**4.3 Bearings**

294 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

periodic characteristics of gearbox vibration signals, sliding window de-noising favorable to retain valuable information as much as possible is employed to extract and identify the fault features in gearbox signals. Experimental validations including the simulation experiments, gear fault diagnosis and normal gear detection prove the effectiveness of the multi-wavelet lifting schemes as compared to various conventional wavelets. In (Saravanan and Ramachandran, 2010) the vibration signals monitored at a bevel gear box in various conditions and fault conditions were processed with DWT. Wavelet features were extracted for all the wavelet coefficients and for all the signals using the Daubechies wavelets "db1" to "db15". ID3 Decision Tree is used for feature selection and artificial neural network were employed for classification of various faults of the gear box. The features selection of various discrete wavelets was carried out and the wavelet having the highest average efficiency of fault classification was chosen as the most appropriate. In (Rafiee et al., 2010) vibration signals recorded from two experimental set-ups were processed for gears and bearing conditions. Four statistical features were selected: standard deviation, variance, kurtosis, and fourth central moment of continuous wavelet coefficients of synchronized vibration signals (CWC-SVS). An automatic feature extraction algorithm is introduced for gear and bearing defects. It also shows that the fourth central moment of CWC-SVS is a proper feature for both bearing and gear failure diagnosis. Standard deviation and variance of CWC-SVS demonstrated more appropriate outcome for bearings than gears. Kurtosis of CWC-SVS illustrated the acceptable performance for gears only. (Wang et al., 2010) proposed a technique to provide accurate diagnosis of gearboxes under fluctuating load conditions. The residual vibration signal, i.e. the difference of time synchronously averaged signal from the average tooth-meshing vibration, is analyzed as source data due to its lower sensitiveness to the alternating load condition. Complex Morlet continuous wavelet transform was used for the vibration signal processing. A fault growth parameter (FGP) was introduced, based on the continuous wavelet transform amplitudes over all transform scales. FPG actually measures the relative CWT amplitude change. This parameter proved insensitive to varying load and can correctly indicate early gear fault. Other features such as kurtosis, mean, variance, form factor and crest factor, both of residual signal and mean amplitude of continuous wavelet transform waveform, were also checked and proved to be influenced by the changing load. The effectiveness of the proposed fault indicator was demonstrated using a full lifetime vibration data history obtained under sinusoidal varying

To overcome the shift-variance deficiency of classical DWT, a novel fault diagnosis method based on the redundant second generation wavelet packet transform was proposed in (Zhou et al., 2010). Initially, the redundant second generation wavelet packet transform (RSGWPT) was constructed on the basis of second generation wavelet transform and redundant lifting scheme. Then, the vibration signals were decomposed by RSGWPT and the faulty features were extracted from the resultant wavelet packet coefficients. In the end, the extracted fault features were given as input to classifiers for identification/classification. The proposed method was applied for the fault diagnosis of gearbox and gasoline engine valve trains. Test results indicate that a better classification performance can be obtained by using the proposed fault diagnosis method in comparison with using conventional second generation wavelet packet transform method. (Wang et al., 2010) employed the dual-tree complex wavelet transform (DTCWT) for the de-noising of vibration signals from gearbox and bearings monitoring. They compared the de-noising via DTCWT with other wavelet-based techniques (DWT and second generation wavelet transform (SGWT)) as well as with fast

load.

The fault diagnosis of rolling element bearings is very important for improving mechanical system reliability and performance in rotating machinery as bearing failures are among the most frequent causes of breakdowns in rotating machinery. When localized fault occurs in a bearing, periodic or non-periodic impulses appear in the time domain of the vibration signal, and the corresponding bearing characteristic frequencies (BCFs) and their harmonics emerge in the frequency domain. However, in the early stage of bearing failures, the BCFs usually carry very little energy and are often suppressed/hidden by noise and higher-level macro-structural vibrations. Consequently an effective signal processing method is of utmost importance in the de-noising of vibration or acoustic emission signals acquired or the extraction of damage sensitive features during the condition monitoring of bearings. Wavelet-based techniques meet this challenge in a variety of applications presented in the following.

(Purushotham et al., 2005) have applied the DWT towards the detection of localized bearing defects. The vibration signals were decomposed up to 4 levels using "db2" mother wavelet. The complex cepstral coefficients for wavelet transformed time windows at Mel-frequency scales constituted the features that trained Hidden Markov Models for the fault detection and classification.

In (Yan and Gao, 2005) the Discrete Harmonic Wavelet Packet Transform (DHWPT) was used to decompose the vibration signals measured from a bearing test bed into a number of frequency sub-bands. Given the harmonic wavelet packet coefficients of a vibration signal x(t), the energy feature in each sub-band was calculated as:

$$Energy(\mathbf{s}, t) = \Sigma\_{k=1}^{N} |h\mathbf{w}pt(\mathbf{s}, t, k)|^{2} \tag{53}$$

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 297

induction motors. The discrete Meyer wavelet was used to decompose the recorded signals in three levels. The defect frequency region was determined, and the coefficient energies in the related nodes were calculated. In comparison with the healthy condition, the energy was found to increase in the nodes related to defect frequency regions, therefore it was used as a diagnostic parameter. (Hu et al., 2007) introduced a methodology for fault diagnosis based on improved wavelet package transform (IWPT), a distance evaluation technique and the support vector machines (SVMs) ensemble. Their method consists of three stages. Firstly, with investigating the feature of impact fault in vibration signals, a biorthogonal wavelet with impact property is constructed via lifting scheme, and the IWPT is carried out for feature extraction from the raw vibration signals. Then, the faulty features can be detected by envelope spectrum analysis of wavelet package coefficients of the most salient frequency band. Secondly, with the distance evaluation technique, the optimal features are selected from the statistical characteristics of raw signals and wavelet package coefficients, and the energy characteristics of decomposition frequency band. Finally, the optimal features are input into the SVMs in order to identify the different abnormal cases. The proposed method was applied to the fault diagnosis of rolling element bearings, and testing results showed that the SVMs ensemble can reliably separate different fault conditions and identify the

(Lei et al., 2009) suggested a method relying on wavelet packets transform (WPT) and empirical mode decomposition (EMD) to preprocess vibration signals and extract fault characteristic information from them. Each of the raw vibration signals is decomposed with "db10" WPT at level 3. From a plethora of features extracted at each sub-band, the most relevant ones were selected via distance evaluation techniques and forwarded into a radial basis function (RBF) network to automatically identify different faults (inner race, outer race, roller) in rolling element bearings. A novel health index called frequency spectrum growth index (FSGI) to detect health condition of gear, based on wavelet decomposition was presented in (Wang et al., 2009). "db9" mother wavelet was chosen for signal decomposition and the maximum wavelet decomposition level is 4. In order to evaluate the performance of the proposed FSGI index various wavelets at various decomposition levels were tested. The results obtained prove that FSGI is insensitive to the selection of wavelet type and decomposition level. Three sets of vibration data collected from a mechanical diagnostics test bed were collected and analyzed in order to validate the method. An anti-aliasing lifting scheme is applied by (Bao et al.,2009) to analyze vibration signals measured from faulty ball bearings and testing results confirm that the proposed method is effective for extracting weak fault feature from a complex background. The simple lifting scheme (or 2nd generation wavelet transform) was altered by discarding the split and merge operations and modifying accordingly the prediction and update operators improving significantly the frequency aliasing issue. Testing results showed that the anti-aliasing lifting scheme performs better than the lifting scheme and the redundant lifting scheme in terms of increasing the accuracy of classification algorithms (ANNs or SVMs) of faulty bearing signals. (Yuan et al., 2009) introduced a new method based on adaptive multi-wavelets via two-scale similarity transforms (TSTs). TSTs are simple methods to construct new biorthogonal multi-wavelets with properties of symmetry, short support and vanishing moments. Based on kurtosis maximization principle, adaptive multi-wavelets were designed to match the transient faults in rotating machinery. Genetic algorithms (GAs) were applied to select the optimal multiwavelets and the method was used to successfully diagnose bearing outer-race faults. (Zhu

severity of incipient faults.

The key features were then used as inputs to neural network classifiers for assessing the system's health status. Comparing to the conventional approach where statistical parameters from raw vibration signals are used, the presented approach enables higher signal-to-noise ratios and consequently, more effective and intelligent use of the available sensor information, leading to more accurate system health evaluation.

(Qiu et al., 2006) assessed the performance of wavelet decomposition-based de-noising versus wavelet filter-based de-noising methods on signals from mechanical defects. The comparison revealed that wavelet filter is more suitable and reliable to detect a weak signature of mechanical impulse-like defect signals, whereas the wavelet decomposition denoising method can achieve satisfactory results on smooth signal detection. In order to select optimal parameters for the wavelet filter, a two-step optimization process was proposed. Minimal Shannon entropy was used to optimize the Morlet wavelet shape factor. A periodicity detection method based on singular value decomposition (SVD) was then used to choose the appropriate scale for the wavelet transform. The experimental results verify the effectiveness of the proposed method.

(Abbasion et al., 2007) studied the condition of an electric motor with two rolling bearings (one next to the output shaft and the other next to the fan) with one normal state and three faulty states each. De-noising via the CWT (Meyer wavelet) was conducted and support vector machines (SVMs) were used for the fault classification task. Results have showed 100% accuracy in fault detection. (Ocak et al., 2007) developed a new scheme based on wavelet packet decomposition and hidden Markov modeling (HMM) for the condition monitoring of bearing faults. In this scheme, vibration signals were decomposed into wavelet packets and the node energies of the 3-level decomposition tree were used as features. Based on the features extracted from normal bearing vibration signals, an HMM was trained to model the normal bearing operating condition. The probabilities of this HMM were then used to track the condition of the bearing. In (Zarei and Poshtan, 2007) WPT was used to process stator current signals in order to detect defective bearings at

usually carry very little energy and are often suppressed/hidden by noise and higher-level macro-structural vibrations. Consequently an effective signal processing method is of utmost importance in the de-noising of vibration or acoustic emission signals acquired or the extraction of damage sensitive features during the condition monitoring of bearings. Wavelet-based techniques meet this challenge in a variety of applications presented in the

(Purushotham et al., 2005) have applied the DWT towards the detection of localized bearing defects. The vibration signals were decomposed up to 4 levels using "db2" mother wavelet. The complex cepstral coefficients for wavelet transformed time windows at Mel-frequency scales constituted the features that trained Hidden Markov Models for the fault detection

In (Yan and Gao, 2005) the Discrete Harmonic Wavelet Packet Transform (DHWPT) was used to decompose the vibration signals measured from a bearing test bed into a number of frequency sub-bands. Given the harmonic wavelet packet coefficients of a vibration signal

������(�� �) = ∑ |����(�� �� �)| � �

The key features were then used as inputs to neural network classifiers for assessing the system's health status. Comparing to the conventional approach where statistical parameters from raw vibration signals are used, the presented approach enables higher signal-to-noise ratios and consequently, more effective and intelligent use of the available

(Qiu et al., 2006) assessed the performance of wavelet decomposition-based de-noising versus wavelet filter-based de-noising methods on signals from mechanical defects. The comparison revealed that wavelet filter is more suitable and reliable to detect a weak signature of mechanical impulse-like defect signals, whereas the wavelet decomposition denoising method can achieve satisfactory results on smooth signal detection. In order to select optimal parameters for the wavelet filter, a two-step optimization process was proposed. Minimal Shannon entropy was used to optimize the Morlet wavelet shape factor. A periodicity detection method based on singular value decomposition (SVD) was then used to choose the appropriate scale for the wavelet transform. The experimental results verify

(Abbasion et al., 2007) studied the condition of an electric motor with two rolling bearings (one next to the output shaft and the other next to the fan) with one normal state and three faulty states each. De-noising via the CWT (Meyer wavelet) was conducted and support vector machines (SVMs) were used for the fault classification task. Results have showed 100% accuracy in fault detection. (Ocak et al., 2007) developed a new scheme based on wavelet packet decomposition and hidden Markov modeling (HMM) for the condition monitoring of bearing faults. In this scheme, vibration signals were decomposed into wavelet packets and the node energies of the 3-level decomposition tree were used as features. Based on the features extracted from normal bearing vibration signals, an HMM was trained to model the normal bearing operating condition. The probabilities of this HMM were then used to track the condition of the bearing. In (Zarei and Poshtan, 2007) WPT was used to process stator current signals in order to detect defective bearings at

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x(t), the energy feature in each sub-band was calculated as:

the effectiveness of the proposed method.

sensor information, leading to more accurate system health evaluation.

following.

and classification.

induction motors. The discrete Meyer wavelet was used to decompose the recorded signals in three levels. The defect frequency region was determined, and the coefficient energies in the related nodes were calculated. In comparison with the healthy condition, the energy was found to increase in the nodes related to defect frequency regions, therefore it was used as a diagnostic parameter. (Hu et al., 2007) introduced a methodology for fault diagnosis based on improved wavelet package transform (IWPT), a distance evaluation technique and the support vector machines (SVMs) ensemble. Their method consists of three stages. Firstly, with investigating the feature of impact fault in vibration signals, a biorthogonal wavelet with impact property is constructed via lifting scheme, and the IWPT is carried out for feature extraction from the raw vibration signals. Then, the faulty features can be detected by envelope spectrum analysis of wavelet package coefficients of the most salient frequency band. Secondly, with the distance evaluation technique, the optimal features are selected from the statistical characteristics of raw signals and wavelet package coefficients, and the energy characteristics of decomposition frequency band. Finally, the optimal features are input into the SVMs in order to identify the different abnormal cases. The proposed method was applied to the fault diagnosis of rolling element bearings, and testing results showed that the SVMs ensemble can reliably separate different fault conditions and identify the severity of incipient faults.

(Lei et al., 2009) suggested a method relying on wavelet packets transform (WPT) and empirical mode decomposition (EMD) to preprocess vibration signals and extract fault characteristic information from them. Each of the raw vibration signals is decomposed with "db10" WPT at level 3. From a plethora of features extracted at each sub-band, the most relevant ones were selected via distance evaluation techniques and forwarded into a radial basis function (RBF) network to automatically identify different faults (inner race, outer race, roller) in rolling element bearings. A novel health index called frequency spectrum growth index (FSGI) to detect health condition of gear, based on wavelet decomposition was presented in (Wang et al., 2009). "db9" mother wavelet was chosen for signal decomposition and the maximum wavelet decomposition level is 4. In order to evaluate the performance of the proposed FSGI index various wavelets at various decomposition levels were tested. The results obtained prove that FSGI is insensitive to the selection of wavelet type and decomposition level. Three sets of vibration data collected from a mechanical diagnostics test bed were collected and analyzed in order to validate the method. An anti-aliasing lifting scheme is applied by (Bao et al.,2009) to analyze vibration signals measured from faulty ball bearings and testing results confirm that the proposed method is effective for extracting weak fault feature from a complex background. The simple lifting scheme (or 2nd generation wavelet transform) was altered by discarding the split and merge operations and modifying accordingly the prediction and update operators improving significantly the frequency aliasing issue. Testing results showed that the anti-aliasing lifting scheme performs better than the lifting scheme and the redundant lifting scheme in terms of increasing the accuracy of classification algorithms (ANNs or SVMs) of faulty bearing signals. (Yuan et al., 2009) introduced a new method based on adaptive multi-wavelets via two-scale similarity transforms (TSTs). TSTs are simple methods to construct new biorthogonal multi-wavelets with properties of symmetry, short support and vanishing moments. Based on kurtosis maximization principle, adaptive multi-wavelets were designed to match the transient faults in rotating machinery. Genetic algorithms (GAs) were applied to select the optimal multiwavelets and the method was used to successfully diagnose bearing outer-race faults. (Zhu

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 299

signals have demonstrated that the proposed methodology can effectively measure the

(Xian, 2010) presented a combined discrete wavelet transform (DWT) and support vector machine (SVM) technique for mechanical failure classification of spherical roller bearing application in high performance hydraulic injection molding machine. The proposed technique consists of preprocessing the mechanical failure vibration signal samples using discrete wavelet transform with 'db2' mother wavelet at the fourth level of decomposition of vibration signal for failure classification. The energy of the approximation and the details was calculated and populated the feature vectors that trained the support vector machine that was built for the classification of mechanical failure types of the spherical roller bearings. In (Yan and Gao, 2010) the generalized harmonic wavelet transform (HWT) was used to enhance the signal-to-noise ratio for effective machine defect identification in rolling bearings that contained different types of structural defects. In harmonic wavelet transform a series of sub-frequency band wavelet coefficients are constructed by choosing different harmonic wavelet parameter pairs. The energy and entropy associated with each subfrequency band are then calculated. The filtered signal is obtained by choosing the wavelet coefficients whose corresponding sub-frequency band has the highest energy-to- entropy ratio. Experimental studies using rolling bearings that contain different types of structural defects have confirmed that the developed new technique enables high signal-to-noise ratio for effective machine defect identification. (Su et al., 2010) developed a new autocorrelation enhancement algorithm including two aspects of autocorrelation and extended Shannon function. This method does not need to select a threshold and can be implemented in an automatic way and is realized in various stages. First, to eliminate the frequency associated with interferential vibrations, the vibration signal is filtered with a band-pass filter determined by a Morlet wavelet whose parameters are optimized by genetic algorithm. Then, the envelope of the autocorrelation function of the filtered signal is calculated. Finally the enhanced autocorrelation envelope power spectrum is obtained. The method is employed to the simulated signal and the real bearing vibration signals under various conditions, such as normal, inner-race fault and outer-race fault. There are only several single spectrum lines left in the enhanced autocorrelation envelope power spectrum. The single spectrum line with largest amplitude is corresponding to the bearing fault frequency for a defective bearing while it is corresponding to the shaft rotational frequency for a normal bearing. (Huang et al., 2010) utilized the lifting-based second generation wavelet packet transform to process vibration signals from a rolling element bearing test. The wavelet packet energy was calculated by the coefficients at the nth node of the wavelet packet. This corresponds to the energy of the coefficients in a certain frequency band. Normalization is applied to minimize possible bias due to different ranges of the wavelet packet energies. The fuzzy c-means method has been used to assess the bearing performance and classify the faulty and the healthy recordings. In (Pan et al., 2010) a new method based on lifting wavelet packet decomposition and fuzzy c-means for bearing performance degradation assessment is proposed. Vibration signals during run-in tests up to bearing failure were processed with lifting wavelet packet. Feature vectors composed of node energies were constructed and fed in a fuzzy c-means expert system for classification of healthy, degraded and failed bearings. (He et al., 2010) proposed a hybrid method which combines Morlet wavelet filter and sparse code shrinkage (SCS) to extract the impulsive features buried in the vibration signal. Initially, the parameters of a Morlet wavelet filter

severity of both inner- and outer-race faults.

et al., 2009) introduced a new method that combines the CWT -through the Morlet waveletand the Kolmogorov–Smirnov test to detect transients contained in the vibrations signals from gearbox as well as faulty bearings. CWT initially decomposed the time domain vibration signals into two dimensional time-scale plane. By removing the Gaussian noise coefficients at all scales in the time-scale plane and then applying the inverse CWT to the noise reduced wavelet coefficients, the signal transients in the time domain were evaluated enhancing thus the difficult task of effective and reliable fault identification. A new robust method relying on the improved wavelet packet decomposition (IWPD) and support vector data description (SVDD) is proposed in (Pan et al., 2009). Node energies of IWPD were used to compose feature vectors. Based on feature vectors extracted from normal signals, a SVDD model fitting a tight hypersphere around them is trained, the general distance of test data to this hypersphere being used as the health index. IWPD is based on the second generation wavelet transform (SGWT) realized by lifting scheme. SVDD is an excellent method of oneclass classification, with the advantages of robustness and high computation. A methodology developed on the combination of these two methods for bearing performance degradation proved effective and reliable when applied to vibration signals from a bearing accelerated life test. (Feng et al., 2009) introduced the normalized wavelet packets quantifiers as a new feature set for the detection and diagnosis of localized bearing defect and contamination fault. The "Wavelet packets relative energy" measures the normalized energy of the wavelet packets node; the "Total wavelet packets entropy" measures how the normalized energies of the wavelet packets nodes are distributed in the frequency domain; the "Wavelet packets node entropy" describes the uncertainty of the normalized coefficients of the wavelet packets node. Unlike the conventional feature extraction methods, which use the amplitude of wavelet coefficients, these new features were derived from probability distributions and are more robust for diagnostic applications. Acoustic Emission signals from faulty bearings of rotating machines were recorded and the new features were calculated via WPT and Daubechies mother wavelets ("db1-db10"). Their study showed that both localized defects and advanced contamination faults can be successfully detected and diagnosed if the appropriate feature was chosen. The Bayesian classifier was also used to quantitatively analyze and evaluate the performance of the proposed features. They also showed that by reducing the Daubechies wavelet order or the length of the signal segment will generally increase the classification rate probability. (Hao and Chu, 2009) presented a novel morphological undecimated wavelet (MUDW) decomposition scheme for fault diagnostics of rolling element bearings. The MUDW scheme was developed based on the morphological wavelet (MW) theory and was applied for both the extraction of impulse components and de-noising. The efficiency of the MUDW was assessed using simulated data as well as monitored vibration signals from a bearing test rig. (Hong and Liang, 2009) presented a new version of the Lempel–Ziv complexity as a bearing fault (single point) severity measure based on the continuous wavelet transform (CWT). The CWT (realized with the Morlet wavelet) was used to identify the best scale where the fault resides and eliminate the interferences of noise and irrelevant signal components as much as possible. Next, the Lempel–Ziv complexity values were calculated for both the envelope and highfrequency carrier signal obtained from wavelet coefficients at the best scale level. As the noise and other un-related signal components have been removed, the Lempel–Ziv complexity value will be mostly contributed by the bearing system and hence can be reliably used as a bearing fault measure. The applications to the bearing inner- and outer-race fault

et al., 2009) introduced a new method that combines the CWT -through the Morlet waveletand the Kolmogorov–Smirnov test to detect transients contained in the vibrations signals from gearbox as well as faulty bearings. CWT initially decomposed the time domain vibration signals into two dimensional time-scale plane. By removing the Gaussian noise coefficients at all scales in the time-scale plane and then applying the inverse CWT to the noise reduced wavelet coefficients, the signal transients in the time domain were evaluated enhancing thus the difficult task of effective and reliable fault identification. A new robust method relying on the improved wavelet packet decomposition (IWPD) and support vector data description (SVDD) is proposed in (Pan et al., 2009). Node energies of IWPD were used to compose feature vectors. Based on feature vectors extracted from normal signals, a SVDD model fitting a tight hypersphere around them is trained, the general distance of test data to this hypersphere being used as the health index. IWPD is based on the second generation wavelet transform (SGWT) realized by lifting scheme. SVDD is an excellent method of oneclass classification, with the advantages of robustness and high computation. A methodology developed on the combination of these two methods for bearing performance degradation proved effective and reliable when applied to vibration signals from a bearing accelerated life test. (Feng et al., 2009) introduced the normalized wavelet packets quantifiers as a new feature set for the detection and diagnosis of localized bearing defect and contamination fault. The "Wavelet packets relative energy" measures the normalized energy of the wavelet packets node; the "Total wavelet packets entropy" measures how the normalized energies of the wavelet packets nodes are distributed in the frequency domain; the "Wavelet packets node entropy" describes the uncertainty of the normalized coefficients of the wavelet packets node. Unlike the conventional feature extraction methods, which use the amplitude of wavelet coefficients, these new features were derived from probability distributions and are more robust for diagnostic applications. Acoustic Emission signals from faulty bearings of rotating machines were recorded and the new features were calculated via WPT and Daubechies mother wavelets ("db1-db10"). Their study showed that both localized defects and advanced contamination faults can be successfully detected and diagnosed if the appropriate feature was chosen. The Bayesian classifier was also used to quantitatively analyze and evaluate the performance of the proposed features. They also showed that by reducing the Daubechies wavelet order or the length of the signal segment will generally increase the classification rate probability. (Hao and Chu, 2009) presented a novel morphological undecimated wavelet (MUDW) decomposition scheme for fault diagnostics of rolling element bearings. The MUDW scheme was developed based on the morphological wavelet (MW) theory and was applied for both the extraction of impulse components and de-noising. The efficiency of the MUDW was assessed using simulated data as well as monitored vibration signals from a bearing test rig. (Hong and Liang, 2009) presented a new version of the Lempel–Ziv complexity as a bearing fault (single point) severity measure based on the continuous wavelet transform (CWT). The CWT (realized with the Morlet wavelet) was used to identify the best scale where the fault resides and eliminate the interferences of noise and irrelevant signal components as much as possible. Next, the Lempel–Ziv complexity values were calculated for both the envelope and highfrequency carrier signal obtained from wavelet coefficients at the best scale level. As the noise and other un-related signal components have been removed, the Lempel–Ziv complexity value will be mostly contributed by the bearing system and hence can be reliably used as a bearing fault measure. The applications to the bearing inner- and outer-race fault signals have demonstrated that the proposed methodology can effectively measure the severity of both inner- and outer-race faults.

(Xian, 2010) presented a combined discrete wavelet transform (DWT) and support vector machine (SVM) technique for mechanical failure classification of spherical roller bearing application in high performance hydraulic injection molding machine. The proposed technique consists of preprocessing the mechanical failure vibration signal samples using discrete wavelet transform with 'db2' mother wavelet at the fourth level of decomposition of vibration signal for failure classification. The energy of the approximation and the details was calculated and populated the feature vectors that trained the support vector machine that was built for the classification of mechanical failure types of the spherical roller bearings. In (Yan and Gao, 2010) the generalized harmonic wavelet transform (HWT) was used to enhance the signal-to-noise ratio for effective machine defect identification in rolling bearings that contained different types of structural defects. In harmonic wavelet transform a series of sub-frequency band wavelet coefficients are constructed by choosing different harmonic wavelet parameter pairs. The energy and entropy associated with each subfrequency band are then calculated. The filtered signal is obtained by choosing the wavelet coefficients whose corresponding sub-frequency band has the highest energy-to- entropy ratio. Experimental studies using rolling bearings that contain different types of structural defects have confirmed that the developed new technique enables high signal-to-noise ratio for effective machine defect identification. (Su et al., 2010) developed a new autocorrelation enhancement algorithm including two aspects of autocorrelation and extended Shannon function. This method does not need to select a threshold and can be implemented in an automatic way and is realized in various stages. First, to eliminate the frequency associated with interferential vibrations, the vibration signal is filtered with a band-pass filter determined by a Morlet wavelet whose parameters are optimized by genetic algorithm. Then, the envelope of the autocorrelation function of the filtered signal is calculated. Finally the enhanced autocorrelation envelope power spectrum is obtained. The method is employed to the simulated signal and the real bearing vibration signals under various conditions, such as normal, inner-race fault and outer-race fault. There are only several single spectrum lines left in the enhanced autocorrelation envelope power spectrum. The single spectrum line with largest amplitude is corresponding to the bearing fault frequency for a defective bearing while it is corresponding to the shaft rotational frequency for a normal bearing. (Huang et al., 2010) utilized the lifting-based second generation wavelet packet transform to process vibration signals from a rolling element bearing test. The wavelet packet energy was calculated by the coefficients at the nth node of the wavelet packet. This corresponds to the energy of the coefficients in a certain frequency band. Normalization is applied to minimize possible bias due to different ranges of the wavelet packet energies. The fuzzy c-means method has been used to assess the bearing performance and classify the faulty and the healthy recordings. In (Pan et al., 2010) a new method based on lifting wavelet packet decomposition and fuzzy c-means for bearing performance degradation assessment is proposed. Vibration signals during run-in tests up to bearing failure were processed with lifting wavelet packet. Feature vectors composed of node energies were constructed and fed in a fuzzy c-means expert system for classification of healthy, degraded and failed bearings. (He et al., 2010) proposed a hybrid method which combines Morlet wavelet filter and sparse code shrinkage (SCS) to extract the impulsive features buried in the vibration signal. Initially, the parameters of a Morlet wavelet filter

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 301

fault have been successfully detected by using the proposed approach. In (Peng et al., 2007) the wavelet transform modulus maximal (WTMM) method was used to calculate the Lipschitz exponents of the vibration signals with different faults. The Lipschitz exponent can give a quantitative description of the signal's singularity. The proposed singularity based parameters proved a set of excellent diagnostic features, which could separate the four kinds of faults very well. The results showed that, with the fault severity increasing, the vibration signals' singularities and singularity ranges increased as well, and therefore one could evaluate the fault severity through measuring the vibration signals' singularities and

(Wu and Liu, 2008) instead of WPT utilized a DWT technique combined with a feature selection of energy spectrum and fault classification using ANNs for analyzing fault signals of internal combustion engines. The features of the sound emission signals at different resolution levels were extracted by multi-resolution analysis and Parseval's theorem. (Niu et al., 2008) applied multi-level wavelet decomposition on transient stator current signals for fault diagnosis of induction motors. After the signal preprocessing using smoothing– subtracting and wavelet transform techniques, features were extracted from each level of detail component of decomposed signals using DWT and "db10" mother wavelet. 21 features in total are acquired from each sensor consisting of the time domain (10 features), frequency domain (three features) and regression estimation (eight features). Totally, two 70 321 features sets are calculated from seven types of signals collected by three current probes at each wavelet decomposition level. The calculated two features sets consisted of the training and test sets respectively and consist of the input in four different classifiers for pattern recognition with quite satisfactory results. (Chen et al., 2008) proposed a methodology based on Wavelet Packet Analysis (WPA) and Kolmogorov-Smirnov (KS) test to analyze monitored vibration signals from the water hydraulic motor to assess the fault degradation of the pistons in water hydraulic motor. The fault detection procedure applied is summarized in the following. First, the time-domain vibration signals were decomposed through the WPT in two levels. The soft-thresholding technique was used in the wavelet and approximation coefficients to get the de-noised coefficients. The reconstructed denoised vibration signal with improved signal-to-noise ratio (SNR) was obtained by reconstructing the de-noised coefficients in the multi-decomposition of the vibration signal. Then the kurtosis of the de-noised signal was calculated and finally the KS test was used to classify the kurtosis statistical probability distribution (SPD) under seven different piston conditions. Thus the piston condition in water hydraulic motor was successfully assessed. (Widodo and Yang, 2008) introduced an intelligent system for faults detection and classification of induction motor using wavelet support vector machines (W-SVMs). W-SVMs were built by utilizing the kernel function using wavelets. Transient current signals were monitored in various damage conditions of the induction motor. The acquired signals were preprocessed through DWT ("db5", 5 levels) and various statistical features were extracted. Principal component analysis (PCA) and kernel PCA were utilized to reduce the dimension of features and to extract the useful features for classification process. Finally the classification process for diagnosing the faults was carried out using W-SVMs and

conventional SVMs based on one against-all multi-class classification.

(Wu and Liu, 2009) proposed a fault diagnosis system for internal combustion engines using wavelet packet transform (WPT) and artificial neural network (ANN) techniques on monitored sound emission signals. In the preprocessing phase, WPT coefficients are used,

singularity ranges.

(center frequency and bandwidth) are optimized by differential evolution (DE) in order to eliminate the interferential vibrations and obtain the fault characteristic signal. Then, to further enhance the impulsive features and suppress residual noise, SCS which is a softthresholding method based on maximum likelihood estimation (MLE) is applied to the filtered signal. The results of simulated experiments and real bearing vibration signals verify the effectiveness of the proposed method in extracting impulsive features from noisy signals in condition monitoring.

(Chiementin et al., 2010) studied the effect of wavelet de-noising and other techniques on acoustic emission signals from faulty bearings. They applied DWT and attempted to optimize the various parameters selection involved in a wavelet-based de-noising scheme. They assessed the different de-noising techniques and concluded that the wavelet approach enhanced the signal kurtosis and crest factor more than the other techniques.
