AL-RAHEEM Khalid F., ROY Asok, RAMACHANDRAN K.P., HARRISON David K., GRAINGER Steven: The Exploitation of Wavelet de-Noising to Detect Bearing Faults (Wykorzystanie akustyki w identyfikacji uszkodzeń łożyska)

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THE EXPLOITATION OF WAVELET DE-NOISING TO

DETECT BEARING FAULTS

WYKORZYSTANIE AKUSTYKI W IDENTYFIKACJI

USZKODZEŃ ŁOŻYSKA

AL-RAHEEM Khalid F.

_{, ROY Asok}

_{, RAMACHANDRAN K.P.}

_,

HARRISON David K.

_{, GRAINGER Steven}

(1, 3) Deptartment of Mechanical and Industrial Eng., Caledonian College of Eng., Oman (2, 4, 5) School of Engineering Science and Design / Glasgow Caledonian University, Scotland, UK

E-mail: (1) khalid@caledonian.edu.om, (2) A.Roy@gcal.ac.uk, (3) ramkp@caledonian.edu.om, (4) D.K.Harrison@gcal.ac.uk, (5) S.grainger@gcal.ac.uk

Abstract. Failure diagnosis is an important component of the Condition Based

Maintenance (CBM) activities for most engineering systems. Rolling element bearings are the most common cause of rotating machinery failure. The existence of the amplitude modulation and noises in the faulty bearing vibration signal present challenges to effective fault detection method. The wavelet transform has been widely used in signal de-noising due to its extraordinary time-frequency representation capability. In this paper, we proposed new approach for bearing fault detection based on the autocorrelation of wavelet de-noised vibration signal through a wavelet base function derived from the bearing impulse response. To improve the fault detection process the wavelet parameters (damping factor and center frequency) are optimized using maximization kurtosis criteria to produce wavelet base function with high similarity with the impulses generated by bearing defects, that leads to increase the magnitude of the wavelet coefficients related to the fault impulses and enhance the fault detection process. The results show the effectiveness of the proposed technique to reveal the bearing fault impulses and its periodicity for both simulated and real rolling bearing vibration signals.

Keywords: Kurtosis maximization, wavelet de-noising, autocorrelation, Bearing fault detection, impulse-response wavelet.safety, dependability, technical object, decision-making support, engineering.

1. Introduction

When a rolling element hits a defect in the raceway of a bearing an impulse of short duration is generated which in turn excites the bearing system resonance frequencies. Therefore, the overall vibration signal measured on the bearing housing shows a pattern consisting of succession of oscillating bursts dominated by the major bearing system resonance frequency. The period of the impulse is extremely short compared with the interval between impulses, and so its energy is distributed at a very low level over a wide range of frequency and hence can be easily masked by noise and low frequency effects. Theses impulses will occur with a frequency determined by the velocity of the rolling element, the location of the defect and the bearing geometry and denoted as bearing characteristic frequencies (BCF) [1]. The rolling elements experience some slippage as the rolling elements enter and leave the bearing load zone, as a consequence the occurrence of the impacts never produce exactly at the same position from cycle to another, more ever when the position of the defect is moving with respect to the load distribution of the bearing the impulses are modulated in amplitude, as a result the periodicity and the amplitude of the impulses experience a certain degree of randomness.

In such cases the signal is not strictly periodic, but cyclo-stationary of second order [2, 3], or quasi-periodic [4]. All theses make being the bearing defects so difficult to detect by conventional FFT analysis that assumes the signal to be analyzed must be strictly periodic. Therefore, effective methods for feature extraction from noisy signals should be used.

The wavelet transform provides a potent multi-resolution analysis in both time and frequency domain and thereby becomes a favoured tool to extract the transitory features of non-stationary vibration signal produced by the faulty bearing [5,6]. The wavelet analysis results in a series of wavelet coefficients, which indicate how close the signal is to the particular wavelet. In order to extract the fault features of the signal more effectively appropriate wavelet base function should be selected [7,8]. To eradicate the effects of the signal noise from the resulting wavelet coefficients a number of initial methods are applied [9,10]. The wavelet de-noising technique included of decomposes the signal using wavelet transform, threshold the resulting coefficients to eliminate the redundant information and further enhance the interested spectral features of the signal, then reconstruct the signal from the threshold wavelet coefficients using inverse wavelet transform [11,12].

In this paper the wavelet de-noising technique is applied for bearing fault detection. The wavelet base function is derived based on the response of the bearing system to the impulse input to be similar to impulses produced by localized bearing fault. The wavelet parameters are optimized using maximum kurtosis criteria. This paper is organized as follows. In the next section the vibration model for rolling bearing with outer and inner-races fault is introduced. Then in section 3 the procedures of the proposed approach is set up. In section 4 the implementation of the proposed approach to the detection of localized ball bearing defects for both simulated and actual bearing vibration signals are presented. Conclusions are finally given in section 5.

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2. The Vibration Model for determining Localized Defects

Each time the rolling element pass over a defect in the raceway or every time a defect in the rolling element hits the raceway an impulse of vibration for short duration is produced which in turn excites the natural frequencies of the bearing system and housing structure. Therefore, the overall vibration signal measured on the bearing housing shows a pattern consisting of a sequence of oscillating bursts dominated by the major resonance frequency of the structure.

The response of the bearing structure (as an under damped second order mass-spring-damper system) to the generated fault impulses can be simulated by:

(1) where ζ is the damping ratio and ωd is the damped natural frequency of the bearing structure.

When the shaft rotates, the bearing impulse response will occurs periodically with an impact frequency depending on the fault location, shaft speed and the bearing geometry and denoted as the Bearing Characteristic Frequency BCF [2, 7]. In reality, there are: a slight random variation for the spacing between impulses, and an impulse amplitude modulation as a result of changing the load angle for each rolling element when it passes through the bearing load zone. Therefore, the overall vibration signal for the rolling bearing with an incipient fault can be simulated by:

(2) where S(t-Ti) is the waveform generated by ith impact at the time Ti, Ti = iT+τi where

T is the average impulses period, τi describe

the random slips of the rolling elements. Ai

is the time varying amplitude-demodulation, and n(t) is an additive background noise which takes into account the effects of the other vibrations in the bearing structure.

The spectrum of such a signal would consist of a harmonic series of frequency components spread at the bearing defect frequency. Theses frequency components are flanked by sidebands if there is an amplitude modulation (in case of inner and rolling element defects).

Figures 1a and 1d show the simulated vibration signals based on equation (2), for both outer and inner-race bearing faults respectively.

Fig. 1. The simulated vibration signal, wavelet de-noised signal and, the autocorrelation

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Fig. 1 (cont.). The simulated vibration signal, wavelet de-noised signal and, the auto-correlation function for outer-race fault (c),

Inner-race fault (d, e and f).

3. The Wavelet De-Noising Technique

The wavelet transform is the inner product of a time domain signal with the translated and dialed wavelet-base function. The ensuing wavelet transform 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 characteristics to the bearing impulse response generated by presence of bearing incipient fault (Eq.1). Base on that the proposed wavelet-base function is denoted as impulse-response wavelet and given by:

(3) where β is the damping factor that control the decay rate of the exponential envelop in the time and hence regulate the resolution of the wavelet, concurrently corresponds to the frequency band width of the wavelet in the frequency domain, the wc determining

the number of significant oscillations of the wavelet in the time domain and correspond to the wavelet centre frequency in the frequency domain, and A is an arbitrary scaling factor.

Figure 2 shows the proposed wavelet and its power spectrum. The proposed wavelet satisfy the admissibility condition:

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Fig. 2. The impulse-response wavelet (a) with its spectrum (b).

where: Cg is the admissibility constant and, ψ^ _{(f) is the Fourier transform of ψ (t). This}

implies that the wavelet has no zero frequency component, ψ^_{(0) =0, or put it}

another way, the wavelet ψ(t) must have a zero mean.

The proposed wavelet De-nosing technique consists of the following steps:

1. Optimize the wavelet shape parameters (β and wc) based on maximization

kurtosis of the signal- wavelet inner product,

2. Perform a wavelet transform of the bearing vibration signal using the optimized wavelet (5):

(5) where <.> indicates the inner product, the superscript asterisk '*_{' stands for 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; 3. Shrink the wavelet coefficients expressed

in Eq.(5) using soft-threshold function proposed by [10] (6):

(6) where ξ>0 is parameter governing the shape of the threshold function; 4. Perform the inverse wavelet transform to

reconstruct the signal using the shrunken wavelet coefficients (7),

(7) 5. Evaluate the auto-correlation function for the de-noised signal to estimate the periodicity of the extracted impulses.

4. The Implementation of wavelet De-Noising for bearing fault Detection

To demonstrate the performance of the proposed wavelet de-noising technique, this section presents several application examples for detection of localized bearing faults. In all the examples, impulse response wavelet has been used as a wavelet base-function. The wavelet parameters (damping factor and centre frequency) are optimized based on maximizing the kurtosis value for the wavelet transform coefficients.

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4.1. Application to the simulation Signals

The performance of the proposed de-noising method was examined by simulation signals based on the vibration model described in section 2. Based on the assumed bearing specification and

operating conditions the calculated BCF for an outer-race fault is 107.36 Hz and for an inner-race fault is 162.18 Hz. Figures 1a and 1b, show the time domain waveform of simulation signal for rolling bearing with outer and inner-race faults. The result of the signal de-noising using proposed wavelet de-noising method for outer and inner race faults are displayed in Figure 1c and 1d respectively. The results show that the noise of the signal has effectively removed and the impulses of the defective bearing are easy to identify. The impulses periodicity is easily extracted through the

auto-correlation of the de-noised signal (see Figures 1e and 1f), which is 0.00975 sec (fo=102.564 Hz) for outer-race fault and

0.006167 sec (fi=162.153 Hz) for inner-race

fault and it is close to the calculated BCF. The optimal values for wavelet parameter damping factor (β) and the center frequency (wc) for outer-race fault signal are shown in

Figure 3a.

4.2. Application to Experimental Data

The collected vibration signals for a deep groove ball bearing with artificial outer race fault collected at different shaft rotational speeds are displayed in Figures: 4a, 5a, and 6a.

The bearing time waveform vibration signal was recorded through B&K 752A12 piezoelectric accelerometer mounted horizontally on the bearing housing. The output from the accelerometer passed to the

PC through B&K controller module type 7536 at sampling rate 12.8 KHz. Based on the bearing parameters the calculated outer race fault BCF as a function of the shaft rotational speed is equal to 0.05115*RPM, see Table.1. 0 0.5 1 1.5 2 0 5 10 15 200 2 4 6 8 10 x 104 Damping Factor figk2-s1outer X: 0.8 Y: 18 Z: 8.211e+004 Centre Frequency (Hz) K u rt o si s 0 0.5 1 1.5 2 0 5 10 15 20 0 0.5 1 1.5 2 x 107

Damping Factor, Beta

figk1-b4s1 X: 0.6 Y: 12 Z: 1.956e+007 Centre Frequency, wc (Hz) K u rt o si s 0 0.5 1 1.5 2 0 5 10 15 200 1 2 3 4 5 x 104

Damping Factor, Beta

figk3-X130 X: 0.9 Y: 17 Z: 4.817e+004 Cetre Frequency,wc (Hz) K u rt o si s a) b) c)

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Fig. 3. The optimized wavelet parameters based on maximization of the kurtosis value for (a)

simulated vibration signal, (b) the measured

signals, and (c) the CWRU signals.

Fig. 4. The time domain vibration signal (a), the wavelet de-noised signal (b), and the auto-correlation function (c), for rolling bearing with

outer-race fault at shaft rotational speed of 983.887 RPM.

Fig. 5. The time domain vibration signal (a), the wavelet de-noised signal (b), and the auto-correlation function (c), for rolling bearing with

outer-race fault at shaft rotational speed of 2080.28 RPM. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -8 -6 -4 -2 0 2 4 6 8 Time (s) ac ce le ra ti o n ( m .s -2) figmo9-b4s1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 X: 0.06383 Y: 0.4148 X: 0.0443 Y: 1.547 Time (s) A cc el er at io n ( m .s -2) figmo8-b4s1 -0.05 -0.04 -0.03-1 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 X: -0.02031 Y: 0.153 X: 0.02031 Y: 0.153 Delay (s) C o rr el at io n c o ef f. figmo7-b4s1 Autocorrelation 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -40 -30 -20 -10 0 10 20 30 40 Time (s) A cc el er at io n ( m .s -2) figmo15-b4s6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -3 -2 -1 0 1 2 3 X: 0.02336 Y: 2.061 X: 0.03242 Y: -1.692 Time (s) A cc el er at io n ( m .s -2) figmo14-b4s6 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 X: -0.009297 Y: 0.2485 Delay (s) C o rr el at io n c o ef f. figmo13-b4s6 X: 0.009297 Y: 0.2485 Autocorrelation

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Table.1 The calculated outer race fault BCF as a function of the shaft rotational speed

The optimized wavelet parameters are depicted in Figure 3b. With application of the proposed wavelet de-noising technique the impulses generated by the outer-race fault are easily defined, Figures 4b - 6b. The periodicity of these impulses is easy to identify through the auto-correlation of the de-noised signal, and it is closed to the calculated BCF, Figures 4c-6c. Through the comparison of Figures 4 - 6, the sensitivity of the proposed de-noising technique to the variation of shaft rotational speed can be concluded.

4.3. Application to Vibration Data

The time domain vibration signals provided by CWRU bearing data centre website for rolling bearings with outer-race & inner-race faults, and the corresponding de-noised signal and its auto-correlation applying the proposed de-noising method are depicted in Figure 7.

Fig. 7(a). The CWRU website vibration signal with the wavelet de-noised signal and the auto-correlation function for bearing with outer-race

fault

The calculated BCF are, 107.36 Hz for outer-race fault and, 162.185 Hz for inner-race fault. The optimized wavelet

parameters are shown in Figure 3c. The auto-correlation function of the de-noised signal shows the periodicity of 0.009333 sec (fo=107.14 Hz) for outer race fault and

0.006167 sec (fi=162.153 Hz) for inner race

fault. 0 0.05 0.1 0.15 0.2 0.25 -4 -3 -2 -1 0 1 2 3 4 Time (s) A cc el er at io n ( m .s -2) figmo3-x130 0 0.05 0.1 0.15 0.2 0.25 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 X: 0.02475 Y: 0.3852 Time (s) A cc el er at io n ( m .s -2) figmo2-x130 X: 0.03392 Y: -0.3649 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 X: -0.009333 Y: 0.3597 X: 0.009333Y: 0.3597 Delay (s) C o rr el at io n c o ef f. figmo1-x130 Autocorrelation

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4.Conclusions

This paper has presented a procedure is for rolling bearing fault diagnosis based on

Fig. 7 (b). The CWRU website vibration signal with the wavelet de-noised signal and the

auto-correlation function for bearing with inner-race

fault.

wavelet de-noising technique with wavelet-base function derived from the bearing system response to the impulse input. The results for both simulated and actual bearing vibration data show the

effectiveness of the proposed approach for extraction the rolling bearing fault impulses hidden in the noisy vibration signal, and evaluate its periodicity.

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