Training at Two Different Rotational Speeds and Testing at Extrapolated

4.3 Simulation Results

4.3.3 Training at Two Different Rotational Speeds and Testing at Extrapolated

In this case the SVM is trained at two rotational speeds and tested at an extrapolated rotational speed. Training rotational speeds and the extrapolated testing rotational speed are tabulated in Table 4.2.

Table 4.2 The training and testing speeds for the fault prediction using the interpolation and the extrapolation

Training speeds (Hz) Testing speed (Hz) Training speed range (Hz) For interpolation speed

10, 15 12.5 5

5 5 5

15, 20 17.5

20, 25 22.5

25, 30 27.5

10, 20 15.0 10

10 10

15, 25 20.0

20, 30 25.0

For extrapolation speed

10, 12.5 15 5

5 5

20, 22.5 25

25, 27.5 30

10, 15 20 5

10 10

15, 20 25

20, 25 30

Optimization of Parameters: For selecting data points the same procedure is considered as used for the intermediate interpolation, but only difference is that for testing the extrapolation speed is considered. The optimized percentage accuracy (all classes) of the different SVM formulation is shown in Table 4.4.

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Table 4.3 Data sets used for parameter estimation and validation

Cases Same speed Interpolation and extrapolation

speed For GSM

Parameters estimation 9 4 270× × 9 4 125

9 8 125 9 4 125

× × 

⇒ × ×

× × 

Final testing or validation 9 4 30× × 9 4 30× ×

For GA and ABCA Training for optimizing the parameters

9 4 180× × 9 4 90

9 8 90 9 4 90

× × 

⇒ × ×

× × 

Testing of optimizing the parameters 9 4 90× × 9 4 45

9 8 45 9 4 45

× × 

⇒ × ×

× × 

Final testing or validation 9 4 30× × 9 4 30× ×

Fault Prediction Ability: After optimization the same strategy for data points selection are considered for the testing as described in the interpolation case, whereas here the extrapolated speed is considered. In the extrapolated speed range of 5 Hz, the prediction accuracy (all classes) is 70.83% to 100%; and in case of 10 Hz range, it is 38.33% to 77.50%. It is observed that at 15 Hz and 30 Hz testing speeds for the case of 5 Hz range the GSM shows marginally good results, which is 1.16% and 0.83% more than the GA and ABCA techniques, respectively. On the other hand for 25 Hz testing speed the ABCA dominates in prediction accuracies (all classes). At 10 Hz range case, the GA and the ABCA shows reasonable results.

For 5 Hz range it gives the best classification accuracy, and it increases gradually from 70.83% to 100% from the lowest to highest rotational speeds. In 10 Hz range also the best accuracy (all classes) increases from 62.50% to 77.50% from the lower to higher rotational speeds.

Table 4.4 Classification of gear faults (all classes) with the training and the testing for interpolation and extrapolation speed (time domain features)

Training speed (Hz)

Testing speed (Hz)

SVM formulation

Training accuracy* (%) Testing accuracy (%)

CV GA ABCA CV GA ABCA

For interpolation speed (range 5 Hz)

10, 15 12.5 C-SVC 92.96 90.00 90.00 88.33 87.50 87.50

ν

-SVC 93.43 91.39 91.94 89.17 89.17 84.17 15, 20 17.5 C-SVC 93.43 91.39 91.39 94.17 93.33 93.33

ν

-SVC 93.80 91.67 92.22 94.17 94.17 93.33 20, 25 22.5 C-SVC 97.22 97.22 96.94 95.83 95.83 95.83

ν

-SVC 97.78 96.94 97.22 96.67 96.67 96.67 25, 30 27.5 C-SVC 98.98 98.61 98.61 100.00 100.00 100.00

ν

-SVC 99.54 99.72 99.72 100.00 100.00 100.00 For interpolation speed (range 10 Hz)

10, 20 15 C-SVC 95.00 95.83 96.11 48.33 46.67 46.67

ν-SVC 96.11 96.11 96.67 59.17 36.67 35.00

15, 25 20 C-SVC 95.46 94.17 94.44 65.00 63.33 65.83

ν-SVC 95.93 93.61 94.44 79.17 63.33 85.83

20, 30 25 C-SVC 96.11 96.11 96.11 90.00 93.33 93.33

ν-SVC 96.85 96.39 96.39 92.50 94.17 91.67 For extrapolation speed (range 5 Hz)

10, 12.5 15 C-SVC 95.00 95.00 94.72 71.67 70.83 74.17

ν-SVC 95.65 95.83 95.83 75.83 73.33 73.33 20, 22.5 25 C-SVC 96.39 95.83 96.11 80.83 80.00 80.00

ν-SVC 97.04 94.72 97.22 83.33 70.00 90.83 25, 27.5 30 C-SVC 99.54 99.72 99.72 98.33 99.17 96.67

ν-SVC 99.63 99.72 100.00 100.00 99.17 95.00 For extrapolation speed (range 10 Hz)

10, 15 20 C-SVC 92.96 90.00 90.00 45.00 38.33 38.33

ν-SVC 93.43 91.39 91.94 62.50 62.50 50.00

15, 20 25 C-SVC 93.43 91.39 91.39 55.00 49.17 49.17

ν-SVC 93.80 91.67 92.22 62.50 62.50 69.17

20, 25 30 C-SVC 97.22 97.22 96.94 71.67 72.50 71.67

ν-SVC 97.78 96.94 97.22 73.33 73.33 77.50

* CV means the cross validation accuracy of the GSM. Bold values represent prediction accuracies of the maximum accuracy.

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Table 4.5 Percentage prediction (individual classes) accuracies at various speeds with the training and the testing (time domain features)

Speed (Hz) CT (%) MT (%) WT (%) ND (%)

For same speed

10 100.00 100.00 100.00 96.67

15 93.33 76.67 100.00 93.33

20 96.67 93.33 100.00 100.00

25 100.00 100.00 100.00 100.00

30 100.00 100.00 100.00 100.00

For interpolation speed (range 5 Hz)

12.5 86.67 96.67 96.67 76.67

17.5 90.00 90.00 100.00 96.67

22.5 96.67 90.00 100.00 100.00

27.5 100.00 100.00 100.00 100.00

For interpolation speed (range 10 Hz)

15 70.00 93.33 40.00 33.33

20 86.67 80.00 86.67 90.00

25 100.00 96.67 93.33 86.67

For extrapolation speed (range 5 Hz)

15 43.33 73.33 100.00 86.67

25 90.00 100.00 100.00 73.33

30 100.00 100.00 100.00 100.00

For extrapolation speed (range 10 Hz)

20 43.33 86.67 96.67 23.33

25 73.33 100.00 90.00 13.33

30 96.67 100.00 96.67 16.67

The best percentage prediction accuracy (individual classes) at various rotational speeds is illustrated in Table 4.5. In the case of speed range of 5 Hz the lowest prediction accuracy (individual classes) is at 15 Hz for the fault detection of CT (i.e., 43.33%). Similarly, in the case of speed range of 10 Hz, the lowest performance is for the ND condition at 25 Hz (i.e.,

13.33%). For more clarity the overall performances are tabulated in Table 4.6. The lowest prediction accuracy (all classes) and highest prediction accuracy (all classes) are the lowest and highest testing accuracy (all classes) found from Table 4.1 and Table 4.4, respectively.

The lowest and highest fault predictions are found from Table 4.5.

Table 4.6 Summary of overall fault prediction performances (time domain features)

Same rotational

speed

Interpolation speed Extrapolation speed Range 5 Hz Range 10 Hz Range 5 Hz Range 10 Hz

Lowest prediction accuracy (all

classes)

88.33% 88.33% 35.00% 70.83% 38.33%

Highest prediction accuracy (all

classes)

100.00% 100.00% 94.17% 100.00% 77.50%

Lowest individual faults prediction

76.67%

(15 Hz, MT case)

76.67%

(12.5 Hz, ND case)

33.33%

(15 Hz, ND case)

43.33%

(15 Hz, CT case)

13.33%

(25 Hz, ND case)

Highest individual faults

prediction

100.00% 100.00% 100.00% 100.00% 100.00%

Initially for each of the four classification cases (CT, MT, WT and ND), the training data was provided at running speeds from 10 Hz to 30 Hz in intervals of 2.5 Hz. Then the multiclass classification capability of two classes of SVM was noted for these running speeds. It is concluded that the SVM has the ability to make perfect classifications if the training data is available for that particular running speed. The classification capability of the SVM for the interpolation and the extrapolation beyond its training data speeds are performed and it is noted that the SVM can still be able to make accurate classification utilizing the training data

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from the beginning and end of a range of running speeds, whereas the test data could be at any of speeds from within that range (interpolation) and out of range (extrapolation). In the interpolation and extrapolation fault predictions, it has been observed that in many cases the GA and the ABCA show similar results as the GSM. On the other hand, the GSM shows its marginal superiority over the GA and ABC results with order of 12.5 %, 1.16% and 0.83%

more for three cases. It is also observed that the prediction accuracy gradually increases with the increase of the rotational speed. This is due to the high signal-to-noise level at higher rotation speeds due to better manifestation of faults in vibration signals. It is also observed that interpolation accuracy is better than the extrapolated one. It is know that the classifier is tested for classification ability within the range and outside the range of speeds. The classifier distinguishes the data on the basis of the classification boundary created by support vectors during the training. In the intermediate speed, the fault frequency and its sidebands are within two trained data fault frequency and its sidebands. It may better influence fault features to be tested than the extrapolated one, in which fault frequencies and its sidebands are outside the trained data fault frequencies and its sidebands.