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Copyright 2003 by The McGraw-Hill Companies, Inc Click Here for Terms of Use
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CHAPTER EIGHTEEN
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Chance failures is a term used generally to describe those failures which occur suddenly and at random during the anticipated useful life of a component or system They may be due to a variety of causes, including inherent weakness, misuse, etc; they are not due to the component having completed its normally anticipated useful life, ie, to the component wearing out Wearout failure is another general term to describe failure due to the wearing out of a component which has more or less completed its anticipated useful life Both chance and wearout failures may be partial or complete
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FAILURE RATE
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The number of failures occurring per unit time is known as the failure rate As with all quantities describing change (speed, acceleration, etc), an average value may be obtained by dividing the total number of failures which have occurred during a time interval by the length of the interval The shorter the interval, the nearer the average value gets to the instantaneous failure rate The instantaneous failure rate at any one time is the slope of the curve plotting failures against time at that particular time If, in determining the failure rate, the number of failures occurring during the time interval is expressed as a proportion of the number of survivors at the beginning of the time interval, then the failure rate obtained is called the proportional failure rate It is denoted by the symbol Unless otherwise stated the words failure rate within this handbook imply the proportional failure rate
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THE BATHTUB DIAGRAM
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A typical graph plotting the percentage failure rate with respect to time is shown in Fig 181 It is often referred to as a bathtub diagram because of its shape During the burn-in period, a high failure rate exists, owing to the presence of substandard components in the sample tested After the weak components have died out, the failure rate stabilizes at an approximately constant value; this period is called the useful-life period
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FIGURE 181 Bathtub diagram
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RELIABILITY, SYSTEMS, AND SAFETY ENGINEERING
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Eventually wearout failures begin to occur, and the failure rate rises again During this wearout period, chance failures may, of course, still be occurring In order to make reasonably accurate predictions of reliability, failures due to chance and wearout must be studied and an analysis made of the times in relation to the type of failure
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CONSTANT-FAILURE-RATE CASE
When the failure rate is constant, reliability prediction is made much easier mathematically, since it is possible to use exponential curves to assist analysis As was shown above, the failure rate may be assumed to be constant when failures are due to chance alone; this can be achieved by correct overhaul schedules, which eliminate wearout failure It is also possible, however, to achieve a constant failure rate, and thus simplify the mathematics involved, by a process of immediate replacement on wearout This latter case is not as obvious but has been conclusively demonstrated It should be noted that replacement on failure is a procedure which cannot always be adopted, since certain systems or components cannot be allowed to fail even temporarily
RELIABILITY EQUATIONS AND CURVES WHEN FAILURE RATE IS CONSTANT
The probability of no failures occurring in a given time can be expressed by the following equation, provided that the failure rate is constant: R e
(181)
where R is the probability of no failures in time t (ie, the reliability), e is the exponent 27183, and is the constant failure rate (This is in fact the no-event term of the Poisson probability function) The unreliability Q is de ned as the probability of total failure It follows logically that R and that Q Q 1 e 1
(182) (183)
where Q is the probability of total failure in time t A graph of R and Q against time yields the familiar exponential curves shown in Fig 182 0, R 1, and Q 0, at time t 1/ , R 037, and Notice that at time t Q 063 (from tables of values of e raised to various powers) A graph of survivors, ie, the number of components still alive at time t against time, will yield the same shape as the reliability curve in Fig 182 A graph of failures against time yields the same shape as the unreliability curve (see Fig 183) The equation of the graph of survivors versus time (survival curve) is Ns N0e
(184)
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