**3. Bearing fault diagnosis using wavelet analysis**

The Wavelet Transform (WT) coefficients are analyzed in both the time and frequency domains. In the time domain the autocorrelation of the wavelet de-noised signal is applied to evaluate the period of the fault pulses using the impulse wavelet as a wavelet base function. However, in the frequency domain the wavelet envelope power spectrum has been used to identify the fault frequencies with the single sided complex Laplace wavelet as the mother wavelet function.

#### **3.1 Wavelet de-noising method**

316 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

The rolling surfaces on the rings are referred to as raceways. The number of balls is defined as *Nb*, their diameter as *Db*. The pitch diameter or the diameter of the cage is designated *Dp*. The point of contact between a ball and the raceways is characterized by the contact angle α,

Fig. 2. Rolling element bearing basic geometry and velocities.

bearing can be loaded in one thrust direction only.

ball bearings have been used in this research for fault detection.

The rolling bearings that support loads perpendicular to their axis of rotation are called radial bearing. However, the bearings which support loads parallel to the axis of rotation are termed thrust bearings (the contact angle exceeding 45o), Figure 1d. Angular contact bearings have one ring shoulder removed; this may be from the inner or outer ring, Figure 1c. This allows a larger ball complement than found in comparable deep groove bearings, giving a greater load capacity. Speed capacity of angular contact bearings is also greater than deep groove ball bearing. The normal angular contact bearings have a contact angle which does not exceed 40o. Angular contact bearings support a combination of radial and thrust loads or heavy thrust loads depending on the contact angle. A single angular contact

Because roller bearings have a greater rolling surface area in contact with inner and outer races, they generally support a greater load than comparably sized ball bearings. The small contact area (point contact) in the ball bearing compared with the roller bearing (line contact) leads to more stress concentration and is more affected by the fatigue failure during the bearing rotation. Moreover, the angular contact ball bearing can easily separate its components (separable) to introduce the artificial faults. Based on that the angular contact

Figure 2.

#### **3.1.1 Impulse wavelet function**

The WT is the inner product of a time domain signal with the translated and dilated waveletbase function. The resulting coefficients reflect the correlation between the signal and the selected wavelet-base function. Therefore, to increase the amplitude of the generated wavelet coefficients related to the fault impulses, and to enhance the fault detection process, the selected wavelet-base function should be similar in characters to the bearing impulse response generated by the presence of a bearing incipient fault. Based on that, the investigated waveletbase function is denoted as the impulse-response wavelet and given by,

$$\varphi(t) \ = A \quad e^{-\frac{\beta}{\sqrt{1-\beta^2}}a\_c t} \quad \sin(\phi\_c t) \tag{1}$$

Where *β* is the damping factor that controls the decay rate of the exponential envelope in time and hence regulates the resolution of the wavelet, simultaneously it corresponds to the frequency bandwidth of the wavelet in the frequency domain, *ωc* determining the number of significant oscillations of the wavelet in the time domain and correspond to the wavelet centre frequency in frequency domain, and *A* is an arbitrary scaling factor. Figure 3 shows the proposed wavelet and its power spectrum.

Fig. 3. (a) the impulse wavelet time waveform, (b) its FFT-spectrum.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 319

(a)

(b)


outerpulse5hist


s1outerhist


s1noisehis

(c)

Fig. 4. (a) The noise signal (kurtosis=3.0843), (b) the overall vibration signal (kurtosis=7.7644), and (c) outer-race fault impulses (kurtosis=8.5312), with the

corresponding intensity distribution curve.

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> -4

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> -5

outerpulse5

s1outer

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> -150

s1noise


#### **3.1.2 The wavelet de-noising autocorrelation technique**

The proposed wavelet de-noising technique consists of the following steps:

a. Optimize the wavelet shape parameters (β and ωc) based on maximization of the kurtosis of the signal- wavelet inner product.

It is possible to find optimal values of *β* and *ω<sup>c</sup>* for a given vibration signal by adjusting the time-frequency resolution of the impulse wavelet to the decay rate and frequency of the impulses to be extracted.

Kurtosis is an indicator that reflects the "peakiness" of a signal, which is a property of the impulses and also it measures the divergence from a fundamental Gaussian distribution. A high kurtosis value indicates a high impulsive content of the signal with more sharpness in the signal intensity distribution. Figure 4 shows the kurtosis value and the intensity distribution for a white noise signal, pure impulsive signal, and impulsive signal mixed with noise.

The objective of the impulse wavelet shape optimization process is to determine the wavelet shape parameters (*β* and *ωc*) which maximize the kurtosis of the wavelet transform output;

$$Optimal\left(\boldsymbol{\beta}, \boldsymbol{o}\_c\right) = \max \left. \frac{\sum^N \mathsf{V} \mathsf{T}^4(\mathbf{x}(t), \boldsymbol{\nu}\_{\boldsymbol{\beta}, \boldsymbol{a}\_c}(t))}{\left[ \sum^N \mathsf{V} \mathsf{T}^2(\mathbf{x}(t), \boldsymbol{\nu}\_{\boldsymbol{\beta}, \boldsymbol{a}\_c}(t)) \right]^2} \right] \tag{2}$$

The genetic algorithm with specifications shown in Table 1 is used to optimize the wavelet shape parameters using Equation 2 as the GA fitness function. A flowchart of the algorithm is shown in Figure 5.


Table 1. The applied GA parameters.

b. Apply the wavelet de-noising technique: which consists of:

1. Perform a wavelet transform for the bearing vibration signal *x(t)* using the optimized wavelet,

$$\mathcal{W}T\{\mathbf{x}(t),a,b\} = <\mathcal{W}\_{a,b}.\\ \mathbf{x}(t) \succ = \frac{1}{\sqrt{a}} \int \mathbf{x}(t) \ \left. \Psi \right|\_{a,b}^\* (t) \ \left. dt \tag{3}$$

where <. > indicates the inner product, and the superscript asterisk '\*' indicates the complex conjugate. The *ψa,b* is a family of daughter wavelets derived from the mother wavelet *ψ(t)* by continuously varying the scale factor *a* and the translation parameter *b*. The factor 1 / *a* is used to ensure energy preservation.

a. Optimize the wavelet shape parameters (β and ωc) based on maximization of the

It is possible to find optimal values of *β* and *ω<sup>c</sup>* for a given vibration signal by adjusting the time-frequency resolution of the impulse wavelet to the decay rate and frequency of the

Kurtosis is an indicator that reflects the "peakiness" of a signal, which is a property of the impulses and also it measures the divergence from a fundamental Gaussian distribution. A high kurtosis value indicates a high impulsive content of the signal with more sharpness in the signal intensity distribution. Figure 4 shows the kurtosis value and the intensity distribution for a white noise signal, pure impulsive signal, and impulsive signal mixed

The objective of the impulse wavelet shape optimization process is to determine the wavelet shape parameters (*β* and *ωc*) which maximize the kurtosis of the wavelet transform output;

1

*N*

*<sup>n</sup> <sup>c</sup> <sup>N</sup>*

Population size *10*  Number of generations *20* 

Termination function *Maximum generation*  Selection function *Roulette wheel*  Cross-over function *Arith-crossover*  Mutation function *Uniform mutation* 

1. Perform a wavelet transform for the bearing vibration signal *x(t)* using the

<sup>1</sup> { ( ), , } . ( ) ( ) ( ) *WT x t a b x t x t t dt a b a b*

where <. > indicates the inner product, and the superscript asterisk '\*' indicates the complex conjugate. The *ψa,b* is a family of daughter wavelets derived from the mother wavelet *ψ(t)* by continuously varying the scale factor *a* and the translation parameter *b*. The factor 1 / *a* is

, ,

*a*

( , ) max.[ ]

1

*n*

The genetic algorithm with specifications shown in Table 1 is used to optimize the wavelet shape parameters using Equation 2 as the GA fitness function. A flowchart of the algorithm

4

,

*c*

(2)

*c*

\*

(3)

2 2 ,

 

[ ( ( ), ( ))]

*WT x t t*

*WT x t t*

( ( ), ( ))

 

**3.1.2 The wavelet de-noising autocorrelation technique** 

kurtosis of the signal- wavelet inner product.

*Optimal*

 

b. Apply the wavelet de-noising technique: which consists of:

impulses to be extracted.

with noise.

is shown in Figure 5.

Table 1. The applied GA parameters.

optimized wavelet,

used to ensure energy preservation.

The proposed wavelet de-noising technique consists of the following steps:

Fig. 4. (a) The noise signal (kurtosis=3.0843), (b) the overall vibration signal (kurtosis=7.7644), and (c) outer-race fault impulses (kurtosis=8.5312), with the corresponding intensity distribution curve.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 321

 (5)

(6)

(7)

(4)

2. Shrink the wavelet coefficients expressed in Equation 3 by soft thresholding,

[ ( ( , )) ] [ ( ( , )) ] *Max WT a b Max WT a b thr e e* 

3/2 ( ) (,) *soft*

c. Evaluate the auto-correlation function Rx (τ) for the de-noised signal x\*(t) to estimate the

\* \* ( ) [ ( ). ( )] *R E xt x t <sup>x</sup>*

where *τ* is the time lag, and *E [ ]* denotes ensemble average value of the quantity in square

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 impulse wavelet has been used as the wavelet base-function. The wavelet parameters (damping factor and centre frequency) are optimized based on maximizing the kurtosis

To evaluate the performance of the proposed method, the autocorrelation functions of the optimized impulse wavelet, impulse wavelet with non-optimized parameters, and the widely used Morlet wavelet are carried out and shown in Figure 7. The comparison of Figures 7a, b and c, shows the increased effectiveness of the optimized impulse wavelet over non-optimized impulse and Morlet wavelets for extraction of the bearing fault impulses and corresponding periodicity. Consequently, the performance of the bearing fault diagnosis

For a rolling element bearing with specifications as given in Table 2, the calculated BCFs (appendix A) for a shaft rotational speed of 1797 rev/min are 107.36 Hz and 162.18 Hz for outer and inner-race faults respectively. Figure 8 (a and d) shows the time domain waveform of the simulated signals for the rolling bearing with outer and inner-race faults based on the bearing vibration mathematical model (Khalid F. Al-Raheem *et al.* 2008). The

*da x t C WT a t <sup>a</sup>* 

3. Perform the inverse wavelet transform to reconstruct the signal using the shrunken

( )( ) *soft WT thr WT sign WT WT thr WT thr*

0

wavelet coefficients.

brackets.

periodicity of the extracted impulses

**3.1.3 Applications for bearing fault detection** 

value for the wavelet coefficients as shown in Figure 6.

process has been improved using the proposed technique.

**(a) Simulated vibration data** 

using soft-threshold function (*thr)* proposed by YANG and REN (2004),

where *ξ*>0 is parameter governing the shape of the threshold function.

\* <sup>1</sup>

*g*

Fig. 5. Wavelet shape parameters optimization process using GA.

Problem encoding: the population is coded into chromosomes (binary representation)

Population Initialization: generate the initial population for *β*, and *ωc*

> Fitness Function Evaluation [ ( ( ) \* ( , )] *wavelet <sup>c</sup> kurtosis WT x t*

Generate a new solution

No

Max. Fitness (Termination)

Decoding

Yes

Crossover Mutation

Fitness Evaluation

Replacement

 

Fig. 5. Wavelet shape parameters optimization process using GA.

Final Solution (optimal β and ωc) 2. Shrink the wavelet coefficients expressed in Equation 3 by soft thresholding,

$$\left| \text{V} \mathbf{V} \right\rangle^{\text{soft}} = \begin{cases} 0 & \left| \text{V} \mathbf{T} \right| < \text{thr} \\ \text{sign}(\text{V} \mathbf{T}) \left( \text{V} \mathbf{T} - \text{thr} \right) & \left| \text{V} \mathbf{T} \right| > \text{thr} \end{cases} \tag{4}$$

using soft-threshold function (*thr)* proposed by YANG and REN (2004),

$$\text{tfhr} = e^{-\left\{\text{Max}\left(\left\|\forall T(a,b)\right\|\right)^{\circ}\right\}} - e^{-\left\{\text{Max}\left(\left\|\forall T(a,b)\right\|\right)^{\circ}\right\}} \tag{5}$$

where *ξ*>0 is parameter governing the shape of the threshold function.

3. Perform the inverse wavelet transform to reconstruct the signal using the shrunken wavelet coefficients.

$$\stackrel{\*}{\text{ax}}(t) = \mathbb{C}\_{\text{g}}{}^{-1} \int\_{-a}^{a} \text{V} \, \text{T}^{\text{soft}} \, \text{ (}a, t\text{)} \quad \frac{da}{a^{3/2}} \tag{6}$$

c. Evaluate the auto-correlation function Rx (τ) for the de-noised signal x\*(t) to estimate the periodicity of the extracted impulses

$$R\_x(\tau) = E\left[\stackrel{\bullet}{x}(t) \stackrel{\bullet}{.} \infty (t + \tau)\right] \tag{7}$$

where *τ* is the time lag, and *E [ ]* denotes ensemble average value of the quantity in square brackets.

#### **3.1.3 Applications 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 impulse wavelet has been used as the wavelet base-function. The wavelet parameters (damping factor and centre frequency) are optimized based on maximizing the kurtosis value for the wavelet coefficients as shown in Figure 6.

To evaluate the performance of the proposed method, the autocorrelation functions of the optimized impulse wavelet, impulse wavelet with non-optimized parameters, and the widely used Morlet wavelet are carried out and shown in Figure 7. The comparison of Figures 7a, b and c, shows the increased effectiveness of the optimized impulse wavelet over non-optimized impulse and Morlet wavelets for extraction of the bearing fault impulses and corresponding periodicity. Consequently, the performance of the bearing fault diagnosis process has been improved using the proposed technique.

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

For a rolling element bearing with specifications as given in Table 2, the calculated BCFs (appendix A) for a shaft rotational speed of 1797 rev/min are 107.36 Hz and 162.18 Hz for outer and inner-race faults respectively. Figure 8 (a and d) shows the time domain waveform of the simulated signals for the rolling bearing with outer and inner-race faults based on the bearing vibration mathematical model (Khalid F. Al-Raheem *et al.* 2008). The

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 323

(a) (b)

(c) Fig. 7. The autocorrelation function of the wavelet de-noised outer-race fault signal using (a) optimized impulse-wavelet, (b) non-optimized impulse-wavelet, and (c) Morlet-wavelet.

Based on the bearing parameters given in Table 2, the calculated outer race fault

51.16 11.9 8 0 3.069 4.930 4.066

Figures 9 to 11 show the application of the proposed wavelet de-noising technique for the rolling bearing with outer-race fault at different shaft rotational speed. The bearing fault impulses and corresponding periodicity are easily discerned in the wavelet de-noised signal and the de-noised autocorrelation function, respectively. Comparison of Figures 9 to 11

Defect Frequencies (multiple of running speed, Hz) Outer-race Inner-race Rolling

element

characteristic frequencies (*FBPO*) for different shaft speeds are shown in Table 2.

α (degree)

Table 2. Bearing specification: Deep groove ball bearing RHP LJT 1 ¼.


figmo1-x130

**Delay (s)**

X: 0.009333 Y: 0.3597

Autocorrelation

X: -0.009333 Y: 0.3597

**(b) Experimental vibration data** 

Nb (ball)

Db (mm)

Dp (mm)


**Correlation coeff.**

result of the wavelet de-noising method (wavelet transform, shrink the wavelet coefficients and take the inverse wavelet transform) for the rolling bearing with outer and inner race faults using the optimized impulse wavelet and the corresponding autocorrelation function are displayed in Figure 8 (b, c e and f). The results show that the signal noise has been diminished and the impulses generated by the faulty bearing are easy to identify in the wavelet de-noised signal. The impulse periodicity of 0.00975 sec (*FBPO*=102.564 Hz) for outer-race fault and 0.006167 sec (*FBPI*=162.153 Hz) for inner-race fault are effectively extracted through the auto-correlation of the de-noised signal and exactly match the theoretical calculation of the BCF.

Fig. 6. The optimal values for Laplace wavelet parameters based on maximum kurtosis for,(a) simulated outer-race fault,(b) the measured outer-race fault,(c) the CWRU vibration data.

(c)

result of the wavelet de-noising method (wavelet transform, shrink the wavelet coefficients and take the inverse wavelet transform) for the rolling bearing with outer and inner race faults using the optimized impulse wavelet and the corresponding autocorrelation function are displayed in Figure 8 (b, c e and f). The results show that the signal noise has been diminished and the impulses generated by the faulty bearing are easy to identify in the wavelet de-noised signal. The impulse periodicity of 0.00975 sec (*FBPO*=102.564 Hz) for outer-race fault and 0.006167 sec (*FBPI*=162.153 Hz) for inner-race fault are effectively extracted through the auto-correlation of the de-noised signal and exactly match the

theoretical calculation of the BCF.

X: 0.8 Y: 18 Z: 8.211e+004

0

figk2-s1outer

0 5

10

**Kurtosis**

**Centre Frequency (Hz)**

data.

0.5

**Cetre Frequency,wc (Hz)**

**Kurtosis**

1

**Damping Factor**

1.5

2

(a) (b)

figk3-X130

0

(c)

Fig. 6. The optimal values for Laplace wavelet parameters based on maximum kurtosis for,(a) simulated outer-race fault,(b) the measured outer-race fault,(c) the CWRU vibration

X: 0.9 Y: 17 Z: 4.817e+004

0.5

1

**Damping Factor, Beta**

1.5

2

**Kurtosis**

0

figk1-b4s1

X: 0.6 Y: 12 Z: 1.956e+007

0 5

**Centre Frequency, wc (Hz)**

0.5

1

**Damping Factor, Beta**

1.5

2

Fig. 7. The autocorrelation function of the wavelet de-noised outer-race fault signal using (a) optimized impulse-wavelet, (b) non-optimized impulse-wavelet, and (c) Morlet-wavelet.

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

Based on the bearing parameters given in Table 2, the calculated outer race fault characteristic frequencies (*FBPO*) for different shaft speeds are shown in Table 2.


Table 2. Bearing specification: Deep groove ball bearing RHP LJT 1 ¼.

Figures 9 to 11 show the application of the proposed wavelet de-noising technique for the rolling bearing with outer-race fault at different shaft rotational speed. The bearing fault impulses and corresponding periodicity are easily discerned in the wavelet de-noised signal and the de-noised autocorrelation function, respectively. Comparison of Figures 9 to 11

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 325

Fig. 9. (a) the collected vibration signal, (b) the corresponding wavelet de-noised signal, and (c) the auto-correlation function, for bearing with outer-race fault at shaft rotational speed of

Fig. 10. (a) the collected vibration signal, (b) corresponding wavelet de-noised signal, and (c) auto-correlation function, for bearing with outer-race fault at shaft rotational speed of

Fig. 11. (a) the collected vibration signal, (b) corresponding wavelet de-noised signal, and (c)

auto-correlation function, for rolling with outer-race fault at shaft rotational speed of

983.887 rev/min.

2080.28 rev/min.

3541.11 rev/min.

shows the sensitivity of the proposed de-noising technique to the variation of the *FBPO* as a result of variation in the shaft rotational speed as listed in Table 3.


Table 3. The calculated and extracted (FBPO) at different shaft rotational speeds.

Fig. 8. The simulated vibration signal, the corresponding wavelet de-noised signal and the auto-correlation function Rx (τ) for bearing with outer-race fault (a, b and c), Inner-race fault (d, e and f) respectively.

shows the sensitivity of the proposed de-noising technique to the variation of the *FBPO* as a

983.887 50.32 0.020310 49.236 2080.28 106.4 0.009297 107.561 3541.11 181.12 0.005391 185.493

Fig. 8. The simulated vibration signal, the corresponding wavelet de-noised signal and the auto-correlation function Rx (τ) for bearing with outer-race fault (a, b and c), Inner-race fault

Table 3. The calculated and extracted (FBPO) at different shaft rotational speeds.

Extracted Period (sec)

Extracted *FBPO* (Hz)

result of variation in the shaft rotational speed as listed in Table 3.

Calculated *FBPO* (Hz)

Shaft Speed (rev/min)

(d, e and f) respectively.

Fig. 9. (a) the collected vibration signal, (b) the corresponding wavelet de-noised signal, and (c) the auto-correlation function, for bearing with outer-race fault at shaft rotational speed of 983.887 rev/min.

Fig. 10. (a) the collected vibration signal, (b) corresponding wavelet de-noised signal, and (c) auto-correlation function, for bearing with outer-race fault at shaft rotational speed of 2080.28 rev/min.

Fig. 11. (a) the collected vibration signal, (b) corresponding wavelet de-noised signal, and (c) auto-correlation function, for rolling with outer-race fault at shaft rotational speed of 3541.11 rev/min.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 327

Fig. 12. The CWRU collected vibration signal, corresponding wavelet de-noised signal and auto-correlation function, respectively for bearing with (a) outer-race fault, and (b) inner-

race fault.

### **(c) CWRU vibration data**

The vibration data for deep groove ball bearings (bearing specification shown in Table 4) with different faults were obtained from the Case Western Reserve University (CWRU) website (Bearing Data Center, seeded fault test data, *http://www.eecs.case.edu/*).


Table 4. Bearing specification: Deep groove ball bearing SKF 6205.

At a shaft rotational speed of 1797 rev/min, the calculated BCF for the bearing specifications given in Table 4, are 107.36 Hz for an outer-race fault and, 162.185 Hz for an inner-race fault. The time course of the vibration signals for bearing with outer and inner race faults, the corresponding wavelet de-noised signal and the auto-correlation function are depicted in Figure 12. The autocorrelation functions of the de-noised signal reveal a periodicity of 0.009333 sec (*FBPO*=107.14 Hz) and 0.006167 sec (*FBPI*=162.153 Hz) for outer and inner race fault respectively, which are very close to the calculated BCF.

### **3.2 The wavelet envelope power spectrum**

To avoid the wavelet admissibility condition (e.g. double sided wavelet function) which is essential in the inverse wavelet transforms (Mallat, 1999). And to be able to use a single side wavelet function which provides more similarity with the bearing fault pulses. A second approach for bearing fault detection based on the analysis of the wavelet coefficients is developed in this section. The WT coefficients using a single-sided function so called Laplace wavelet have been analyzed in frequency domain using a novel wavelet envelope power spectrum technique.
