In spite of the apparent complexity of the equation, the method itself is fairly easy to use. We need to track the run-lengths of equipment, and to record the failures and failure modes. Recording of preventive repair or replacement of components before the end of their useful life is not too demanding. These, along with the time of occurrence (or, if more appropriate, the number of cycles or starts), are adequate for Weibull (or other) analysis. We can obtain such data from the operating records and maintenance management systems.
Such analysis is carried out at the failure modes level. For example, we can look at the failures of a compressor seal or bearing. We need five (or more) failure points to do Weibull analysis. In other words, if we wished to carry out a Weibull analysis on the compressor seal, we should allow it to fail at least five times! This may not be acceptable in practice, because such failures can be costly, unsafe, and environmentally unacceptable. Usually, we will do all we can to prevent failures of critical equipment. This means that we cannot collect enough failure data to improve the preventive maintenance plan and thus improve their performance. On items that do not matter a great deal—for example, light bulbs, vee-belts, or guards—we can obtain a lot of failure data. However, these are not as interesting as compressor seals. This apparent contradiction or conundrum was first stated by Resnikoff8.
3.10 DETERMINISTIC AND PROBABILISTIC DISTRIBUTIONS
Information about the distribution of time to failures helps us to predict failures. The value of the Weibull shape parameter β can help determine the sharpness of the curve. When β is 3.44, the pdf curve approaches the normal or Gaussian distribution. High β values, typically over 5, indicate a peaky shape with a narrow spread. At very high values of β, the curve is almost a vertical line, and therefore very deterministic. In these cases, we can be fairly certain that the failure will occur at or close to the η value. Figure 3.16 shows a set of pdf curves with the same n value of 66.7 weeks we used earlier, and different β values. Figure 3.17 shows the corresponding survival probability or reliability curves. From the latter, we can see that when β is 5, till the 26th week, the reliability is 99%.
On the other hand, when we can be fairly sure about the time of failure,that is, with high Weibull β values, time-based strategies can be effective. If the failure distribution is exponential, it is difficult to predict the failures using this information alone, and we need additional clues. If the failures are evident, and we can monitor them by measuring some deviation in performance such as vibration levels, condition based strategies are effective and will usually be cost-effective as well.
If the failures are unrevealed or hidden, a failure-finding strategy will be effective and is likely to be cost-effective. Using a simplifying assumption that the failure distribution is exponential, we can use expression 3.13 to determine the test interval. In the case of failure modes with high safety consequence, we can use a pre-emptive overhaul or replacement strategy, or design the problem out altogether.
When β values are less than 1, this indicates premature or early failures. In such cases, the hazard rate falls with age, and exhibits the so-called infant mortality symptom. Assuming that the item has survived so far, the probability of failure will be lower tomorrow than it is today. Unless the item has already failed, it is better to leave it in service, and age-based preventive maintenance will not improve its performance. We must address the underlying quality problems before considering any additional maintenance effort. In most cases, a root cause analysis can help identify the focus area.
Figure 3.16 Probability density functions for varying beta values.
Sometimes it is difficult to assess the reliability of the equipment either because we do not have operating experience, as in the case of new designs, or because data is not available. In such cases, initially we estimate the reliability value based on the performance of similar equipment used elsewhere, vendor data, or engineering judgment. We choose a test interval that we judge as being satisfactory based on this estimate. At this stage, it is advisable to choose a more stringent or conservative interval. If the selected test interval reveals zero or negligible number of failures, we can increase it in small steps. In order to use this method, we have to keep a good record of the results of tests. It is a trial and error method, and is applicable when we do not have access to historical data. This method is called age-exploration.
In order to evaluate quantitative risks, we need to estimate the probability as well as the consequence of failures. Reliability engineering deals with the methods used to evaluate the probability of occurrence.
Figure 3.17 Survival probability for varying beta values.
We began with failure histograms and probability density curves. In this process we developed the calculation methodology with respect to survival probability and hazard rates, using numerical examples. Constant hazard rates are a special case and we examined their implications. Thereafter we derived a simple method to compute the test intervals in the case of constant hazard rates, quantifying the errors introduced by using the approximation.
Reliability analysis can be carried out graphically or using suitable software using data held in the maintenance records. The Weibull distribution appears to fit a wide range of failures and is suitable for many maintenance applications. The Weibull shape factor and scale factors are useful in identifying appropriate maintenance strategies.
We discussed age-exploration, and how we can use it to determine test intervals when we are short of historical performance data.
REFERENCES
1.Nowlan F.S. and H.F.Heap. 1978. Reliability-Centered Maintenance. U.S. Department of Defense. Unclassified, MDA 903-75-C-0349.
2.Knezevic J. 1996. Inherent and Avoidable Ambiguities in Data Analysis. UK: SRD Association, copyright held by AEA Technology, PLC. 31-39 .
3.Allen, Timothy. RCM: The Navy Way for Optimal Submarine Operations. http://www.reliabilityweb.com/artO6/rcm_navy.htm
4.Hoyland A. and M.Rausand. 2004. System Reliability Theory, 2nd ed. John Wiley and Sons, Inc. ISBN:978-0471471332.
5.Edwards, A.W.F. 1992. Likelihood. Johns Hopkins University Press. ISBN 0801844452
6.Weibull W. 1951. A Statistical Distribution of Wide Applicability. Journal of Applied Mechanics, 18: 293-297.
7.Davidson J. 1994. The Reliability of Mechanical Systems. Mechanical Engineering Publications, Ltd. ISBN 0852988818.22-33
8.Resnikoff H.L. 1978. Mathematical Aspects of Reliability Centered Maintenance. Dolby Access Press.