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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 23 (2009) 1554–1572

0888-32

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Use of autocorrelation of wavelet coefficients for fault diagnosis

J. Rafiee a,�, P.W. Tse b

a Department of Mechanical, Aerospace and Nuclear Engineering, Jonsson Engineering Center, 110 8th Street, Rensselaer Polytechnic Institute, Troy,

NY 12180-3590, USAb Smart Engineering Asset Management Laboratory (SEAM), Department of Manufacturing Engineering and Engineering Management (MEEM),

City University of Hong Kong, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:

Received 19 January 2008

Received in revised form

11 February 2009

Accepted 17 February 2009Available online 26 February 2009

Keywords:

Condition monitoring

Fault detection and diagnosis

Pattern recognition

Wavelet

Autocorrelation

Sinusoidal approximation

Mother wavelet

Daubechies

Gearbox

db44

70/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ymssp.2009.02.008

responding author. Tel.: +1518 276 6351; fax:

ail addresses: [email protected], krafiee81@gmai

a b s t r a c t

This paper presents a novel time–frequency-based feature recognition system for gear

fault diagnosis using autocorrelation of continuous wavelet coefficients (CWC).

Furthermore, it introduces an original mathematical approximation of gearbox vibration

signals which approximates sinusoidal components of noisy vibration signals generated

from gearboxes, including incipient and serious gear failures using autocorrelation of

CWC. First, the drawbacks of the continuous wavelet transform (CWT) have been

eliminated using autocorrelation function. Secondly, the autocorrelation of CWC is

introduced as an original pattern for fault identification in machine condition

monitoring. Thirdly, a sinusoidal summation function consisting of eight terms was

used to approximate the periodic waveforms generated by autocorrelation of CWC for

normal gearboxes (NGs) as well as occurrences of incipient and severe gear fault (e.g.

slight-worn, medium-worn, and broken-tooth gears). In other words, the size of

vibration signals can be reduced with minimal loss of significant frequency content by

means of the sinusoidal approximation of generated autocorrelation waveforms of CWC

as reported in this paper.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Fault detection and diagnosis of gearboxes [1,2] is one of the most common and intricate challenges in industries as aresult of frequent gear defects in machines [3,4]. Vibration signal processing of gears [5] is categorized as a reliable methodin condition monitoring. To analyze vibration signals, various techniques such as time (e.g. [6,7]), frequency (e.g. [8]), andtime–frequency domain (e.g. [9]) have been extensively studied. Among these, wavelet transform [10–13] has progressed inthe last two decades, and outweighs the other time–frequency methods, although it is lacking in a few aspects as well.

The prime concern in machine fault diagnosis is to find a proper pattern with the characteristics of small-sizedconfiguration and convincing classification ability. One of the influential approaches is based upon continuous wavelettransform (CWT) because of its minimal loss of information and maximal resolution of the signals. However, CWTgenerating continuous wavelet coefficients (CWC) suffers from a deficiency. The CWC encompasses too much data in eachscale. When resampling for fault identification systems, the results may produce a loss of information.

On the other hand, the obvious issue in rotating machines is that vibration signals contain a wide range of natural anddefect frequencies because of periodic behaviors of the machine, and extracting significant frequencies within a small-sizedpattern for fault diagnosis is still a challenge in signal processing. Moreover, finding a suitable feature applicable to avariety of datasets can be complicated. Since most of the vibration signals generated from machines are non-stationary

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Nomenclature

a scaling parameterARS adjusted R-Squareb shifting parameter/bearingB number of bearingsDbi number of inner race defectsDbb number of rolling element defectsE[] expected valuef summation sinusoidal functionG number of gearslbd vibrations of inner race defectsMSE mean squared errorm number of fitted coefficientsn number of response valuesN number of sinusoidal termsObd vibrations of outer race defectsp interpolatorPp piecewise cubic Hermite interpolationP0p first derivative of piecewise cubic Hermite

interpolationP00p second derivative of piecewise cubic Hermite

interpolation

Ps piecewise cubic spline interpolationP00s second derivative of piecewise cubic spline

interpolationRMSE root mean squared errorRxx autocorrelation functionR number of epicyclic gear trainsS number of fixed-axis shaftsSSE sum of the squares of the errorSSR sum of squares of the regressionSST total sum of squares of the total errort timev degrees of freedomvs vibrations of shaftvsg vibrations of gearvr vibrations of epicyclic gear trainsx signalx0 mean value of populationrbd vibrations of rolling element defectst time delays standard deviations2 variancerxx population autocorrelation coefficientsrxx population autocorrelation coefficients

J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (2009) 1554–1572 1555

[14,15], it is critical to have a proper feature which adapts for different datasets. In previous research, several papers (e.g.[16–26]) have been documented in this area, particularly based on wavelet transform [27]. The improvement on time-domain analysis of wavelet transform is contributed by Halim et al. [28], who implemented time synchronous average andwavelet transform to extract the periodic waveforms of gear vibration signals at different scales.

To minimize the above-mentioned deficiencies, an original technique based on time-series analysis of CWT wasdesigned and tested in this research. Vital features were obtained from the autocorrelation of CWC of gearbox vibrationsignals. The autocorrelation of CWC is able to reduce the size of data without information loss in significant frequencycontent. The down-sampling improved upon the work done by Halim et al. A sinusoidal summation function is presentedapproximating the periodic trends of autocorrelation of CWC with satisfactory preciseness. The simple sinusoidalsummation function could approximate the behaviors of vibration signals for different incipient and serious faults.

In wavelet analysis, signal decomposition (scale) is another issue which needs to be considered. In experiments, thehigh-noise vibration signals were divided into 24 sub-signals (24 scales) in fourth level of decomposition by CWT. In such away, the complex signals are converted into simplified signals with more resolution in time and frequency domains. Then,autocorrelation is applied to reduce the length of the sub-signals (series of wavelet coefficients) containing significantfrequencies. These frequencies were found to be different from one condition of the gearbox to another. For example,different classes of faulty signals produced different amplitudes in their dominant frequencies and harmonics as well asrelated sidebands.

1.1. Nature of gearbox vibrations

In machine construction, there is a frequent need to change the rotational speed between the motor and the workingmachine. Hence, geared systems are extensively used. Gearboxes are essential sources of vibrations because of discretetransfer of load by the successive meshing teeth. Simulated models of gearbox vibrations use the sum of vibratingcomponents (e.g. gear, bearings, shafts) modified by the transmission path effects. These include the sum of the vibrationfor fixed-axis shafts, meshing points of their mounted gears, epicyclic gear trains and bearing defects. In general, thegearbox vibration is defined as follows [29]:

vðtÞ ¼XR

r¼1

vrðtÞ þXS

s¼1

vsðtÞ þXGs

g¼1

vsgðtÞ

!þXB

b¼1

XDbi

d¼1

lbdðtÞ þXDbo

d¼1

ObdðtÞ þXDbb

d¼1

rbdðtÞ

!

where R is the number of epicyclic gear trains, S is the number of fixed-axis shafts, Gs is the number of gears on the shaft s, B

is the number of bearings, Dbi is the number of inner race defects on bearing b, Dbo is the number of outer race defects onbearing b, Dbb is the number of rolling element defects on bearing b, vs(t) is the vibration of shaft s, vsg(t) is the vibration ofgear g on shaft s, vr(t) is the vibration of epicyclic gear train r, lbd(t), Obd(t), and rbd(t) are the vibrations of inner race, outerrace, and rolling element defects d on bearing b, respectively.

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J. Rafiee, P.W. Tse / Mechanical Systems and Signal Processing 23 (2009) 1554–15721556

Periodically variable numbers of gear teeth in the unit cases is one of the important causes of the parametric vibrationswith characteristics modulating effects. The structure of gear vibrations is complex because it includes vibrational effectscaused by manufacturing errors and assembly faults. A typical spectrum of gearboxes includes shaft rotational frequency,gear natural and mesh frequencies as well as sidebands.

2. A novel methodology for feature extraction

The development of our algorithm is outlined in the following steps:

1.
Raw vibration signals were recorded from a motorcycle gearbox system. Three types of gear defects were selected andtested. The conditions of the gearbox consisted of slight-worn gear (SW), medium-worn gear (MW), broken-tooth gear(BT), and normal gearbox (NG).
2.
Piecewise cubic spline interpolation (PCSI) [30] was used to synchronize the vibration signals. Note that a ‘‘samplesignal’’ is defined as a segmented signal with the length of one complete revolution of the input shaft as shown in Fig. 1.
3.
Continuous wavelet coefficients of synchronized vibration signals (CWC-SVS) were calculated at the fourth level ofdecomposition (24 scales for each sample). These were calculated with 324 selected mother wavelet functions fromdiverse families.
4.
At the fourth level of decomposition, variance of CWC-SVS was calculated for 324 mother wavelets. Among them, themost similar one to gear vibration signals, the Daubechies 44 (db44) [31], was selected. The db44 has the highest valuesfor wavelet coefficients compared to the other 323 mother functions, and subsequently provided the proper similarity tothe gear vibration signals.
5.
Autocorrelation of CWC-SVS was determined. Then, the power spectrum density (PSD) of CWC-SVS and autocorrelatedCWC-SVS were compared to observe the advantage of applying autocorrelation to CWC-SVS.
6.
Frequency attributes of autocorrelated CWC-SVS were calculated using the PSD to identify proper features for classifyinggearboxes operating under normal and faulty conditions.
7.
The sinusoidal summation function was used to approximate the autocorrelated CWC-SVS to verify the preciseness ofreconstructing the original signal patterns to both normal and faulty gear patterns.
1000

100Normal

100

0

100

Slight-worn

100

0

100

Medium-worn

0 200 400 600 800 1000 1200 1400 1600 1800 2000400

300

200

100

0

100

200

300

400

Broken-tooth

Fig. 1. Raw vibration signals of four gearbox conditions.

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2.1. Experimental set-up

To record vibration signals, a gearbox of a four-speed motor-driven system was running during data recording. Thesystem consists of a driving motor with a constant nominal rotation speed of 1420 rpm, a load mechanism including afriction wheel to make inconsistent rotations, four shock absorbers under the bases of the test-bed, and the gearboxincluding 24 teeth for the pair driving gear and 29 teeth (tested gear) for the driven gear. A schematic diagram of the gearedsystem in the neutral state is shown in Fig. 2 [25]. In this figure, gears A4 and B4 are a pair of driving and driven gears.Gears A2 and A4 mounted on the output shaft and B1 and B3 mounted on the input shaft were fixed in the gearbox. The restof the gears move axially across the shafts depending on the specific speed. As the rotation speed of the motor (input shaft)is 24.05 Hz (fr); according to Fig. 2, the rotation speed of the output shaft is 29.06 Hz and the meshing frequency is 697.5 Hz.

For collecting vibration data, a multi-channel ‘Pulse’ analyzer system, a triaxial accelerometer and a tachometer wereused. The vibration signals were recorded by mounting the accelerometer on the outer surface of the gearbox’s case nearthe input shaft of the gearbox. Three different fault conditions were selected as slight-worn, medium-worn, and broken-tooth of a spur gear. To evaluate the precision of the technique, two very similar models of worn gear were taken intoaccount with partial difference. Also, a serious failure of a broken-tooth gear was considered to show the reliability of thetechnique for different faulty signals. The real rotational speed of the motor was measured by the tachometer. The samplingrate was set at 16,384 Hz as well. More detail is addressed to Rafiee et al. [32].

2.2. Synchronization of raw vibration signals

The number of data-points per each shaft revolution change in our gearbox because the shaft speed fluctuates (e.g. seeFig. 3). To overcome this flaw, the PCSI was exploited to resample the data to a regular time base before signal analysis.

For interpolation purpose, on each sub-interval x(t), kptpk+1, suppose P(x) be the interpolant of the given valueshaving certain slopes at the two end points. Between each two adjacent data sites x(k) and x(k+1), x(t) is a polynomial. Forpiecewise cubic Hermite interpolation (PCHI), Pp(x) indicates the interpolator. The first derivative, P0p(x), is continuous, butP00p(x) is not necessarily continuous, which is a drawback of PCHI. The function Ps(x) supplied by the PCSI is constructed sothat the slopes at the x(k) are chosen to make P00s(x) continuous. Therefore, this process makes Ps(x) smoother and moreaccurate. Thus, PCSI was considered to synchronize the vibration signals. The length of sample signals which were notequal in gear dataset was synchronized by PCSI with minimal loss of information.

2.3. Continuous wavelet transform

Basic theory of CWT as well as potential applications in machine condition monitoring can be found in several papers(e.g. [27]). The result of the CWT, wavelet coefficients, shows how well a mother wavelet function correlates with aparticular signal. If the signal has a major frequency component corresponding to a particular scale, then the motherwavelet at that scale (daughter wavelet) is similar or close to the signal at a particular location where the frequencycomponent occurs. As a result, the CWC have a large value at that location and scale.

CWT may process the gearbox vibration signals better than discrete wavelet transform (DWT) because down-samplingof the signals using DWT would lead to the loss of significant information. Wavelet decomposition is divided into two mainbranches: pyramid and packet decompositions. In both methods, signals are divided into approximation (low frequency)and detail (high frequency) in the first level. In pyramid decomposition, after the first level only approximations are

Fig. 2. Schematic of the gearbox in neutral state [25].

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5 10 15 20 25 30 35 40 45 50677678679680681682683684685

No.

of d

atap

oint

s in

sa

mpl

e si

gnal

No. of revolution of the driven shaft

Fig. 3. Number of data-points vs. the revolution of the input shaft for 50 segmented signals.


decomposed into higher levels. However, in packet decomposition both approximation and detail are decomposed intofurther levels. Therefore, packet decomposition offers richer contents of the signals. However, what is commonly known aswavelet packet transform (WPT) is the discrete transform with packet decomposition. That means down-sampling of thewavelet coefficients while increasing the decomposition level and consequently the loss of information. CWT, which meanscontinuous shifting through the time, was used with packet decomposition through the scales in this research. Based onthis idea, better resolution on frequency domain is achieved by means of packet decomposition as well as no loss ofinformation throughout the time-domain signals.

Therefore, after synchronizing the raw vibration signals, the CWT and autocorrelation function were applied to thesynchronized signals and generated continuous wavelet coefficients of synchronized vibration signals as mentioned above.To find the most suitable mother wavelet, 324 candidate mother wavelet functions were studied form various familiesincluding Haar, Daubechies (db), Symlet, Coiflet, Gaussian, Morlet, complex Morlet, Mexican hat, bio-orthogonal, reversebio-orthogonal, Meyer, discrete approximation of Meyer, complex Gaussian, Shannon, and frequency B-spline. The mostsimilar mother wavelet for analyzing the gear vibration signal was selected based on the following steps:

1.
Raw vibration signals were recorded and synchronized. The feature vector is: the variance of CWC for each of the 24
scales calculated by each of the 50 segmented signals in each gearbox condition. The average of the feature vector in the50 segmented signals was computed for each gearbox condition.

2.
Variances of the mentioned average of the four gearbox conditions were determined for each scale (24 elements). Thefive highest values of the calculated vector were selected as the feature because the more variance we have, the greaterthe ability to properly classify failures.
3.
The summation of the five elements, called ‘‘SUMVAR’’ for simplicity, was compared with those obtained from the other323 candidate mother wavelets (a total of 324 mother wavelets). The one that had the highest SUMVAR was selected asthe most similar function to our vibration signals.
Fig. 4a shows the decision-making flow chart for selecting the most similar mother wavelet. Among the 324 motherwavelets, the SUMVAR of Daubechies 44 was the highest. The db44 is the most similar function to gear vibration signals inthis research. As illustrated in Fig. 4b, the shape of db44 has a near-symmetric characteristic that the shapes of other highorder db do not have.

2.4. Autocorrelation

The autocorrelation function is an important diagnostic tool for analyzing time series in the time domain.Autocorrelation plots, called correlograms, present a better understanding of the evolution of a process through timeusing the probability of the relationship between data values separated by a specific number of time steps (lags). Thecorrelogram plots autocorrelation coefficients on the vertical axis, and lag values on the horizontal axis.

For a signal x(t), the autocorrelation function [33] is the average value of the product xðtÞxðt þ tÞ, where t is time delay.Formally, the autocorrelation function, Rx(t), is defined as

RxðtÞ � E½xðtÞxðt þ tÞ� ¼ LimT!1

Z T

0xðtÞxðt þ tÞdt (1)

Mean and variance of this function are independent of time. Therefore,

E½xðtÞ� ¼ E½xðt þ tÞ� ¼ x0 (2)

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Raw vibration signals

50 segmented & synchronized signals

by PCSI (SW)

50 Segmented & synchronized signals

by PCSI (BT)


by PCSI (MW )


by PCSI (NG)

Segmentation of signals into sample signals

A= Variance of CWC-SVS in each scale for each 50 segmented signals (for 4 gearbox conditions)

324 mother

F = Variance of the calculated average in gearbox conditions in each scale

Selecting one mother wavelet, one scale, and one sample signal in a gearbox condition

16 scales (in 4th level )

Calculated A

Average of A in 50 samples (SW)

Average of A in 50 samples (BT)

Average of A in 50 samples (MW)

Average of A in 50 samples (NG)

SUMVAR = Sum of five F elements with higher values (out of 16)

Max SUMVAR

wavelet functions

Fig. 4. (a) Algorithm of the most similar mother wavelet function and (b) Daubechies 44.


and

s2xðtÞ ¼ s2

xðtþtÞ ¼ s2x ¼ E½x2ðtÞ� � x02 (3)

The autocorrelation coefficients can be defined as

rxxðtÞ �Ef½xðtÞ � x0�½xðt þ tÞ � x0�g

s2x

(4)

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which can be expanded as follows:

rxxðtÞ �Ef½xðtÞxðt þ tÞ� � x0E½xðt þ tÞ� � x0E½xðtÞ� þ x02g

s2x

(5)

Substitution of Eqs. (1) and (2) into Eq. (5) leads to an expression that relates the autocorrelation function to itscoefficient:

rxxðtÞ �RxðtÞ � x02

s2x

(6)

or

RxðtÞ ¼ rxxðtÞs2x þ x02 (7)

Some limits can be placed on the value of Rx(t). Because �1rxxðtÞ1; RxðtÞ is bounded as

�s2x þ x02RxðtÞs2

x þ x02 (8)

The variance of x can be expressed in terms of the expectation of x2, E[x2], and the square of the mean of x, x02:

s2 ¼ E½x2� � x02 (9)

Based upon Eq. (1), in Eq. (8) with regard to Eq. (9) we have

Rxð0Þ ¼ E½x2� ¼ s2 þ x02 (10)

Therefore, the maximum value that Rx(t) can have is E[x2]. That is, the maximum value of Rx(t) occurs at t ¼ 0. UsingEq. (9) and the definition of the autocorrelation coefficients as in Eq. (6),

rxxð0Þ ¼ 1 (11)

Further, as t-N there is a lesser correlation between x(t) and x(t+t) because x(t) is the signal of a random variable. Thatis,

rxxðt!1Þ ¼ 0 (12)

which indicates that

Rxðt!1Þ ¼ x02 (13)

a value equal to the limiting value which signifies that there is no correlation.Autocorrelated CWC-SVS will help us to classify different types of gear health conditions in a small-size structure with

acceptable performance. Loss of information in preprocessing of non-stationary signals is a challenging problem andautocorrelated CWC-SVS allows for reduction of size with minimum information loss. This point distinguishes this researchfrom prior proposed techniques.

2.5. Sinusoidal approximation

For approximation purposes, we used a summation sinusoidal function defined as follows:

f ðxÞ �XN

i¼1

ai sinðbixþ ciÞ (14)

where ai, bi, and ci are constant coefficients. In this paper, we used trial and error to determine an N value of 8. Fittedcoefficients were obtained based on the nonlinear least-square method. It is important to note that the main purpose invibration machine monitoring is to present a small-structure pattern for different conditions. It is obvious that the higherthe N value, the better the approximation will be. However, f(x) consisting of eight terms approximates all gearboxconditions with high accuracy and, with consideration to three constant coefficients this function can compress the sub-signals (series of wavelet coefficients for each scale) to a meaningful and reliable approximation with 3�8 elements. Thiswould reduce the data to 16�3�8 coefficients for each gearbox condition.

To verify approximation accuracy, the following common statistical criteria were used in this research:SSE: The sum of the squares of the error measures the total deviation of the response values from the fit to the response

values:

SSE �XN

i¼1

ðxi � xciÞ2 (15)

A value closer to 0 indicates that the approximation has a smaller random error component.

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20 40 60 80 100 120-1

-0.5

0

0.5

1

Lags

Autocorrelation of SVS (Normal-Gearbox)

20 40 60 80 100 120-1

-0.5

0

0.5

1

Lags

Autocorrelation of SVS (Slight-Worn)

20 40 60 80 100 120-1

-0.5

0

0.5

1

Lags

Autocorrelation of SVS (Medium-Worn)

20 40 60 80 100 120-1

-0.5

0

0.5

1

Lags

Autocorrelation of SVS (Broken-Tooth)

Fig. 5. (a) Autocorrelation plot of a synchronized vibration signal for normal gearbox, (b) autocorrelation plot of a synchronized vibration signal for slight-

worn gear, (c) autocorrelation plot of a synchronized vibration signal for medium-worn gear and (d) autocorrelation plot of a synchronized vibration

signal for broken-tooth gear.

0v0 Synchronized segmented signal

4,1 4,2 4,3 4,4 4,5 4,13 4,7 4,8 4,9 4,10 4,11 4,12 4,6 4,14 4,15 4,0

3,1 3,2 3,3 3,4 3,5 3,7 3,6 3,0

2,1 2,2 2,3 2,0

1,1 1,0 Level 1

Level 2

Level 3

Level 4

Fig. 6. Decomposition tree of the wavelet transform.


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(4,1)

(4,2) (4,3)

(4,4) (4,5)

(4,6) (4,7)

(4,8) (4,9)

(4,10) (4,11)

(4,12) (4,13)

(4,14) (4,15)

(4,0)

(4,0)

(4,1)

(4,2) (4,3)

(4,4) (4,5)

(4,6) (4,7)

(4,8) (4,9)

(4,10) (4,11)

(4,12) (4,13)

(4,14) (4,15)

Fig. 7. (a) Autocorrelation of a CWC-SVS for normal gearbox condition (X-axis: lags ¼ 125, Y-axis: �1 to 1, scale in title), (b) autocorrelation of a CWC-SVS

for slight-worn gear (X-axis: lags ¼ 125, Y-axis: �1 to 1, scale in title), (c) autocorrelation of a CWC-SVS for medium-worn gear (X-axis: lags ¼ 125, Y-axis:

�1 to 1, scale in title) and (d) autocorrelation of CWC-SVS for broken-tooth gear (X-axis: lags ¼ 125, Y-axis: �1 to 1, scale in title).


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(4,0) (4,1)

(4,2) (4,3)

(4,4) (4,5)

(4,6) (4,7)

(4,8) (4,9)

(4,10) (4,11)

(4,12) (4,13)

(4,14) (4,15)

(4,8) (4,9)

(4,10) (4,11)

(4,12) (4,13)

(4,14) (4,15)

(4,0) (4,1)

(4,2) (4,3)

(4,4) (4,5)

(4,6) (4,7)

Fig. 7. (Continued)


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RS: R-square is defined as the ratio of the sum of squares of the regression (SSR) and the total sum of squares (SST):

SSR �XN

i¼1

wiðyi � yÞ2 (16)

SST is also called the sum of squares about the mean, and is defined as

SST �XN

i¼1

wiðyi � yÞ2 (17)

where SST ¼ SSR+SSE. Given these definitions, R-square is expressed as

RS ¼SSR

SST¼ 1�

SSE

SST(18)

R-square can take on any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of varianceis accounted for by the approximation.

ARS: Degrees of freedom adjusted R-square uses the R-square and adjusts it based on the residual degrees of freedom.The residual degree of freedom is determined as

v ¼ n�m (19)

where n is the number of response values and m is the number of fitted coefficients. The residual degree of freedomindicates the number of independent pieces of information including the n data-points that are required to determine thesum of squares. The degrees of freedom are increased by the number of such parameters

ARS ¼ 1�SSEðn� 1Þ

SSTðvÞ(20)

ARS can take on any value less than or equal to 1, with a value closer to 1 indicating a better fit. Negative values can occurwhen the model contains terms that do not help to predict the response.

0 1000 2000 3000 4000 5000 6000 7000 80000

200

400

600

800

1000

1200

Frequency (Hz)

Power spectral density (Normal-Gearbox)

f 2f 3f

4f

5f 6f

10f9f

7f

8f

650 700 750 8000

20

40

60

80

100

120

Frequency (Hz)

SidebandsSidebands

fPSD of NG

Fig. 8. (a) The PSD of raw vibration signal with f as tooth meshing frequency and 2f, 3f, etc., as its harmonics (normal gearbox) and (b) the zoomed PSD at

the tooth meshing frequency, f, and its sidebands (normal gearbox).

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RMSE: Root mean squared error is also known as the fit standard error, which can estimate the standard deviation of therandom components in the data:

RMSE ¼ffiffiffiffiffiffiffiffiffiffiMSEp

(21)

where MSE is the mean square error.By using the above statistical evaluation methods, one can determine whether the employed approximation function is

suitable for estimating a particular signal.

0 1000 2000 3000 4000 5000 6000 7000 80000

200

400

600

800

1000

1200

1400

Frequency (Hz)

Power spectral density (Slight-Worn)

4f

7f

2f

5f

f

3f6f 8f

0 1000 2000 3000 4000 5000 6000 7000 80000

200

400

600

800

1000

1200

Frequency (Hz)

Power spectral density (Medium-Worn)

f 2f

3f

4f

6f

9f 10f5f 8f

7f

0 1000 2000 3000 4000 5000 6000 7000 80000

200

400

600

800

1000

Frequency (Hz)

Power spectral density (Broken-Tooth)

2f

8f

3f

9f

4f

10f

7f

6f

Fig. 9. (a) The PSD of raw vibration signals (slight-worn), (b) the PSD of raw vibration signals (medium-worn) and (c) the PSD of raw vibration signals

(broken-tooth).

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3. Results and discussion

3.1. Use of autocorrelated CWC-SVS for gear fault classification

In autocorrelation function of the signal, which literally correlates the signal with itself, essential information can begathered by examining how the amplitude of the signal’s time history record at one point compares to its amplitude at

0 2000 4000 6000 80000

1000

2000

3000

(4,0)PSD of CWC-SVS

0 2000 4000 6000 80000

0.5

1

1.5

(4,0)PSD of Autocorrelation

8f6f9f5f

7f

4f4f

5f

6f

7f

8f

9f10f

0 2000 4000 6000 80000

2000

4000

6000

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Fig. 10. (a) A comparison of the PSDs generated by autocorrelated CWC-SVS and those generated by CWC-SVS for normal gearbox and (b) a comparison of

the PSDs generated by autocorrelated CWC-SVS and those generated by CWC-SVS for broken-tooth gear.

Fig. 11. Autocorrelated CWC-SVS in 50 revolutions of the shaft [slight-worn gear/scale (4,0)].

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another point in time. Hence, the periodic behaviors of the signals can be revealed by autocorrelation. The autocorrelationplots of synchronized vibration signals show that the periodic trends exist in all gearbox conditions (see Figs. 5a–d). If theautocorrelation dies out quickly the series is considered stationary. If the autocorrelation dies out slowly this indicates thatthe process is non-stationary, as with the worn gears shown in Figs. 5b and c.

From the viewpoint of time-domain signals, it is difficult to spot the periodic impacts of the defect gears, particularlyworn ones. Recognizing any meaningful features from the autocorrelation of raw signals in all conditions seems to be ademanding effort. Hence, the signals needed to be divided into aforementioned sub-signals using CWC-SVS.

Afterward, the autocorrelations of each scale of CWC-SVS were determined for all four gearbox conditions, includingnormal, slight-worn, medium-worn and broken-tooth. The calculation of autocorrelation was extended to the fourth levelof decomposition as illustrated in the decomposition tree of Fig. 6. In the fourth level, the 16 decomposition plotsrepresenting the 16 scales from (4,0) to (4,15) of the calculated autocorrelation of CWC-SVS under the gear normalcondition are displayed in Fig. 7a. Similarly, the 16 decomposition plots of the gears in the slight-worn, medium-worn, andbroken-tooth conditions are shown in Figs. 7b–d, respectively. From these decomposition plots, one can clearly recognizethe variation in the decomposed components of the signals under different gear health conditions. Hence, the differencebetween each condition is easily obvious in these sub-signals and can be used for the classification of different gear healthconditions.

In applying vibration-based gear fault diagnosis, traditional frequency domain methods, such as determination of toothmeshing frequency, its harmonics, and sidebands are usually used to identify the gear faults. The PSD of raw signalsrecorded from the gearbox operating under normal condition is depicted in Fig. 8a with f as tooth meshing frequency and2f, 3f, etc., as its harmonics. Fig. 8b shows the zoomed PSD at the tooth meshing frequency and its sidebands. The PSDs ofslight-worn, medium-worn and broken-tooth gears are shown in Figs. 9a–c, respectively. To verify that there is no

Fig. 12. Automatic frequency extraction of vibration signals using PSD of autocorrelated CWC-SVS distributed on 125 lags (X-axis) for 10 segmented

signals of each condition (Y-axis).

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significant loss of information after applying autocorrelation to the CWC-SVS, the PSDs generated at every quarterly scale ofthe fourth level are shown in Figs. 10a and b to compare to those generated by CWC-SVS in normal and broken-toothconditions.

In machine fault diagnosis, one of the central questions is: ‘‘Is the pattern used for recognition applicable to differentsignals extracted at various times from the machine given that the signals are embedded with non-stationary attributes?’’The autocorrelation of CWC-SVS is a powerful tool for pattern recognition because the results generated have minimalfluctuations from one sample signal to another even though the raw signals are non-stationary. Fig. 11 shows the resultsobtained from 50 revolutions of rotating when the gearbox was under slight-worn condition. Note that the variation issmall. Hence, the proposed method is robust even when applied to non-stationary signals.

Autocorrelation function has proven its reliability for checking the randomness of data. For the gearbox vibration data,the number of lags can be limited to less than 30. Usually, such a lag value is sufficient to verify the randomness containedin the data. In our experiments, we set the number of lags to 125. This number was obtained by trial and error. We selectedsuch a large value so that it is capable of not only checking the randomness of the data, but also reducing the size of CWC-SVS by almost one-sixth. The observation is that the larger the number of lags, the better the accuracy. Nevertheless, asmentioned above, our goal is to maintain a small-size feature pattern for machine fault diagnosis. The large value of lagswill lead to large-size feature.

Fig. 12 shows the 16 PSD plots generated by the 16 scales of the fourth level of decomposition of the autocorrelatedCWC-SVS’s results. Note that the X-axis is the number of lags (125). It is mandatory to further explain that autocorrelationof CWC-SVS will present 125-element vector for each scale. Therefore, it is more logical that each PSD plot is presented for

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Fig. 13. The approximation of autocorrelated CWC-SVS for medium-worn gearbox (original values of the autocorrelation of CWC-SVS—data points; the

approximated values-continuous curves).

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125 frequency points rather than up to 8 kHz. Instead of distributing the frequency from 0 to 8000 Hz as in Figs. 8a and9a–d, the X-axis has been distributed evenly from 1 to 125 (the number of lags) units. In other words, the X-axis isindirectly proportional to time. The Y-axis represents different health conditions of the gearbox including 10 sample signalsfor each of the four conditions. The Z-axis is the magnitude of autocorrelations of CWC-SVS shown as different color scales.As observed in Fig. 12, at high scale values (low frequency) from (4,8) to (4,15), the dominant frequency as well as its

Table 1Statistical criteria to determine the appropriateness of approximation for normal gearbox.

(a)

Statistical criteria (4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7)

SSE 0.75737 0.39793 0.20623 0.24983 0.6297 0.2323 0.05468 0.2731

RS 0.91681 0.97938 0.98217 0.97771 0.9307 0.9696 0.9977 0.9857

ARS 0.89787 0.97469 0.97811 0.97263 0.9149 0.9626 0.9971 0.9825

RMSE 0.086595 0.062769 0.045187 0.049735 0.07896 0.04796 0.02327 0.052

(b)


SSE 0.09919 0.2591 0.01034 0.09233 0.000782 0.00223 0.01933 0.2017

RS 0.9931 0.9771 0.999 0.9923 0.9999 0.9998 0.9986 0.9835

ARS 0.9915 0.9719 0.9987 0.9906 0.9999 0.9998 0.9983 0.9798

RMSE 0.03134 0.05065 0.01012 0.03023 0.002783 0.004699 0.01384 0.04469

Table 2Statistical criteria to determine the appropriateness of approximation for slight-worn gearbox.

(a)


SSE 0.39302 0.08932 0.17174 0.30342 0.16068 0.25093 0.40206 0.02817

RS 0.97085 0.99685 0.98564 0.99206 0.99465 0.98542 0.95741 0.9974

ARS 0.96421 0.99613 0.98236 0.99025 0.99343 0.9821 0.94771 0.9968

RMSE 0.06238 0.02974 0.04124 0.054811 0.03989 0.04984 0.063093 0.0167

(b)


SSE 0.000307 0.02128 0.01507 0.0178 0.00018 0.00568 0.000429 0.007449

RS 0.99997 0.9976 0.9979 0.9973 0.99998 0.9996 0.99998 0.9997

ARS 0.99996 0.997 0.9975 0.9966 0.99998 0.9996 0.99998 0.9996

RMSE 0.001746 0.01451 0.01221 0.01328 0.00133 0.00749 0.00206 0.008588

Table 3Statistical criteria to determine the appropriateness of approximation for medium-worn gearbox.

(a)


SSE 0.33929 0.14011 0.11539 0.064327 0.086767 0.23801 0.24163 0.0072315

RS 0.97226 0.99478 0.98812 0.99779 0.99466 0.98076 0.98411 0.99921

ARS 0.96594 0.99359 0.98542 0.99729 0.99345 0.97638 0.98049 0.99904

RMSE 0.05796 0.037246 0.0338 0.025237 0.02931 0.048544 0.048912 0.0084616

(b)


SSE 0.020759 0.025864 0.10668 0.000609 0.004405 0.009086 0.000855 0.021126

RS 0.99752 0.99637 0.98815 0.99995 0.99965 0.99935 0.99995 0.99865

ARS 0.99696 0.99555 0.98545 0.99994 0.99956 0.99921 0.99993 0.99834

RMSE 0.014336 0.016002 0.0325 0.002456 0.006604 0.009485 0.002909 0.014463

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Table 4Statistical criteria to determine the appropriateness of approximation for broken-tooth gearbox.

(a)


SSE 0.63214 1.0327 0.38321 0.3347 0.11525 0.36788 0.47478 0.041554

RS 0.84039 0.827 0.91235 0.94801 0.98529 0.95184 0.89918 0.9942

ARS 0.80404 0.7876 0.89239 0.93617 0.98194 0.94088 0.87622 0.99288

RMSE 0.079113 0.10112 0.061596 0.057566 0.03378 0.060352 0.068562 0.020284

(b)


SSE 0.036982 0.23491 0.30835 0.00147 0.007434 0.005850 0.001333 0.037677

RS 0.99498 0.96459 0.95444 0.99982 0.99923 0.99946 0.99988 0.99612

ARS 0.99384 0.95652 0.94407 0.99978 0.99906 0.99933 0.99985 0.99524

RMSE 0.019135 0.04823 0.05525 0.00380 0.008579 0.007611 0.003633 0.019314


sidebands of each health condition can be revealed. However, classifying the four health conditions using these scales,particularly the scales (4,11) to (4,15), are difficult as their features are located adjacently. Although, in low scales (highfrequency) such as (4,2), the features of each of the four health conditions are easily distinguished. The ability to identifythe characteristic frequencies will be useful in the future for making an automatic feature extraction algorithm so that theprocess of fault detection and classification can be automated.

3.2. The sinusoidal approximation for reconstructing the gearbox vibration signals

Using the aforementioned statistical evaluation methods, the preciseness of the sinusoidal approximation was verifiedby all four conditions of the gearbox. The original values of the autocorrelation of CWC-SVS (displayed as continuouscurves) as compared to the approximated values (displayed as data points) for some arbitrary selected scales under themedium-worn health condition are shown in Fig. 13. Note that the approximation can follow the waveforms ofautocorrelated CWC-SVS closely, particularly at higher scales. The results of the statistical evaluation are tabulated inTables 1–4. From the results shown in Fig. 13 and Tables 1–4, the sinusoidal summation function with eight termsapproximates the waveforms generated by autocorrelation of CWC-SVS for all the four health conditions.

By observing the results in the tables, the statistical methods show much better fitness in comparison with low scales.The reason is that autocorrelation of CWC-SVS in high scales possesses a greater variety of frequency contents compared tothose in low scales (see Figs. 7a–d). In these figures, the frequency components of autocorrelation of CWC-SVS in highscales are higher than those in low scales. Therefore, the assessment of preciseness of sinusoidal approximation in lowscales is better observed as stated in Tables 1–4. Although the fit is satisfactory when using a sinusoidal summationfunction with eight terms, the number of sinusoidal terms (N) in Eq. (14) can be increased for more complex waveforms.The proposed approximation function can also be applicable to other defects, such as bearing defects, because the bearingfaulty signals are impulsive in nature, similarly to gear faulty signals.

4. Conclusion

Based on our proposed algorithms and the experimental results used in evaluating the effectiveness of the algorithms,we can summarize our findings as follows:

1.
Autocorrelation of CWC-SVS has been introduced as a suitable feature for non-stationary signals in machine conditionmonitoring.
2.
A simple sinusoidal summation function can approximate the waveforms generated by autocorrelation of CWC-SVS fornormal gearboxes as well as other defective gears with satisfactory performance. The function achieved properapproximation even though the waveforms are different from one condition to another as they possess differentfrequency contents of vibration signals. The proposed simple algorithm can be the base of feature extraction in machinecondition monitoring such that the meaningful approximation coefficients with the small-size attribute can be realized.
3.
The authors believe that the proposed techniques can be applied to other faulty vibration signals, even bearing faultysignals. Further research could be conducted to confirm the effectiveness of the proposed techniques using a variety ofsignals collected from industrial machines. Further research on the mother wavelet function could be conducted tooptimize db44 for specific purposes.

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Acknowledgements

The research was supported by the Research Grants Council of Hong Kong SAR, China (project no. CityU 120506), theVibration and Modal Analysis Lab at University of Tabriz, Iran, and the Department of Mechanical, Aerospace & NuclearEngineering at Rensselaer Polytechnic Institute, USA.

The authors would like to write in memoriam of a dedicated mentor, Professor James Li at RPI, whodevoted his life to enlightening a myriad of students as well as making noteworthy contributions toseveral aspects of machine condition monitoring. The authors appreciate the very constructivecomments of the anonymous reviewers and would also like to offer special thanks to them forspending their valuable time to review the current research. They also extend their appreciation toDiane V. Michaelsen, for assistance in editing and preparing this paper.

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