Multiscale slope feature extraction for rotating machinery fault diagnosis using wavelet analysis

Measurement 46 (2013) 497–505

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Multiscale slope feature extraction for rotating machinery fault diagnosis using wavelet analysis Peng Li, Fanrang Kong, Qingbo He ⇑, Yongbin Liu Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, Anhui 230026, PR China

a r t i c l e

i n f o

Article history: Received 9 January 2012 Received in revised form 6 June 2012 Accepted 27 August 2012 Available online 17 September 2012 Keywords: Discrete wavelet transform Rotating machinery Multiscale slope feature Fault diagnosis

a b s t r a c t This paper proposes a multiscale slope feature extraction method using wavelet-based multiresolution analysis for rotating machinery fault diagnosis. The new method mainly includes three following steps: the discrete wavelet transform (DWT) is first performed on vibration signals gathered by accelerometer from rotating machinery to achieve a series of detailed signals at different scales; the variances of multiscale detailed signals are then calculated; finally, the wavelet-based multiscale slope features are estimated from the slope of logarithmic variances. The presented features reveal an inherent structure within the power spectra of vibration signals. The effectiveness of the proposed feature was verified by two experiments on bearing defect identification and gear wear diagnosis. Experimental results show that the wavelet-based multiscale slope features have the merits of high accuracy and stability in classifying different conditions of both bearings and gearbox, and thus are valuable for machinery fault diagnosis. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The rotating machinery is one of the most important equipments in modern industrial applications. Unexpected failures of rotating machinery may endanger normal machine operation and productivity, and thus cause significant economic losses. Therefore, condition monitoring and fault diagnosis of rotating machinery play a valuable role in terms of system maintenance and process automation. For this purpose, vibration analysis is usually used since vibration signals are easy to gather and are highly correlative with the working conditions of rotating machinery. In past decades, many methods have been proposed to extract features from vibration signals, such as time-domain analysis [1], frequency-domain analysis [2], short-time Fourier transform [3], wavelet transform [4], empirical mode decomposition [5], and manifold learning [6]. Through combining pattern recognition methods, including neural network [7], fuzzy inference [8], Gaussian ⇑ Corresponding author. Tel.: +86 551 3607985; fax: +86 551 3607894. E-mail address: [email protected] (Q. He). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.08.007

mixture model network [9], Bayesian classifier [10], and support vector machines [11], ant colony classifier [12], etc., fault diagnosis system can realize automatic identification and correct diagnosis of faults. The quality of generated features would determine the accuracy of fault diagnosis and the proper analysis method is the prerequisite of effective feature extraction. Multiscale analysis is more suited for extracting information from measured data than single scale analysis [13,14]. This is because measured data from most processes are inherently multiscale in nature owing to contributions from events occurring at different locations and with different localization in time and frequency. A large scale can be used to decompose low frequency components and a small scale can be employed to analyze high frequency components. Many multiscale feature extraction methods have been developed. Li et al. [14] gave a thorough exploration of the capacity of multiscale morphological filters for gear fault detection. Loutridis [15] reported a kind of multiscale gear fault diagnostics based on local scaling analysis. Zhang et al. [16] discussed a novel approach to fault diagnosis using multiscale morphology analysis to extract

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impulsive features from the signals with strong background noise. Yoo et al. [17] proposed a dynamic monitoring method for multiscale fault detection and diagnosis in the wastewater treatment process. These papers display good performance of multiscale methods for feature extraction. In recent years, wavelet-based multiscale analysis has been increasingly employed in analyzing both stationary and non-stationary signals. This is due to the merits of wavelet transform (WT), which can analyze time and frequency localized features simultaneously with high resolution. The WT has become a powerful alternative for analyzing non-stationary signals, such as transient signals, and hence is suited for rotating machinery fault diagnosis. Many studies have been reported on application of wavelet analysis for multiscale feature extraction in the fault diagnosis area. Lou and Loparo [8] presented a new scheme for diagnosis of localized defects in ball bearings using the WT to generate multiscale feature vectors. Zhang et al. [18] suggested a diagnostic technique for identifying structural defects in spindles based on multiscale enveloping via analytic wavelet. Li et al. [19] developed an approach for fault diagnosis of rolling bearing using wavelet-based fractal analysis to extract multiscale features of different working conditions. Yoon and MacGregor [20] presented an approach to process monitoring and fault diagnosis by principal component analysis models of WT-based multiscale data. These works indicate wavelet-based methods are very suited for multiscale feature analysis from vibration signals of rotating machinery. In this paper, a multiscale feature, called multiscale slope feature, is explored based on wavelet analysis for identifying the working conditions of rotating machinery. The developed method selects the discrete wavelet transform (DWT) to analyze the mutilscale feature of vibration signals. The decomposed wavelet coefficients at multiple scales carry important information of vibration signals. They would be quite different when the working condition of a rotating machine changes. However, this difference is difficult to measure for large quantity of the coefficients and complex relations among the coefficients in different scales. In this study, a novel methodology is developed based on the analysis of the power spectrum of vibration signals. Specifically, to extract multiscale slope features, the DWT is first performed on vibration signals gathered by accelerometer from rotating machinery and a series of detailed signals at different scales are achieved. Then the variances of multiscale detailed signals are calculated to form a feature vector. Finally, the wavelet-based multiscale slope features are estimated from the slopes of logarithmic variances. Experimental results demonstrate the effectiveness of the proposed method for pattern representation and classification of rotating machinery. This paper is organized as follows. The background is introduced in Section 1. In Section 2, the principle of wavelet-based multiscale slope feature extraction is presented. In Section 3, the proposed method is verified by analyzing vibration signals gathered from bearings and a gearbox. Finally, conclusions and discussions are provided in Section 4.

2. Multiscale slope feature extraction 2.1. Discrete wavelet transform The WT has proved its great capabilities in decomposing, de-noising, and analyzing non-stationary signals. It can detect transient components as traditional Fourier transform is inept to perform because the WT could characterize both time and frequency information. Mathematically, the WT is defined as the integral of the raw signal x(t) multiplied by scaled, shifted versions of a basic wavelet function w(t):

CWTða; bÞ ¼

Z R

  1 tb dt; xðtÞ pffiffiffi w a a

a 2 Rþ  f0g;

b 2 R: ð1Þ

where a is the scaling parameter, b is the time localization parameter, w is an analyzing wavelet, and w is the complex conjugate of w. If a and b vary continuously, the WT is called continuous WT (CWT). The DWT is derived from the discretization of the CWT to avoid intractable computations when operating at every scale of the CWT. The DWT is more efficient than and just as accurate as the CWT. In the DWT, the scaling parameter a and the time localization parameter b are defined as follows:

a ¼ 2j ;

j2Z

b ¼ k2j ;

ð2Þ

j; k 2 Z

ð3Þ

where Z = {0, ±1, ±2, . . .}. The above discretization of the scaling and time parameters leads to the DWT as defined as below:

DWTðj; kÞ ¼ 2j=2

Z

xðtÞw ð2j t  kÞdt

ð4Þ

R

Through the discretization, the continuous wavelet function becomes the discrete wavelet function and the scaling function as below:

wj;k ðtÞ ¼ 2j=2 wð2j t  kÞ

ð5Þ

/j;k ðtÞ ¼ 2j=2 /ð2j t  kÞ

ð6Þ

Then by the DWT, the wavelet coefficients of a signal x(t) can be defined by the following equations:

a2j ðkÞ ¼

d2j ðkÞ ¼

Z Z

xðtÞ/j;k ðtÞdt

ð7Þ

xðtÞwj;k ðtÞdt

ð8Þ

Consequently, the signal can be decomposed into a hierarchical structure with wavelet details and approximations at various scales. At high frequencies, the wavelet reaches at a high time resolution but a low frequency resolution, contrarily at low frequencies, high-frequency resolution and low time resolution can be obtained. Such adaptive ability of time-frequency analysis reinforces the important role of the WT in the fault diagnostics field. In

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the physical interpretation, the modulus of the WT coefficients shows how the energy of the signal varies with time and frequency [21]. 2.2. Principle of multiscale slope feature Based on the DWT, a new feature extraction approach is developed for rotating machinery fault diagnosis. The principle of this approach is mainly based on the characteristics of rotating machine vibrations. The power spectrum of a typical vibration signal of a rotating machine is demonstrated in Fig. 1, which corresponds to the structure resonance region caused by a defect on the machine. The typical frequency contents are due to the modulation of resonance frequency and defect frequency. This is a special phenomenon in rotating machinery vibrations related to defect-induced vibration and makes the spectrum of the vibration signal have the typical form as shown in Fig. 1. The spectrum of the vibration signal can be divided into two parts by the resonance frequency as indicated in Fig. 1. It can be seen that the part B of the spectrum nearly satisfies the process of 1/f, which is generally defined as processes whose empirical power spectra are of the following form [22]:

SðxÞ 

d

:

The variance of wavelet coefficients in different scales is proportional to the energy in corresponding frequency band. Thus, the S(x) in Eq. (9) can be replaced by the variance of the detail signals d2j , which can be estimated as [25,26]

ð9Þ

xb

where x is the frequency, r2 is the variance of the original signal, and b is the spectral component (the slope that gets the spectral density over several decades of frequency). Based on the description of 1/f process, the power spectrum of vibration signal of rotating machinery may be expressed as:

SðxÞ 

2.3. Estimation of multiscale slope feature

Varðd2j Þ 

2

8 <

of linearity. The DWT method has been proposed to estimate the slope b according to Eq. (9) [19,23,24]. After the DWT, the energy information is divided into different scales, each with a specific narrower frequency band. The wavelet coefficients variance at multiple scales could be then taken for a multiscale variance fitting analysis. Due to the typical spectrum structure for the rotating machinery vibration signals as shown in Fig. 1 and expressed in Eq. (10), the condition pattern of rotating machinery can thus be characterized by the slopes b1 and b2 estimated by the DWT-based multiscale variance analysis. In the following, two multiscale slope features according to these two typical parts in power spectrum of the rotating machinery vibrations are investigated to represent the vibration pattern.

2

r

ðxx0 Þb1 2

r

ðx0 xÞb2

;

when x P x0

;

when x < x0

ð10Þ

where the x0 is the resonance frequency in the power spectrum. According to Eq. (10), the logarithmic power spectrum in the above two parts will indicate the property

r2 ð2j Þb

ð11Þ

where j represents the scale, b represents the slope which measures the variance progression over the scales. The variance of the detail signal at scale j is given as

Varðd2j Þ ¼

2 1 X d2j  Eðd2j Þ Nj  1

ð12Þ

where E() indicates the mean of the detail coefficients, and Nj represents the number of samples of the detail coefficients at the jth scale. The logarithm of the variance at scale j, denoted by lj, is calculated as:

lj ¼ log2 ½Varðd2j Þ

ð13Þ

The slope b can then be estimated by the multiscale variance analysis using the least square method (LSM):

Extremum

Part A

PSD (Db)

Part B

Resonance Frequency

Frequence (Hz) Fig. 1. Typical spectrum of a faulty vibration signal of a rotating machine.

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P  P P  Ns Ns Ns Ns j¼1 sj lj  j¼1 sj j¼1 lj b¼ P  P 2 Ns 2 Ns  Ns j¼1 sj j¼1 sj

ð14Þ

where sj represents the selected scale, lj represents the logarithm of the wavelet coefficient variance in corresponding scale sj, and Ns is the number of the scales used to fit the line. The slope of the logarithmic variance in different scales would be able to estimate the slope of the logarithmic power spectrum. Two slopes, b1 and b2, are calculated respectively according to two power spectrum parts divided by the resonance frequency as illustrated in Fig. 1. These multiscale slope features make a convenient estimate of the signal structure which depends on the rotating machinery working conditions. The algorithm of multiscale slope feature extraction is outlined in Fig. 2, where the detailed steps are described as follows. 1. Raw vibration signals are recorded from bearings and an automobile transmission gearbox. The bearings are tested at three health conditions including healthy, outer race wearing defect and rolling element wearing defect. The gearbox is tested at three health conditions such as healthy gear, slight wearing gear, and severe wearing gear. 2. Usually the signal recorded by accelerometer is noisy and have some irrelative components outside the resonance band, such as the gear meshing frequency in gearbox vibration. Thus, a pretreatment is necessary to filter these irrelative components out and to highlight the resonance information. In this study, the empirical mode decomposition (EMD) [27] is selected

Rotating Machine

Vibration signal acquisition

Pretreatment

DWT

Calculate the logarithm of the coefficients variance in each scale

Turning point selection

Line fitting by least square method

Multiscale slope features

Fig. 2. The algorithm of multiscale slope feature extraction.

to deal with this processing. It is to be noted that the pretreatment is not requested for all kind of signals. In this paper, bearing vibration signals do not need this pretreatment since there are not obvious irrelative components like the meshing frequency component in gearbox vibration signals. 3. The DWT is conducted on vibration signals to achieve J levels of decomposition. The coefficient variance of the detailed signal at scale j is calculated by Eq. (12), and the logarithm of the variance is calculated as Eq. (13). Meanwhile, the mother wavelets are selected experimentally and the selection of level J will be stated in Section 3. 4. The maximum point in the log-variance vs. scale curve is chosen as the turning point. Two spectrum parts are divided by the turning point. Two slopes of the logarithmic variance progression, b1 and b2, are estimated for two parts by using the LSM as expressed in Eq. (14), respectively. 5. The slopes b1 and b2 are then explored as the multiscale slope features to characterize different conditions of bearings and gearbox. 3. Experimental verifications The effectiveness of the proposed multiscale slope feature is verified by the following two experiments on bearings and a gearbox, respectively. 3.1. Bearing defect identification The bearing experiments in this study were carried out on the technical bearing test system as showed in Fig. 3a. Two accelerometers with the type of Vibxpert6.142 were placed on the YVS-2 system and signals in two channels (in horizontal and in vertical) were collected. The tested bearings are rolling bearings with the type of N203, which consist of two concentric rings with a set of rolling elements running in their tracks. To be tested, the bearing is fixed between the axis driven by a motor and the fixed mount. The contact terminals of accelerometers were slightly touched to the surface of outer ring of the tested bearing so that the accelerometers could record the vibration signals when the motor works. The defects to be studied include outer race wear as shown in Fig. 3b and the rolling element wear as shown in Fig. 3c. The bearings were tested when motor rotated at 1750 rpm and the sampling frequency was set to be 65,536 Hz, which gave a sufficient range of frequency for analysis. The power spectra of typical signals with three conditions are shown in Fig. 4 and the difference among three spectra is hard to be identified directly. Then the proposed multiscale slope features are analyzed as follows. The acceleration signals were decomposed into eight levels using the Daubechies wavelet [28] with the order N = 6. In this study, the mother wavelet and its order are selected by experience. Fig. 5 shows the results of a group of signals after the decomposition based on the DWT. Although deeper decomposition can provide more scales for subsequent treatment, the signals are decomposed into 8 levels here. This is based on the

P. Li et al. / Measurement 46 (2013) 497–505

501

Fig. 3. The setup of bearing test: (a) YVS-2 system for bearings vibration signal collection, (b) tested bearing with outer-race wearing defect, and (c) tested bearing with rolling element wearing defect.

400

(a)

300 200 100 0

0

0.5

1

1.5

2

2.5

3

4

PSD (m 2·s -3 )

x 10 400

(b)

300 200 100 0

0

0.5

1

1.5

2

2.5

3

4

x 10 400

(c)

300 200 100 0

0

0.5

1

1.5

2

Frequency (Hz)

2.5

3

4

x 10

Fig. 4. Spectra of bearing vibration signals with three defect types: (a) normal condition, (b) outer race wear and (c) rolling element wear.

consideration that the frequency components represented in larger scales are lower than 128 Hz and are easily polluted by low frequency noise, which usually cause the frequency components meaningless for fault diagnosis. The least-square fitting results for different working conditions are indicated in Fig. 6 where it can be seen that scale 4 corresponds to the obvious turning point, which is just consistent with the truth that the peak frequency of the spectrum in Fig. 4 is in the frequency band according to scale 4. The logarithmic variance shows good linearity

for scales 1–4 and 4–8, respectively. The fitting lines of different working conditions can then be distinguished easily by their slopes. In the case shown in Fig. 6, the slopes b1 was calculated to be 3.21 for normal bearing, 1.57 for outer race wearing bearing and 2.34 for rolling element wearing bearing. In addition, the slopes b2 of three conditions are 0.59, 0.58 and 1.18, respectively. To confirm the stability of the presented method, the slopes were calculated by a temporal window covering 4096 data points for each raw signal. This process was repeated in the next temporal window with an overlap of 2048 sampling points. Total 30 samples were tested for each condition and the results are shown in Fig. 7. It can be seen that the signals recorded from different bearings are distinguished successfully. The mean values and variance of the calculated temporal window slope features are also estimated as listed in Tables 1 and 2. It can be seen that, for all conditions, the variances of slope feature values are less than 0.01, which indicates the great stability property of the proposed feature. 3.2. Gear wear diagnosis Next, a typical gearbox was tested to validate the proposed feature extraction method. The experimental setup as shown in Fig. 8 was designed to conduct a fatigue test of an automobile transmission gearbox. The gearbox can load 5 forward speeds and one backward speed. The vibration signals were acquired by an accelerometer, which was mounted on the outer case of the gearbox when it is loaded with the third speed gearbox. The working parameters of

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Signal d1

5 0 -5

d2

10 0 -10

d3

50 0 -50

d4

50 0 -50

d5

20 0 -20

d6

10 0 -10

d7

10 0 -10

d8

Normal 20 0 -20

10 0 -10

Outer race wear

0

0.125

0

Rolling element wear

0.125

Time (s)

0

0.125

Time (s)

Time (s)

Fig. 5. Detailed signals from scales 1 to 8 achieved by DWT for three types of bearings.

7 0 5 Normal

-0.2 3

Rolling element wear

2

-0.4

Slope β

log2 [(Var(dj ))]

Outer race wear

1

-0.6 -0.8

-1 -1 Normal -3

Outer race wear

-1.2

Rolling element wear -5

1

2

3

4

5

6

7

8

Scale j

-1.4 1.4

1.6

1.8

2

2.2

2.4

2.6

Slope β

2.8

3

3.2

3.4

1

Fig. 6. The log-variance-scale plot of the wavelet coefficients variances and the fitting lines generated by the LSM.

Fig. 7. Scatter plot of vertical vibration signals using two multiscale slope features.

the third speed gears are shown in Table 3. The rotating speed is 1600 rpm and the corresponding meshing frequency of the third speed is calculated to be 500 Hz. The sampling frequency was set to be 3000 Hz. Three different working conditions of the third speed driving gear, including faultless, slight wear, and severe wear, are selected for the following signal analysis and fault diagnosis. A pretreatment is first performed on each condition signal to extract the resonance band near 250 Hz by using the EMD technique. Fig. 9 shows the power spectra of the preprocessed vibration signals with different working conditions. It can be seen that the energy near 250 Hz rises

with the increase of gear wear. Such a frequency band can be considered as a characteristic band that reflects the fault development. It can be also seen that the energy distribution in frequency is quite different for three working conditions of the gearbox, but all of them follow the distribution form as represented in Fig. 1. The multiscale slope feature analysis is then adopted with the experiential mother wavelet selection of Daubechies wavelet with order N = 8. Here, the sampling frequency in gearbox test is much lower than that in the above bearing test and the frequency band represented in the fifth level is lower than 100 Hz. Lower frequency components in higher levels are

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P. Li et al. / Measurement 46 (2013) 497–505 Table 1 The multiscale slopes of temporal window test in vertical. Bearings working condition

Slope b1

Normal Outer race wear Rolling element wear

Slope b2

Mean

Variance

Mean

Variance

3.2472 1.5889 2.3445

0.0050 0.0020 0.0019

0.5922 0.5912 1.1783

0.0069 0.0081 0.0025

Table 2 The multiscale slopes of temporal window test in horizontal. Bearings working condition

Slope b1

Normal Outer race wear Rolling element wear

(a) input shaft

Z27

Z24

Z32

Slope b2

Mean

Variance

Mean

Variance

3.3111 1.6253 2.2744

0.0070 0.0065 0.0024

0.611 0.586 1.222

0.0071 0.0066 0.0035

Z28

Z32

(b)

Z26

forth speed

output shaft third speed second Z25 speed Z18

counter shaft

first speed

fifth speed Z42

Z13 Accelerometer reverse speed

Fig. 8. The automobile transmission gearbox: (a) structure of the gearbox and (b) gearbox setup.

Table 3 Working parameters of the third speed gears.

Driving gear Driven gear

150

Number of teeth

Rotating frequency (Hz)

Meshing frequency (Hz)

25

20

500

27

18.5

500

usually polluted and meaningless for gearbox condition monitoring, so the gearbox acceleration signals are decomposed into five levels. Fig. 10 shows the typical coefficients achieved by the wavelet decomposition. The logarithmic variance plot is then indicated in Fig. 11. In the gearbox test, scale 3 is selected as the turning point for its obvious highest position in the logarithmic variance plot according to the peaks of spectra indicated in Fig. 9. As provided in Fig. 11, the log-variance vs. scale plot of the wavelet coefficients variances shows good linear features from scales 1 to 3 and from scales 3 to 5, which corresponding to two slope features b1 and b2, respectively. The results confirm that the least-square fitting lines indicate the difference of three conditions. Temporal window test is also executed, but only 1024 data points are contained in each window and 512 data

(a)

100 50 0

PSD (m 2·s-3)

Gear

0

500

1000

1500

150

(b)

100 50 0

0

500

1000

1500

150

(c)

100 50 0

0

500

1000

1500

Frequency (Hz) Fig. 9. Spectra of gearbox vibration signals with different working conditions: (a) normal condition, (b) slight wear and (c) severe wear.

points are overlapped between adjacent windows. Total 15 steps were executed for each condition, and the distributions of all slope feature values are shown in Fig. 12. It can be seen that the two slope features display clear

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Normal

Slight wear

Severe wear

Signal

2 0 -2

d1

0.1 0 -0.1

d2

1 0 -1

d3

5 0 -5

d4

2 0 -2

d5

2 0 -2

0

0.25

0.5

0

0.25

Time (s)

0.5

0

0.25

Time (s)

0.5

Time (s)

Fig. 10. Detailed signals of gearbox vibration signals with different conditions achieved by DWT.

2

0

0

-0.5

-2

2

2

Slope β

log [Var(d j )]

-1 -4 -6 -8

-1.5 -2

-10

-2.5

-12

-3

Normal Sligth wear Severe wear

-14 -16

1

2

3

4

-3.5 6.5

Normal Slight wear Severe wear 6.6

6.7

Scale j Fig. 11. The log-variance-scale plot of the wavelet coefficients and the fitting lines generated by the LSM for gearbox signals.

clustering results, which indicates the effectiveness of the proposed feature in characterizing different working conditions. Statistics of the multiscale slope features, including mean value and variance, are calculated as listed in Table 4 to further validate the merits of proposed features. The results demonstrate the new presented feature extraction approach can play a good performance and has great value in rotating machinery fault diagnosis.

6.8

6.9

7

7.1

7.2

7.3

Slope β

5

1

Fig. 12. Representation by two multiscale slope features for testing samples of gearbox signal.

Table 4 The results of temporal window test for gearbox. Working stage

Normal Slight wear Severe wear

Slope b1

Slope b2

Mean

Variance

Mean

Variance

7.134 6.778 6.715

0.0073 0.0082 0.0093

1.277 2.513 0.521

0.0079 0.0094 0.0075

P. Li et al. / Measurement 46 (2013) 497–505

4. Conclusions and discussions In this paper, a new approach is developed for rotating machinery feature extraction from the vibration signals by using wavelet analysis. The principle of this approach is mainly based on the analysis of spectrum structure of rotating machinery vibrations. The new developed feature is called multiscale slope feature. Two typical rotating machines, the bearing and gearbox, are analyzed to verify the approach. These two examples have great similarity in signal structure and their log-variance-scale plot both show two linear parts, which locate at two sides of the peak frequency in the power spectrum. In these two parts of spectrum, vibration signals generally show important information which can reveal the working conditions of machines. Hence, the slope features characterize the health information embedded in the vibration signals. To estimate the wavelet-based multiscale slope features, we used the detailed signals decomposed by DWT from scales 1 to 8 whose frequency band is approximately 1–32,768 Hz for bearing vibration signal processing, and scales 1–5 whose frequency band is approximately 1–1500 Hz for gearbox signal analysis. The turning points in log-scale-variance plots in these two experimental examples are located at scale 4 and scale 3, whose frequency bands are [1048, 4096] Hz and [187.5, 375] Hz, respective, just corresponding to the region where the resonance frequencies are. Temporal window tests are executed. Good clustering and low variance of the results confirm the validity of this method. In conclusion, the experiments demonstrate that the slope features can be used to characterize different working conditions of the bearing and gearbox. The approach explored in this paper provides an alternative for rotating machinery condition monitoring and fault diagnosis. In addition, it is easy for fast estimation which is very important for real-time monitoring of device. By combining with pattern identification theory, this approach has a great foreground for fault diagnosis of rotating machinery in practice. Acknowledgements This work has been partly supported by the National Natural Science Foundation of China (51005221 and 51075379), and the Startup Funding for New Faculty of University of Science and Technology of China. The constructive comments from the anonymous reviewers are sincerely appreciated. References [1] B. Sreejith, A.K. Verma, A. Srividya, Fault diagnosis of rolling element bearing using time-domain features and neural networks, in: Proc. 3rd IEEE Int. Conf. Ind. Inf. Syst., Kharagpur, India, 2008, pp. 1–6. [2] Y. Lei, Z. He, Y. Zi, A new approach to intelligent fault diagnosis of rotating machinery, Expert Systems with Applications 35 (2008) 1593–1600. [3] T.P. Banerjee, S. Das, J. Roychoudhury, A. Abraham, Implementation of a new hybrid methodology for fault signal classification using short-time Fourier transform and support vector machines, Advances in Intelligent and Soft Computing 73 (2010) 219–225.

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