Classification using fuzzy logic

The advantages of the fuzzy logic approach include the possibility to change the linguistic rules into decisions copying the procedure and thinking of a human analyser. The rules that include uncertainty and inaccuracy are changed into numbers describing the severity of the fault or the probability of a fault. The rules and membership functions can be tuned so that the sensitivity of the system is good. On the other hand, when the fuzzy logic requires tuning of parameters, it is possible that it is tuned to find faults from test data, but the logic remains case specific. In this thesis, the purpose of introducing fuzzy logic is to demonstrate its possibilities as part of an automatic fault classification; the purpose is not to tune parameters in order to maximise the reliability of the system. The fuzzy logic has been used in machinery vibration analysis by for example Mechefske (1998).

In order to demonstrate the applicability of fuzzy logic two different models are created. The first model uses the 16 dimensional feature vectors presented in the previous chapter. Features of each fault type are tested separately. Therefore, the inputs include the magnitudes of the envelope spectrum components at the expected ball pass frequency and their three nearest har-monic frequencies separately for all the following faults: outer race fault o, inner race fault i, rolling element fault b and cage fault c. The input sets consist of the sets low, medium, high and very high. The membership functions are triangle shaped. In the fuzzy rules, the minimum method is used for and- rules and the maximum method for the or- rules. The weakness of these

methods is that they do not take into account of the all input memberships. These methods are only used for their simplicity. The output sets are healthy, suspicious, broken and seriously damaged describing the degree of the fault. The centroid of membership function for the set healthy is at 0.1, for the set suspicious is at 0.2, for the set broken is at 0.5. For the set seriously damaged the membership increases linearly from the value 0.4.

The classification results with the test data introduced in the previous chapter are presented in Table 8-3. All of the broken cases are classified to the class broken and the healthy case is clas-sified to the classes healthy or suspicious.

The other model introduces new parameters in order to have more reliable results. It can be supposed that the risk of a wrong decision is minimised when the degree of the fault is evalu-ated as well as the probability of a specific fault. Then the output of logic describes better the risk of the continuing the use of a motor without service. This risk can be described with the multiplication of the degree of the fault and the qualitative probability of a certain fault. The probability is calculated as an error between the estimated and measured quantities. Also, the fuzzy logic can handle the outputs of quantitative and qualitative logics. Then the following rules are used:

1. If the magnitudes of the outputs (for different faults) of previous logic are high the degree of the fault is high.

2. If the RMS amplitude of the vibration is high and the magnitudes of the outputs (for different faults) of previous logic are low the condition of the machine is ab-normal.

3. If there are high peaks in the envelope spectrum at non- predicted frequencies the condition is classified as other abnormal situation.

4. If the distance to a certain fault is low using the statistical classification of the pre-vious chapter or if the spectrum peaks are found near to predicted frequencies the probability of the fault is great.

5. If there are peaks near to the harmonic frequencies of the supposed bearing pass peak frequency or if there are rotational frequency side bands found at predicted frequencies the probability of the fault is great.

The ouputs of previous rules can be used in new rules such as

1. If the probability of a certain fault is great and the degree of the fault is high the output is act fast.

2. If the probability of a certain fault is great and the degree of the fault is not small the output is act.

3. If the probability of a certain fault is low and the degree of the fault is high the out-put is act.

4. If the probability of a certain fault is low and the degree of the fault is small the output is wait.

5. If the non-normal situation holds then the output is warning.

In Figure 8-12, the envelope spectrum of a motor with a broken bearing is presented. The input features concerning the spectrum values are marked in the figure. The quantitative variables include the existence of rotational sidebands, the frequency difference between calculated fre-quencies and peaks in the spectrum, the ratios between the peaks, the peaks at the unpredicted frequencies and the statistical distance between prototypes and test data. The results of the logic are illustrated in Tables 8-3, 8-4 and 8-5. The results clearly demonstrate the advantages of introducing both quantitative and qualitative features in the calculation as well as combining the statistical classification and fuzzy logic.

0 100 200 300 400 500 600

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

f [Hz]

expected bearing pass frequencies * rotational side band frequencies o other maximum value x

Fig. 8-12. The envelope spectrum of a motor with a broken bearing with the selected features marked on the spectrum peaks.

Table 8-3. Classification results using simple fuzzy logic for the determination of fault degree.

test data outer race inner race ball spin cage

healthy 0.1774 0.1575 0.1736 0.1832

outer race 0.4723 0.166 0.1153 0.1032 inner race 0.1774 0.4711 0.1736 0.1832 ball spin 0.1774 0.1575 0.5011 0.1832

cage 0.1774 0.1575 0.1736 0.495

Table 8-4. Fuzzy logic for abnormality in the measurement data.

test data abnormality action

healthy 0.19 no

outer race 0.19 no

inner race 0.2 no

ball spin 0.19 no

cage 0.18 no

Table 8-5. Classification results using fuzzy logic that estimates the fault degree as well as the probability of the faults. Action can be formed with fuzzy logic or multiplication of the fields of Table 8-3 with the probabilities of this table. The actions are selected with trigger levels.

test data outer race inner race ball spin cage outer race inner race ball spin cage

healthy 0.18 0.17 0.17 0.17 0.03 0.03 0.03 0.03

outer race 0.55 0.175 0.625 0.17 0.26 0.03 0.07 0.02

inner race 0.32 0.3 0.16 0.16 0.06 0.14 0.03 0.03

ball spin 0.32 0.16 0.46 0.16 0.06 0.03 0.23 0.03

cage 0.32 0.16 0.16 0.56 0.06 0.03 0.03 0.28

Fault probability Action

The previous logic did not consider the rotational speed estimation error (directly). If the rota-tional speed is not measured it should be taken into account in the probability calculations. On the other hand the rotational speed can be estimated before the fault classification. There are several methods (Ishida, 1984), (Vas, 1993), (Kim, 1994), (Kubota, 1994), (Blaschke, 1996), (Rajashekara, 1996) but most methods require a current and/or voltage measurement. On the other hand, the rotational side bands of the bearing pass frequency can be used in the iterative estimation using the envelope spectrum of the vibration signal. Then, if rotational side bands are found the result can be used in the bearing pass frequency calculations and probability variables can be increased and vice versa. The flow diagram of a rotational speed estimator is presented in Figure 8-13.

CALCULATE CHARACTERISTIC FREQUENCY fc

NO

NO

YES

SEARCH FOR LARGEST PEAK AND PEAKS NEAR TO CALCULATED CHARACTERISTIC FREQUENCY IN SPECTRUM

INITIAL GUESS OF RPM

RPM, f or 2xf SIDEBANDS FOUND

RECALCULATION OF RPM NEEDED

RECALCULATE RPM

NO YES

PEAKS AT MULTIPLE FREQUENCIES OF fc ?

Fig. 8-13. The flow diagram of a rotational speed estimator using the envelope spectrum of an accelera-tion signal.