63 Equation 3-39 shows how the residual time of the system could be calculated from our
model. In Chapter 4, we fit the model we had developed with the observed data. We then show all the results and attempt to compare those results with a previous study
conducted by Zhang 2004.
3.10 Summary
This chapter proposed a model to calculate the residual time of the monitored items by defining a N state space of the system. The concept of a Hidden Markov Model is
applied to a two-stage failure process. We explained this process and formulated the transition probabilities between stages and their relationship between the monitoring
data and the state of the system. Using the filtering technique, the probability of the state given its monitoring history is calculated. Two parameter estimation techniques
are used, namely the maximum likelihood estimator and EM algorithms. Both methods produced the desired results given that the failure information was used.
64
4 CHAPTER 4: EARLY FAULT IDENTIFICATION
– A CASE STUDY USING VIBRATION DATA
4.1 Introduction
In this chapter, the model introduced in Chapter 3 is fitted to a set of vibration data obtained from a laboratory experiment Wang, 2002; see Figure 3-4 in Chapter 3. Also,
an attempt has been made to compare the results from this model with those of a past study using a Statistical Process Control SPC chart conducted by Zhang 2004.
4.2 Numerical Results
Here, we seek to identify the initial point of a random defect or an abnormal stage, as explained in Chapter 3, from observed vibration data. To start with, we need to estimate
the parameters used in our model from the dataset. To do that, we randomly selected three sets of observed vibration data Gu-b2, Gu-b5 and Gu-b1 for parameter
estimations and reserved the other three Gu-b3, Gu-b4 and Gu-b6 for model testing. Using this approach, we could verify our model later. Given all the formulations for
estimating model parameters see Chapter 3, and based on observed vibration data, we
carried out the calculations and produced the results depicted in Table 4-1 below.
ˆ
ˆ bˆ
1
ˆ
2
ˆ 0.4314
3.635 0.0983
0.0089 0.0093
Table 4-1: Estimated parameters from vibration data
The estimated parameters are only point estimates; hence, to show the amount of uncertainty of these estimates, a variance-covariance matrix was produced based on the
log-likelihood, equation 3-28. The variance-covariance matrix of the estimated
parameters is shown in Table 4-2 below.
ˆ
ˆ
bˆ
1
ˆ
2
ˆ
ˆ 4.593E-5
-3.032E-4 -4.926E-6 -2.149E-6 1.179E-8
ˆ
-3.032E-4 8.481E-4 4.942E-5
1.492E-5 -2.316E-7
bˆ -4.926E-6 4.942E-5
2.813E-6 -1.058E-6 1.11E-8
1
ˆ
-2.149E-6 1.492E-5
-1.058E-6 1.542E-6 -1.787E-8
65
2
ˆ
1.179E-8
-2.316E-7 1.11E-8 -1.787E-8 2.013E-5
Table 4-2: Variance and covariance matrix
The variance of each parameter is relatively small, which indicates that our estimation is relatively good. The covariance also shows that there is little correlation between
parameters. These results allow the model to be used with confidence.
Using these estimated parameters, the |
i i
x P
model was calculated. From the model,
the starting point of defect stage 2 is easily identified when
| 2
i i
X P
has a probability 1. Two of the results are shown in
Table 4-3
and
Table 4-4
below.
time vib. in rms
| 1
i i
X P
| 2
i i
X P
9 1.6012
0.9535 4.65E-02
25.5 1.7306 0.9448
5.52E-02 35
1.8917 0.9680
3.19E-02 50
2.4231 0.9587
4.13E-02 57
2.5086 0.9749
2.50E-02 74
3.4853 0.6641
0.335847
85 5.3865
0.0000 1
102 13.0421
0.0000 1
105 23.7722
0.0000 1
Table 4-3: Case 1:
|
i i
x P
and the starting point of the abnormal stage for Gu-b3
66 time
vib. in rms
| 1
i i
X P
| 2
i i
X P
6.5 2.5478
0.9422 0.0577
17.5 2.7402
0.8472 0.1527
30 2.857
0.7633 0.2366
40 3.0721
0.6356 0.3643
50 3.2599
0.4231 0.5768
61 3.342
0.2420 0.7579
74.5 3.7751
0.0257 0.9742
86 3.9638
0.0009 0.9990
94 4.0591
0.0000 0.9999
106 4.3209
0.0000 1
. .
. .
. .
. .
. .
. .
. .
. .
269.5 13.9153
0.0000 1
280.5. 19.6412 0.0000
1
Table 4-4: Case 2:
|
i i
x P
and the starting point of the abnormal stage for Gu-b6
The figures show different paths of bearing failures but share the same properties in
which they stay flat at an early stage and then increase rapidly before a failure. This justifies the two-stage failure process defined by Christer et al., 1984. Determining the
changing point is very important for maintenance personnel to enable suitable maintenance decisions to be made. For case 1, the defective state is picked up with
probability 1 at time = 85 with vibrations reading 5.3865 in rms; in case 2 the defective state is detected at time = 106 with vibrations reading 4.3209 in rms. According to the
study using SPC with the same dataset, Zhang 2004 recommended that a vibration level around 5 in rms seems to be a reasonable point at which to say that the bearings
67 are in a defective state. This suggests that our model performed well, reaching nearly
the same conclusion. The full comparison of results using the SPC chart method and the model developed is shown in Section 4.4.
4.3 Testing the Model