AdvancedTopic 88 of 104Engineering Mathematics
Reliability and failure distributions
Exponential and Weibull models predict component life and maintenance intervals.
In depth
Reliability R(t) is the probability of survival to time t. The exponential model with constant failure rate λ gives R=e^{-λt} and MTBF=1/λ. The Weibull distribution with shape β captures infant mortality (β<1), random (β=1) and wear-out (β>1) failures.
Key points
- Bathtub curve
- Series systems multiply reliabilities
- Parallel redundancy improves reliability
ExampleTwo components in parallel with R=0.9 each have system reliability 1-0.1²=0.99.
What is Reliability and failure distributions?
Exponential and Weibull models predict component life and maintenance intervals.
Why is it important to understand reliability and failure distributions?
Failure models predict how long components last and when to service them, so they guide maintenance plans and warranty costs.
What is a Weibull distribution used for?
It models failure times with an increasing, constant or decreasing failure rate, which covers early-life, random and wear-out failures in one family.
What is a bathtub curve?
A plot of failure rate over life, showing high early failures, a long constant-rate period and rising wear-out failures.
How is Reliability and failure distributions used in practice?
Typical applications include estimating the expected life of a bearing under load, choosing a replacement interval for a pump seal and comparing early-life and wear-out failure patterns in a fleet.
What should I learn before Reliability and failure distributions?
Study "Regression analysis" first (topic 87 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Reliability and failure distributions?
Bathtub curve. Series systems multiply reliabilities. Parallel redundancy improves reliability.
Can you give an example of Reliability and failure distributions?
Two components in parallel with R=0.9 each have system reliability 1-0.1²=0.99.
What comes after Reliability and failure distributions?
The next topic is "Monte Carlo simulation", which builds directly on this one.
How difficult is Reliability and failure distributions?
It is classed as Advanced in Engineering Mathematics (topic 88 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.