**3.2.1 Laplace wavelet function**

The Laplace wavelet is a complex, single side damped exponential function formulated as an impulse response of a single mode system to be similar to data features commonly encountered in health monitoring tasks. It has been applied to the vibration analysis of an aircraft for aerodynamic and structural testing (Lind and Brenner, 1998), and to diagnose the wear of the intake valve of an internal combustion engine (Yanyang *et al*., 2005).

The Laplace wavelet is a complex, analytical and single-sided damped exponential given by,

$$\begin{array}{ll} \Psi(t) &= A \, \, \, \epsilon \left( \frac{\rho}{\sqrt{1-\rho^2}} + j \right)^{\alpha\_c t} & \text{if } \qquad \qquad \qquad \qquad \qquad \qquad t \ge 0 \\\\ \Psi(t) &= 0 & \text{where} \qquad \qquad t < 0 \end{array} \tag{8}$$

Where *β* is the damping factor and *ω<sup>c</sup>* is the wavelet centre frequency*.* Figure 13 shows the Laplace wavelet, its real part, imaginary part, and spectrum. The wavelet shape parameters *β and ωc* have been optimized using GA based on the maximization of the kurtosis value, Equation 2.

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)

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

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

The Laplace wavelet is a complex, single side damped exponential function formulated as an impulse response of a single mode system to be similar to data features commonly encountered in health monitoring tasks. It has been applied to the vibration analysis of an aircraft for aerodynamic and structural testing (Lind and Brenner, 1998), and to diagnose the

The Laplace wavelet is a complex, analytical and single-sided damped exponential given by,

<sup>2</sup> <sup>1</sup> ( ) <sup>0</sup>

(8)

() 0 0

Where *β* is the damping factor and *ω<sup>c</sup>* is the wavelet centre frequency*.* Figure 13 shows the Laplace wavelet, its real part, imaginary part, and spectrum. The wavelet shape parameters *β and ωc* have been optimized using GA based on the maximization of the kurtosis value,

*t where t*

wear of the intake valve of an internal combustion engine (Yanyang *et al*., 2005).

 

*<sup>c</sup> j t t Ae if t*

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

website (Bearing Data Center, seeded fault test data, *http://www.eecs.case.edu/*).

39.04 7.94 9 0 3.5858 5.4152 4.7135

**(c) CWRU vibration data** 

Db (mm) Nb (ball) α (degree)

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

fault respectively, which are very close to the calculated BCF.

**3.2 The wavelet envelope power spectrum**

power spectrum technique.

Equation 2.

**3.2.1 Laplace wavelet function** 

Dp (mm)

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) innerrace fault.

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 329

Fig. 14. The Kurtosis distribution for the wavelet transforms scales using (a) Morlet wavelet,

The vibration signal of a faulty rolling bearing can be viewed as a carrier signal at a resonant frequency of the bearing housing (high frequency) modulated by a decaying envelope. The frequency of interest in the detection of bearing defects is the modulating frequency (low frequency). The goal of the enveloping approach is to replace the oscillation caused by each impact with a single pulse over the entire period of the

The WT of a finite energy signal *x(t)*, with the mother wavelet *ψ(t)*, is the inner product of

\*

( )

(9)

*i t*

, ,

*a b a b*

*a WT a b j WT a b A t e*

2 2 *A t EWT a b WT a b WT a b* ( ) ( , ) {Re[ ( , )]} {Im[ ( , )]} (10)

Re[ ( , )] Im[ ( , )] ( )

The time-varying function *A(t)* is the instantaneous Enveloped Wavelet Transform (*EWT*) which extracts the slow time variation of the signal (modulating frequency) is given by,

To extract the frequency content of the enveloped correlation coefficients, the scale Wavelet

a,b . Since the analytical and complex wavelet is employed to calculate the wavelets transform,

<sup>1</sup> { ( ), , } ( ), ( ) ( ) ( )

*WT x t a b x t t x t t dt*

and (b) Laplace wavelet.

impact.

**3.2.2 Enveloped wavelet power spectrums** 

*x(t)* with a scaled and conjugate wavelet ψ\*

the result of the wavelet transform is also an analytical signal,

Power Spectrum (*WPS*) (energy per unit scale) is given by,

Fig. 13. (a) the Laplace wavelet, (b) the real part, (c) the imaginary part, and (d) wavelet spectrum.

To show the effectiveness of the proposed Laplace wavelet over the widely used Morlet wavelet, Figure 14 shows the scale-kurtosis distribution of the wavelet transform using Morlet and Laplace wavelets respectively, for different bearing conditions. The comparison of the two wavelets indicates the high sensitivity of the Laplace wavelet over the Morlet wavelet for bearing fault diagnosis.

Fig. 13. (a) the Laplace wavelet, (b) the real part, (c) the imaginary part, and (d) wavelet

To show the effectiveness of the proposed Laplace wavelet over the widely used Morlet wavelet, Figure 14 shows the scale-kurtosis distribution of the wavelet transform using Morlet and Laplace wavelets respectively, for different bearing conditions. The comparison of the two wavelets indicates the high sensitivity of the Laplace wavelet over the Morlet

spectrum.

wavelet for bearing fault diagnosis.

Fig. 14. The Kurtosis distribution for the wavelet transforms scales using (a) Morlet wavelet, and (b) Laplace wavelet.

#### **3.2.2 Enveloped wavelet power spectrums**

The vibration signal of a faulty rolling bearing can be viewed as a carrier signal at a resonant frequency of the bearing housing (high frequency) modulated by a decaying envelope. The frequency of interest in the detection of bearing defects is the modulating frequency (low frequency). The goal of the enveloping approach is to replace the oscillation caused by each impact with a single pulse over the entire period of the impact.

The WT of a finite energy signal *x(t)*, with the mother wavelet *ψ(t)*, is the inner product of *x(t)* with a scaled and conjugate wavelet ψ\* a,b .

Since the analytical and complex wavelet is employed to calculate the wavelets transform, the result of the wavelet transform is also an analytical signal,

$$\begin{aligned} \text{WIT} \{ \mathbf{x}(t), a, b \} &= < \mathbf{x}(t), \; \boldsymbol{\nu}\_{a,b}(t) > = \frac{1}{\sqrt{a}} \int \mathbf{x}(t) \; \boldsymbol{\Psi}\_{a,b}^\*(t) \; dt \\ &= \text{Re} [\mathsf{VMT}(a, b)] + j \, \text{Im} [\ \mathsf{VMT}(a, b)] = A(t) \, e^{i \; \theta(t)} \end{aligned} \tag{9}$$

The time-varying function *A(t)* is the instantaneous Enveloped Wavelet Transform (*EWT*) which extracts the slow time variation of the signal (modulating frequency) is given by,

$$A(t) = \text{EVIT}(a, b) = \sqrt{\left[\text{Re}[\text{VVT } (a, b)]\right]^2 + \left[\text{Im}[\text{ VVT } (a, b)]\right]^2} \tag{10}$$

To extract the frequency content of the enveloped correlation coefficients, the scale Wavelet Power Spectrum (*WPS*) (energy per unit scale) is given by,

$$\text{WPS}(a, o) = \int\_{-\infty}^{o} \left| \text{SEV} \nabla (a, o) \right|^2 \, dco \tag{11}$$

Wavelet Analysis and Neural Networks for Bearing Fault Diagnosis 331

Fig. 15. The wavelet-level power spectrum using (a) Morlet-wavelet, (b) Laplace-wavelet for

new and outer-race defective bearing.

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

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

$$TVPS = \int \left| \mathbf{x}(t) \right|^2 \, dt \quad = \frac{1}{2\pi} \int VPS(a, ao) \, da \tag{12}$$
