Méthodologie de Recherche

   Gearbox fault diagnosis using ensemble empirical mode

decomposition (EEMD) and residual signal  

Author: Bouaouiche Karim

Abstract –This paper presents the application of new time frequency method, ensemble empirical mode

decomposition (EEMD), in purpose to detect localized faults of damage at an early stage. EEMD is a

self adaptive analysis method for non-linear and non-stationary signals and it was recently proposed by

Huang and Wu to overcome the drawbacks of the traditional empirical mode decomposition (EMD). The

vibration signal is usually noisy. To detect the fault at an early stage of its development, generally the

residual signal is used. There exist different methods in literature to calculate the residual signal, in this

paper we mention some of them and we propose a new method which is based on EEMD. The results given

by the different methods are compared by using simulated and experimental signals.

Keywords:Ensemble empirical mode decomposition (EEMD) / residual signal / gearbox fault diagnosis /

fault detection / rotating machines

INTRODUCTION [Modifier]

Fault diagnosis of gearboxes has shown a great devel-

opment in techniques based on the analysis of vibration

signals [1–6], because vibration signals carry a great deal

of information, which can be used to detect early faults in

rotating machines. However, vibrations signals are influ-

enced by vibration from many sources. Thus, the resulting

signals are non stationary and nonlinear. To analyze such

signals, time-frequency analysis has been applied to fault

diagnosis of gearboxes in order to combine the advantages

of both time and frequency domains.

Empirical Mode Decomposition (EMD) is a time-

frequency analysis method, recently proposed by Huang

et al. for the study of ocean waves [7, 8]. The method

has been developed and has been widely used [9–13].

In the field of fault diagnosis of rotating machines, the

EMD method has also been widely applied for identifica-

tion of faults [5, 14–20].

EMD is based on the local characteristic time scale of

a signal and could decompose the complicated signal into

a set of elementary signals called Intrinsic Mode Func-

tions (IMFs). The IMFs represent the nature oscillation

mode embedded in the signal and are determined by the

signal itself. Thus, EMD is a self adaptive signal process-

ing method and acts as a filter bank [21]. However, the

original EMD has some drawbacks [22,23]. One of major drawbacks of the original EMD is the frequent appearance

of mode mixing [22, 24], which is defined as a single in-

trinsic mode function (IMF), either consisting of signals

of widely disparate scales, or a single of a similar scale

residing in different IMF components. To overcome the

problem of mode mixing in EMD, a noise assisted data

analysis (NADA) method was proposed by Huang and

Wu [25], it was called ensemble empirical mode decompo-

sition EEMD. This method defines the true IMF compo-

nents as the mean of an ensemble of trials, each consisting

of the signal plus a white noise of finite amplitude. How-

ever, the measured signal is generally contaminated by

noise that hides the information which is in direct rela-

tion with faults and may increase the amplitude of noise

used by EEMD.

To alleviate these difficulties, in this work we use the

EEMD method to calculate the residual signal (RS). The

RS is obtained by removing some IMFs which represent

the noise, the harmonics of the tooth meshing frequency

and the regular signal.

By applying EEMD method in the calculation of the

residual signal, we can decompose the signal at different

levels and the change in the vibration signals caused by

the localized fault is even more visible and the damage

can be early identified.

The structure of the paper is as follows: Section 2

introduces the basic of EMD. Section 3 is dedicated

to EEMD method. Section 4 presents the method and

the procedure of the residual signal based on EEMD.   

2.Empirical mode decomposition

(EMD) method

2.1 Review stage

The empirical mode decomposition EMD is basically

the output of an iterative algorithm [7, 8], it admits no

analytical definition. The signal x(t) can be decomposed

as follows:

1. Identify all the local extrema, and then connect all

the local maxima by a cubic spline line as the upper

envelope.

2. Repeat the procedure for the local minima to produce

the lower envelope. The upper and lower envelopes

should cover all the data between them.

3. The mean of upper and lower envelopes value is desig-

nated as m1(t), and the difference between the signal

x(t) and m1(t) is h1(t)

h1(t) = x(t) − m1(t) (1)

4. If h1(t) is an IMF, then h1(t) is the first component

of x(t).

5. If h1(t) is not an IMF, h1(t) is treated as the original

signal and repeat steps (1–3); we got:

h11(t) = h1(t) − m11(t) (2)

in which, m11(t) is the mean of upper and lower en-

velopes value of h1(t).

6. After repeated sifting process K times, h1k(t) becomes

an IMF, that is

h1k(t) = h1(k−1)(t) − m1k(t) (3)

then, it is designated c1(t) = h1k(t) as the first IMF

component from the original data. c1(t) should contain

the finest scale or the shortest period component of the

signal.

7. Separate c1(t) from x(t), we could get:

r1(t) = x (t) − c1(t) (4)

8. r1(t) is treated as the original data, and repeat the

above processes, the second IMF component c2(t) of

x(t) could be got.

9. Let us repeat the process as described above for n

times, then n-IMFs of signal x(t) could be got. Then,

r1(t) − c2(t) = r2(t)

rn−1(t) − cn(t) = rn(t). 

3 Ensemble empirical mode decomposition

(EEMD) method

The major drawback of the original EMD is the mode

mixing [24, 25], which is the consequence of signal inter-

mittence. The intermittence could cause the aliasing prob-

lem and makes the physical meaning of the IMF inclear.

To overcome the mode mixing separation problem, a new

noise-assisted data analysis (NADA) method is proposed.

This method is named the Ensemble EMD (EEMD) [25],

it defines the true IMF components as the mean of an en-

semble of trials, each consisting of the signal plus a white

noise of finite amplitude.

The proposed EEMD is defined as follows:

1. Add a white noise y(t) to the original signal x(t) to

generate a new signal:

xk(t) = x(t) + βk.y(t) (7)

βk is a fraction of the standard deviation of the origi-

nal signal x(t).

2. Use the EMD to decompose the generated signals

xk(t) into n IMFs cjk(t), j = 1, . . . , n, where cjk(t)

is the jth IMF of the kth trial.

3. Repeat steps (1) and (2) K times with different white

noise series each time to obtain an ensemble of IMFs:

cjk(t), k = 1,...,K.

4. Determine the ensemble mean of the K trials for each

IMF as the final result:

cj (t) = lim K→∞

K

k=1

cjk(t), j = 1,...,n (8)

To remove the influence of some indesired components

as the noise and to show clearly the signal components

generated by the crack damage, we propose to use the

residual signal.

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