Classification using the Mahalanobis distance calculation and minimum

A bearing fault feature vector consists of PSD components of a signal envelope at selected frequencies. The selected components represent the characteristic fault frequencies and their three nearest harmonic frequencies for all the following faults: outer race fault o, inner race

fault i, rolling element fault b and cage fault c. The frequencies are calculated according equa-tions 3.1-3.4. These PSD components X form a 16 dimensional feature vector as follows

[

(3 ) (4 ) ( ) (2 ) (3 ) (4 )

]

Also the procedure that analyses all the damages separately using only a four-dimensional fea-ture space is tested. The values of prototype vectors are selected to represent 40 different levels of failure. The impulse response of the motor frame to the bearing impulse is a series of resonat-ing waves as described in Chapter 3.1. The envelope of the vibration is not sinusoidal.

Therefore, the harmonic waves of the bearing pass frequency are found in spectrum but are attenuated compared to the bearing pass frequency. The values of prototype vectors are selected bearing this in mind. The prototype vectors are done without testing their discrimination power using a big value for the characteristic frequency and then a smaller value to the harmonics in descending order. The features not influenced by a certain fault are set to one, the bearing pass frequency component to five, the first harmonic to four, the second to three and the third to two.

Then the values are altered to represent 40 different levels of damage and 20 % random varia-tion is added to the values.

Formation of a test vector. From the envelope spectrum of real measurement the twelve compo-nents in the neighbourhood of the calculated characteristic frequency (frequency resolution of PSD ∆f ≈ 0.2 Hz) were selected for pre-processing, in which, the maximum value of these five was selected as a feature. Envelope spectra for the outer race fault were formed using actual measurements. In the case of other faults the spectra of the healthy case were altered artificially near characteristic frequencies.

Classification. In the first test the distance between both the test vector and the vectors repre-senting the different fault types of the healthy case were calculated according to Equation 4.11 using the jacknified Mahalanobis distance calculation. Then a minimum value was selected and the test vector was classified to belong to a corresponding class. In the second test the distances were calculated with reduced feature spaces so that only the features representing a certain fault class were selected to a feature vector. The distances between the four test vectors and the cor-responding fault prototypes as well as healthy class prototypes were calculated separately. If in one or more calculations the distance to a faulty class was less than to the healthy class the corresponding fault was classified. This procedure makes it possible to recognise multiple si-multaneous fault cases. On the other hand, the possibility of an incorrect estimate is greater than in previous test because the values not changing due to a fault are not taken into account (weaker covariance information).

Results. The envelope spectrum of a healthy motor is illustrated in the Figure 8-9 (upper dia-gram). The spectrum components that are selected for the test feature vector are marked with *.

This vector is illustrated in the lower figure with the (mean) prototype vectors of fault classes.

The corresponding curves in the case of the outer race fault and inner race fault are presented in Figures 8-10 and 8-11. The classification results are presented in Tables 8-1 and 8-2. Table 8-1 presents the classification results using a 16 dimensional feature space, covering all four fault types and Table 8-2 presents the classification results using four dimensional feature space.

Both indicate only correct classification results. The statistical distance between healthy and broken cases is bigger when four- dimensional feature space is used. On the other hand, there is more chance of misclassification if the shape of the test feature vector change. It is important to

bear in mind, firstly, that the correct classification was obtained without any tuning of the proto-type vectors, and secondly, that there can be many other fault modes that were not taken into account and much more research work should be done with various fault types and motors be-fore jumping to conclusions that generalise the result obtained with these tests.

0 100 200 300 400 500 600

0 1 2 3 4 5 6x 10^-3

f [Hz]

0 2 4 6 8 10 12 14 16

0 1 2 3 4 5x 10^-3

Outer race

Inner race

rolling

element cage

test vector

features

Fig. 8-9. The envelope spectrum of a healthy motor is illustrated in the upper figure. The spectrum com-ponents that are selected for the test feature vector are marked with *. This vector is illustrated in the lower figure with the prototype vectors of the outer race fault (∧ ), the inner race fault (∨ ), the ball spin fault (o ) and the cage fault ( ). The vector of healthy case is a unity vector.

0 100 200 300 400 500 600

Fig. 8-10. The envelope spectrum of a motor with an outer race faulted bearing is illustrated in the upper figure. The spectrum components that are selected for the test feature vector are marked with *.

This vector is illustrated in the lower figure with the prototype vectors of the outer race fault (∧), the inner race fault (∨ ), the ball spin fault (o ) and the cage fault ( ). The vector of healthy case

Fig. 8-11. The envelope spectrum of a motor with an inner race faulted bearing is illustrated in the upper figure and features are marked similarly as in the previous figure.

Table 8-1. Classification results using 16 dimensional feature space covering all four fault types. The bold cases are selected with the minimum distance classifier.

healthy outer race inner race ball spin cage

healthy 0.6981 1.3607 2.7905 1.6052 2.3892

outer race 5.3817 1.2018 7.0394 8.7226 7.5465

inner race 3.7498 5.5534 1.666 4.43 5.7718

ball spin 6.5451 5.3318 8.2085 1.0579 6.7352

cage 0.6952 0.883 1.1753 1.3041 0.2143

Table 8-2. Classification results using four dimensional feature space. The distance between test vector and any fault prototypes are calculated separately. The bold cases are selected with the minimum distance classifier.

test data distance to outer race inner race ball spin cage

healthy healthy 0.1767 0.148 0.1107 0.138

broken 0.781 4.235 1.2339 1.3203

outer race healthy 3.9542 0.2946 0.2454 0.2164

broken 0.0394 6.5133 1.4492 1.4779

inner race healthy 0.2531 2.0595 0.137 0.0961

broken 0.9665 0.7749 1.4533 1.5183

ball spin healthy 0.2327 0.3109 4.4202 0.1176

broken 0.9328 4.1784 0.2219 1.7265

cage healthy 0.2827 0.3915 0.2275 4.0238

broken 1.0563 5.0148 1.7709 0.073

In document On the Condition Monitoring of Induction machines (sivua 118-122)