5.1. Varying Time Experiments on the Gear-Rotor Platform

As shown in Figure 6, the experiments simulate the vibration signal from a gear-rotor experimental platform (51 and 23 tooth gears) operating at different speeds. The LabView NI9401 DI/O was used as the interface for input signals to the circuit board, which dynamically controls the frequency to achieve the automation of speed variation. The operating process was as follows: The speed was dropped from 1600 rpm to 800 rpm within two seconds, and then increased to the final value of 1500 rpm. The non-stationary speed curve is shown in Figure 7. The NI9234 DAQ simultaneously acquired the vibration signals and speed signals at a sampling frequency of 6400 Hz.

In this experiment, non-stationary signals are measured under 8 different kinds of conditions, with 20 samples taken for each condition for a total of 160 samples. In the time-domain, the signals of all samples were subjected to STFT. Thus, the first 200 orders of the total signal characteristics in the time-frequency order spectrum are obtained through the non-stationary time-frequency extrac-

Table 1. Gear-rotor experimental platform failure conditions.

Figure 6. Gear speed change simulation platform.

Figure 7. Non-stationary speed curve.

tion technique.

5.2. Non-Stationary Signal Feature Extraction

Table 1 shows the conditions of eight different vibration signals used in the experiment. Figures 8(a) and (e) respectively show conditions A and B of the platform of the gear-rotor signal in the time domain for all conditions. Figures 8(b) and (f) respectively show the spectra of the fast Fourier transform frequency, and Figures 8(c) and (g) respectively show the frequency of spectrum images obtained from STFT. The spectrum images were taken every 0.1 second so that the time-frequency order spectrum is obtained by dividing the instant time-frequency spectrum by the instant speed frequency. All instant time-frequency order spectra were superimposed and then averaged to obtain the time-frequency order spectrum, represented respectively in Figures 8(d) and (h). The operational statuses for signals C ~ H were similarly measured and analyzed, and are respectively depicted in Figures 9-11. For the eight calculated time-frequency order spectra, the conditions of B and C at the 23^{th} order display significant amplitude, respectively indicating broken teeth and wear on the gear. Condition D at the 51^{th} order shows significant amplitude, indicating significant gear imbalance. Lastly, the rotor conditions E to H generate varying amplitudes in order 1^{st }~ 3^{rd}.

5.3. Back-Propagation Neural Network Diagnosis Results

This study used STFT with non-stationary signal extraction techniques to obtain the time-frequency order spectrum. Next, the value of the extracted features was inputted to the back-propagating neural networks to obtain the failure diagnosis as follows:

1) Figure 12 illustrates the structure of back-propagating neural networks used in this study. The time-frequency order spectrum is used as input, and the input neuron count a is defined as 200 while the hidden level neuron count is 250.

2) In the experiment, conditions A ~ H (Table 1) can be used in the neural network, with different respective assembled outputs as follows: [1 0 0 0 0 0 0 0], [0 1 0 0 0

Figure 8. Non-stationary signal feature extraction for conditions A and B. (a) Time-domain signal (condition A); (b) Frequency spectrum (condition A); (c) STFT time-frequency spectrum (condition A); (d) STFT time-frequency order spectrum (condition A); (e) Time-domain signal (condition B); (f) Frequency spectrum (condition B); (g) STFT time-frequency spectrum (condition B); (h) STFT time-frequency order spectrum (condition B).

Figure 9. Non-stationary signal feature extraction for conditions C and D. (a) Time-domain signal (condition C); (b) Frequency spectrum (condition C); (c) STFT time-frequency spectrum (condition C); (d) STFT time-frequency order spectrum (condition C); (e) Time-domain signal (condition D); (f) Frequency spectrum (condition D); (g) STFT time-frequency spectrum (condition D); (h) STFT time-frequency order spectrum (condition D).

Figure 10. Non-stationary signal feature extraction for conditions E and F. (a) Time-domain signal (condition E); (b) Frequency spectrum (condition E); (c) STFT time-frequency spectrum (condition E); (d) STFT time-frequency order spectrum (condition E); (e) Time-domain signal (condition F); (f) Frequency spectrum (condition F); (g) STFT time-frequency spectrum (condition F); (h) STFT time-frequency order spectrum (condition F).

Figure 11. Non-stationary signal feature extraction for conditions G and H. (a) Time-domain signal (condition G); (b) Frequency spectrum (condition G); (c) STFT time-frequency spectrum (condition G); (d) STFT time-frequency order spectrum (condition G); (e) Time-domain signal (condition H); (f) Frequency spectrum (condition H); (g) STFT time-frequency spectrum (condition H); (h) STFT time-frequency order spectrum (condition H).

Figure 12. Time-frequency order spectrum neural network training.

Table 2. STFT time-frequency order spectrum + BPNN recognition accuracy.

0 0 0], [0 0 1 0 0 0 0 0], [0 0 0 1 0 0 0 0], [0 0 0 0 1 0 00], [0 0 0 0 0 1 0 0], [0 0 0 0 0 0 0 0 1 0] and [0 0 0 0 0 0 0 0 0 1].

3) Of the 20 samples each for A ~ H (160 samples total), sample assemblies were randomly chosen for training and testing, with the four assemblies as follows: First, 20% training and 80% testing (32 training samples, 128 test samples); second, 40% training and 60% testing (64 training samples, 96 test samples); third, 60% training and 40% testing (96 training samples, 65 test samples) and finally, 80% training samples and 20% testing samples (128 training samples, 32 test samples).

4) Diagnosis results: To verify the STFT time-frequency order spectrum network output values, this study took the values of 0.7, 0.8 and 0.9 as the criteria for the determination value, as shown in Table 2. Among all these numbers, if the threshold value of 0.7 is correct, the accuracy is high but the low threshold would make it prone to error. The threshold of 0.9 is too high, resulting in low accuracy. Therefore, this study recommends the setting threshold value as 0.8.

6. Conclusions

Combining the time-frequency order spectrum with neural networks obtains good recognition results for all types of non-stationary signals. The conclusions as follows:

1) In the proposed method, the time-frequency order 2) Under different conditions, the proposed time-frequency order spectrum can fix features such that speed will not change during the time period. Thus, the magnitude of changes in feature order can be observed in nonstationary signals, allowing for the recognition of anomalies.

3) The proposed combination of the time-frequency order spectrum and back-propagation neural networks diagnoses failures with an accuracy rate of 93% or higher using a ratio of 20:80 for training to test samples. All the testing samples reduce the required training time to 196 seconds.

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NOTES

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