**3.2.3 Implementation of WPS for bearing fault detection**

To demonstrate the performance of the proposed approach, this section presents several application examples for the detection of localized bearing defects. In all the examples, the Laplace wavelet is used as a WT base-function. The wavelet parameters (damping factor and centre frequency) are optimized based on maximizing the kurtosis value for the wavelet coefficients.

#### **(a) Simulated vibration data**

Using a rolling element bearing with specification shown in Table 2, the scale-wavelet power spectrum comparison for the Laplace-wavelet and widely used Morlet wavelet was carried out, Figure 15. It can be found that the amplitude of the power spectrum is greater for the faulty bearing than the normal one, and the power spectrum is concentrated in the scale interval of [15-20] for the Laplace-wavelet compared with the distributed power spectrum over a wide scale range for the Morlet wavelet. That shows the improved effectiveness of the Laplace wavelet over the Morlet wavelet for bearing fault impulses extraction.

The FFT-Spectrum, envelope spectrum using Hilbert Transform and the Laplace wavelet transform envelope spectrum for the simulated outer-race, inner-race and rolling element faults vibration signals at rotational speed of 1797 rev/min, are shown in Figure 16. The results show that the BCFs are unspecified in the FFT-Spectrum and are not clearly defined in the envelope power spectrum but are clearly identified in the Laplacewavelet power spectrum for both outer, inner race and rolling element faults, Figure 17.

The TWPS effectively extracts the fault frequencies of 105.5 Hz, 164.1 Hz and 141.4 with their harmonics for outer-race, inner-race and rolling element faults, respectively, which are very close to the calculated frequencies (*FBPO*= 107.364 Hz, *FBPI*= 162.185 Hz and *2FB*= 141.169 Hz). The side bands at the rotational speed can be recognized for inner race and rolling element faults as a result of amplitude modulation.

To evaluate the robustness of the proposed technique to extract the BCF for different signal to noise ratio (SNR) , and randomness in the impulses period (*τ*) as a result of slip variation, Figure 18 shows the *TWPS* for outer-race fault simulated signals for different values of SNR, and τ as a percentage of the pulse period (*T*).

<sup>2</sup> *WPS a SEWT a d* (, ) (, )

<sup>2</sup> <sup>1</sup> ( ) (, ) <sup>2</sup> *TWPS x t dt WPS a da*

To demonstrate the performance of the proposed approach, this section presents several application examples for the detection of localized bearing defects. In all the examples, the Laplace wavelet is used as a WT base-function. The wavelet parameters (damping factor and centre frequency) are optimized based on maximizing the kurtosis value for the wavelet

Using a rolling element bearing with specification shown in Table 2, the scale-wavelet power spectrum comparison for the Laplace-wavelet and widely used Morlet wavelet was carried out, Figure 15. It can be found that the amplitude of the power spectrum is greater for the faulty bearing than the normal one, and the power spectrum is concentrated in the scale interval of [15-20] for the Laplace-wavelet compared with the distributed power spectrum over a wide scale range for the Morlet wavelet. That shows the improved effectiveness of the Laplace wavelet over the Morlet wavelet for bearing fault impulses

The FFT-Spectrum, envelope spectrum using Hilbert Transform and the Laplace wavelet transform envelope spectrum for the simulated outer-race, inner-race and rolling element faults vibration signals at rotational speed of 1797 rev/min, are shown in Figure 16. The results show that the BCFs are unspecified in the FFT-Spectrum and are not clearly defined in the envelope power spectrum but are clearly identified in the Laplacewavelet power spectrum for both outer, inner race and rolling element faults, Figure

The TWPS effectively extracts the fault frequencies of 105.5 Hz, 164.1 Hz and 141.4 with their harmonics for outer-race, inner-race and rolling element faults, respectively, which are very close to the calculated frequencies (*FBPO*= 107.364 Hz, *FBPI*= 162.185 Hz and *2FB*= 141.169 Hz). The side bands at the rotational speed can be recognized for inner race and rolling

To evaluate the robustness of the proposed technique to extract the BCF for different signal to noise ratio (SNR) , and randomness in the impulses period (*τ*) as a result of slip variation, Figure 18 shows the *TWPS* for outer-race fault simulated signals for different values of

 

(12)

(11)

where *SEWT (a, ω)* is the Fourier Transform of *EWT(a,b)*.

**3.2.3 Implementation of WPS for bearing fault detection** 

element faults as a result of amplitude modulation.

SNR, and τ as a percentage of the pulse period (*T*).

The total energy of the signal *x(t)*,

coefficients.

extraction.

17.

**(a) Simulated vibration data** 

Fig. 15. The wavelet-level power spectrum using (a) Morlet-wavelet, (b) Laplace-wavelet for new and outer-race defective bearing.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 333

Fig. 17. The Laplace envelope spectrum of the simulated vibration signal for bearing with

(a) outer-race fault, (b) inner-race fault, and (c) rolling element fault, at speed of

1797 rev/min.

Fig. 16. The simulated vibration signal power spectrum, envelope power spectrum, and Laplace-wavelet transform power spectrum respectively, for rolling bearing with (a) Outerrace fault and, (b) Inner-race fault.

Fig. 16. The simulated vibration signal power spectrum, envelope power spectrum, and Laplace-wavelet transform power spectrum respectively, for rolling bearing with (a) Outer-

race fault and, (b) Inner-race fault.

Fig. 17. The Laplace envelope spectrum of the simulated vibration signal for bearing with (a) outer-race fault, (b) inner-race fault, and (c) rolling element fault, at speed of 1797 rev/min.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 335

TWPS, the power spectrum peak values at the location of the outer-race characteristic frequency and its harmonics are easily defined and match the calculated *FBPO*, Figure 19. Applied to different shaft rotational speed, Figure 20 shows that the TWPS is sensitive to the variation of the fault frequencies as a result of variation in the shaft rotational speeds, Table 5. The TWPS for bearings with inner and rolling element faults are shown in Figures 21 and 22, respectively. The fault frequencies are clearly extracted at 126 Hz for inner race fault and 140.1 Hz for rolling element fault which are very close to the calculated fault frequencies.

**Outer-Race Fault** Shaft Speed, rev/min Calculated FBPO, Hz TWPS peak , Hz 90.96 91 113.70 112 136.44 135 159.18 166 181.92 182 227.40 226

Table 5. The calculated and extracted (FBPO) at different shaft rotational speeds.

Fig. 19. The measured bearing vibration signal (a) the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with outer race fault at speed of 2000 rev/min

(calculated FBPO=181.92 Hz).

#### **(b) Experimental vibration data**

For angular contact ball bearings with specifications as given in Table 2, the calculated fault frequencies for different shaft rotational speeds are shown in Table 3.With application of the

Fig. 18. The TWPS for Bearing with outer-race fault for different (a) SNR, and (b) slip variation (τ).

For angular contact ball bearings with specifications as given in Table 2, the calculated fault frequencies for different shaft rotational speeds are shown in Table 3.With application of the

**(b) Experimental vibration data** 

TWPS, the power spectrum peak values at the location of the outer-race characteristic frequency and its harmonics are easily defined and match the calculated *FBPO*, Figure 19. Applied to different shaft rotational speed, Figure 20 shows that the TWPS is sensitive to the variation of the fault frequencies as a result of variation in the shaft rotational speeds, Table 5.

The TWPS for bearings with inner and rolling element faults are shown in Figures 21 and 22, respectively. The fault frequencies are clearly extracted at 126 Hz for inner race fault and 140.1 Hz for rolling element fault which are very close to the calculated fault frequencies.


Table 5. The calculated and extracted (FBPO) at different shaft rotational speeds.

Fig. 19. The measured bearing vibration signal (a) the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with outer race fault at speed of 2000 rev/min (calculated FBPO=181.92 Hz).

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 337

(d) 1250 rev/min

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -50

Time (Sec)

GCUCH6-2

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -25

Time (Sec)

GCUCH5-2


> -20 -15 -10 -5 0 5 10 15 20 25

Acceleration (m.sec-2)

Acceleration (m.sec-2)

Power Spectrum

X: 112 Y: 9.047

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

gcuchecked-outer-1250y6-bcf113.18-GCUCH5

Frequency (Hz)

gcuchecked-out-1000y5-bcf92.55-GCUCH6

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

Frequency (Hz)

(e) 1000 rev/min

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Power Spectrum

X: 91 Y: 0.8633

Fig. 20. (cont.) (a-e) the bearing vibration signals and the corresponding Laplace envelope

spectrum column for bearing with outer race fault at different rotational speeds.

Fig. 20. (a-e) the bearing vibration signals and the corresponding Laplace envelope spectrum column for bearing with outer race fault at different rotational speeds.

(a) 2500 rev/min

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -150

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -100

Time (Sec)

GCUCH4-2

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -60

Time (Sec)

Time (Sec)

GCUCH3-2

GCUCH2-2

> -80 -60 -40 -20 0 20 40 60 80 100

Acceleration (m.sec-2)

Acceleration (m.sec-2)

Power Spectrum

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

gcuchecked-outer-2500y8-bcf=229.67-GCUCH2

Frequency (Hz)

gcuchecked-1750-y4-bcf168.22-GCUCH3

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

Frequency (Hz)

gcuchecked-out-1500-y7-bcf136.25-GCUCH4

<sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>900</sup> <sup>1000</sup> <sup>0</sup>

Frequency (Hz)

X: 499 Y: 2.696

X: 226 Y: 46.63

X: 87 Y: 28.98

> X: 166 Y: 7.335 X: 333 Y: 6.645

X: 135 Y: 7.905

X: 31 Y: 9

(b) 1750 rev/min

Power Spectrum

(c) 1500 rev/min Fig. 20. (a-e) the bearing vibration signals and the corresponding Laplace envelope spectrum

Power Spectrum

X: 50 Y: 3

column for bearing with outer race fault at different rotational speeds.



0

Acceleration (m.sec-2)

20

40

60

Fig. 20. (cont.) (a-e) the bearing vibration signals and the corresponding Laplace envelope spectrum column for bearing with outer race fault at different rotational speeds.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 339

Fig 22. The bearing vibration signal (a), the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with rolling element fault at speed of 1500 rev/min

corresponding TWPS are shown in Figures 23 to 26, respectively.

The time course of the vibration signals for a normal bearing and bearings with outer race, inner race and rolling element faults at a shaft rotational speed of 1797 rev/min with its

The TWPS for the vibration data shows spectral peaks at 106.9 Hz, 161.1 Hz and 141.166 Hz and their harmonics for outer race, inner race and rolling element faults, respectively. The sidebands at shaft speed (30 Hz) as a result of amplitude modulation are shown for inner

(the calculated, 2FB=142.74 Hz).

**(c) CWRU vibration data** 

and rolling element faults.

Fig. 21. The bearing vibration signal (a), the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with inner race fault at speed of 1000 rev/min (the calculated FBPI=125.70 Hz).

Fig. 21. The bearing vibration signal (a), the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with inner race fault at speed of 1000 rev/min

(the calculated FBPI=125.70 Hz).

Fig 22. The bearing vibration signal (a), the FFT spectrum (b) and the Laplace wavelet envelope spectrum (c) for bearing with rolling element fault at speed of 1500 rev/min (the calculated, 2FB=142.74 Hz).
