Abstract
Typical time course of failure rate of unrepaired objects, called bathtub curve, is shown and its main stages are explained: period of early failures, useful life, and period of aging and deterioration. Attention is paid to the useful-life period, where the failure rate is constant and the distribution of times to failure (or between failures) is exponential. Illustrative examples are included.
Keywords
- Failure rate
- bathtub curve
- early failures
- steady-state operation
- period of aging
- exponential distribution
Failure rate, as defined in Chapter 3, can change with time. Figure 1 shows the time course of λ(
Stage I. Failure rate λ(
Stage II. Failure rate λ is low and approximately constant. In contrast to early failures, caused by the inherent weakness of the object, the failures during stage II occur mostly due to external reasons, such as overloading, collision with another object, weather or natural catastrophes, hidden defects, and mistakes of the personnel. (In the case of people, the reasons for the “failures” during this stage are traffic accidents, diseases, wars, and murders.) Depending on the object and conditions, failure rates for various objects can be very different. Stage II represents the major part of the life and is called the
Stage III. Failure rate λ(
Figure 1 shows the general shape of the time course of failure rate. In reality, various patterns of λ(
Also, stage III, the wear-out period, can be avoided for more complex objects if their technical condition is monitored and the critical parts approaching stage III are replaced in time by new ones. This case belongs to repairable objects. The “bathtub curve” here consists only of periods I and II (early failures and useful life) or even only period II (steady-state operation).
Remark: The failures from external reasons can happen at any time; the instantaneous resultant failure rate equals the sum of failure rates from all reasons.
This is a very important case, as constant failure rate can often be assumed (approximately) for the prevailing period of useful life (stage II in Fig. 1). With λ =
The reliability (i.e. the fraction of serviceable objects) decreases with time as
The distribution of times to failure is exponential with the probability density
and the mean value
Vice versa, the failure rate of some kind of components can be obtained from the mean time to failure,
The time course of reliability may thus also be expressed as
note that the argument in the exponential function is nondimensional.
The mean time to failure (and also the mean time between failures) can be calculated by Equation (4). With λ =
The empirical determination of the mean time to failure is based on the testing or monitoring of a group of components of the same kind and measuring their times to failure,
the summation is done for all
In design, the knowledge of failure rate λ of a component, found from the manufacturer’s catalog or by measurement, enables the determination of the mean time to failure, which is important for the determination of the overall reliability of more complex systems (cf. Chapter 5).
Exponential distribution is typical of systems consisting of many elements, where failures happen from various reasons, as usual in electric or electronic appliances. However, one should not forget that the period with constant failure rate often becomes dominant only after some time
Note: One must always keep in mind that the mean time between failures, calculated as the reciprocal value of failure rate, has nothing in common with the mean time to failures caused by aging or fatigue. Failure rate given in catalogs is determined from the period of steady-state operation. For example, a high-quality component has a failure rate λ = 10–6 h1. However, this does not mean that these components will work until
A device should work 2 h without failure, and such operation should be 99% guaranteed. (There may be only 1% probability of failure during this time.) Assume that you can choose from various devices available in the market. What are the demanded failure rate and the mean time to failure of a suitable device? Assume exponential distribution of the time to failure.
Solution. The probability of failure-free operation is
For the demanded
A ventilator (air fan) has exponential distribution of times to failure with the mean time
Solution.
Probability of not failing:
Probability of failure: