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Monday, December 20, 2021

12 Measurement reliability and safety systems

 12.1 Reliability

The reliability of measurement systems can be quantified as the mean time between faults occurring in the system. In this context, a fault means the occurrence of an unexpected condition in the system that causes the measurement output to either be incorrect or not to exist at all. The following sections summarize the principles of reliability theory that are relevant to measurement systems. A fuller account of reliability theory, and particularly its application in manufacturing systems, can be found elsewhere (Morris, 1997).

 

12.1.1 Principles of reliability

The reliability of a measurement system is defined as the ability of the system to perform its required function within specified working conditions for a stated period of time. Unfortunately, factors such as manufacturing tolerances in an instrument and varying operating conditions conspire to make the faultless operating life of a system impossible to predict. Such factors are subject to random variation and chance, and therefore reliability cannot be defined in absolute terms. The nearest one can get to an absolute quantification of reliability are quasi-absolute terms like the mean-timebetween-failures, which expresses the average time that the measurement system works without failure. Otherwise, reliability has to be expressed as a statistical parameter that defines the probability that no faults will develop over a specified interval of time.

In quantifying reliability for a measurement system, an immediate difficulty that arises is defining what counts as a fault. Total loss of a measurement output is an obvious fault but a fault that causes a finite but incorrect measurement is more difficult to identify. The usual approach is to identify such faults by applying statistical process control techniques (Morris, 1997).

Reliability quantification in quasi-absolute terms

Whilst reliability is essentially probabilistic in nature, it can be quantified in quasi-absolute terms by the mean-time-between-failures and the mean-time-to-failure parameters. It must be emphasized that these two quantities are usually average values calculated over a number of identical instruments, and therefore the actual values for any particular instrument may vary substantially from the average value.

The mean-time-between-failures (MTBF) is a parameter that expresses the average time between faults occurring in an instrument, calculated over a given period of time. For example, suppose that the history of an instrument is logged over a 360 day period and the time intervals in days between faults occurring was as follows:

               11 23 27 16 19 32 6 24 13 21 26 15 14 33 29 12 17 22

The mean interval is 20 days, which is therefore the mean-time-between-failures. An alternative way of calculating MTBF is simply to count the number of faults occurring over a given period. In the above example, there were 18 faults recorded over a period of 360 days and so the MTBF can be calculated as:

                                      MTBF = 360/18 = 20 days

Unfortunately, in the case of instruments that have a high reliability, such in-service calculation of reliability in terms of the number of faults occurring over a given period of time becomes grossly inaccurate because faults occur too infrequently. In this case, MTBF predictions provided by the instrument manufacturer can be used, since manufacturers have the opportunity to monitor the performance of a number of identical instruments installed in different companies. If there are a total of F faults recorded for N identical instruments in time T, the MTBF can be calculated as MTBF D TN/F. One drawback of this approach is that it does not take the conditions of use, such as the operating environment, into account.

The mean-time-to-failure (MTTF) is an alternative way of quantifying reliability that is normally used for devices like thermocouples that are discarded when they fail. MTTF expresses the average time before failure occurs, calculated over a number of identical devices.

The final reliability-associated term of importance in measurement systems is the mean-time-to-repair (MTTR). This expresses the average time needed for repair of an instrument. MTTR can also be interpreted as the mean-time-to-replace, since replacement of a faulty instrument by a spare one is usually preferable in manufacturing systems to losing production whilst an instrument is repaired.

The MTBF and MTTR parameters are often expressed in terms of a combined quantity known as the availability figure. This measures the proportion of the total time that an instrument is working, i.e. the proportion of the total time that it is in an unfailed state. The availability is defined as the ratio:

                         Availability = MTBF/MTBF + MTTR

In measurement systems, the aim must always be to maximize the MTBF figure and minimize the MTTR figure, thereby maximizing the availability. As far as the MTBF and MTTF figures are concerned, good design and high-quality control standards during manufacture are the appropriate means of maximizing these figures. Design procedures that mean that faults are easy to repair are also an important factor in reducing the MTTR figure.

 

Failure patterns

The pattern of failure in an instrument may increase, stay the same or decrease over its life. In the case of electronic components, the failure rate typically changes with time in the manner shown in Figure 12.1(a). This form of characteristic is frequently known as a bathtub curve. Early in their life, electronic components can have quite a high rate of fault incidence up to time t1 (see Figure 12.1(a)). After this initial working period, the fault rate decreases to a low level and remains at this low level until time t2 when ageing effects cause the fault rate to start to increase again. Instrument manufacturers often ‘burn in’ electronic components for a length of time corresponding to the time t1. This means that the components have reached the high-reliability phase of their life before they are supplied to customers.

Mechanical components usually have different failure characteristics as shown in Figure 12.1(b). Material fatigue is a typical reason for the failure rate to increase over the life of a mechanical component. In the early part of their life, when all components are relatively new, many instruments exhibit a low incidence of faults. Then, at a later stage, when fatigue and other ageing processes start to have a significant effect, the rate of faults increases and continues to increase thereafter.

Complex systems containing many different components often exhibit a constant pattern of failure over their lifetime. The various components within such systems each have their own failure pattern where the failure rate is increasing or decreasing with time. The greater the number of such components within a system, the greater is the tendency for the failure patterns in the individual components to cancel out and for the rate of fault-incidence to assume a constant value.


Reliability quantification in probabilistic terms

In probabilistic terms, the reliability R (T) of an instrument X is defined as the probability that the instrument will not fail within a certain period of time T. The unreliability or likelihood of failure F (T) is a corresponding term which expresses the probability that the instrument will fail within the specified time interval. R(T) and F(T) are related by the expression:

                              F (T) = 1 - R (T)

To calculate R (T) , accelerated lifetime testingŁ is carried out for a number (N) of identical instruments. Providing all instruments have similar conditions of use, the times of failure, t1, t2 ...tn will be distributed about the mean time to failure tm. If the probability density of the time-to-failure is represented by f t , the probability that a particular instrument will fail in a time interval υt is given by f t υt, and the probability that the instrument will fail before a time T is given by:

                                                   F (T) =  

The probability that the instrument will fail in a time interval  following T, assuming that it has survived to time T, is given by:

                                F (T + ) - F (T)/R (T)

where R (T) is the probability that the instrument will survive to time T. Dividing this expression by T gives the average failure rate in the interval from T to T + T as:

                           F (T + T) - F (T)/ TR T

In the limit as T ! 0, the instantaneous failure rate at time T is given by:

                     θf = d[F (T) ]/dt ,1/R (T) = F0 (T) R (T)

If it is assumed that the instrument is in the constant-failure-rate phase of its life, denoted by the interval between times t1 and t2 in Figure 12.1, then the instantaneous failure rate at T is also the mean failure rate which can be expressed as the reciprocal of the MTBF, i.e. mean failure rate = θf = 1/tm.

Differentiating (12.1) with respect to time gives F0 (T) = -R0 (T) . Hence, substituting for F0 (T) in (12.2) gives:

                                 θf = - R0 (T)/R (T)

This can be solved (Miller, 1990) to give the following expression:

                           R (T) = exp(-θf T)

Examination of equation (12.3) shows that, at time t D 0, the unreliability is zero. Also, as t tends to 1, the unreliability tends to a value of 1. This agrees with intuitive expectations that the value of unreliability should lie between values of 0 and 1. Another point of interest in equation (12.3) is to consider the unreliability when T = MTBF, i.e. when T = tm. Then: F (T) = 1 - exp (-1) = 0.63, i.e. the probability of a product failing after it has been operating for a length of time equal to the MTBF is 63%.

Further analysis of equation (12.3) shows that, for T/tm ≤ 0.1:

                                                 F (T)  T/tm

This is a useful formula for calculating (approximately) the reliability of a critical product which is only used for a time that is a small proportion of its MTBF.

Example 12.1

If the mean-time-to-failure of an instrument is 50 000 hours, calculate the probability that it will not fail during the first 10 000 hours of operation

Solution

From (12.3), R (T) = exp (-θfT) = exp (-10 000/50 000) = 0.8187

Example 12.2

If the mean-time-to-failure of an instrument is 80 000 hours, calculate the probability that it will not fail during the first 8000 hours of operation.

Solution

In this case, T/tm = 80 000/8000 = 0.1 and so equation (12.4) can be applied, giving R (T) = 1 - F (T)  1 - T/tm  0.9. To illustrate the small level of inaccuracy involved in using the approximate expression (12.4), if we calculate the probability according to (12.3) we get R (T) = exp (-0.1) = 0.905. Thus, there is a small but finite error in applying (12.4) instead of (12.3).



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