Twelve features have been extracted from four different methods [186]. Not all features are sensitive to the change of the bearing condition. Consequently, the
method only uses five of the most sensitive features as the input to the MSET. The five features are (1) kurtosis extracted from accelerometer signal, f1(t) [96]; (2) kurtosis
extracted from the piecewise aggregate approximation (PAA) combined with circular analysis, f2(t) [96]; (3) kurtosis extracted from the wavelet decomposition, f3(t) [96]; (4)
kurtosis extracted from the EMD, f4(t) [96]; and (5) LLE feature obtained from the
largest Lyapunov exponent algorithm, f5(t) [186]. The five features were extracted over
the 139 days test period. The plot of the aforementioned features is shown in Figure 5.2. These features are used to establish the data matrix P. Thus, the dimension of data matrix P in Eq. (5.1) is 5 by 139, where 5 is the number of features and 139 is the number of days.
To detect the incipient slew bearing defect (Step 1), the MSET algorithm calculates the normal residual matrix (RN) and the monitored residual matrix (RM) (see Figure 5.1). In Step (1), the following inputs are required: training matrix T, memory matrix
D, remaining training matrix L and observation matrix Pobs. In this paper, T of 50 days was used. In the computation, the sizes of matrices, D and L have to be half of
T i.e. 25 days each. The reason why 50 days was used is that coal dust was not injected until the 58th day. In other words, during the first 50 days, the vibration signal
had been acquired when the condition of the bearing was still normal. In practice, matrix Pobs is obtained from daily observation or measurement. The number of measurement days from normal to failure has been set as 139 days. Pobs is the matrix from the 51st to 139th day.
0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40 t, days f 1 ( t) , C ir cu la r ku rt o si s 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40 t, days f2 ( t) , T im e d o m a in ku rt o si s 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40 t, days f 3 ( t) , W D ku rt o si s 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40 t, days f 4 ( t) , E M D ku rt o si s 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 35 40 t, days f5 ( t) , L L E f e a tu re
Figure 5.2 (a) Circular kurtosis; (b) Time-domain kurtosis; (c) Wavelet kurtosis; (d) EMD
kurtosis; (e) The LLE feature. Note: t = dayth.
a
b
c
d
After D and L have been determined, the weight of normal state vector (w1) is then calculated using Eq. (5.3) by employing the Euclidean distance as the non-linear operator. Vector w1 represents the weight of normal data points in the training data.
1
w is used to calculate the normal estimate matrix (Lest) using Eq. (5.5). And the normal residual matrix (RN) is estimated by taking the difference between Lest and
L, as shown in Eq. (5.7). It should be noted that prior to the calculation of the statistical properties, matrix RN is normalized with a mean of zero. This is necessary for the next process in the SPRT method because under normal bearing conditions the mean value of the normal residual matrix (RN) is expected to be 0. The result of normalized matrix (RN) is presented in Table 5.1. A similar procedure is also employed to obtain the monitored residual matrix (RM). Eq. (5.4) is used to calculate the weight of monitored state vector (w2), Eq. (5.6) is then used to calculate the monitored estimate matrix (Pest) and Eq. (5.8) to calculate the monitored residual matrix (RM).
Matrix RM is then normalized within a range between maximum value and minimum value of the normalized normal residual matrix (RN). The calculation of RN and RM
is the final step of MSET. Both results are then fed into the SPRT.
In SPRT, statistical characteristics such as mean value, standard deviation and the range between maximum and minimum values are calculated from each column of the normalized matrix, RN. These statistical properties belong to the normal condition of the bearing and are used as the reference normal data. These properties also form the set of hypotheses in SPRT. For the monitored state, prior to the calculation of the mean value and the standard deviation, the monitored residual matrix, RM is normalized within the range between the maximum and minimum value of the normal state. The mean and standard deviation values of the normal state (up to days 25) are presented in Table 5.2 and Table 5.3, respectively. The detection of the incipient slew bearing defects is obtained by comparing the statistical characteristics, the mean value and the standard deviation) of the monitored state to the statistical characteristics of the normal state.
This comparison is repeated from the 51st to 139th columns of the
obs
reference mean values (Table 5.2) or standard deviation values (Table 5.3), then the incipient defect is considered to have occurred. This step is part of the SPRT procedure. To make a comparison, the list of hypotheses must be determined first including the normal condition (H0) represented by mean valueMnormal and the standard deviationnormal. Alternative hypotheses, H1 to H4, indicate the abnormal conditions. The set of hypotheses is given in Table 5.4.
Table 5.1 Normalized RN matrix.
day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10 day 11 day 12 day 13 0.044 0.042 0.042 0.042 0.042 0.043 0.042 0.043 0.044 0.042 0.042 0.042 0.043 0.025 0.021 0.021 0.022 0.022 0.023 0.021 0.024 0.025 0.022 0.022 0.021 0.022 0.016 0.015 0.015 0.015 0.015 0.016 0.015 0.016 0.016 0.015 0.015 0.015 0.015 0.018 0.017 0.017 0.017 0.017 0.018 0.017 0.018 0.018 0.017 0.017 0.017 0.018 -0.955 -0.957 -0.957 -0.957 -0.957 -0.956 -0.957 -0.956 -0.955 -0.957 -0.957 -0.957 -0.957 day 14 day 15 day 16 day 17 day 18 day 19 day 20 day 21 day 22 day 23 day 24 day 25
0.043 0.042 0.043 0.043 0.042 0.043 0.044 0.042 0.042 0.042 0.042 0.042 0.022 0.021 0.023 0.023 0.022 0.023 0.024 0.021 0.021 0.022 0.022 0.021 0.015 0.015 0.016 0.016 0.015 0.016 0.016 0.016 0.015 0.015 0.015 0.015 0.018 0.017 0.018 0.018 0.017 0.018 0.018 0.017 0.017 0.018 0.018 0.017 -0.957 -0.957 -0.956 -0.956 -0.957 -0.956 -0.955 -0.957 -0.957 -0.957 -0.957 -0.957
Table 5.2 Mean (M) values of normal state.
day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10 day 11 day 12 day 13 -0.169 -0.172 -0.171 -0.171 -0.171 -0.171 -0.172 -0.170 -0.170 -0.171 -0.171 -0.172 -0.171 day 14 day 15 day 16 day 17 day 18 day 19 day 20 day 21 day 22 day 23 day 24 day 25
-0.171 -0.172 -0.170 -0.171 -0.171 -0.171 -0.170 -0.172 -0.172 -0.171 -0.171 -0.172 Note: Mmin = -0.172, Mmax = -0.169
Table 5.3 Standard deviation () values of normal state.
day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10 day 11 day 12 day 13 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 day 14 day 15 day 16 day 17 day 18 day 19 day 20 day 21 day 22 day 23 day 24 day 25
0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 0.439 Note: normal0.439
Table 5.4 Hypotheses for SPRT.
Hypothesis Statistical properties
Mean (M) Standard deviation (σ) 1. Normal condition (H0) Mmin M Mmax normal
2. Abnormal condition 1 (H1) MMmax normal
3. Abnormal condition 2 (H2) MMmin normal
4. Abnormal condition 3 (H3) Mmin M Mmax normal
5. Abnormal condition 4 (H4) Mmin M Mmax normal
To mathematically measure the comparison, the ratio of the probability of alternative hypotheses (H1 to H4) and the probability of the null hypothesis (H0) are calculated as in Eq. (5.10). Once probability ratio Li is obtained, the SPRT index can be calculated by taking the natural logarithm of probability ratio Li. The result of Step (1) is shown in Figure 5.3. It can be seen from the monitored data from the 51st to 89th day
that the bearing is still in normal condition. The impending deterioration of the bearing is identified on the 90th day. Note that the impending deterioration is required in Step
(2) to predict the future state and to estimate the RUL estimation. A sample calculation of Li for one day (the 51st and 90th day) is presented in Table 5.5. When the condition
of the bearing is still normal, the probability of statistical properties si that fall within the alternative hypotheses Hz, Pr(si|Hz)is much lower than the probability of statistical properties si of the null hypothesis H0, Pr(si |H0). To illustrate the normal condition (the 50th day), the natural logarithmic of
i
L is given in Table 5.5 and shown in
Figure 5.3. On the contrary, on the 90th day, the condition of the bearing started to
deteriorate. When the bearing start to deteriorate, the probability of statistical properties si that fall within the hypotheses, Hz, Pr(si|Hz)will increase and the
probability of statistical properties si that fall in the null hypothesis H0, Pr(si|H0)will
0 20 40 60 80 100 120 140 -3.3 -3.2 -3.1 -3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 X: 90 Y: -2.506 t, days S P R T in d e x
Figure 5.3 SPRT index result for detecting impending bearing deterioration.
Table 5.5 Example calculation of Li in one day observation. dayth observation Pr( | ) z i H s Pr(si|H0) Li (Eq. 5.10) day 51 0.04 0.99 -3.209 day 90 0.08 0.98 -2.506
Note: Pr(si|Hz) is probability of statistical properties si given Hz is true and Pr(si|H0) is probability of statistical properties si given H0 is true.