Introduction to the second series of experiments

we decide to try also a different approach regarding which is the best range of frequency to be used at the aim of the classification.

We decide to enlarge the range of frequency to be analyzed, i.e., we decided to work in the frequency range [1,300] Hz instead of the previously used frequency range [1,250] Hz. Thus, we have a total of 300 features describing each sample for each accelerometer.

The Fourier spectrum related to the faultless class considering the third accelerometer and the first 300 features is represented in Fig. 5.5. Figure 5.6 shows the Fourier spectrum related to each type of de- fects and severity, considering the third accelerometer and the first 300 features.

Advanced vibration analysis for CBM programs 79 Table 5.16 Classification of C2 and C3.1. Classifier fusion. Confusion matrix for the test set

Estimated Labels Total C2 C3.1 True C2 489 42 531 labels C3.1 18 513 531 Total 507 555 1062

Figure 5.5 Fourier spectrum for the signal belonging to the fault- less class C1 in the frequency range [1,300] Hz for the third ac- celerometer.

As the classifiers are concerned, whenever possible, simple classifiers, such as LDC and QDC, are used to perform both feature selection and classification. On the other hand, to solve particularly difficult classifi- cation problems we adopt Multi-Layer Perceptron neural networks and multi-classifier systems. In particular, three different types of meth- ods of classifier fusion are analyzed: maximum, minimum and average. The proposed method shows a good sensitivity since it provides accu- racy higher than 99% in all the experiments related to the recognition of different defects and different severity levels of the defects.

As there are four accelerometers, up to 300 × 4 = 1200 features (obtained by concatenating the four groups of 300 features related to

80 Classification and diagnosis of rolling element bearings

(a)Signals belonging to class C2 (inden- tation on the inner raceway)

(b)Signals belonging to class C3.1 (light indentation on the roll)

(c)Signals belonging to class C3.2 (medium indentation on the roll)

(d)Signals belonging to class C3.3 (high indentation on the roll)

(e)Signals belonging to class C4 (sand- blasting of the inner raceway)

(f)Signals belonging to class C5 (unbal- anced cage)

Figure 5.6 Fourier spectra for the signals belonging to each type of defects and severity levels in the frequency range [1,300] Hz for the third accelerometer.

Advanced vibration analysis for CBM programs 81 the four accelerometers) can be used to represent each signal (Fig. 5.7). In other words, each signal can be represented in Rn with n ≤ 1200.

To reduce the computational time, however, we decided to consider each accelerometer singularly.

Figure 5.7 Organization of the features considering the four ac- celerometers.

5.2.2 Classification of C1 and C6

In this experiment we try to classify the signals into two classes: fault- less bearings C1 and damaged bearings C6 (C2, C3, C4, C5). As previ- ously stated, the set of all damaged bearings will be referred to as class C6. We consider the four accelerometers separately from each other. For each accelerometer, we look for the best DFs. This step is per- formed again using the FFS algorithm. Also this time we adopt LDC and QDC (with the regularization parameter set to 0) to perform both feature selection and classification of the signals represented through the selected features.

We use one LDC and one QDC for each accelerometer.

Fixing a maximum number of features equal to 10 so as to keep the computational complexity at a low level, we repeated the two-class experiment 10 times for each accelerometer.

Then, for each accelerometer and each classifier, we selected the best m (m ≤ 10) features that are exactly the same (and in the same order) for all the trials. We will refer to these feature as stable features (SFs). Then using the SFs we computed the accuracy over 10 more trials. We then compared the four accelerometers based on the classification accu- racy, expressed in the form (mean ± standard deviation) (Table 5.17). Please note that N.A. stands for “Not Applicable”, which means that, in the specific case, no SFs have been selected.

82 Classification and diagnosis of rolling element bearings Table 5.17 Classification of C1 and C6

Acc.

LDC QDC

Number Accuracy Number Accuracy of SFs (Mean±Std.Dev.) of SFs (Mean±Std.Dev.) 1 1 (96.04±0.97)% 1 (98.16±0.42)%

2 1 (85.43±0.68)% 0 N.A.

3 1 (94.77±0.43)% 2 (99.69±0.14)%

4 1 (92.21±0.68)% 2 (96.63±0.25)%

the third accelerometer’s features achieves the best accuracy with a very small standard deviation. Actually, with just two features we obtain a good trade-off between accuracy and complexity. Table 5.18 shows the two stable features for the third accelerometer, listed in the order in which the FFS algorithm found them.

Table 5.18 Classification of C1 and C6. List of the SFs QDC, Accelerometer 3

277, 296

With the chosen SFs we obtain a very high accuracy (99.69%) with a very low standard deviation (0.14%). This means that possible ad- ditional features, changing during the experiments, bring a negligible improvement. Thus, considering only two features we succeed in reduc- ing the complexity to an optimal level from R1200 to R2 . Fig. 5.8 shows the separation of the two classes performed by QDC using the stable features 277 and 296 of the third accelerometer.

In all our experiments we decided, based on heuristic reasoning, to consider “good” an accuracy of at least 99.00% and a standard devia- tion close to 0.10%. In particular, such a small range of acceptance for the standard deviation is justified by the fact that we want not only a good classifier (high accuracy) but also a stable classifier (low standard deviation). Taking these two thresholds into account, we can affirm that, for this classification problem, the choice of the two stable fea- tures is an optimal choice and thus we will not investigate further this

Advanced vibration analysis for CBM programs 83

Figure 5.8 Separation of C1 (gray dots) and C6 (black crosses) for the test set performed by QDC using the SFs of the third ac- celerometer.

classification problem.