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.

1. 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), Goodness-of-Fit 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 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.

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 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.

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.

1. Give a definition of reliability that you think is more precise than other available definition and explains why that is so?
   1. 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?

1. 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?

1. 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.
2. Explain the relationship between reliability and the following concepts: availability and maintainability, quality, risk, minimum life cycle cost and optimum life of assets/products?
3. 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 drive-train 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 95-percent design life (2 marks)

5. The reliability for the next 10,000 hours if it has survived the first 10,000 hours (4 marks)

1. Plotting the hazard function  (3marks)
2. 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. Motor-oil 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 R² 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)

1. 3.  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 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 start-ups, the following times in hours until the line shut down were recorded: