Automatic bearing fault classification combining statistical classification and fuzzy logic T. Lindh, J. Ahola, P. Spatenka, A-L Rautiainen
[email protected] Lappeenranta University of Technology Lappeenranta, FINLAND
Abstract - In this paper, a new automatic analysis method for the detection of cyclic bearing faults is introduced. The method uses a multivariate statistical fault classification and fuzzy logic. Features are extracted from an envelope spectrum of the frame acceleration of a motor frame. Qualitative and quantitative measures of the features are utilized. Keywords: bearing fault classification
I. INTRODUCTION The research of condition monitoring of electric motors has been wide for several decades. The research and development at universities and in industry has provided means for the predictive condition monitoring. Many different devices and systems are being developed and are widely used in industry, transportation and in civil engineering. Recently, in order to achieve the automatic analysis of faults, many methods have been developed and reported in scientific arenas. In this paper, the guidelines for the statistical fault classification of the envelope spectrum components using the Mahalanobis distance calculation and deterministic classification with multi- valued fuzzy logic are presented. The main idea is the following: As humans, the algorithm searches patterns in the spectrum, analyses the magnitude of peaks found and estimates the probability of a false triggering with qualitative factors of extracted features of the spectrum. The Mahalanobis metrics is superior in the cases where the features are dependent on each other as is the case with the spectrum components of an acceleration signal. Based on the distance calculations and a minimum distance classifier, possibly with a trigger level for the distance from the healthy case, a fully automatic system can be constructed. The main problems are, firstly, how the prototypes of faulty spectra are created and, secondly, how the trigger levels from a healthy case representing the abnormal situation are selected. In addition, the rotational speed of the motor must be taken into account in the system by assuming known operation conditions or by rotational speed adaptation of the spectrum. Being so that only the healthy case spectrum is available, a-priori-knowledge of the faulty case patterns must be included in the system artificially. Another possibility is to use broken case spectra obtained from another motor with a broken bearing. The advantages of the fuzzy logic approach include the possibility to change the linguistic rules into decisions by copying the procedure and thinking of a human analyser. The rules that include uncertainty and inaccuracy are changed into numbers describing the severity 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 paper, the purpose of introducing fuzzy logic is to demonstrate its possibilities as a 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 [1] and Lindh [2].
II. FEATURE EXTRACTION AND CLASSIFICATION The features are created from the coefficients of the envelope spectra calculated from the motor frame acceleration signal. The expected bearing pass frequencies (the characteristic fault frequencies) [3] for different fault types are calculated and the peaks near to these calculated values are selected as features. The envelope spectra with selected components are presented in Figures 4, 5 and 6. A bearing fault feature vector consists of 16 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. These PSD components X form a 16dimensional feature vector as follows x = [ X ( f o ) X ( 2 f o ) X (3 f o ) X ( 4 f o ) X ( f i ) X ( 2 f i ) X ( 3 f i ) X ( 4 f i ) X ( f b ) X ( 2 f b ) X ( 3 f b ) X ( 4 f b ) X ( f c ) X ( 2 f c ) X ( 3 f c ) X (4 f c )]
(1) The common measure for the statistical distance between samples is the Mahalanobis distance r 2 = (x − m x ) ' C x−1 (x − m x ) ,
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where r is the Mahalanobis distance between feature vector x, Cx is a covarariance matrix containing variances between features and mx is a mean vector of features [4]. The proposed system consists of the following training phase actions • The creation of artificial spectra for the cases of an outer race defect, an inner race defect, a cage defect and a rolling element defect. • The creation of an over all acceleration level feature for low, normal, high and very high levels of vibration. • Calculation of a normal case prototype vector using several measurements from a healthy case. And of test phase actions using statistical distance calculation • The minimum distance classification using the calculation of distance between a test vector and healthy and all broken case prototype vectors. • Classification of abnormal versus normal using trigger level distances from a healthy case vector. And test phase actions using fuzzy logic including • Calculation of memberships in the output sets that describe the degree of fault for every fault type under consideration. • Calculation of memberships in the output sets that describe the correct classification probability of the fault for every fault type under consideration.
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As a combination of the previous memberships, the selection between a healthy and faulty case with fault type information and instructions for the action. The values of prototype vectors are selected to represent 40 different levels of failure. The bearing impulse increases the vibration at characteristic frequencies of the motor frame. Therefore the impulse response of the motor frame to the bearing impulses is a series of resonating waves. The envelope of this vibration is not sinusoidal. Therefore, the harmonics of the bearing pass frequency are found in the spectrum but are attenuated compared to the bearing pass frequency. The values of the 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 smaller values 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 variation is added to the values. A set of prototype vectors for inner race fault is presented in Figure 1. 9
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covariance information). In order to demonstrate the applicability of fuzzy logic two different models are created. The first model (A) uses the 16 dimensional feature vectors presented. 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 harmonic 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 all of the input memberships. These methods are only used because of 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 second 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 evaluated 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. Then the following rules are used:
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Fig.1. The prototype vectors for the inner race fault class. Formation of a test vector. From the envelope spectrum of a real measurement (on running 15 kW induction motor) the twelve components 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 was selected as a feature. Envelope spectra for the outer race fault and inner 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 tests, the distances between the test vector and the vectors representing any of the fault types (broken classes) and of the healthy case were calculated according to Equation 2. The jacknified Mahalanobis distance calculation, where the vector under test is not included into calculation of covariance and mean, was used. Then a minimum distance 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 corresponding 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 simultaneous 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
If the magnitudes of the outputs (for different faults) of previous logic (model A) are high, the degree of the fault is high. • 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 abnormal. • If there are high peaks in the envelope spectrum at nonpredicted frequencies the condition is classified as other abnormal situation. • If the distance to a certain fault is low using the statistical classification or if the spectrum peaks are found near to predicted frequencies, the probability of the fault is great. • 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 outputs of previous rules can be used in new rules such as • If the probability of a certain fault is great and the degree of the fault is high, the output is act fast. • If the probability of a certain fault is great and the degree of the fault is not small, the output is act. • If the probability of a certain fault is low and the degree of the fault is high, the output is act. • If the probability of a certain fault is low and the degree of the fault is small, the output is wait. • If the non-normal situation holds then the output is warning. The features that are extracted from the envelope spectrum are illustrated in Figure 2 and the flow chart of the classification system is illustrated in Figure 3.
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Fig.4. The envelope spectrum of a healthy motor 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 is a unity vector.
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Fig.3. The block diagram of the classification system.
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III. RESULTS Results. The envelope spectrum of a healthy motor is illustrated in the Figure 4 (upper diagram). 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 5 and 6. The classification results are presented in Tables 1 and 2. Table 1 presents the classification results using a 16- dimensional feature space, covering all four fault types. Table 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 fourdimensional feature space is used. On the other hand, there is a bigger risk of misclassification if the shape of the test feature vector changes. It is important to bear in mind, firstly, that the correct classification was obtained without any tuning of the prototype 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 before jumping into conclusions that generalize the result obtained with these tests. The classification results using the fuzzy logic for the degree of fault are presented in Table 3. All of the broken cases were classified to the class broken and the healthy case was classified to the classes healthy or suspicious. In the Table 4 the outputs of the fuzzy logic representing the fault probability are presented. These values are multiplied with the fault degree outputs forming the memberships of the output act. 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.
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Fig.5. 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 is a unity vector.
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The problem of prototype vector creation was solved here in the most straightforward way. The characteristic bearing frequency peaks and their three nearest harmonics were selected as features. In some cases, the harmonics can be attenuated (sinusoidal envelope). Then the risk of misclassification is obvious. In this, the fuzzy logic will correct this if the amplitude of the peak at the bearing pass frequency is high.
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Trigger levels are not required when minimum classifier is used. However, for fuzzy logic requires sets that have limits. The trigger levels, on the other hand are not as strict as with Boolean logic and therefore a priori knowledge of the behavior of different fault types can be used. However, the limits are case specific and are the causes for the possible misclassification.
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Fig.6. 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 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 5.3817 2.9616 6.5451 0.7758
outer race 1.3607 1.2018 3.2331 5.3318 1.4425
inner race 2.7905 7.0394 1.0653 8.2085 0.8656
ball spin 1.6052 8.7226 2.7675 1.0579 1.7417
cage 2.3892 7.5465 2.7684 6.7352 0.1666
Table 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 healthy outer race inner race ball spin cage
distance to outer race healthy 0.1767 broken 0.781 healthy 3.9542 broken 0.0394 healthy 0.556 broken 1.385 healthy 0.2327 broken 0.9328 healthy 0.2827 broken 1.0563
inner race 0.148 4.235 0.2946 6.5133 2.4608 0.3182 0.3109 4.1784 0.3915 5.0148
ball spin 0.1107 1.2339 0.2454 1.4492 0.441 1.6345 4.4202 0.2219 0.2275 1.7709
V. CONCLUSIONS
cage 0.138 1.3203 0.2164 1.4779 2.0332 2.656 0.1176 1.7265 4.0238 0.073
Table 3. Classification results using simple fuzzy logic for the determination of fault degree. test data healthy outer race inner race ball spin cage
outer race 0.18 0.47 0.03 0.18 0.18
Fault degree inner race ball spin 0.17 0.17 0.42 0.18 0.03 0.36 0.17 0.50 0.16 0.17
cage 0.18 0.10 0.32 0.18 0.50
Table 4. 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 3 with the probabilities of this table. The actions are selected with trigger levels. test data healthy outer race inner race ball spin cage
outer race 0.18 0.49 0.32 0.32 0.32
Problems.
Fault probability inner race ball spin 0.17 0.16 0.3 0.3 0.35 0.17 0.17 0.17 0.17 0.16
cage 0.17 0.2 0.16 0.16 0.58
Action outer race inner race ball spin 0.03 0.03 0.03 0.23 0.13 0.05 0.01 0.13 0.01 0.06 0.03 0.09 0.06 0.03 0.03
The risk of a misclassification increases in situations where there exists a high peak value in the envelope spectrum at some characteristic bearing fault frequency and is not caused by a bearing fault. This kind of situation is found in Figure 6 where the peak value caused by vibration at a rotational frequency is misinterpreted to be the one caused by a cage fault. In this case, however, the covariance between other features of cage fault remains low so that the misclassification was avoided. On the other hand, this case also illustrates the usefulness of using the Mahalanobis metrics in the distance calculations. However, with the reduced feature space of Table 2 the difference between the distance to inner race fault and the distance to cage fault is quite small because of the weaker covariance information than with the classification with 16 dimensional feature space of Table 1. In addition, this example demonstrates the usefulness of using the deterministic qualitative measures and fuzzy logic: In this case, the significance of that peak was reduced because of the missing harmonics and rotational side bands.
cage 0.03 0.02 0.05 0.03 0.29
The method where the Mahalanobis distance based statistical classification results were used as one input in fuzzy logic was introduced. This input and other qualitative features of the spectrum were taken into account in the fuzzy logic which evaluated the probability of a certain fault. The other fuzzy logic evaluated the degree of certain fault and by combining the outputs of these logics, the final suggestion of the state and required action were given. Based on the tests, this approach improves the reliability of the fault analysis when the reliability of the analysis means the minimisation of the wrong action, continuing the use or repairing the motor. In many researches the neural network has been used as a pre- processor for the fuzzy logic or as postprocessor for time-frequency representations of signals [5,6]. The presented approach is alternative to that approach. The statistical Mahalanobis distance based classifier as used in this paper may give equal results to the neural network. However, due to the exact input output relationship, the controllability of the classification is better when the Mahalanobis distance classifier is used. The test data was rather limited (real data from outer and inner race faults, simulations for other faults) and jumping to certain conclusions should not be made before the tests are done with many motors with different bearing faults. On the other hand, by discussing the possibilities of an automatic fault analysis, the guidelines for the future work are given. Furthermore, the presented method relies on the known properties of envelope spectrum peaks (for example [7]), if the peaks appear, the method would classify properly any fault types represented in the feature vectors.
References [1] Mechefske C. K., “Objective machinery fault diagnosis using fuzzy logic”. Mechanical systems and signal processing, 1998, No.12, pp. 855-862 [2] Tuomo Lindh, On the condition monitoring of induction machines. Dissertation, Acta Universitatis Lappeenrantaensis 174, Lappeenranta University of Technology, Finland 2003, ISBN 951-764-841-3. [3] Schiltz, R. L., “Forcing frequency identification of rolling element bearings”, Sound and vibration, pp. 16-19, May 1990. [4] P. C. Mahalanobis, Proc. Natl. Institute of Science of India, 2, 49, 1936.
[5] Staszewski W. J., Worden K., Tomlison G. R., “TimeFrequency Analysis in Gearbox Fault Detection Using the Wigner-Ville Distribution and Pattern Recognition”, Mechanical systems and signal processing, 1997, No.11, pp. 673-692 [6] Paya P. A., Esat I. I. “Artificial neural network based fault diagnostics of rotating machinery using Wavelettransforms as a preprocessor”, Mechanical systems and signal processing, 1997, No.11, pp. 751-765 [7] Wang Y. F., Harrap M. J., “Condition monitoring of ball bearings using envelope autocorrelation technique”. Machine Vibration, 1996, No.5, pp. 34-44
