1. Understand and apply reliability concepts and terminology.
Subject: ENGG961: Systems Reliability Engineering
Assignment 1: Defining the reliability concept and modelling reliability data
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.
 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).
Learning Guides
In order to complete the assignment tasks, you need to read the related reading provided in elearning (Moodle) and Chapter 4, Chapter 12 and Chapter 15 given that you have knowledge of Chapter 1 to Chapter 3; and you are required to have knowledge of Statistical Tests in Chapter 16 in the prescribed textbook. You are not asked to master mathematical logic or derivation of the covered distribution but should be able to identify them, their differences and most important their potential application. While you are reading, you need to understand or be aware of the following points:
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), GoodnessofFit test.
2) Be familiar with distribution models such as exponential, normal, lognormal, Weibull, gamma distribution.
3) Learn to use and become very familiar with Excel Templets (spreadsheets) provided with the textbook which will be made available in elearning (Moodle). These templets are your main tool for calculation and drawing distributions.
4) Know how to calculate reliability, failure rate, pdf: substitute value in equation and use excel templets for calculation and drawing
5) Understand definitions of MTTF and MTBF, how to calculate each of them using excel templets.
6) Understand the relationship between reliability and failure rate?
7) Learn how to calculate reliability if a distribution model is given? Use excel templets
8) Understand TypeI and TypeII tests and data characteristics obtained under each test
9) Understand LeastSquares estimate of model parameters.
10) Understand maximum likelihood estimates of model parameters.
11) Understand why emphasis on the use of Weibull distribution is great for engineering asset management.
12) How to select an optimal model for a test data set? To be familiar with GoodnessofFit 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.
Tasks: Task 1 (40 marks)
As a reliability engineer answer the following questions as brief and concise as possible: Although each acceptable answer is worth 5 marks, students may differ on putting emphasis some questions more than others while answering questions and therefore a wide range of different answers to the questions are acceptable.
 Give a definition of reliability that you think is more precise than other available definition and explains why that is so?
 Why engineering assets/products fail? And how is that related to system engineering reliability?
3. Why is failure rate important for system engineering reliability? Can you elaborate the impact of failure rates on system engineering reliability?
 How can you measure/quantify reliability? Is reliability a probability? Can it be predicted and if yes what are the possible basis for its prediction?
5. Why MTBF was popularly applied since World War II? Should we continue to use it? Explain why?
 Is reliability lower if failure rate is increased? 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.
 Explain the relationship between reliability and the following concepts: availability and maintainability, quality, risk, minimum life cycle cost and optimum life of assets/products?
 Can you illustrate why system reliability engineering is important for engineering asset management?
Task 2 (20 marks)
Gearbox is a very critical component in power drivetrain system in a wind turbine. The gearboxes must pass the required reliability testing before they are applied to wind turbines. Three suppliers are providing gearboxes for wind turbine manufacturers. Extensive reliability testing has resulted in the determination of the failure distribution for each vendor’s gearbox, see below:
Vendor

Failure Distribution

ZF Wind Power Gearbox

Weibull distribution with scale parameter θ = 100,000 operating hours and shape parameter
β = 1.2.

GE Gearbox

Lognormal distribution with median time tmed
= 60,000 operating hours and s = 0.8 (s is shape parameter, see Page 81 of the prescribed
textbook)

Zollern Gearbox

R(t) = 1 – 2t/a + t2/a2 where 0 ≤ t ≤ a (measured in operating hours, a = 290,000)

Compare each vendor’s product by finding:
1. R(10,000 hr) (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 95percent design life (2 marks)
5. The reliability for the next 10,000 hours if it has survived the first 10,000 hours (4 marks)
 Plotting the hazard function (3marks)
 Whether the hazard function is DFR, CFR, or IFR (2 marks)
Task 3 (20 marks)
Fifty automobiles using a new type of motor oil were monitored over a period of several months 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. Motoroil failures are believed to follow a Weibull distribution.
1770

2034

2876

3200+

2390

5700

553+

1450

2319

682

2220

2200+

654+

1855

1393

480

1526

4030+

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

Answer the following questions:
1. Use Least Squares estimate to obtain the models’ parameters from the given data set ignoring the censored data and compare R^{2} value of the following distribution models: exponential, Weibull, normal, lognormal and extreme value and find the best fit model from among these. Also, find the mean time to failure (MTTF) and median time to failure (8 marks)
For normal distribution:
By using the excel sheet, calculate the mean and the standard deviation of the data
2. Using the trendline on excel, plot the Least Square estimate and show the equation and R ^{2} value on the plot. (6 marks)
 3. Derive the maximum likelihood estimates and determine a replacement interval in miles based upon a 95percent design life. Compare this to the mean time to failure (MTTF) and median time to failure obtained in 1. (assume multiply censored data) (8 marks)
For exponential distribution
Use the equation 15.20 page 379 to get
For replacement interval see example 15.15 page 380
For Weibull distribution:
Find and by using equation 15.21 and 15.22 in the book (page 379) by using excel See example 15.15 page 380
Task 4 (20 marks)
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 56 line startups, the following times in hours until the line shut down were recorded:
T

0.1

0.2

0.3

0.4

0.45

0.5

0.6

0.8

1.0

1.1

1.1

1.3

1.5

1.8

M

A

A

A

A

A

A

A

A

A

A

A

A

A

A

T

1.9

2.1

2.3

2.5

2.8

4.0

5.7

8.7

9.4

10.0

10.3

11.7

13.7

15.1

M

A

A

A

A

A

A

A

A

A

A

A

A

A

A

T

15.3

19.1

19.3

19.6

21.3

23.2

24.9

25.2

32.7

34.4

47.3

59.9

64.8

65.5

M

A

A

A

B

A

A

A

A

A

A

B

B

B

B

T

73.1

86.8

93.0

99.0

103.1

115.5

118.3

122.7

134.4

147.6

160.4

163.4

180.5

192.6

M

B

B

B

B

B

B

B

B

B

B

B

B

B


Answer the following questions:
1. From among the exponential, Weibull, normal and lognormal distributions find the best fit for each failure mode and for both modes based upon the Least Squares R^{2} value. (4 marks)
2. From among the exponential, Weibull, normal and lognormal distributions find the best fit for Mode A failure based upon ChiSquare GoodnessofFit test. (4 marks)
3. From among the exponential, Weibull, normal and lognormal distributions find the best fit for Mode B failure based upon ChiSquare GoodnessofFit test. (4 marks)
4. Use the Least Squares parameter estimates 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. (4 marks)
5. What conclusion can be reached concerning the operational failures? (4 marks)
Overall Assessment:
Task 1

/40

Task 2

/20

Task 3

/20

Task 4

/20

General Comments

Overall MARK/100

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