you are required to have knowledge of Statistical Tests in
Chapter 16 in the prescribed textbook. While you are reading, you need to understand or answer the
following questions:
1) Be familiar with terminologies used in reliability engineering such as MTTF, median time to
failure, CDF, pdf, hazard function, failure rate, mode of a distribution, confidence interval,
Quantiles, Least Square estimate, maximum likelihood estimate (MLE), Goodness-of-Fit test.
2) Be familiar with distribution models such as exponential, normal, lognormal, Weibull, gamma
distribution.
3) Know how to calculate reliability, failure rate, pdf.
4) Understand definitions of MTTF and MTBF, how to calculate each of them.
5) Understand why MTBF was popularly applied since World War II? Need to consider if it should
be utilised?
6) Is reliability lower if failure rate is higher?
7) How to calculate reliability if a distribution model is given?
8) Understand Type-I and Type-II tests and data characteristics obtained under each test.
9) Understand Least-Squares estimate of model parameters.
10) Understand maximum likelihood estimates of model parameters. Reliability
Time 0 t 1 t 2
4
ENGG961 Tieling and Richard _ Autumn 2015
11) Get known of Weibull plot and Normal plot, i.e., plot data on Weibull probability paper (WPP)
and Normal probability paper.
12) How to select an optimal model for a test data set? To be familiar with Goodness-of-Fit test
(Chapter 16).
13) Learn to know what are AIC, AICc and BIC, and how to use them.
(http://en.wikipedia.org/wiki/Akaike_information_criterion;
http://en.wikipedia.org/wiki/Bayesian_information_criterion )
14) Learn how to calculate 5% and 95% confidence level given a data point.
15) Learn to know how to calculate the lower and upper bound for a 95% confidence interval in
parameter estimate.
16) Learn to use and become very familiar with JMP and Excel.