Haar wavelet for machine fault diagnosis

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 1773–1786 www.elsevier.com/locate/jnlabr/ymssp

Haar wavelet for machine fault diagnosis Li Lia,, Liangsheng Qub, Xianghui Liaoa a

Institute of Mechanical and Material Engineering, China Three Gorges University, 443002, Yichang, Hubei Province, PR China b Research Institute of Diagnostics and Cybernetics, Xian Jiaotong University, 710049 Xi’an, PR China Received 30 November 2005; received in revised form 10 July 2006; accepted 10 July 2006 Available online 7 September 2006

Abstract Continuous wavelet transform (CWT) is a kind of time–frequency analysis method commonly used in machine fault diagnosis. Unlike Fourier transform, the wavelet in CWT can be selected flexibly. In engineering application, there is a problem of how to select a suitable wavelet. At present, the selecting method mainly depends on the waveform similarity between the signal required to extract and the wavelet. This method is imperfect. For example, Haar wavelet possesses the rectangular waveform in its supporting field and dissimilarity to any component in the machine signal. It is rarely used in machine diagnosis. However, the time–frequency periodicity of Haar wavelet continuous wavelet transform (HCWT) should be useful in revealing the features in signals. In addition, Haar wavelets under different scales have good low-pass filter characteristic in frequency domain, particularly under larger scales, and that can allow HCWT to detect the lower frequency signal. These merits are presented in this paper and applied to diagnose three types of machine faults. Furthermore, in order to verify the effect of Haar wavelet, the diagnosis information obtained by HCWT is compared with that by Morlet wavelet continuous wavelet transform (MCWT), which is popular in machine diagnosis. The results demonstrate that Haar wavelet is also a feasible wavelet in machine fault diagnosis and HCWT can provide abundant graphic features for diagnosis than MCWT. r 2006 Elsevier Ltd. All rights reserved. Keywords: Machine fault diagnosis; Continuous wavelet transform (CWT); Haar wavelet; Time-scale periodicity; Frequency characteristic

1. Introduction Wavelet transform has an excellent time–frequency localised property. It can make the interested component submerged in an original signal become distinct under certain scales. So, it has become popular in machine fault diagnosis [1–4]. Unlike Fourier transform that only has a sin basis, continuous wavelet transform (CWT) can be performed based on many types of wavelet basis, which may lead to differences in results when using them to process the same signal. Therefore, an important problem when using wavelet transform is how to select a suitable wavelet. Selecting a wavelet with the highest feature detecting capability by comparing the detecting effects of different wavelets using the simulating signal is not available in machine diagnosis, because the simulation of the machine signal is difficult or even impossible under most machine running conditions. Corresponding author. Tel.: 086 0717 6397560; fax: 086 0717 6397559.

E-mail address: [email protected] (L. Li). 0888-3270/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2006.07.006

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However, the conventional method of according to the waveform similarity between wavelet and interested signal is also imperfect for its neglect of wavelet frequency domain property. In order to prove the argument, the paper selects Haar wavelet as the analysis wavelet, because Haar wavelet possesses the rectangular waveform in its supporting field and dissimilarity to any component in the machine signal. Firstly, the frequency characteristics of some wavelets are discussed. Different wavelets have different filter characteristics in frequency domain. But Haar wavelet has good low-pass filter characteristic, particularly under larger scales. Then the time-scale periodicity of Haar wavelet continuous wavelet transform (HCWT) is explained mathematically and graphically. As the applications, Haar wavelet is used to diagnose the faults of three types of machines, which are rotor, gearbox and rolling bearing. Furthermore, Morlet wavelet, which prevailed in machine fault diagnosis, is selected to be a comparative wavelet. Both HCWT and Morlet wavelet continuous wavelet transform (MCWT) are applied to extract the fault information of the same machine signals.The results demonstrate that HCWT can extract the harmonic and the impulse features, which are the dominating components in machine fault signals and provide abundant graphic features for diagnosis than MCWT. 2. Common wavelet frequency domain characteristics 2.1. CWT If there is a wavelet cðtÞ 2 L2 ðCÞ; where L2 ðCÞ is a complex function space where the functions are square integrable. For a time-variable signal x(t), CWT’s definition is   Z   1 tb W x ða; bÞ ¼ pffiffiffi xðtÞc (1) dt ¼ xðtÞ; cab ðtÞ , a a where   1 tb cab ðtÞ ¼ pffiffiffi c a a

(2)

are the shifted and dilated forms of the wavelet basis and  similar  to a set of variable window functions.  a (a40) is the scale parameter, b is the time parameter, c t  b=a is the complex conjugate of c t  b=a . hdi represents the inner product operator. The wavelet basis in CWT is only required to satisfy the following admissibility condition: Z 1 jCðoÞj2 doo1, (3) Cc ¼ joj 1 R where CðoÞ ¼ cðtÞejot dt. Expression (1) in frequency domain is pffiffiffi Z a X ðoÞC ðaoÞejob do, (4) W x ða; bÞ ¼ 2p R where X ðoÞ ¼ xðtÞejot dt. It can be seen that if CðoÞ is a band-pass filtering function with a concentrated amplitude–frequency characteristic, CWT will have the capability of frequency domain localised. From the view of frequency domain, a set of wavelets with different scales is equal to a set of filters. So, CWT provides the adaptive resolution for the time–frequency decomposition of a signal. 2.2. Wavelet frequency domain filter characteristic Since the wavelets under different scales are equal to a set of filters in frequency domain, it is important to study how the filter characteristics affect the signal processing results. Fig. 1 shows the filter characteristics of several wavelets commonly used in machine diagnosis. These figures indicate that not all wavelets have the band-pass filter characteristic (see Table 1), and the changing trend and the separability of characteristic curves are also different. Naturally, the processing results with different wavelets are not the same. In addition, for Meyer, Mexican Hat and Morlet wavelets, the amplitudes of the characteristic curves degrade sharply

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Fig. 1. Wavelet frequency domain characteristic (a) Haar, (b) Meyer, (c) Mexican Hat, and (d) Morlet. Table 1 Wavelet frequency domain filter characteristic

Characteristic

Haar

Meyer

Mexican hat

Morlet

Low pass

Low pass

Band pass

Band pass

when the scale is larger than 30 (normalisation). It means the capability of using these wavelets to detect the lower frequency signal is not good. We can see the effect in the following applications. 3. Time-scale periodicity of Haar wavelet Standard Haar wavelet can be expressed as 8 0pto12; > < 1; cðtÞ ¼ 1; 12pto1; > : 0 otherwise

(5)

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or cj

1 ½0;2Þ

¼ cj 1

½2;1Þ

¼ 1.

(6)

It is a unit rectangle in the supporting field of [0,1] and its waveform is shown in Fig. 2. Since Haar wavelet is discontinuous in amplitude, it is regarded as a poor feature detector and is rarely used in machine diagnosis. 3.1. Time-scale periodicity of Haar wavelet If x(t) is an integrable function on [0, T], then T is the period. A paper [5] proved that generalised Haar wavelet (a large class of Haar-type wavelets) transform has the periodicity in both time and scale. According to the paper, a generalised Haar wavelet of degree M is a wavelet when there exists MAR (real number set) and siAR such that c ½si ;siþ1 Þ ¼ ci 2 C (natural number set) and MsiAC for all iAZ. For a T-periodic and integrable on [0,T] function, its CWT of a generalised Haar wavelet of degree M is T-periodic in time b and MT-periodic in scale a. So, the standard Haar wavelet is of degree 2 and its time-scale periodicity can be expressed as follows. HCWT of x(t) has the period T in time and the period 2T in scale. Or more formally, x(t) is an integrable function on [0,T] with the period T, c(t) denotes standard Haar wavelet. HCWT of x(t) has the following properties: W x ðb; n2T þ aÞ ¼ W x ðb; aÞ,

(7)

W x ðnT þ b; aÞ ¼ W x ðb; aÞ,

(8)

where n ¼ 1,2,y.. If a set of wavelets are cj½si1 ;si Þ ¼ cj½si ;si þ1Þ ¼ 1; si  si1 ¼ siþ1  si ¼ b;

i 2 Z,

b 2 R; then

W x ðb; n2T þ aÞ  W x ðb; aÞ

Fig. 2. Haar wavelet waveform.

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Z

Z

si ðn2TþaÞþb

si1 ðn2TþaÞþb Z si n2T

si n2T

Z

ðsiþ1 si Þn2T

xðtÞ dt  0

Z

Z

bn2T

xðtÞ dt 

¼ 0

xðtÞ dt si aþb

xðtÞ dt

si1 n2T Z ðsi si1 Þn2T

¼

siþ1 aþb

xðtÞ dt þ si1 aþb

si ðn2TþaÞþb siþ1 n2T

Z

Z

si aþb

xðtÞ dt 

xðtÞ dt 

¼

Z

siþ1 ðn2TþaÞþb

xðtÞ dt 

¼

1777

xðtÞ dt 0

bn2T

xðtÞ dt ¼ 0. 0

We have proved Eq. (7). Similarly, Eq. (8) can be deduced by analogy. 3.2. Graphics interpretation of HCWT The time-scale periodicity of HCWT is also illustrated graphically. Let us see an example. If a periodic sin signal xðtÞ ¼ sinð2p ftÞ, f ¼ 50 Hz or the period T ¼ 0.02 s. The sampling frequency is 1000 Hz. Its HCWT is shown in Fig. 3. From the graphics, we can see that the period in time is T ¼ 0.02 s (note: the coefficients take the absolute values, so two lines represent a period) and the period in scale is 2T ¼ 0.04 s. The relation between scale and period is period ¼ scale=ð2  sampling frequencyÞ, for this example, T ¼ 40=2  1000 ¼ 0:02 s. The reasons for periodicity in time are illustrated by Fig. 4. For a fixed scale, moving Haar wavelet by full periods across time does not change the inner product of HCWT. That forms the time periodicity. Fig. 5 illustrates the periodicity in scale. The area with shaded lines expresses the inner product cancellations of HCWT. When Haar wavelet changes full periods, the inner product stays unchanged for the cancellations that lead to the periodicity in scale. The inner product values of HCWT across a period are different. They depend on the type of signal. That leads to the distinct graphics features shown in Fig. 6 to Fig. 7. Fig. 6 is the HCWT of a harmonic signal. It presents the stripe graphics and is similar to Fig. 3. Fig. 7 gives the HCWT of simulating modulated signals. Fig. 7a expresses a signal with the equal impulse period of 0.0125 s (80 Hz), which simulates an amplitude-modulated signal. The expression is e400t sinð2p  1400  tÞ and its graphics presents the symmetric rhombus. Fig. 7b expresses an impulse signal with frequency-modulated and the expression is e400 t sin½2p  1400  t þ cosð2p  20  tÞ. The rhombus in Fig. 7b becomes asymmetrical and produces a little deformation marked by two dash-dotted lines. Note that the HCWT of the impulse signals are performed

Fig. 3. HCWT graphics of sin signal.

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Fig. 4. Time periodicity.

Fig. 5. Scale periodicity.

with their envelope signals (Hilbert transform). From these graphics features we can see HCWT is very successful in diagnosing different types of machine faults. 4. HCWT for machine diagnosis Harmonic and impulse are the dominant components of machine fault signal. So the graphics feature of HCWT can be exploited to diagnose machine faults. In this part, three types of machine signals are processed by HCWT in order to reveal the diagnosis capability of Haar wavelet. 4.1. Rotor imbalance diagnosis A rotor imbalance fault can be characterised by appearance of the rotating frequency or vibration amplitude increasing of the rotating frequency. Fig. 8 illustrates the HCWT of vibration displacement signal on an air-compressor rotor. The rotating speed is 10,000 rpm (corresponding period/frequency is

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Fig. 6. Graphics of harmonic signal.

Fig. 7. Graphics of impulse signal with (a) amplitude modulated and (b) frequency modulated.

0.006 s/166.7 Hz). In the figure, the scale range is [5,90] and the corresponding frequency band is 44–800 Hz. We can see the graphics feature is very clear and the periods are 0.006 s/166.7 Hz in time and 0.012 s in scale. So, the rotor unbalance is diagnosed successfully. 4.2. Gear fault diagnosis The gear signals are acquired from the life test of an automobile transmission gearbox shown in Fig. 9. The transmission path is







 Z28 Z20 Z30 Z15 input ! ! ! ! ! output: Z48 Z44 Z36 Z42 The speed of the input shaft is 1600 rpm (corresponding frequency is 26.7 Hz). The transmission ratio is 12.65 and the speed frequency of the output shaft is 2.1 Hz. The accelerometer is mounted on the

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Fig. 8. Rotor imbalance graphics of HCWT.

48

36

30

47

Input

28

34

22

38

36 20

38

15

17

44

43

42 30

output

45

Fig. 9. An automobile transmission gearbox sketch.

outer case and the sampling frequency is 5000 Hz. Since the gear pair of Z15/42 in gearbox is the slowest one and endures the largest torque during the test, the faults of fatigue crack and tooth breakage occurs on the gear of Z42. Fig. 10 shows the waveforms of normal, crack and breakage signals. The impulse of the breakage signal is clear in Fig. 10c and the period is near 4.2 Hz or 0.25 s, the double of the speed frequency of the output shaft, which is caused by two symmetric broken teeth in the gear of Z42. However, the crack signal, which should contain the impulse, is buried in noise or other higher energy disturbances and confused with the normal signal. Obviously, the weak feature caused by the crack teeth cannot be detected by the traditional method. The same signals are processed by HCWT and the results are shown in Fig. 11, where the scale range is [5,1800] and the corresponding frequency band is 5.6–1000 Hz. The impulse in the crack signal, period is 0.25 s (4.2 Hz), is detected. The rhombus with a little deformation, marked by two dash-dotted lines, presents the frequency modulation or the stiffness changing caused by the crack. The rhombus of the breakage signal, however, is just the rhombic graphics. The impulse period is 0.25 s (4.2 Hz). The analysis results demonstrate that HCWT can diagnose both the faults and the fault types of gears effectively.

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Fig. 10. Gear signals waveforms (a) normal, (b) tooth crack, and (c) tooth breakage.

Fig. 11. HCWT of gear signals (a) normal, (b) tooth crack, and (c) tooth breakage.

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4.3. Rolling bearing fault diagnosis The signals are collected from a rolling bearing test rig, which consists of a rotating shaft driven by an a.c. motor through a coupling. The rolling bearing of NSK308 is held at the end of shaft. An accelerometer is attached to the bearing seat. The fault bearings can be replaced to get faulty signals. The related parameters are listed in Table 2. Table 2 Parameters of test and bearing The dimensions of the bearing

Natural frequency

Sampling frequency

Sampling number

Rotating frequency

Number of balls: n ¼ 8 Pitch diameter: Dr ¼ 65 mm Ball diameter: d ¼ 15 mm

3827 Hz

20 kHz

8192

26.2 Hz

Table 3 Diagnosable information with MCWT and HCWT Fault type

Characteristic frequency (Hz)

MCWT (Hz)

HCWT (Hz)

Gear crack Gear breakage Outer race spalling Inner race spalling Ball flaking

4.2, stiffness changing 4.2 80.6 129.0 53.7, 10

4.2 4.2 80 132, 26.3 No

4.2, stiffness changing 4.2 80.6 131, 26.8 6.8

Fig. 12. Bearing signals (a) normal (b) outer race spalling, (c) inner race spalling, and (d) ball flaking.

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The local faults in rolling bearings will create periodic impulses with characteristic frequencies listed in Table 3 (Section 5.2). The characteristic frequencies are calculated as follows [6]. Outer race fault frequency: f o ¼ ðn=2Þ 1  ðd=Dr Þ f r ¼ 80:6 Hz;. where fr denotes rotating frequency.   Inner race fault frequency: f i ¼ ðn=2Þ 1 þ ðd=Dr Þ f r ¼ 129:0 Hz. 2 Rolling ball fault frequency: f b ¼ ðDr =2dÞ 1  ðd=D   r Þ f r ¼ 53:7 Hz. 1 Rolling ball rotation frequency: f i ¼ 2 1  ðd=Dr Þ f r ¼ 10:0 Hz. Fig. 12 gives the waveforms of the bearing signals, which are normal, outer race spalling, inner race spalling and ball flaking. The waveforms of the normal and the ball flaking are not very different. Fig. 13 is the HCWT of these signals. The analysis frequency band is 3500–4500 Hz. We can see that the graphic feature is distinctive and the impulses are the dominating components of the signals. The scale ranges are [5,900] in Figs. 13a and b, [5,1800] in Figs. 13c and [5,4940] in d, which correspond to the frequency bands of 44–8000, 22–8000 and 8–8000 Hz, respectively. In Fig. 13b the time period is 0.0124 s (80.6 Hz) corresponding to the characteristic frequency of the outer race spalling. In Fig. 13c the time periods are 0.0076 s (131 Hz) and 0.037 s (27 Hz) corresponding to the characteristic frequencies of the inner race spalling and the shaft unbalance, respectively. In Fig. 13d the half-period in scale is 0.147 s (6.8 Hz) that may correspond to the characteristic frequencies of the ball flaking. The bearing sliding effect may cause the deviation between 10 and 6.8 Hz (the calculated characteristic frequency). The

Fig. 13. HCWT of bearing signals (a) normal, (b) outer race spalling, (c) inner race spalling, and (d) ball flaking.

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Fig. 14. MCWT of gear signals (a) normal, (b) tooth crack, and (c) tooth breakage.

results demonstrate that HCWT can extract the fault features of rolling bearing successfully, and the features are obtained in the time as well as in the scale, which means that HCWT can provide much more redundancy diagnosis information. 5. Comparisons In order to prove that wavelet frequency characteristics really affect signal processing results, the Morlet wavelet, which is considered a good wavelet in machine diagnosis, is applied to analyse the above gear and rolling bearing signals. 5.1. Gear diagnosis with Morlet wavelet Morlet wavelet continuous transform (MCWT) is used to analyse the gear signals in Section 4.2. Fig. 14 is the graphics of MCWT, where the scale range is [10,40]. MCWT can identify the weak impulse caused by the crack teeth, but it cannot display other differences from the breakage teeth.

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Fig. 15. MCWT of bearing signals (a) normal, (b) outer race spalling, (c) inner race spalling, and (d) ball flaking.

5.2. Rolling bearing diagnosis with Morlet wavelet MCWT is also used to analyse the bearing signals in Section 4.3. Fig. 15 is the graphics of MCWT, where the scale range is [1,30]. In Figs. 15b and c, the periodic impulses are displayed, which correspond to the characteristic frequencies of outer and inner faults. In Fig. 15d, the feature of balling fault is not exhibited due to the poor ability of the low-frequency detection of MCWT. Table 3 lists the diagnosable information by both MCWT and HCWT. 6. Conclusions CWT is popular in machine fault diagnosis and its wavelet basis can be selected flexibly. Currently the selecting rule, as it depends on the waveform similarity between the interested signal and the wavelet, is imperfect. The paper validated the argument. Haar wavelet, which has the unit rectangular waveform and the dissimilarity to any component in machine signal, is selected and exploited to diagnose machine signals. The research work obtains some useful results as follows: 1) Wavelets have different filter characteristics in frequency domain. Some are of the low-pass filter characteristic and some are of the band-pass filter characteristic. For the wavelets, such as Morlet and

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Meyer, when the scale exceeds a certain value, the filter characteristics degrade sharply. This means that their capability of detecting lower frequency components is not feasible. However Haar wavelet has good low-pass filter characteristic, particularly under larger scales, that can allow HCWT to detect the lower frequency signal. 2) HCWT has the time-scale periodicity that can extract the features either in time or in scale. The periodicity also makes the harmonic and the impulse signal present special graphic features. These properties were exploited perfectly to extract the feature components from the machine fault signals and diagnose the type of machine fault. 3) Morlet wavelet that prevailed in machine diagnosis is selected to be the comparative wavelet. Although both HCWT and MCWT can extract most fault features from the signals, the feature of the stiffness changing caused by the gear crack cannot be revealed by MCWT. Also, for the rolling bearing signals, MCWT failed to detect the ball fault. Moreover, the graphics features of HCWT are more distinct and sensitive than that of MCWT. 4) The successful application of Haar wavelet shows Haar wavelet is also a feasible wavelet in machine fault diagnosis. The research of HCWT extends the selection range of wavelet and deepens the reorganisation to wavelet.

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