EML 4550: Engineering Design Methods

1. EML 4550: Engineering Design Methods Probability and Statistics in Engineering Design: Reliability Class Notes Hyman: Chapter 5

2. Reliability • Reliability • Probability that an item will perform its stated function without failure under stated conditions of use for a stated measure of the variable (time, distance, batch, etc.) • In the context of reliability we will use the term ‘time’ for the variable (but it may be miles traveled, number of takeoffs and landings, etc.) • Reliability is based only on the ‘available’ time interval and can be expressed as a function of time, R(t)=Ns(t)/No=(Total # survived til time t)/(Total #) • The unreliability is the opposite of R(t) and can be defined as the probability of failure, F(t)=Nf(t)/No =(Total # failed til time t)/(Total #) • R(t)+F(t)=1

3. Mathematical formulation • Probability of system failing by time t is the integral of the failure probability density function f(t) can be considered as the instantaneous failure rate R(t)=1-F(t) 1.0 f(t) F(t) t

4. Per-unit failure rate • Suppose we have a population of 10,000 transistors under test/operation going out at a rate of 50 failures/hour dNf/dt • If we had 1,000 transistors under the same conditions (possibly coming from a subset of the 10,000 above), we would expect 5 failures/hour • It is convenient to define a “per-unit failure rate” or “hazard rate” of, in this case • Nf(t) = number of objects that have failed by time t • Ns(t) = number of objects that have survived by time t

5. Reliability as a function of the per unit failure rate Relatively constant for an extended period of time

6. Constant per-unit failure rate • For systems with constant per-unit failure rate the reliability decreases exponentially with time • Example: 2000 items tested for 500 hours. Per-unit failure rate is 0.002 per hour, how many will survive after 500 hours? R(500)=exp-(0.002*500)=exp(-1)=0.37, 0.37*2000=740 will survive after 500 hours

7. Mean Time Between Failures (MTBF) Mean Time To Failure (MTTF) • Mean Time To Failure (MTTF): The mean of the survival time for all components. This is usually applied to parts that are not repairable, such as light bulbs, pens, etc.. This can also be applied to a system with many components • Mean Time Between Failures (MTBF) is the mean of all intervals between failures provided a large enough sample is taken; usually applied to systems that are to be repaired, such as compressor unit in a power plant, etc.. It is also useful for a system with multiple components (m) and each of which is immediately replaced on failure. • Example: a system has 2 components; one with a MTTF of 2 years, the other has a MTTF of 3 years. What is the MTBF?

8. Mean time to failure (MTTF) • MTTF = Expected, on average, value of the time when failure occurs

9. Example: Reliability & MTTF • A manufacturing process has an established per unit failure rate of 10-5 failures per day. (a) Determine the reliability for a period of 100 days? (b) If the process produces 10,000 parts all together, how many failures we should expect within 100 days? (c) What is the Mean Time To Failure (MTTF)

10. Average per unit failure rate • The average hazard rate, , can sometimes be estimated by performing test on a specific number of samples (n), recording the total number of failures (m), total time span (T), and individual times to failures (ti).

11. Example: Normal Failure Analysis • It is known that an air freshener has a mean lifetime of 800 h with a standard deviation of 40 h (normally distributed). Determine what is the reliability of a freshener at 700 h and 850 h?