Application of the wavelet transform in machine condition monitoring ...

ARTICLE IN PRESS

Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 199–221 www.elsevier.com/locate/jnlabr/ymssp

Review

Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography Z.K. Peng, F.L. Chu* Department of Precision Instruments, Tsinghua University, Beijing 100084, People’s Republic of China Received 26 November 2002; received in revised form 22 April 2003; accepted 25 April 2003

Abstract The application of the wavelet transform for machine fault diagnostics has been developed for last 10 years at a very rapid rate. A review on all of the literature is certainly not possible. The purpose of this review is to present a summary about the application of the wavelet in machine fault diagnostics, including the following main aspects: the time–frequency analysis of signals, the fault feature extraction, the singularity detection for signals, the denoising and extraction of the weak signals, the compression of vibration signals and the system identification. Some other applications are introduced briefly as well, such as the wavelet networks, the wavelet-based frequency response function, etc. In addition, some problems in using the wavelet for machine fault diagnostics are analysed. The prospects of the wavelet analysis in solving non-linear problems are discussed. r 2003 Elsevier Ltd. All rights reserved.

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200

2.

Wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

3.

An application overview of wavelet in fault diagnosis 3.1. Time–frequency analysis of signals . . . . . . . 3.2. Fault feature extraction . . . . . . . . . . . . 3.3. Singularity detection . . . . . . . . . . . . . . 3.4. Denoising and extraction of the weak signals . 3.5. Vibration signal compression . . . . . . . . . . 3.6. System and parameter identification . . . . . . 3.7. Other applications . . . . . . . . . . . . . . .

202 202 206 208 209 211 212 214

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

*Corresponding author. E-mail address: chufl@pim.tsinghua.edu.cn (F.L. Chu). 0888-3270/04/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0888-3270(03)00075-X

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

ARTICLE IN PRESS Z.K. Peng, F.L. Chu / Mechanical Systems and Signal Processing 18 (2004) 199–221

200 4.

Prospects of wavelet in fault diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

214

5.

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216

1. Introduction Condition monitoring and fault diagnostics is useful for ensuring the safe running of machines. Signal analysis is one of the most important methods used for condition monitoring and fault diagnostics, whose aim is to find a simple and effective transform to the original signals. Therefore, the important information contained in the signals can be shown; and then, the dominant features of signals can be extracted for fault diagnostics. Hitherto, many signal analysis methods have been used for fault diagnostics, among which the FFT is one of the most widely used and well-established methods. Unfortunately, the FFT-based methods are not suitable for non-stationary signal analysis and are not able to reveal the inherent information of nonstationary signals. However, various kinds of factors, such as the change of the environment and the faults from the machine itself, often make the output signals of the running machine contain non-stationary components. Usually, these non-stationary components contain abundant information about machine faults; therefore, it is important to analyse the non-stationary signals [1]. Because of the disadvantages of the FFT analysis, it is necessary to find supplementary methods for non-stationary signal analysis. Time–frequency analysis is the most popular method for the analysis of non-stationary signals, such as the Wigner–Ville distribution (WVD) [2] and the short time Fourier transform [3] (STFT). These methods perform a mapping of one-dimensional signal xðtÞ to a two-dimensional function of time and frequency TFRðx : t; oÞ; and therefore are able to provide true time–frequency representations for the signal xðtÞ: But each of the time– frequency analysis methods has suffered some problems. It is no doubt that the WVD has good concentration in the time–frequency plane. However, even support areas of the signal do not overlap each other, interference terms will appear on the time–frequency plane. This will mislead the signal analysis. In order to overcome these disadvantages, many improved methods have been proposed, such as Choi–Willams distribution (CWD) and cone-shaped distribution (CSD), etc. Without exception, however, elimination of one shortcoming will always lead to the loss of other merits. For example, the reduction of interference terms will bring the loss of time–frequency concentration [4]. The problem with STFT is that it provides constant resolution for all frequencies since it uses the same window for the analysis of the entire signal. This means that if we want to obtain a good frequency resolution using wide windows, which is desired for the analysis of low-frequency components, we would not be able to obtain good time resolution (narrow window), which is desired for the analysis of high-frequency components. Therefore, the STFT is suitable for the quasistationary signal analysis (stationary at the scale of the window but not the real stationary signals). Moreover, there exist no orthogonal bases for STFT, therefore it is difficult to find a fast and effective algorithm to calculate STFT.


201

2. Wavelet transform Over the past 10 years, wavelet theory has become one of the emerging and fast-evolving mathematical and signal processing tools for its many distinct merits. The novel concept of wavelet was first put forward definitely by Morlet in 1984. However, at that time, Morlet faced much criticism from his colleagues. Later, under the help of Grossman, Morlet formalised the continuous wavelet transform (CWT), shown as Eq. (1), and devised the inverse transform Z tb 1=2 xðtÞc dt; ð1Þ Wx ða; b; cÞ ¼ a a where a is the scale parameter, b is the time parameter, cðtÞ is an analysing wavelet, and the c ðdÞ is the complex conjugate of cðdÞ: In 1985, Meyer constructed a beautiful orthogonal wavelet base with very good time and frequency localisation properties. In the next year, Meyer and Mallat, a graduate student at Upenn, developed the idea of multi-resolution analysis (MRA) that made it very easy to construct other orthogonal wavelet bases. A more important event was that the MRA led to the famous fast wavelet transform—a simple and recursive filtering algorithm to compute the wavelet decomposition of the signal from its finest scale approximation. Before long, Daubechies constructed orthogonal wavelet bases compactly supported in a simple but ingenious way. In addition, Daubechies has done many research on wavelet frames that allow more liberty in the choice of the basis wavelet functions at a little expense of some redundancy, and the ‘‘Ten Lectures on Wavelets’’ by Daubechies has also been playing an important role for the popularisation of the wavelet. Daubechies, along with Mallat, is therefore credited with the development of the wavelet from continuous to discrete signal analysis. In the discrete wavelet formalism (DWT), the scale a and the time b are discretised as following: a ¼ am 0;

b ¼ nam 0 b0 ;

ð2Þ

where m and n are integers. So the continuous wavelet function ca;b ðtÞ in Eq. (1) become the discrete wavelets given by m=2

cm;n ðtÞ ¼ a0

cðam 0 t nb0 Þ:

ð3Þ

The discretisation of the scale parameter and time parameter leads to the discrete wavelet transform, defined as Z m=2 ð4Þ Wx ðm; n; cÞ ¼ a0 xðtÞc ðam 0 t nb0 Þ dt: In 1992, Coifman, Meyer and Wickerhauser developed the wavelet packet, which is a natural extension of the MRA. Different from the STFT, the wavelet transform can be used for multi-scale analysis of a signal through dilation and translation, so it can extract time–frequency features of a signal effectively. Therefore, the wavelet transform is more suitable for the analysis of non-stationary signals [4]. Now, the wavelets have obtained great success in machine fault diagnostics for its many distinct advantages, not only for its ability in the analysis of non-stationary signals. Table 1 gives a comparison of performances of CWT, STFT, WVD, CWD, and CSD.


202

Table 1 Comparison of the performances of the different methods Methods

Resolution

Interference term

Speed

CWT

Good frequency resolution and low time resolution for low-frequency components; low frequency resolution and good time resolution for high-frequency components Dependent on window function, good time or frequency resolution Good time and frequency resolution Good time and frequency resolution Good time and frequency resolution

No

Fast

No

Slower than CWT

Severe interference terms

Slower than STFT

Less interference terms than WVD Less interference terms than CWD

Very slow

STFT WVD CWD CSD

Very slow

3. An application overview of wavelet in fault diagnosis 3.1. Time–frequency analysis of signals The wavelet transform is a linear transform, whose physical pattern is to use a series of oscillating functions with different frequencies as window functions ca;b ðtÞ to scan and translate the signal of xðtÞ; where a is the dilation parameter for changing the oscillating frequency. Although the wavelet transform is similar to the STFT in a certain sense, differences between them exist. Compared with the STFT, whose time–frequency resolution is constant, the time– frequency resolution of the wavelet transform depends on the frequency of the signal. At high frequencies, the wavelet reaches at a high time resolution but a low frequency resolution, whereas, at low frequencies, high-frequency resolution and low time resolution can be obtained. Such adaptive ability of time–frequency analysis reinforces the important status of the wavelet transform in the fault diagnostics field. In the physical interpretation, the modulus of the wavelet transform shows how the energy of the signal varies with time and frequency. In engineering applications, the square of the modulus of the CWT is often called as scalogram, defined as Eq. (5), which has been widely used for fault diagnostics SGx ða; b; cÞ ¼ jWx ða; b; cÞj2 :

ð5Þ

As early as 1990, Leducq [5] had used the wavelet to analyse the hydraulic noise of the centrifugal pump. It was maybe the first paper about the use of the wavelet in diagnostics. In 1993, Wang and McFadden [6] applied wavelet to analyse the gear vibration signals, and found that the wavelet is able to detect the incipient mechanical failure and to detect different types of faults simultaneously. Further research was also performed by them [7,8]. In about 1994, Newland published several papers successively [9–12], in which the wavelet transform was introduced systematically, and the basic theory, the methods, application examples about the use of the wavelet in vibration signal analysis were given. Moreover, he proposed a new wavelet—harmonic


203

wavelet and discussed its properties and applications. In 1999, he used the harmonic wavelet to identify the ridge and phase of the transient signals successfully [13]. Newland’s work made the wavelet popular in engineering applications, especially for vibration analysis; and later on, the wavelets prevailed in the machine fault diagnostics. Gears, as one kind of the most important components in machines, were probably the most exploited objects by wavelets, which were pioneered by Wang and McFadden [6]. Boulahbal et al. [14] used the scalogram on the residual vibration signal of gears. Some distinctive features of the cracked tooth were obtained and the precise location of a crack was detected. Wang et al. [15] experimentally investigated the sensitivity and robustness of the currently well-accepted techniques for gear damage monitoring, including the wavelet transform, and the results show that the wavelet transform is a reliable technique for gear health condition monitoring, which is more robust than other methods. Dalpiaz [16] has studied a gear pair affected by a fatigue crack. Yesilyurt and Ball [17] used the wavelet to detect the weakened gear teeth caused by bending fatigue cracks and to assess its severity. Many other applications included tooth defects in gear systems [18–20], planetary gear train [21,22] and spur gear [23], etc. The cracks in rotor systems or in structures were another important objects for the application of the wavelets. Adewusi and Al-Bedoor [24] analysed the start-up and steady-state vibration signals of the rotor with a propagating transverse crack by scalograms and space-scale energy distribution graphs. The start-up results showed that the crack reduced the critical speed of the rotor system. The steady-state results showed that the propagating crack caused changes in vibration amplitudes with the frequencies corresponding to 1X, 2X and 4X harmonics. The vibration amplitude with the frequency 1X may increase or decrease depending on the location of the crack and the side load. However, the amplitude with the frequency corresponding to 2X increases continuously as the crack propagates. Wavelets were also used for crack detection. Examples include the edge cracks in cantilever beams [25], cracks in the rotors [26], cracks in beam structures [27] and in smart structures [28], damages in structures [29,30], cracks in metallic structures [31], cracks in composite plates [32,33], etc. Staszewski [34] made a review on structural and mechanical damage detection using wavelets. Besides gears and cracks, many other objects have been the clients of wavelets. Dalpiaz and Rivola [35] assessed and compared the effectiveness and reliability of different vibration analysis techniques for fault detection and diagnostics in cam mechanisms, including wavelets. Tse and Peng [36] compared the effectiveness of the wavelet and the envelope detection (ED) method using for rolling element bearing fault diagnosis, and the results showed that both the wavelet and ED methods are effective in finding the bearing fault, but the wavelet method is less time expensive. Peng et al. [37] analysed three kinds of typical faults: rub-impact, oil whirl and coupling misalignment, which often occur in rotating machines, by scalograms. Further research on the rub-impact in the rotor system was carried out with scalograms and wavelet phase spectrums [38]. Fig. 1 shows an example of the wavelet scalogram derived from the vibration response in an industrial machine. Here, a two-dimensional contour plot of the scalogram is used, together with the classical time and frequency domain representations. It can be seen that the scalogram can better exhibit the non-stationary of the analysed signal whose frequency components change with the time. When the complex wavelet is used, the wavelet transform can provide the amplitude and phase information of the signals simultaneously. The phase spectrum can be calculated easily from the

ARTICLE IN PRESS 204

Z.K. Peng, F.L. Chu / Mechanical Systems and Signal Processing 18 (2004) 199–221

Fig. 1. A non-stationary vibration signal (a), its frequency representation (b) and wavelet scalogram (c).

Fig. 2. A simulation signal (a) and its wavelet phase map (b) (at the time of 1 s, the signal’s frequency changed and started to increase with time).

wavelet transform, shown as following: 1 Im½Wx ða; b; cÞ WPx ða; b; cÞ ¼ tg : Re½Wx ða; b; cÞ

ð6Þ

Compared with the scalograms, the phase spectrum of the wavelet transform is much more difficult to be interpreted. However, it can also provide useful information in particular about signal discontinuities and impulses. It has a very distinct property: for every discontinuity in the signal, there will be a taper direct to it accurately, which is formed by the bands with constant phases. It is an inherent property for the wavelet phase spectrum no matter what the chosen wavelet function is. Fig. 2 shows the phase spectrum of the wavelet transform that can be used to detect the signal discontinuities. Staszewski and Tomlinson [39] used the wavelet phase to detect the damaged tooth in a spur gear. Boulahbal et al. [14] used both the scalograms and the phase maps in conjunction to assess the condition of an instrumented gear test rig, and the phase map was found to be able to display distinctive features in the presence of a cracked tooth. Wong and Chen [40] investigated the nonlinear and chaotic behaviour of structural systems by scalograms and phase spectrums.


205

Good ability in time–frequency analysis makes the wavelet qualify for the transient process analysis. Chancey and Flowers [41] used the harmonic wavelets to identify transient vibration characteristics and found a relationship between the transient vibration patterns and the absolute value of the wavelet coefficients. Kang and Birtwhistle [42] developed a wavelet-based technique to characterise the vibration burst signals of the power transformer on-load tap-changer (OLTC). This technique can identify the delays between bursts, the number of bursts and the strengths of bursts well, all of which are important for the condition assessment of OLTC. Yacamini et al. [43] proposed a method to detect shaft torsional vibrations in AC motors and generators from their stator currents, and the wavelet was used to deal the transient conditions, under which the measured stator current was non-stationary. Wang [44] applied the time–frequency-scale wavelet map to detect the transients from different mechanical systems. Al-Khalid et al. [45] used the wavelet transform to detect the fatigue damage in structures, which was modelled as a random impulse in the input signal. Gaberson [46] used the wavelet transform to identify the location and magnitude of the transient events in machinery vibration signals. Without doubt, for the adaptive time–frequency analysis ability, the wavelet is generally able to perform better than other methods, such as the FFT and STFT, etc. Therefore, the wavelet has been widely used for fault diagnostics, which can be seen through the abundant applications as mentioned above. However, the wavelet also has its shortcomings, but rarely taken into account in the applications. For example, it will always suffer the effects of the border distortion and energy leakage, and the phase spectrum of the wavelet is not robust to noise, and therefore once a signal is contaminated by noise, its phase spectrum will change greatly. Moreover, since the definition of wavelet transform is essentially based on the convolution, the occurrence of the overlapping is inevitable. The overlapping will cause undesirable frequency aliasing and bring the interference terms to the scalograms under certain conditions. It can be seen that the scalogram shown by Fig. 1(c) has many interference terms. The overlapping and interference terms will mislead our analysis of signals. To overcome these problems of the wavelets, Tse and Yang have made lot of efforts. They had proposed a simple but not very accurate algorithm, which would employ the singularity detection method, to deal with the overlapping problem occurring in the CWT [47]; and also presented a new family of DWTs, which mainly consisted of a series of Butterworth filter banks, to lighten the overlapping problem in the DWT case [48]. The reassigned method [37] can reduce the interference terms and improve the readability of the scalograms. However, the computing of the reassigned scalogram is time expensive. The reassigned scalogram of the case in Fig. 1(c) is shown as Fig. 3. It can be seen that there are few interference items in the reassigned scalogram.

Fig. 3. A reassigned scalogram.



3.2. Fault feature extraction Apart from the original intention of the wavelet transform for the analysis of non-stationary signals, another very important and successful application of the wavelet in machine fault diagnostics is fault feature extraction. Due to the compact support of the basis functions used in the wavelet transforms, wavelets have good energy concentration properties. Most coefficients cmn are usually very small, and can be discarded without causing a significant error for signal’s presentation. Therefore, the wavelet transform can present the signal with a limited number of coefficients. These coefficients usually can be directly used as the fault features. The key problem is which coefficients should be selected as the fault features and can best describe the fault. There were already many solutions to this problem, among which the thresholding method is a typical one, in which the wavelet coefficients are set to zero according to the threshold function ( cmn ; cmn > y; ð7Þ Aðcmn Þ ¼ 0; cmn py; where y is a threshold. Chen et al. [49,50] decomposed dynamic transient signals by the discrete wavelet transform and selected the wavelet coefficients by the hard-thresholding method; that is, keeping those coefficients that are bigger than the constant thresholding and discarding other smaller coefficients. Then, those coefficients, as being fault features, were inputted into an ART net for fault classification. This method has been applied to a refinery fluid catalytic cracking process successfully. Lin and Qu [51] used the wavelet entropy as a rule to optimise parameters of the wavelet function. Then, the vibration signals from the rolling bearing and the gearbox were decomposed with the wavelet function. Finally, an improved soft-thresholding method was used to extract the impulse component as fault features from vibration signals. Yen and Lin [52] decomposed the vibration signals with the wavelet packet transform, and selected the coefficients as fault features with the aid of a statistics-based criterion. Goumas et al. [53] used discrete wavelet transform to analyse the transient signals of the vibration velocity in washing machines and fault features were extracted from the wavelet coefficient. Then minimum distance Bayes classifiers were used for classification purposes and such a method was used for product quality control. Similar investigation on washing machines was carried by Stavrakaki et al. [54] but they used Karhunen Loeve transform to select features from wavelet coefficients and several classifiers’ performance was compared. Lu and Hsu [55] presented a wavelet-based method to detect the existence and location of structural damage. They found that the changes in the wavelet coefficients of the vibration signals were very sensitive to minor localised damage and the maximum change of the wavelet coefficients was often corresponding to the location of the damage. Liu et al. [56] proposed a wavelet packet-based method for fault diagnostics. Wavelet packet coefficients were used as features. Ball bearings were studied, and the results showed that the coefficients had a high sensitivity to faults. Momoh and Dias [57] used both the FFT and the wavelet transform as feature extractors to diagnose the type and location of faults in the power distribution system. They concluded that features extracted from wavelet transforms gave better results. Ye and Wu [58] calculated the features with wavelet packet decomposition coefficients of the stator current to detect the induction motor rotor bar breakage. Momoh et al. [59] compared


207

the performances of feature extractors for DC power system faults, including the FFT, the Hartley transform and the wavelet transform; and the conclusions showed that the wavelet exhibited a superior performance. Altmann and Mathew [60] presented a novel method, which was based on an adaptive network-based fuzzy inference system, to select the wavelet packets of interest as fault features automatically. It was proved that the method could enhance the detection and diagnostics of low-speed rolling-element bearing faults. Akbaryan and Bishnoi [61] used PCA technique to reduce the size of the feature space extracted from wavelet coefficients. Mufti and Vachtsevano [62] fuzzed the fault features extracted by the wavelet transform; and then, a fuzzy inference was applied. Aminian [63] developed an analog circuit fault diagnostic system based on Bayesian neural networks using wavelet transform, normalisation and principal component analysis as preprocessors. Essawy et al. [64] presented an automated integrated predictive diagnostics method for the monitoring of the health of complex helicopter gearboxes. In this method, the neuro-fuzzy algorithm and the sensor fusion were used, and the wavelets were used to analyse the vibration data and to prepare them for neural network inputs. The applications of wavelet coefficients can be found in many other works [65,66]. Besides the wavelet coefficients, many other wavelets-based fault features were presented, such as Xu et al. [67] calculated the singularity exponents of the envelops of vibration signals with the wavelet transform-based method and used those singularity exponents as features to diagnose the breakers’ fault. Hambaba and Huff [68] decomposed the vibration signals from gears in a helicopter with discrete wavelet transform, and then approximated the wavelet-transformed signals at each level. Finally, the probability density functions (PDFs) of the residual errors were expanded into Hermite polynomial and the coefficients of this expansion were used as fault features for the detection of the early fatigue cracks in gears. Zheng, Li and Chen used the feature energy of the time-averaged wavelet spectrum as fault information to detect the gear fault in a gear-box [69]. Yen and Lin [70] used the wavelet packet node energy selected by Fisher criterion function as fault feature and the network as classifier. Seven types of faults of gearboxes in a helicopter were investigated by this method. A comparison was made with the Fourier-based features, and the results showed that the wavelet packet-based method was more robust to the white noise. Liu and Ling extended the Mallat and Zhang’s matching pursuit [71] to machine diagnostics. The wavelets were treated as features directly for the detection of diesel engine malfunctions. The results showed that both the sensitivity and the reliability of this method were very good [72,73]. Osypiw et al. [74] developed a fast Gaussian wavelet algorithm with very narrow band-pass filtering technique to extract some main features from the vibration signals, such as the significant frequencies, etc. Ren et al. [75] took the wavelet modulus maximum as the fault features to detect and diagnose the faults in a control system. Chen and Wang used the instantaneous scale distribution of the wavelet transform for quantifying pattern features, then a multi-layer perceptron pattern classifier was used to identify gearbox faults [76]. Peng et al. [77] used the number of wavelet modulus maxima lines and the singularity exponents as features to identify the shaft centre orbits of the rotating machines. In Ref. [78], Shibata, Takahash and Shira used the wavelet transform to analyse the sound signals generated by bearings in the time– frequency domain, and the signal component indicative of a fault was identified. In addition, a symmetrised dot pattern method was also described, which can visualise the sound signals in a diagrammatic representation, and so, it was possible to distinguish differences between normal and faulty bearings.



In conclusion, based on the wavelet transform, many kinds of fault features can be obtained, all of which can be classified as the wavelet coefficients based, wavelet energy based, singularity based and wavelet function based, etc. roughly. Since the wavelet coefficients will highlight the changes in signals, which often predicate the occurrence of the fault, the wavelet coefficients-based features are relatively suitable for early fault detection. However, because the slight changes in signals often have small energy these changes will be easily masked in the wavelet energy-based features. Therefore, the wavelet energy-based features are often not able to detect the early faults. The singularity-based features will easily suffer from the influence of noise, even the slight noise will cause the remarkable change in the singularities, and therefore how to lighten the influence of noise is worth great research efforts when using the singularity based features. 3.3. Singularity detection Most information of a signal is often carried by the singularity points, such as the peaks, the discontinuities, etc. Moreover, at the moment when faults occur, the output signals usually contain jump points that often are singularity points. Therefore, singularity detection has played an important role in fault diagnostics. The polynomial trends in the signals could mask the local weak singularities in signals and this caused some methods failing to detect those singularities. On the other hand, the wavelet function can be chosen as the orthogonal to polynomial behaviour of arbitrarily high order, and therefore can remove the polynomial trends and highlight the singularity points in signals, thus the singularity points can be detected easily by the wavelet-based methods. The wavelet modulus maxima method has been almost a standard method for the detection of singularity points [79], in which the wavelet modulus maxima lines play an important role. The modulus maxima line consists of the points that are local maxima in the time–scale . plane, and whenever the analysed signal xðtÞ has a local Holder exponent hðt0 ÞoN (the vanishing moment of the wavelet function c), there is at least one modulus maxima line pointing towards t0 ; along which the wavelet coefficients have the scaling behaviours as follows: jWx ða; t0 ; cÞjBahðt0 Þ ;

ð8Þ

. where the hðdÞ are the Holder exponents, which are also often used as fault features. Fig. 4 shows a vibration signal sampled from a rotor with serious rub-impact malfunction and its wavelet modulus maxima.

Fig. 4. A vibration signal (a) and its wavelet modulus maxima (b).


209

Now, the wavelet transforms are often used for the detection of the singularity points in output signals sampled from the machines, furthermore, for fault diagnostics. Sun and Tang applied wavelet transform modulus maxima to detect abrupt changes in the vibration signals obtained from operating bearings being monitored [80]. Ruiz, Nougues and Calderon et al used the wavelets to determine the singularities of the transients and to reduce the dimensionality of the data. Then the processed signals were input into an ANN for the fault classification. This method was applied to a batch chemical plant [81]. Tang and Shi [82] combined the dyadic wavelet transform method and the singularity analysis method to separate the weak reflected signals from the defects from the noise, and, furthermore, to detect the weak bonding defects occurring in solid-phase welded joints. Dong et al. [83] introduced five kinds of applications of the wavelets in power system fault signal analysis, including fault location identification through the singularity detection technique. Jia et al. [84] employed the singularity detection with wavelets to obtain the polarities and magnitudes of the abrupt change of voltage and current caused by the fault, and determined the faulty circuit through the comparison of the polarities and magnitudes. This method was applied for single-phase-to-ground fault analysis. Chen and Lu [85,86] introduced a method that used the wavelet transform to detect the singular signal and its singularity and extended this method to fault diagnostics for the electro-hydraulic-servo system. Lin et al. [87] used the similar method to analyse the vibration signals of the reciprocating compressor valve. Zhang et al. [88] applied the wavelet-based singularity detection method to detect the position of the rub fault occurring in the rotating machines. Undoubtedly, the wavelet transforms are very successful in singularity detection, but when using the wavelet transforms to detect singularity, regularity could be an important criterion in the selection of a wavelet function. Usually, the selected wavelet must be sufficiently regular, which implies a long filter impulse response; otherwise some singularities would be overlooked. Additionally, the noise will influence the performance of the wavelet greatly, therefore before the singularity is detected, the signal preprocessing must be carried out. 3.4. Denoising and extraction of the weak signals Signal preprocessing is an important step to enhance the data’s reliability and, thereby, to improve the accuracy of the signal analysis. The core of signal preprocessing is to increase the signal-to-noise ratio (SNR), that is, to remove the noise and to highlight the signals interested. However, the noise is generally unavoidable, which is usually introduced into signals by various disturbances, such as the disturbance from the exotic environment, and from testing instrument self, etc. Denoising and extraction of the weak signals are very important for fault diagnostics, especially for early fault detection, in which cases features are often very weak and masked by the noise. The noises are often stochastic signals with broadband, whose frequency band will overlap with the interested signals’. Therefore, it is difficult to eliminate the noise from the signals effectively with general filtering methods. In addition, traditional methods require some information and assumptions about the signals that one wants to extract from the noise, such as which class the signal belongs to. With wavelets, it is enough to know that a signal belongs to a much family, which often includes many more classes, but to know anything more. Just as Donoho [89] said, ‘‘you do as well as someone who makes correct assumptions, and much better than someone who makes wrong assumptions.’’ An orthogonal wavelet transform can compress



the ‘‘energy’’ of the signal in a relatively small number of big coefficients, while the energy of the white noise will be dispersed throughout the transform with relatively small coefficients. It gives us more options to select some sample methods to eliminate the noise. Now, a lot of wavelet-based methods for the denoising have been available, for example, the soft-thresholding method [90] by Donoho, and the wavelet shrinkage denoising by Zheng and Li [91]. The super merits of the wavelet in the denoising make it to be used widely for signal preprocessing in the fault diagnostics field. Altmann and Mathew [92] used the discrete wavelet packet analysis-based multiple band-pass filtering to deal with the vibration signals from a lowspeed rolling-element bearing and good results were obtained with a significantly improved SNR compared to its high-pass counterpart. Littler and Morrow [93] applied the discrete wavelet transform for the denoising for power system disturbance signals, and the transient fault signals were thus enhanced. Yang and Liao [94] proposed a wavelet-based denoising approach, in which the threshold of eliminating the noises will be adjusted adaptively according to the background noises. This method was used in a power quality monitoring system to achieve the purpose of the easy and correct detection and localisation of the disturbances in the power systems. Menon et al. [95] used the wavelet-based method to eliminate the background operational noises, which were troublesome in using the acoustic emission technology to detect small fatigue cracks in rotor head components. Shao et al. [96] used the wavelet to preprocess the fault signals, the noises and the spikes were removed, and then the wavelet coefficients obtained were used as the inputs of the non-linear PCA algorithm for the process performance monitoring. Pineyro et al. [97] compared the performances of three methods in the detection of the localised defects in rolling element bearings, including the second-order power spectral density, the bispectral technique and the wavelet. The wavelet was proved to be useful in the short transient detection for the reason that it could eliminate the background noise. Lin [98] applied the wavelet-based method to remove the noise from the machine sound, and, furthermore, to extract the fault features for diagnostics. Watson and Addison [99] employed the wavelet transform modulus maxima filtering to the nondestructive testing signals of the piled foundations. It was proved that the technique allowed for the effective partitioning of sonic echo signal and noise. Bukkapatnam et al. [100] coupled the neighbourhood method and the wavelet method for signal separation of the chaotic signals. Its capability was proved by numerical studies and the vibration signal analysis sampled from a wear tool in machining. Krishnan and Rangayyan [101] presented a novel wavelet-based denoising method to improve the SNR of the knee joint vibration signals. Hu and Zhou [102] applied the wavelet transform modulus maxima method to eliminate the noise from the residual signal and, therefore, to improve the robustness of the fault detection. This method was applied to the structure fault detection in the fighter. Liu et al. [103] used the wavelet to preprocess the diesel cylinder vibration signals, mainly focused on the denoising. Duan and Zhang [104] used the wavelet to filter the noise to purify the centre orbit of a rotor. The principle of the wavelet for denoising is different from that of the traditional filter-based method. In brief, in the filter-based methods, the frequency components outside a certain range are often set to zero, which may cause some useful fault information to be lost since some burst faults often appear as impulses in signals and these impulses always cover a wide frequency range, therefore the filter-based denoising methods will smooth some impulses. On the contrary, the wavelet-based methods are often to set some small wavelet transform coefficients to zero, which can retain the impulses in signals well because those impulses are often represented as some big


211

wavelet coefficients in the wavelet transform. Especially, the wavelet transform modulus maxima method can do very well in retaining the useful fault information meanwhile denoising. Therefore, in conclusion, the wavelet-based methods are more suitable for the preprocessing of fault signals than the filter-based methods. 3.5. Vibration signal compression For rotating machines, in order to acquire enough information to assure the diagnostics accuracy, a fault diagnostics system must sample signals through many channels simultaneously with high sampling speed, and therefore, the data is often massive. Thus, for fault diagnostics systems, especially for those online systems, it is difficult but necessary to save the real-time data on hard disks for long time. Additionally, with the development of the internet-based remote fault diagnostics technique, high-performance data compression algorithms are needed, which will be useful to solve the bottleneck of the massive data transmission, to reduce the cost of the data transmission, and further to improve the performance of the remote fault diagnostics system. Data compression maybe the most successful application of the wavelet transform, including onedimensional signal compression and two-dimensional image compression. Also due to the compact support of the basis functions used in the wavelet transform, wavelets have good energy concentration properties. Most wavelet coefficients are therefore very small, and they can be discarded without causing a significant error in the reconstruction stage, then data compression is achieved. The data compression and decompression algorithms essentially consist of five major steps: transform, thresholding, quantisation/encoding, decoder and reconstruction, shown as the following block diagram (Fig. 5). The performance of the compression algorithm can be evaluated by two parameters [105]. The first one is the compression ratio defined as the ratio of the number of bits of the original data xðtÞ # to the number of bits of the compressed data xðtÞ: The second parameter is the normalised meansquare error given as N 100 X ðxi x# i Þ2 ; MSEðxÞ ¼ Ns2x i¼1

ð9Þ

where sx is the standard deviation of the xðtÞ and N is the number of the sample points in the analysed data. The wavelet transform has been used for vibration signal compression successfully. Staszewski has studied the performances of the wavelet-based compression methods for different types of signals. The results indicated that the wavelet-based methods are more suitable for non-stationary vibration signal compression. Usually, the more non-stationary the vibration signal is, the better the compression performance will be. The criterions for the selection of the wavelet function used x(t)

DWT

Thresholding

Encoder Sparse Matrix Storage

xˆ ( t )

IDWT

Decoder

Fig. 5. Block diagram of compressor/decompressor.



for vibration signal compression were given. In general more compactly supported, and therefore less smooth wavelet functions, are better suitable for non-stationary and irregular signals, such as transient signals. Less compactly supported, and therefore more smooth wavelet functions, are better suitable for stationary and regular signals, such as periodic data. Actually, this criterion is suitable for feature extractions as well [105]. Staszewski [106] combined the wavelet with the genetic algorithm to compress signals. The method was applied to the compression of the gearbox vibration spectra showing a potential for storage, transmission and fault feature selection for condition monitoring. Makoto et al. [107] tried to use several different Daubechies wavelet functions to compress the vibration signals sampled from motor bearing rings. Moreover, they constructed a requantiser to optimise bit allocation under the given permissible distortion. The comparison with discrete cosine transform (DCT)-based methods showed that the wavelet-based method was more effective. Shen and Gao [108] used the wavelet method to compress the mechanical vibration signals and the compression ratios of 10–20 were obtained. In Ref. [109], Ma and Lu introduced a vibration signal compression method in detail, which was based on the wavelet and the embedded coding of zero tree. The performances of the method were demonstrated through three kinds of typical vibration signal compression. Xu et al. [110] reviewed the properties of vibration signals in rotating machines and the relation between the wavelet coefficients and the singularities of the signals. On this basis, they proposed a new data compression algorithm, in which wavelet coefficients were used to present the singularities of the signals and the frequency components to present the normal characteristics of the signals. Application results showed that this method could achieve high compression ratio with a reservation of good local characteristics. Actually, the data compression principles are similar to the denoising, that is, setting some small coefficients to zero so that we can use a few bits to represent the signal during the encoding stage. Therefore, the use of the wavelet-based method can often obtain high compression ratio meanwhile retaining the singularities of signals, which often contain most of the fault information. While it is difficult to achieve these purposes for the Fourier transform (FT)-based method, for example, the DCT-based method, which suits to the regular signal compression but the irregular signals [107]. In conclusion, compared with the FT-based method, the wavelet transforms have prominent advantages in vibration signal compression because the wavelet transforms can retain more fault information during the compression. 3.6. System and parameter identification Machine faults can be reflected by the changes of the system parameters or modal parameters, such as the natural frequency, damping, stiffness, etc. Therefore, people extended the system parameter identification to the fault diagnostics field [111]. Different identification methods have been developed. These methods can be classified based on the time and the frequency domains. The thought of the time domain techniques is basically to fit the impulse response function of a mechanical system. They will be affected by the noise greatly and be with large amount of computation cost. The frequency domain methods are based on the frequency response functions (FRF), which will often give significant errors resulting from the influences of the energy leakage and the spectrum overlap. In order to overcome all these problems, filtering operations are inevitable in system and parameter identifications. The wavelet transform has dominant


213

advantages in signal filtering, which, plus other merits such as time–frequency presentation, the compactly support base, etc., makes the wavelet transform perform well at the parameter identifications. Staszewski [112] presented a wavelet-based mode decoupling procedure, in which three different damping estimation procedures for linear systems were presented. One based on the wavelet transform cross-sections, which can be obtained directly from the frequency domain of wavelet formula pffiffiffi Z # ðaf Þej2pfb df ; ð10Þ Wx ða; b; cÞ ¼ a xðf Þc a;b # ðdÞ is the complex conjugate of cðdÞ: # where c The second method recovered the impulse response function for a single mode i from the wavelet transform by using Eq. (11) Z tþaDtc 1 da ð11Þ Wx ða; b; cÞ ; xðtÞ ¼ Cci taDtc a where Dtc and Dfc are the duration and bandwidth of the basic wavelet function, and Z fi þDfc =a # jcðf Þj2 Cci ¼ df : jf j fi Dfc =a

ð12Þ

The third method used the wavelet ridges and skeletons. Staszewski [113] extended the wavelet ridges and skeletons methods for the identification of non-linear systems and good results were obtained. Further experimental studies can be found in Ref. [114]. Akhmetshin and Sendetski [115] introduced the wavelet packet algorithm into freeoscillation testing method for the estimation of fault parameters of structures. Robertson et al. [116,117] presented a wavelet transform-based method to extract the impulse response characteristics from the measured disturbances and response histories of linear structural dynamic systems. They [118,119] also used the discrete wavelet transforms for the identification of structural dynamics models. With these methods, the structural modes, mode shapes and damping parameters were extracted. Freudinger et al. [120] used the Laplace wavelet to decompose a signal into impulse responses of single mode subsystems, and, thereby, to identify the model parameters of a flutter. Similar efforts were made by He and Zi [121] for the identification of the natural frequency of the hydro-generator axle. Yu et al. [122] discussed the application of the wavelet in structural system identification. Detailed steps for the determination of the impulse response and the estimation of the response function were given. Their researches also included the modulated Gaussian wavelet for modal parameter identifications [123] and the wavelet for time-varying modal parameter identification [124]. Ruzzene et al. [125] used the wavelet transform as a time– frequency representation for system identification purposes, and the results showed that wavelet analysis of the free response of a system allowed the estimation of the natural frequencies and viscous damping ratios. Liu et al. [126] addressed a wavelet-based method for the identification of dense modal parameters of a flexible space structure. Ma et al. [127] presented a wavelet-based method on identifying the dynamic characteristics of bearings. Experiment results showed that this method was well robust to the influence of the measurement error. Many other applications



of the wavelet in identifications included hydro-generator axles’ viscous damping coefficients [128] and FRF [129]. 3.7. Other applications Compared with the traditional networks, the wavelet neural networks, in which the basis functions are drawn from a family of orthonormal wavelets, have better localisation characteristics both in the time and frequency domains. These networks allow hierarchical and multi-resolution learning of input–output maps and therefore have strong ability in the function approximation and good resolution when using for pattern recognition. Wavelet neural networks have been widely used for the fault diagnostics [130–134]. Moyo and Brownjohn applied the wavelet power spectra and the cross-wavelet power spectra to characterise the structural response [135]. Staszewski and Giacomin [136] extended the concept of the FRF to the time–scale domain, and presented the concept of the wavelet-based FRF, which reflected the ratio of output to input in the time–scale domain and thus fully characterising time-variant physical systems. The method was used for the analysis of the vehicle road data. The wavelet has often been used to estimate the PDF to the process monitoring [137–139]. The wavelet can do very well for trend analysis and condition prediction as well [140,141].

4. Prospects of wavelet in fault diagnostics With the development for about 10 years, the wavelet transform has been widely used in fault diagnostics. But, compared with the FT, the applications of the wavelets have still not achieved a standard status. Many reasons have caused the current status of wavelets in fault diagnostics. For example, many functions can be used as the wavelet basis, but there is no a standard or a general method to select the wavelet function for different tasks. It is an obstacle for the popularisation of the wavelet transform. Some people have paid attention to this problem [105]. Additionally, an ignored problem is how to determine the range scales used in the wavelet transform. The solution to this problem is important. Wavelet transforms with scales out of this range would bring some meaningless information, which will mislead the signal analysis. Some discussion about this problem can be found in Ref. [142]. Unlike the FT, the results of the wavelet transform have no straightforward physical implications, and therefore it is difficult to obtain useful information directly from the results of the wavelet transform. Furthermore, different wavelet functions may result in different analysis results, which will make not only many engineers but also some new learners of wavelets become confused. Therefore, although the wavelet transform has many merits in signal processing, almost all engineers are still favorite with the FT. It is, therefore, very necessary to find an easily understood way to present the results of the wavelet transforms. The wavelet scalograms, which are similar to the spectrums of the STFTs, can be understood with relatively easy, so it may be able to remedy the regret of difficulty to understand the results of the wavelet transforms in a certain extent. Solving all those problems mentioned above will promote the popularisation of the wavelets. Non-linear problem analysis, which is unavoidable in the fault diagnostics field, maybe is another field in which the wavelet will achieve success. Wong and Chen [38] have investigated the


215

non-linear and chaotic behaviours of the structural system by wavelet transforms. Jubran et al. [143] used the modulated Gaussian wavelet to analyse the chaotic behaviour of the flow-induced vibration of a single cylinder in cross-flow. Pernot and Lamarque [144,145] presented a waveletGalerkin procedure to investigate time-periodic systems, including transient vibration and stability analysis. Suh and Chan [146] developed an innovative wavelet-based diagnostic methodology to perform real-time detection of mechanical chaos occurring in high-speed, highperformance rotor-dynamic systems. Zheng et al. [147] have studied the bifurcation and chaos phenomena with wavelets. It can be expected that the wavelet transform would enjoy greater success in non-linear problem analysis. Hitherto, almost all applications of the wavelet are limited to find some new phenomena or some new fault features from the sampled signals, based on which some novel and useful fault detection methods have been presented. But it is a pity that few further researches have been carried out to discuss the reasons for those new phenomena found by the wavelet-based methods, and, actually, such efforts will be very helpful for the investigation of the fault reasons. Additionally, there is still a pity about the applications of the wavelet, that is, until now, the wavelets are always used to analyse a single signal and rarely used to analyse two or more signals simultaneously to find the relationships between them. It is well known that in order to achieve accurate fault diagnostics, many different signals have to be sampled from machines. There exist some relationships between these sampled signals. Undoubtedly, raveling those relationships will give more useful information for fault diagnostics. The wavelet cross-scalogram can carry out the correlation analysis between two signals on the time–frequency plane and therefore can give more information than the traditional correlation analysis only in the time domain. It can be expected to be a useful tool for fault diagnostics.

5. Concluding remarks The applications of the wavelet analysis have covered almost every aspect of the fault diagnostics. A review on all of them in a couple of pages is certainly not possible. In this review, all applications were divided into several main aspects, including the time–frequency analysis of signals, the fault feature extraction, the singularity detection for signals, the denoising and extraction of the weak signals, the compression of vibration signals and the system identification. Some other applications are introduced briefly as well, such as the wavelet networks, the waveletbased frequency response function, etc. In addition, some problems occurred in the use of the wavelet to fault diagnostics are analysed. The prospect about using the wavelet to solve non-linear problems is discussed.

Acknowledgements This research is supported financially by National Key Basic Research Special Fund (No. G1998020309) and Natural Science Foundation of China (Grant No. 50105007).



References [1] M.C. Pan, P. Sas, Transient analysis on machinery condition monitoring, International Conference on Signal Processing Proceedings, ICSP 2 (1996) 1723–1726. [2] P.C. Russell, J. Cosgrave, D. Tomtsis, A. Vourdas, L. Stergioulas, G.R. Jones, Extraction of information from acoustic vibration signals using Gabor transform type devices, Measurement Science and Technology 9 (1998) 1282–1290. [3] I.S. Koo, W.W. Kim, Development of reactor coolant pump vibration monitoring and a diagnostic system in the nuclear power plant, ISA Transactions 39 (2000) 309–316. [4] A. Francois, F. Patrick, Improving the readability of time–frequency and time–scale representations by the reassignment method, IEEE Transactions on Signal Processing 43 (1995) 1068–1089. [5] D. Leducq, Hydraulic noise diagnostics using wavelet analysis, Proceedings of the International Conference on Noise Control Engineering, 1990, pp. 997–1000. [6] W.J. Wang, P.D. McFadden, Application of the wavelet transform to gearbox vibration analysis, American Society of Mechanical Engineers, Petroleum Division (Publication) PD 52 (1993) 13–20. [7] W.J. Wang, P.D. McFadden, Application of wavelets to gearbox vibration signals for fault detection, Journal of Sound and Vibration 192 (1996) 927–939. [8] W.J. Wang, P.D. McFadden, Application of orthogonal wavelets to early gear damage detection, Mechanical Systems and Signal Processing 9 (1995) 497–507. [9] D.E. Newland, Wavelet analysis of vibration, Part I: theory, Journal of Vibration and Acoustics, Transactions of the ASME 116 (1994) 409–416. [10] D.E. Newland, Wavelet analysis of vibration, Part 2: wavelet maps, Journal of Vibration and Acoustics, Transactions of the ASME 116 (1994) 417–425. [11] D.E. Newland, Some properties of discrete wavelet maps, Probabilistic Engineering Mechanics 9 (1994) 59–69. [12] D.E. Newland, Progress in the application of wavelet theory to vibration analysis, American Society of Mechanical Engineers, Design Engineering Division (Publication) 84 (1995) 1313–1322. [13] D.E. Newland, Ridge and phase identification in the frequency analysis of transient signals by harmonic wavelets, Journal of Vibration and Acoustics, Transactions of the ASME 121 (1999) 149–155. [14] D. Boulahbal, G.M. Farid, F. Ismail, Amplitude and phase wavelet maps for the detection of cracks in geared systems, Mechanical Systems and Signal Processing 13 (1999) 423–436. [15] W.Q. Wang, F. Ismail, M.F. Golnaragh, Assessment of gear damage monitoring techniques using vibration measurements, Mechanical Systems and Signal Processing 15 (2001) 905–922. [16] G. Dalpiaz, A. Rivola, R. Rubini, Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears, Mechanical Systems and Signal Processing 14 (2000) 387–412. [17] I. Yesilyurt, A.D. Ball, Detection and severity assessment of bending fatigue failure in spur gear teeth using the continuous wavelet transform, VTT Symposium 171 (1997) 303–312. [18] C.K. Sung, H.M. Tai, C.W. Chen, Locating defects of a gear system by the technique of wavelet transform, Mechanism and Machine Theory 35 (2000) 1169–1182. [19] J.H. Suh, S.R.T. Kumara, S.P. Mysore, Machinery fault diagnostics and prognosis: application of advanced signal processing techniques, CIRP Annals—Manufacturing Technology 48 (1999) 317–320. [20] M.J. Brennan, M.H. Chen, A.G. Reynolds, Use of vibration measurements to detect local tooth defects in gears, S V Sound and Vibration 31 (1997) 12–17. [21] P. Samuel, D.J. Pines, Health monitoring and damage detection of a rotorcraft planetary gear train system using piezoelectric sensors, Proceedings of SPIE 3041 (1997) 44–53. [22] P. Samuel, D.J. Pines, Vibration separation methodology for planetary gear health monitoring, Proceedings of SPIE 3985 (2000) 250–260. [23] W.J. Staszewski, G.R. Tomlinson, Application of the wavelet transform to fault detection in a spur gear, Mechanical Systems and Signal Processing 8 (1994) 289–307. [24] S.A. Adewusi, B.O. Al-Bedoor, Wavelet analysis of vibration signals of an overhang rotor with a propagating transverse crack, Journal of Sound and Vibration 246 (2001) 777–793.


217

[25] L.X. Zhang, Z. Li, X.Y. Su, Crack, detection in beams by wavelet analysis, Proceedings of SPIE 4537 (2001) 229–232. [26] J. Zou, J. Chen, Y.P. Pu, P. Zhong, On the wavelet time–frequency analysis algorithm in identification of a cracked rotor, Journal of Strain Analysis for Engineering Design 37 (2002) 239–246. [27] S.T. Quek, Q. Wang, L. Zhang, K.-K. Ang, Sensitivity analysis of crack detection in beams by wavelet technique, International Journal of Mechanical Sciences 43 (2001) 2899–2910. [28] K.J. Jones, Wavelet crack detection algorithm for smart structures, Proceedings of SPIE 4328 (2001) 306–313. [29] Q. Wang, X.M. Deng, Damage detection with spatial wavelets, International Journal of Solids and Structures 36 (1999) 3443–3468. [30] X.M. Deng, Q. Wang, Crack detection using spatial measurements and wavelet analysis, International Journal of Fracture 91 (1998) L23–L28. [31] C. Bieman, W.J. Staszewski, C. Boller, G.R. Tomlinson, Crack detection in metallic structures using piezoceramic sensors, Key Engineering Materials 167 (1999) 112–121. [32] X.M. Deng, Q. Wang, V. Giurgiutiu, Structural health monitoring using active sensors and wavelet transforms, Proceedings of SPIE 3668 (1999) 363–370. [33] W.J. Staszewski, S.G. Pierce, K. Worden, B. Culshaw, Cross-wavelet analysis for lamb wave damage detection in composite materials using optical fibres, Key Engineering Materials 167 (1999) 373–380. [34] W.J. Staszewski, Structural and mechanical damage detection using wavelets, Shock and Vibration 30 (1998) 457–472. [35] G. Dalpiaz, A. Rivola, Condition monitoring and diagnostics in automatic machines: comparison of vibration analysis techniques, Mechanical Systems and Signal Processing 11 (1997) 53–73. [36] P.W. Tse, Y.H. Peng, R. Yam, Wavelet analysis and envelope detection for rolling element bearing fault diagnosis—their effectiveness and flexibilities, Journal of Vibration and Acoustics 123 (2001) 303–310. [37] Z. Peng, F. Chu, Y. He, Vibration signal analysis and feature extraction based on reassigned wavelet scalogram, Journal of Sound and Vibration 253 (2002) 1087–1100. [38] Z. Peng, Y. He, Q. Lu, F. Chu, Feature extraction of the rub-impact rotor system by means of wavelet, Journal of Sound and Vibration 259 (2003) 1000–1010. [39] W.J. Staszewski, G.R. Tomlinson, Application of the wavelet transform to fault detection in a spur gear, Mechanical System and Signal Processing 8 (1994) 289–307. [40] L.A. Wong, J.C. Chen, Nonlinear and chaotic behavior of structural system investigated by wavelet transform techniques, International Journal of Non-Linear Mechanics 36 (2001) 221–235. [41] V.C. Chancey, G.T. Flowers, Identification of transient vibration characteristics using absolute harmonic wavelet coefficients, JVC/Journal of Vibration and Control 7 (2001) 1175–1193. [42] P. Kang, D. Birtwhistle, Condition assessment of power transformer on-load tap-changers using wavelet analysis, IEEE Transactions on Power Delivery 16 (2001) 394–400. [43] R. Yacamini, K.S. Smith, L. Ran, Monitoring torsional vibrations of electro-mechanical systems using stator currents, Journal of Vibration and Acoustics, Transactions of the ASME 120 (1998) 72–79. [44] W.J. Wang, Wavelets for detecting mechanical faults with high sensitivity, Mechanical Systems and Signal Processing 15 (2001) 685–696. [45] A. Al-Khalidy, M. Noori, Z. Hou, R. Carmona, S. Yamamoto, A. Masuda, A. Stone, Study of health monitoring systems of linear structures using wavelet analysis, American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) 347 (1997) 49–58. [46] H.A. Gaberson, Wavelet use for transient event detection in machinery vibration signals, Proceedings of the International Modal Analysis Conference—IMAC 2 (2001) 998–1004. [47] P.W. Tse, W.Y. Yang, The practical use of wavelet transforms and their limitations in machine fault diagnosis, Proceedings of the International Symposium on Machine Condition Monitoring and Diagnosis—Mechanical Engineering Congress, Tokyo, Japan, 2002, pp. 9–16. [48] P.W. Tse, L.S. He, Can wavelet transforms used for data compression equally suitable for the use of machine fault diagnosis? Proceedings of the DETC’01, Pittsburgh, PA, 2001, vib-21647. [49] B.H. Chen, X.Z. Wang, S.H. Yang, C. McGreavy, Application of wavelets and neural networks to diagnostic system development, 1, feature extraction, Computers and Chemical Engineering 23 (1999) 899–906.



[50] X.Z. Wang, B.H. Chen, S.H. Yang, C. McGreavy, Application of wavelets and neural networks to diagnostic system development, 2, an integrated framework and its application, Computers and Chemical Engineering 23 (1999) 945–954. [51] J. Lin, L.S. Qu, Feature extraction based on Morlet wavelet and its application for mechanical fault diagnostics, Journal of Sound and Vibration 234 (2000) 135–148. [52] G.G. Yen, K.-C. Lin, Conditional health monitoring using vibration signatures, Proceedings of the IEEE Conference on Decision and Control 5 (1999) 4493–4498. [53] S. Goumas, M. Zervakis, A. Pouliezos, et al., Intelligent on-line quality control using discrete wavelet analysis features and likelihood classification, Proceedings of SPIE 4072 (2000) 500–511. [54] G.S. Stavrakakis, S.K. Goumas, M.E. Zervakis, Classification of washing machines vibration signals using discrete wavelet analysis for feature extraction, IEEE Transactions on Instrumentation and Measurement 51 (2002) 497–508. [55] C.-J. Lu, Y.-T. Hsu, Application of wavelet transform to structural damage detection, Shock and Vibration Digest 32 (2000) 50. [56] B. Liu, S.-F. Ling, Q.F. Meng, Machinery diagnostics based on wavelet packets, JVC/Journal of Vibration and Control 3 (1997) 5–17. [57] J.A. Momoh, L.G. Dias, Solar dynamic power system fault diagnostics, NASA Conference Publication 10189 (1996) 19. [58] Z. Ye, B. Wu, N. Zargari, Online mechanical fault diagnostics of induction motor by wavelet artificial neural network using stator current, IECON Proceedings 2 (2000) 1183–1188. [59] J.A. Momoh, W.E.J. Oliver, J.L. Dolce, Comparison of feature extractors on DC power system faults for improving ANN fault diagnostics accuracy, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics 4 (1995) 3615–3623. [60] J. Altmann, J. Mathew, Multiple band-pass autoregressive demodulation for rolling-element bearing fault diagnostics, Mechanical Systems and Signal Processing 15 (2001) 963–977. [61] F. Akbaryan, P.R. Bishnoi, Fault diagnostics of single-variate systems using a wavelet-based pattern recognition technique, Industrial and Engineering Chemistry Research 40 (2001) 3612–3622. [62] M. Mufti, G. Vachtsevanos, Automated fault detection and identification using a fuzzy-wavelet analysis technique, AUTOTESTCON (Proceedings), Atlanta, 1995, pp. 169–175. [63] F. Aminian, M. Aminian, Fault diagnostics of analog circuits using Bayesian neural networks with wavelet transform as preprocessor, Journal of Electronic Testing: Theory and Applications (JETTA) 17 (2001) 29–36. [64] M.A. Essawy, S. Diwakar, S. Zein-Sabbato, M. Bodruzzaman, Helicopter transmission fault diagnostics using neuro-fuzzy techniques, Intelligent Engineering Systems Through Artificial Neural Networks 7 (1997) 661–666. [65] B.A. Paya, I.I. Esat, M.N.M. Badi, Artificial neural network based fault diagnostics of rotating machinery using wavelet transforms as a preprocessor, Mechanical Systems and Signal Processing 11 (1997) 751–765. [66] Y. Wu, R. Du, Feature extraction and assessment using wavelet packets for monitoring of machining process, Mechanical Systems and Signal Processing 10 (1996) 29–53. [67] X.-G. Xu, Y.-C. Ji, W.-B. Yu, The application of wavelet singularity detection in fault diagnostics of high voltage breakers, IECON Proceedings 3 (2001) 490–494. [68] A. Hambaba, E. Huff, Multiresolution error detection on early fatigue cracks in gears, IEEE Aerospace Conference Proceedings 6 (2000) 367–372. [69] H. Zheng, Z. Li, X. Chen, Gear fault diagnosis based on continuous wavelet transform, Mechanical Systems and Signal Processing 16 (2002) 447–457. [70] G.G. Yen, K.-C. Lin, Wavelet packet feature extraction for vibration monitoring, IEEE Transactions on Industrial Electronics 47 (2000) 650–667. [71] F. Bergeaud, S.G. Mallat, Matching pursuit of images, Proceedings of SPIE 2491 (1995) 2–13. [72] B. Liu, S.F. Ling, R. Gribonval, Bearing failure detection using matching pursuit, NDT and E International 35 (2002) 255–262. [73] B. Liu, S.F. Ling, On the selection of informative wavelets for machinery diagnosis, Mechanical Systems and Signal Processing 13 (1999) 145–162.


219

[74] D. Osypiw, M. Irle, G.Y. Luo, Vibration modelling with fast Gaussian wavelet algorithm, Advances in Engineering Software 33 (2002) 191–197. [75] Z. Ren, J. Chen, X.J. Tang, W.S. Yan, Fault feature extracting by wavelet transform for control system fault detection and diagnostics, IEEE Conference on Control Applications—Proceedings 1 (2000) 485–489. [76] D. Chen, W.J. Wang, Classification of wavelet map patterns using multi-layer neural networks for gear fault detection, Mechanical Systems and Signal Processing 16 (2002) 695–704. [77] Z. Peng, Y. He, Z. Chen, F. Chu, Identification of the shaft orbit for rotating machines using wavelet modulus maxima, Mechanical Systems and Signal Processing 16 (2002) 623–635. [78] K. Shibata, A. Takahashi, T. Shirai, Fault diagnostics of rotating machinery through visualisation of sound signals, Mechanical Systems and Signal Processing 14 (2000) 229–241. [79] S. Mallat, W.L. Hwang, Singularity detection and processing with wavelets, IEEE Transactions on Information Theory 38 (1992) 617–643. [80] Q. Sun, Y. Tang, Singularity analysis using continuous wavelet transform for bearing fault diagnosis, Mechanical Systems and Signal Processing 16 (2002) 1025–1041. [81] D. Ruiz, J.M. Nougues, Z. Calderon, et al., Neural network based framework for fault diagnostics in batch chemical plants, Computers and Chemical Engineering 24 (2000) 777–784. [82] W. Tang, Y.W. Shi, Detection and classification of welding defects in friction-welded joints using ultrasonic NDT methods, Insight: Non-Destructive Testing and Condition Monitoring 39 (1997) 88–92. [83] X.Z. Dong, Z.X. Geng, Y.Z. Ge, et al., Application of wavelet transform in power system fault signal analysis, Proceedings of the Chinese Society of Electrical Engineering 17 (1997) 421–424 (in Chinese). [84] Q.-Q. Jia, L.-G. Liu, Y.-H. Yang, J.-H. Song, Abrupt change detection with wavelet for small current fault relaying, Proceedings of the Chinese Society of Electrical Engineering 21 (2001) 78–82 (in Chinese). [85] Z.W. Chen, Y.X. Lu, Signal Singularity Detection and Its Application, Journal of Vibration Engineering 10 (1997) 148–155 (in Chinese). [86] Z.W. Chen, Research on fault diagnostics of the electro-hydraulic servo-system, China Mechanical Engineering 11 (2000) 544–546 (in Chinese). [87] J. Lin, H.X. Liu, L.S. Qu, Singularity detection using wavelet and its application in mechanical fault diagnostics, Signals Processing 13 (1997) 182–188 (in Chinese). [88] Z.Y. Zhang, J.L. Gu, W.B. Wang, Application of Continuous wavelet analysis to singularity detection and Lipeschitz exponents calculation of a rub-impact signal, Journal of Vibration, Measurement and Diagnostics 21 (2001) 112–115 (in Chinese). [89] B.H. Barbara, The Word According to Wavelet: the Story of a Mathematical Technique in the Making, A.K. Peters Ltd., Wellesley, MA, 1998. [90] D.L. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory 41 (1995) 613–627. [91] Y. Zheng, D.B.H. Tay, L. Li, Signal extraction and power spectrum estimation using wavelet transform scale space filtering and Bayes shrinkage, Signal Processing 80 (2000) 1535–1549. [92] J. Altmann, J. Mathew, Multiple band-pass autoregressive demodulation for rolling-element bearing fault diagnostics, Mechanical Systems and Signal Processing 15 (2001) 963–977. [93] T.B. Littler, D.J. Morrow, Signal enhancement and de-noising of power system disturbances using the wavelet method, Proceedings of the Universities Power Engineering Conference 2 (1996) 590–593. [94] H.-T. Yang, C.-C. Liao, A de-noising scheme for enhancing wavelet-based power quality monitoring system, IEEE Transactions on Power Delivery 16 (2001) 353–360. [95] S. Menon, J.N. Schoess, R. Hamza, D. Busch, Wavelet-based acoustic emission detection method with adaptive thresholding, Proceedings of SPIE 3986 (2000) 71–77. [96] R. Shao, F. Jia, E.B. Martin, A.J. Morris, Wavelets and non-linear principal components analysis for process monitoring, Control Engineering Practice 7 (1997) 865–879. [97] J. Pineyro, A. Klempnow, V. Lescano, Effectiveness of new spectral tools in the anomaly detection of rolling element bearings, Journal of Alloys and Compounds 310 (2000) 276–279. [98] J. Lin, Feature extraction of machine sound using wavelet and its application in fault diagnostics, NDT&E 34 (2001) 25–30.



[99] J.N. Watson, P.S. Addison, A. Sibbald, De-noising of sonic echo test data through wavelet transform reconstruction, Shock and Vibration 6 (1999) 267–272. [100] S.T.S. Bukkapatnam, S.R.T. Kumara, A. Lakhtakia, P. Srinivasan, The neighborhood method and its coupling with the wavelet method for signal separation of chaotic signals, Signal Processing 82 (2002) 1351–1374. [101] S. Krishnan, R.M. Rangayyan, Denoising knee joint vibration signals using adaptive time–frequency representations, Canadian Conference on Electrical and Computer Engineering 3 (1999) 1495–1500. [102] S.-S. Hu, C. Zhou, W.-L.A Hu, new structure fault detection method based on wavelet singularity, Journal of Applied Sciences 18 (2000) 198–201 (in Chinese). [103] S.D. Liu, L.H. Zhang, Z.H. Wang, Application of wavelet denoising in diesel fault diagnostics, Journal of Mechanical Strength 23 (2001) 134–137 (in Chinese). [104] J.A. Duan, X.D. Zhang, Feature extraction from the orbit of the center of a rotor based on wavelet transform, Journal of Vibration, measurement and Diagnostics 17 (1997) 29–32 (in Chinese). [105] W.J. Staszewski, Wavelet based compression and feature selection for vibration analysis, Journal of Sound and Vibration 211 (1998) 735–760. [106] W.J. Staszewski, Vibration data compression with optimal wavelet coefficients, genetic algorithms in engineering systems, Innovations and Applications (Conference Publication) 446 (1997) 186–189. [107] M. Tanaka, M. Sakawa, K. Kato, et al., Application of wavelet transform to compression of mechanical vibration data, Cybernetics and Systems 28 (1997) 225–244. [108] D.M. Shen, W. Gao, Research on wavelet multiresolution analysis based vibration signal compress, Turbine Technology 42 (2000) 193–197 (in Chinese). [109] Y.K. Ma, S.Y. Lu, Research on vibration data compress using wavelet decomposition and embedded coding of zero trees of wavelet coefficients, Turbine Technology 43 (2001) 87–92 (in Chinese). [110] M.Q. Xu, J.Z. Zhang, G.B. Zhang, W.H. Huang, Method of data compressing for rotating machinery vibration signal based on wavelet transform, Journal of Vibration Engineering 13 (2000) 531–536 (in Chinese). [111] F. Chu, W. Lu, Determination of the rubbing location in a multi-disk rotor system by means of dynamic stiffness identification, Journal of Sound and Vibration 248 (2001) 235–246. [112] W.J. Staszewski, Identification of damping in MDOF systems using time–scale decomposition, Journal of Sound and Vibration 203 (1997) 283–305. [113] W.J. Staszewski, Identification of non-linear systems using multi-scale ridges and skeletons of the wavelet transform, Journal of Sound and Vibration 214 (1998) 639–658. [114] W.J. Staszewski, J.E. Chance, Identification of nonlinear systems using wavelets—experimental study, Proceedings of IMAC 1 (1997) 1012–1016. [115] A.M. Akhmetshin, A.S. Sendetski, Estimation of fault parameters of stratified structures based on wavelet packet algorithm in free-oscillation testing method, Proceedings of SPIE 3078 (1997) 162–169. [116] A.N. Robertson, K.C. Park, K.F. Alvin, Extraction of impulse response data via wavelet transform for structural system identification, Journal of Vibration and Acoustics, Transactions of the ASME 120 (1998) 252–260. [117] A.N. Robertson, K.C. Park, K.F. Alvin, Extraction of impulse response data via wavelet transform for structural system identification, American Society of Mechanical Engineers, Design Engineering Division (Publication) 84 (1995) 1323–1334. [118] A.N. Robertson, K.C. Park, K.F. Alvin, Identification of structural dynamics models using wavelet-generated impulse response data, Journal of Vibration and Acoustics, Transactions of the ASME 120 (1998) 261–266. [119] A.N. Robertson, K.C. Park, K.F. Alvin, Identification of structural dynamics models using wavelet-generated impulse response data, American Society of Mechanical Engineers, Design Engineering Division (Publication) 84 (1995) 1335–1344. [120] L.C. Freudinger, R. Lind, M.J. Brenner, Correlation filtering of modal dynamics using the Laplace wavelet, Proceedings of IMAC 2 (1998) 868–877. [121] Z.J. He, Y.Y. Zi, Laplace wavelet and its engineering application, Journal of Engineering Mathematics 18 (2001) 87–92 (in Chinese). [122] K.P. Yu, J.X. Zou, G.X. Shi, Application of wavelet analysis for structure system identification, Mechanics and Practices 20 (1998) 42–44 (in Chinese).


221

[123] K.P. Yu, J.X. Zou, B.Y.A Yang, method of the wavelet transform for modal parameter identification, Journal of Astronautics 20 (1999) 72–77 (in Chinese). [124] J.X. Zou, K.P. Yu, B.Y. Yang, Methods of time-varying structural parameter identification, Advances in Mechanics 30 (2000) 370–377 (in Chinese). [125] M. Ruzzene, A. Fasana, L. Garibaldi, B. Piombo, Natural frequencies and damping identification using wavelet transform: application to real data, Mechanical Systems and Signal Processing 11 (1997) 207–218. [126] Y.W. Liu, H.H. Zhang, H.X. Wu, Identification based wavelets for closely spaced modes parameters of flexible space structures, Chinese Space Science and Technology 21 (2001) 1–10 (in Chinese). [127] J.K. Ma, Y.F. Liu, Z.J. Zhu, et al., Study on identification of dynamic characteristics of oil-bearings based on wavelet analysis, Chinese Journal of Mechanical Engineering 36 (2000) 81–83 (in Chinese). [128] Z.M. Lu, J.W. Xu, Identification method of dynamic natural frequencies and damping for large-scale structure, Chinese Journal of Mechanical Engineering 37 (2001) 72–75 (in Chinese). [129] Y.Y. Kim, J.-C. Hong, N.-Y. Lee, Frequency response function estimation via a robust wavelet de-noising method, Journal of Sound and Vibration 244 (2001) 635–649. [130] W.R. Chen, Q.Q. Qian, X.R. Wan, Wavelet neural network based transient fault signal detection and identification, Proceedings of ICICS 3 (1997) 1377–1381. [131] G. Vachtsevanos, P. Wang, N. Khiripet, et al., An intelligent approach to prognostic enhancements of diagnostic systems, Proceedings of SPIE 4389 (2001) 1–14. [132] J.S. Zhao, B.Z. Chen, J.Z. Shen, Multidimensional non-orthogonal wavelet-sigmoid basis function neural network for dynamic process fault diagnostics, Computers qnd Chemical Engineering 23 (1998) 83–92. [133] P. Wang, G. Vachtsevanos, Fault prognostics using dynamic wavelet neural networks. Artificial intelligence for engineering design, Analysis and Manufacturing 15 (2001) 349–365. [134] B.R. Bakshi, G. Stephanopoulos, Wave-net: a multiresolution, hierarchical neural network with localized learning, A.I.Ch.E. Journal 39 (1999) 57–81. [135] P. Moyo, J.M.W. Brownjohn, Detection of anomalous structural behavior using wavelet analysis, Mechanical Systems and Signal Processing 16 (2002) 429–445. [136] W.J. Staszewski, J. Giacomin, Application of the wavelet based FRFs to the analysis of nonstationary vehicle data, Proceedings of IMAC 1 (1997) 425–431. [137] A. Ikonomopoulos, A. Endou, Wavelet application in process monitoring, Nuclear Technology 125 (1999) 225–234. [138] A.A. Safavi, J. Chen, J.A. Romagnoli, Wavelet-based density estimation and application to process monitoring, A.I.Ch.E. Journal 43 (1997) 1227–1241. [139] M.J. Desforges, P.J. Jacob, J.E. Cooper, Applications of probability density estimation to the detection of abnormal conditions in engineering, Proceedings of the Institution of Mechanical Engineers, Part C 212 (1998) 687–703. [140] T. Chen, L.S. Qu, The theory and application of multiresolution wavelet network, China Mechanical Engineering 08 (1997) 57–59 (in Chinese). [141] K. Mori, N. Kasashima, T. Yoshioka, Y. Ueno, Prediction of spalling on a ball bearing by applying the discrete wavelet transform to vibration signals, Wear 195 (1996) 162–168. [142] B.A. Paya, I.I. Esat, M.N.M. Badi, Artificial neural network based fault diagnostics of rotating machinery using wavelet transforms as a preprocessor, Mechanical Systems and Signal Processing 11 (1997) 751–765. [143] B.A. Jubran, M.N. Hamdan, N.H. Shabaneh, Wavelet and chaos analysis of flow induced vibration of a single cylinder in cross-flow, International Journal of Engineering Science 36 (1998) 843–864. [144] C.-H. Lamarque, S. Pernot, An overview on a wavelet multi-scale approach to investigate vibrations and stability of dynamical systems, Nonlinear Analysis, Theory, Methods and Applications 47 (2001) 2271–2281. [145] S. Pernot, C.-H. Lamarque, A wavelet-Galerkin procedure to investigate time-periodic systems: transient vibration and stability analysis, Journal of Sound and Vibration 245 (2001) 845–875. [146] C.S. Suh, A.K. Chan, Wavelet-based technique for detection of mechanical chaos, Proceedings of SPIE 4056 (2000) 267–274. [147] J.B. Zheng, H.S. Gao, Y.H. Guo, Application of wavelet transform to bifurcation and chaos study, Applied Mathematics and Mechanics (English Edition) 19 (1998) 593–599.

Application of the wavelet transform in machine condition monitoring ...

Application of the wavelet transform in machine condition monitoring ...

Suggest Documents

application of wavelet transform for monitoring of operation of high ...

Application of Discrete Wavelet Transform in Watermarking

application of the wavelet transform to filtering

application of wavelet transform and wavelet thresholding in ... - eurasip

Application of the cross wavelet transform and wavelet coherence to ...

Machine Condition Monitoring = Improved Manufacturing

Application of Wavelet Transform for Fault

Application of Wavelet Transform Analysis to ADCs

Application of wavelet transform in the ... - Academic Journals

Oil Transmission Pipelines Condition Monitoring Using Wavelet ...

Application of Short Time Fourier Transform and Wavelet Transform for

The Application of the Wavelet Transform of Travelling Wave ... - IPST

the application wavelet transform algorithm in testing adc effective ...

Application of the Lipschitz exponent and the wavelet transform to ...

Application of the wavelet transform for analysis of precipitation and ...

Application of the wavelet transform for analysis of precipitation and ...

Application of wavelet transform in reflection seismic data analysis

Application of wavelet Transform in power Quality - Semantic Scholar

Application of Improved Wavelet Transform in Biological Particle

application of the second generation wavelet transform for ... - J-Stage

The Application of Discrete Wavelet Transform with Improved ... - MDPI

Application of the Haar wavelet transform to solving ... - CiteSeerX

Application of the Haar Wavelet Tree Transform to

APPLICATION OF THE 2-D WAVELET TRANSFORM TO ... - CiteSeerX