73
i i
t i
t
dl l
f dl
dl l
f t
l l
l f
1 1
1 1
2 2
2 1
2 1
1
i i
i i
t l
t l
t i
t
dl dl
l f
l f
dl dl
l f
t l
l l
f
1 2
2 2
1 1
1 2
2 2
1 2
1 1
1 1
, |
,
1 2
1 2
i i
i i
t L
L x
t L
L E
150.80 76.34
147.34
134.69 73.04
131.29 126.03
71.13 124.28
113.96 68.05
112.07 109.13
66.54 108.07
99.62 62.46
87.14
94.99 59.28
59.28
89.78 53.05
53.05 89.05
51.76 51.75
Table 4-10:
, |
,
1 2
1 2
i i
i i
t L
L x
t L
L E
with Weibull distribution for Gu-b3
To summarize, we have shown how the expected time to first reach stage 2 or fault and failure can be calculated. Using exponential distributions for the two-stage failure
process would yield a constant rate of the condition expected, which is inaccurate. Hence, to show an example to work this out, we choose a Weibull distribution with
given parameters suitable for the two-stage failure process. For the case of the initialisation of random defects,
Table 4-8
shows at first checking time = 9, the expected time to first reach stage 2 is 69.29; and Table 4-10 shows the expected time to first
reach failure stage is 147.34. This process is continued until time = 85, where the expected time to first reach stage 2 is 0 the second stage is reached, and the expected
time to first reach failure stage is 59.28. Using this analysis it could aid maintenance managers to plan and schedule their maintenance actions.
4.4 Comparison with a Statistical Process Control SPC Chart
Approach
The Statistical Process Control SPC has been formally defined as a methodology for monitoring a process to identify special causes of variations that signal the need to take
corrective actions when it is appropriate Evans and Lindsay, 2005. In the case of condition-based maintenance, the SPC model was developed to monitor the process of
observing variables before any maintenance action can be taken. This section aims to summarize a study by Zhang 2004, which applied the SPC chart approach Wetherill
and Brown, 1991 to the same dataset we used here. In her work, Zhang proposed three
74 SPC chart models, which are useful for identifying the initial point of a random defect.
The models are:
3. Shewhart average level
4. Adaptive moving average and moving range charts
5. Adaptive Shewhart average level
Without any discussion of those methods cited above, a comparison will be made here of the Zhang‘s results with those obtained from our state prediction model, as the same
data is used to achieve the same objectives. The results are shown in Table 4-11 below. Set of vibration data
Techniques used
Gu-b1 Gu-b2
Gu-b3 Gu-b4
Gu-b5 Gu-b6
SPC – Shewhart
average level 6.6828
5.8982 5.3865
5.3062 5.1302
8.3994 SPC-
Adaptive moving average
6.6828 5.4781
5.3865 3.7751
5.1302 2.6598
SPC – Adaptive
moving range 6.6828
8.2242 5.3865
6.3926 5.1302
3.8376 SPC
– Adaptive Shewhart
average level 4.0744
3.8286 5.3865
3.9638 5.1302
8.3994
State prediction model
6.6828 4.6055
5.3865 4.0591
6.4194 8.3994
Table 4-11: Starting point of the abnormal stage using SPC techniques and state prediction model
The results in Table 4-11 shows the starting point abnormal of vibration readings,
which indicate that, the system may be in an abnormal state. It is clearly shown that the state prediction model produce almost the same result as the SPC models discussed by
Zhang 2004. Briefly, using the Shewhart average level to detect a fault, we need to have a threshold level indicating a warning limit, which can be established from
experience by an expert, or from a manufacturer. The adaptive moving average and adaptive moving range models also need a threshold value and averaging the process
will increase the delays in responding to sudden jumps. The adaptive Shewhart average level model performs well without a threshold value but is not suitable with one-at-a-
time data unless several such data can be grouped together using a predetermined interval size. Again, by averaging the process, the delays in responding to a sudden
jump will depend on the average difference of interval. See Zhang 2004 for details of
75 the techniques used. In contrast to the SPC models, the model that we developed, which
we called a state prediction model, does not set any threshold values and does not use the averaging concept. In fact, it uses the weight of each state to predict the future state,
which is more appropriate.
From this comparison of the techniques used, the advantages of state prediction can be summarized as:
1. Useful as a complete decision model for CRT, to determine the initial point of default and predict the residual time of the item monitored. A previous study
had used SPC as a technique to identify random faults before it was possible to use the CRT model.
2. Can be extended to predict any state defined in a system, while SPC is limited to two states.
3. Given the structure of the state prediction model, maintenance actions could be carried out directly according to the state.
4. SPC cannot provide the probability prediction of state one, which is needed for other modelling, as explained in Section 4.3.
To summarize, we could say that the state prediction model enriches the range of existing techniques for fault identification.
4.5 Summary
