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:

Compare each vendor's product by finding

1. R(10 years) (1 year =365 days)
2. The MTTF and median time to life
3. The mode of each distribution model by plotting pdf
4. The 95-percent design life 
5. The reliability for the next 5 years if it has survived the first 10 years 
6. Plotting the hazard function 
7. Whether the hazard function is DFR, CFR, or IFR

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.

Answer the following questions

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.

2. Use Least Squares estimate to find the model parameters and give R2 value of the distribution model obtained for the given data set.

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).

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:

Answer the following questions

1) From among the exponential, Weibull, normal and lognormal distributions find the best fit for each failure mode based upon the Least Squares R2 value.

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.

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.

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) 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?

6) What conclusion can be reached concerning the operational failures?

4. A reliability engineer is asked to answer the following questions. Can you help answer them?

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.

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.