**(a) The CWRU vibration data**

The smallest fault diameter which introduce smallest fault pulse amplitudes is selected in this study at two different shaft rotational speeds of 1797 rev/min (with no motor load condition) for training data and 1772 rev/min (with 1 HP motor load) for the test data.

The neural network input feature vectors consist of five groups representing the different bearing conditions, a total of 3856 segments of 1000 samples each. The data sets were split between training and test (unseen) sets of size 1928 samples each. The parameters of the applied BP neural network are listed in Table 6.


Table 6. Applied neural network architecture and training parameters.

The distribution of the extracted features (normalized between 0 and 1), time domain features (RMS and kurtosis) on x-axis and frequency domain features (*fmax/frpm* and *Amax/sum(A*)) on the y-axis , for the most dominant scales of the Laplace wavelet transform for different rolling bearing fault conditions is shown in Figure 30a. It is clear that the normalized feature values for the bearing with outer-race fault are the highest as a result of the high energy fault pulses compared with the less energy pulses generated by inner-race and roller faults as a consequence of amplitude modulation.

The result of the learning process of the developed NN is depicted in Figure 30b, which shows that the training with 300 iterations met the MSE stopping criteria (MSE less than 10E-20). The NN test process for unseen vibration data of the trained ANN combined with the ideal output target values are presented in Figure 30d, which indicates the high success classification rate (≈ 100 %) for rolling bearing fault detection and classification.

The smallest fault diameter which introduce smallest fault pulse amplitudes is selected in this study at two different shaft rotational speeds of 1797 rev/min (with no motor load condition) for training data and 1772 rev/min (with 1 HP motor load) for the test

The neural network input feature vectors consist of five groups representing the different bearing conditions, a total of 3856 segments of 1000 samples each. The data sets were split between training and test (unseen) sets of size 1928 samples each. The parameters of the

**Neural Network architecture** 

No. of output Hidden nodes

**NN Training parameters** 

Training Algorithms Learning rate Training Stop Criteria

The distribution of the extracted features (normalized between 0 and 1), time domain features (RMS and kurtosis) on x-axis and frequency domain features (*fmax/frpm* and *Amax/sum(A*)) on the y-axis , for the most dominant scales of the Laplace wavelet transform for different rolling bearing fault conditions is shown in Figure 30a. It is clear that the normalized feature values for the bearing with outer-race fault are the highest as a result of the high energy fault pulses compared with the less energy pulses generated by inner-race

The result of the learning process of the developed NN is depicted in Figure 30b, which shows that the training with 300 iterations met the MSE stopping criteria (MSE less than 10E-20). The NN test process for unseen vibration data of the trained ANN combined with the ideal output target values are presented in Figure 30d, which indicates the high success classification rate (≈ 100 %) for rolling bearing fault detection and

Table 6. Applied neural network architecture and training parameters.

and roller faults as a consequence of amplitude modulation.

LM 0.52 Max. epoch MSE

Hidden layer nodes

1000 10E-20

No. of input nodes

Sigmoid Linear 5 4 4

 **(a) The CWRU vibration data** 

Transfer Function

Layer

classification.

applied BP neural network are listed in Table 6.

Output Layer

data.

Fig. 30. (a) the extracted features distribution, (b) ANN learning process, (c) ANN classification MSE, (d) ANN Training/Test process, for the CWRU bearing vibration data.

#### **(b) Simulated vibration data**

Using the same bearing specifications but CWRU data with 0.6 dB signal to noise ratio and random slip of 10 percent the period T. Figure 31 shows the Wavelet-ANN bearing fault training/classification process for the simulated bearing vibration signal. The results show that the Wavelet-ANN training process reached the specified stopping criteria after 67 epochs, with overall classification MSE less than 6.0E-9, and 100% classification rate.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 349

Fig. 32. (a) the extracted features distribution, (b) ANN learning process, (c) ANN classification MSE, and (d) ANN Training /Test process, for the experimental bearing

The results for both the simulated and real bearing vibration data show the effectiveness of the combined wavelet-ANN technique for rolling bearing fault pattern detection and classification, and that the Laplace wavelet analysis is an effective approach in fault feature

The novelty of this chapter is concerned with the applications of the wavelet analysis in two

vibration data.

**5. Conclusions** 

extraction for the NN classifier.

different new approaches:

Fig. 31. (a) the extracted features distribution, (b) ANN learning process, (c) ANN classification MSE, and (d) ANN Training /Test process, for the simulated bearing vibration data.

#### **(c) Experimental vibration data**

The ANN training sets have been prepared using an acquired vibration signal at a shaft speed of 1000 rev/min, and the ANN testing set at a shaft speed of 1250 rev/min. Figure 32 shows the Wavelet-ANN bearing fault training/classification process for the measured bearing vibration signals. The results show that the Wavelet-ANN training process achieved the specified stopping criteria after 28 epochs, with overall classification MSE less than 6.0E-5 with 100 % classification rate.

Fig. 31. (a) the extracted features distribution, (b) ANN learning process, (c) ANN

data.

**(c) Experimental vibration data** 

5 with 100 % classification rate.

classification MSE, and (d) ANN Training /Test process, for the simulated bearing vibration

The ANN training sets have been prepared using an acquired vibration signal at a shaft speed of 1000 rev/min, and the ANN testing set at a shaft speed of 1250 rev/min. Figure 32 shows the Wavelet-ANN bearing fault training/classification process for the measured bearing vibration signals. The results show that the Wavelet-ANN training process achieved the specified stopping criteria after 28 epochs, with overall classification MSE less than 6.0E-

Fig. 32. (a) the extracted features distribution, (b) ANN learning process, (c) ANN classification MSE, and (d) ANN Training /Test process, for the experimental bearing vibration data.

The results for both the simulated and real bearing vibration data show the effectiveness of the combined wavelet-ANN technique for rolling bearing fault pattern detection and classification, and that the Laplace wavelet analysis is an effective approach in fault feature extraction for the NN classifier.
