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Timeenergy density analysis based on wavelet transform

Cheng Junsheng*, Yu Dejie, Yang Yu

College of Mechanical and Automotive Engineering, Hunan University, Changsha 410082, Peoples Republic of China

Received 21 January 2005; accepted 6 February 2005

Available online 17 March 2005

Abstract

Energy is an important physical variable in signal analysis. The distribution of energy with the change of time and frequency can show the

characteristics of a signal. A timeenergy density analysis approach based on wavelet transform is proposed in this paper. This method cananalyze the energy distribution of signal with the change of time in different frequency bands. Simulation and practical application of the

proposed method to roller bearing with faults show that the timeenergy density analysis approach can extract the fault characteristics from

vibration signal efficiently.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Wavelet transform; Timeenergy density; Roller bearing; Fault characteristic

1. Introduction

The main goal of signal analysis lies in finding a simple

and effective signal transform method to present the main

characteristics of a signal. Usually, we analyze a signal in

time and frequency domain. Time domain analysis studies

the changing regulation of the signal forms with the change

of time, while frequency domain analysis studies the

changing regulation of signal energy or power with the

change of frequency. However, as for non-stationary

signals, a method that can combine time domain analysis

and frequency domain analysis together is expected. FFT

can only provide the energy distribution with the change of

time or frequency respectively [1,2]. The windowed Fourier

transform (WFT) display a time signal on a joint time

frequency plane. However, once the window function is

chosen, its size of the timefrequency window is fixed, so,the time and frequency resolution are same for all signals

including different time scales [3,4]. Wavelet is a time

frequency analysis method with adjustable window. With

the character of reflecting the localized information in time

and frequency domain simultaneously, wavelet transform

has been is extensively applied in signal analysis [57].

However, if the time and frequency domain information of

the results of wavelet transform and wavelet package

decomposing is going to be extracted, these results, that is,

the time domain waveform in certain frequency band,

should be re-handled in order to get needed time domain or

frequency domain results [8].

Energy is an important physical variable, whose

distribution with the change of time and frequency can

reflect the main characteristics of the signal. However, with

the limitation of Heisenberg Uncertainty Principle, we

cannot discuss such as the instantaneous energy density

jf(t)j2 and jF(u)j2 at a certain point in phase space (u,t), for

conceptually, to say that the energy with certain frequency

at certain time makes no sense [9,10]. While when we

come to frequency analysis, we always want to know the

distribution of signals energy with the change of time, and

as for non-stational signals we even want to know thedistribution of signals energy in some frequency bands with

the change of time. In this paper, the timeenergy density

analysis method based on wavelet transform is proposed.

The proposed method can analyze the distribution of

signals energy at each frequency band with the change of

time. According to the characteristic of the fault vibration

signals of roller bearings, the timeenergy density analysis

method has been applied to the roller bearing fault

diagnosis. Simulation and experiments of roller bearing

with faults show that the approach can extract the fault

characteristic of signal efficiently.

NDT&E International 38 (2005) 569572

www.elsevier.com/locate/ndteint

0963-8695/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ndteint.2005.02.002

* Corresponding author. Tel.:C 86 7318821744; fax:C86 7318711911.

E-mail address: signalp@tom.com (C. Junsheng).

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2. The timeenergy density analysis approach based

on wavelet transform

Given: j(t)2L2(R)hL(R) and j0Z0, then the func-

tion family {ja,b(t)} is produced as following [9,10]

ja;btZ jajK1=2

jtKb

a

a; b2R; as0 (1)

ja,b(t) is called analyzing wavelet or continuous wavelet;j(t) is called basic wavelet or mother wavelet and ju is

Fourier transform ofj(t); a in Eq. (1) is the scale parameter

and b is the time parameter.

If the function f(t)2L2(R) owns finite energy, the

continuous wavelet transform of function f(t) is defined as

Wfa; bZ hft;ja;btiZ jajK1=2

R

ftjtKb

a

dt (2)

The wavelet transform is isometric, that is to say the

wavelet transform of f(t) is energy conservation, and then

the following formula can be obtained:R

jftj2

dtZ1

Cj

R

R

jWfa; bj2 dadb

a2(3)

where

CjZ

R

jjuj2

jujdu!N

is taken as the admissibility condition.

From the isometric character of wavelet [see formula

(3)], we get following:

hft;ftiZR

jftj2dtZ 1Cj

R

aK2daR

jWfa; bj2db (4)

where, because of the limitation of the Heisenberg

Uncertainty Principle, jWf(a,b)j2/Cja

2 cannot be taken as

instantaneous density. However, jWf(a,b)j2/Cja

2 can be

taken as the energy density function in plane (a,b). That is to

say that jWf(a,b)j2/Cja

2 gives the energy in space (aGDa,

bGDb). Thus, formula (4) can be put as:R

jftj2dtZ

R

Ebdb (5)

where,

EbZ1

Cj

R

jWfa; bj2

a2 da (6)

Formula (6) gives the energy value of the signal in the

time span bGDb. E(b) is called timeenergy density

function. It reflects the distribution of signals energy at

all frequency bands with the change of time parameter b.

The following formula (7) shows the distribution of signals

energy in integrating range [a1, a2] with the change of time

parameter b.

E0bZ

1

Cj

a2

a1

jWfa; bj2a2 da (7)

E0(b) is called local timeenergy density function and it

shows all energy of the signal in the range from scale

(frequency) a1 to scale (frequency) a2. By fetching different

values for a1 and a2, the distribution of the signal energy in

different bands with the change of time can be obtained.

3. The application of timeenergy density analysis

approach in fault diagnosis of roller bearing

The high frequency vibration caused by local fault of therotating roller bearing can inspire the resonant frequency of

the bearing vibration system. Given the pulse force as the

input of the bearing system, and the vibration signals picked

up by the sensor on the bearing seat as output, the vibration

signals of roller bearing with fault can be presented as:

ftZXLlZ1

t

KN

rtqtat eKsltKt cos ultKtdt (8)

where, r(t) is the pulse force, q(t) is the loads distribution

function of the roller bearing, a(t) shows the structure

character of transfer path between the pulse action position

and the sensor, sl and ul is the intrinsic character of thesystem and Lis the quantity of inspired resonant frequencies

of the system.

Fig. 1 is a typical time course of roller bearing vibration

signal with local fault. The non-stationary character of the

signal is obviously showed in this figure. Around the pulse

action time, abrupt change of the signal inspires the resonant

Table 1

The resonant frequency of simulation signal (unit: Hz)

u1 u2 u3 u4 u5 u6 u7 u8 u9 u10

3000 3500 4000 4500 5000 5500 6000 6500 7000 7500

Fig. 1. Time course of roller bearing vibration signal with typical local

fault.

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frequency of the bearing vibration system, and the intrinsic

vibration components will attenuate rapidly because of

damp. Whereas, at the time without pulse action, vibration

frequency of signal is relatively low and the vibration signal

consists of mainly noises and interfering signals, which,

perhaps, are caused by non-centering or non-balance of the

rotating elements.

According to formula (8), f(t), the vibration signal of

roller bearing with fault is modulated by the pulse force r(t),

and the fault characteristic frequency cannot be defined by

direct Fourier spectrum analysis. If the timeenergy density

analysis approach proposed by this paper is applied to this

vibration signal, we would choose appropriate scale

parameters a1 and a2 in formula (7), and let integrating

range locate within the range of resonant frequency of the

bearing system, because the inherent vibration components

centralize around the pulse action time, the value of E0(b)

will be relatively larger; while low-frequency interfering

and noises are mainly away from pulse action time, the

value ofE0(b) will be relatively smaller. Thus, the frequency

of E0(b) with the change of time is equal to the action

frequency of pulse. In this way, spectrum analysis of E0(b)

can be done, and the fault characteristic frequency of roller

bearing can be received according to the peak value of the

spectrum. Compared with sample frequency, the fault

character frequencies of roller bearing are quite low, in

order to improve the resolution of the timeenergy density

spectrum, zoom-spectrum analysis of low frequency range

of E0(b) is required.

3.1. Analysis of the simulation signal

Given, under certain operating condition, the outer-race

fault characteristic frequency of a roller bearing is 128 Hz;

sample frequency is 16,384 Hz; and let LZ10, 10 intrinsicfrequencies are given in Table 1; r(t), q(t), and a(t) are

fetched by Ref. [3], then we get Fig. 2 as the time domain

wave form of simulation signals of a roller bearing with

outer-race fault, and the background interfering is noises

with low-frequency.

Though some of the impact characters can be shown in

Fig. 2, the fault characteristic frequency (128 Hz) cannot befound in its Fourier spectrum due to the modulation of the

signal amplitude. By choosing appropriate scale parameters

a1Z1.5, a2Z2 to make integrating range locate within high

frequency band, the simulation signal is carried out

continuous wavelet transform with daubechies 10(D10)

wavelet base to get local timeenergy density E0(b), and

zoom-spectrum analysis is applied to E0(b) to get spectrum

as Fig. 3. From Fig. 3, spectrum peak value is easily defined

at the position of the fault characteristic frequency (128 Hz)

and its harmonic wave.

3.2. Application

Fig. 4 shows an actual acceleration time course of

vibration signal of a 6311-type roller bearing with outer-

race fault. The sample frequency is 16,384 Hz and the outer-

race fault characteristic frequency is f1oc Z76 Hz. With the

application of the timeenergy density approach, the local

timeenergy density E0(b) can be obtained. By carrying out

the zoom-spectrum analysis to E0(b), we get Fig. 5, in which

Fig. 3. Spectrum of simulation signals local timeenergy density of a roller

bearing with outer-race fault.

Fig. 2. The time domain waveform of simulation signal of a roller bearing

with outer-race fault.

Fig. 5. Zoom-spectrum of local timeenergy density of roller bearing with

outer-race fault.

Fig. 7. Zoom-spectrum of local timeenergy density of roller bearing with

inner-race fault.

Fig. 6. Accelerative vibration signal of roller bearing with inner-race fault.

Fig. 4. Accelerative vibrationsignal of a roller bearing with outer-race fault.

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the spectrum peak value at the position of the outer-race

fault characteristic frequency f1oc 76 Hz is easily defined.

Figs. 6 and 7 are the actual acceleration time course of

vibration and the zoom-spectrum of local timeenergy

density of 6311-type roller bearing with inner-race fault,

respectively. The inner-race fault characteristic frequency is

f2

ic Z99:2 Hz. In Fig. 7, the spectrum peak value at theposition of the inner-race fault characteristic frequency f2icZ99:2 Hz is easily defined.

4. Conclusion

As we all know, time scale and the corresponding energy

distribution are two most important parameters of a signal in

signal processing. For non-stational signals we want to

know the timefrequency energy distributions of it. FFT

could only give average energy of the signal in the time or

frequency domain respectively and could not give con-

sideration to the whole and local feature in the two domains

at the same time. Wavelet analysis is an effective tool for

signal processing and feature extraction. It can provide the

local time and frequency information simultaneously and is

especially suitable for non-stational signal processing.

However, if the time and frequency domain information

of the results of wavelet transform and wavelet package

decomposing is going to be extracted, these results, that is,

the time domain waveform in certain frequency band,

should be re-handled in order to get needed time domain or

frequency domain results. In this paper, timeenergy

density analysis approach based on wavelet transform is

proposed. It can analyze the distribution of the signalsenergy at different frequency bands with the change of time,

and extract the character of the signals. According to the

characteristics of the roller bearing vibration signals with

faults, the proposed method has been applied to the roller

bearing fault diagnosis. Simulation and experiments of

roller bearing with faults show that the timeenergy density

analysis approach can extract the fault character of signal

efficiently. This is of great practical significance in

mechanical fault diagnosis.

Acknowledgements

The support for this research under Chinese National

Science Foundation Grant (No. 502755050) is gratefully

acknowledged.

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