**4. Applications overview of wavelets in condition based maintenance**

#### **4.1 Wavelet-based de-noising**

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

that db44 is the most similar function across both gear and bearing vibration signals. The drawback of the db44 function is that the high-order db functions take more CPU time than most others. In another work (Rafiee et al., 2009) utilized genetic algorithms (GAs) to optimize the selection of mother wavelet function (among several members of the Daubechies family), the number of the decomposition levels of the wavelet packet transform (WPT) as well as the number of neurons in the ANNs hidden layers used for the fault classification, resulted in a high-speed, effective two-layer ANN with a small-sized structure. "db11", level 4 and 14 neurons have been selected as the best values for Daubechies order, decomposition level, and the number of nodes in hidden layer, respectively. In (Gketsis et al., 2009) the optimum wavelet choice criterion is the maximization of the cross-correlation between the signal of interest and the wavelet. In an application of condition monitoring in electrical machines, they tested several wavelet functions, namely Haar, Daubechies 2, 4, 8, Symlet 2, 3, 4, 8 and Coiflet 3 and concluded to "db2". (Saravanan and Ramachandran, 2009) found that among the 15 members of Daubechies wavelet, "db1" and "db5" gave the maximum classification efficiency of an

Other researchers prefer more qualitative explanations. (Xu and Li, 2008) support that in the common family of wavelet bases i.e. Morlet, Haar, Shannon, Symmlets, Coiflets and Daubechies wavelets, etc., the most popular is the Daubechies wavelet, as it bears the shortest compactly supported scaling function in all of orthogonal wavelets when given exponent number of vanishing moment. Moreover, it gives the best overall performance in the respect of both mean squared error between reconstruction signal and original signal, and maximizing the SNR improvement. Therefore, the Daubechies wavelet is applied and others are for comparison in this case. (Jazebi et al., 2011) state that one specific mother wavelet is best suited for a particular application. For this purpose, mother wavelet type and decomposition level have been chosen based on experience and trial and error. The research includes detecting and analyzing low amplitude, short duration, fast decaying, and oscillating type of current signals. For this purpose, Daubechies's mother wavelet seems to be an appropriate choice. In comparison with Haar wavelet, Daubechies are best suited for feature extraction due to their low-pass and highpass filters. On the other hand because of its inherent orthogonality, it satisfies Parseval theorem, not like biorthogonal wavelets such as Coiflet and Meyer wavelets . db4 mother wavelet over level d4 has been chosen because the maximum energy localization in details

(Daviu et al., 2007) supports that the Daubechies family is well suited for application of DWT in condition monitoring due to its interesting inherent properties. An important fact they observed when using the Daubechies family, was the overlap between the frequency bands (frequency aliasing) associated with the DWT decomposition of their signals. This is due to the non-ideal filtering process performed by the wavelet signals, a fact that makes that the signal components, included within a certain frequency band and placed in the proximity of its limits, overlap partially with the adjacent band. When using a high-order Daubechies wavelet for signal decomposition, this effect is less intense than when using a low-order one. In other words, high-order wavelets behave as more ideal filters. Maximization of statistical features such as kurtosis or crest factor can be utilized as a criterion for the choice of mother wavelet within a family or among various families. In an

expert system (Decision Tree) at around 98.7%.

(1–4) was obtained using these parameters.

Wavelet based de-noising is a very interesting and important application of wavelets in the processing of signals from condition monitoring. It is very widely adopted in many studies as it is ideal to extract hidden diagnostic information and enhance the impulsive components of complex, non-stationary signals with strong background. Wavelet thresholding is based on the idea that the energy of the signal is concentrated in a few wavelet coefficients, while the energy of noise spreads throughout all the resulted wavelet coefficients. Similarity between the mother wavelet and the signal to be analyzed plays a very important role, making it possible for the signal to concentrate on fewer coefficients and thus its choice is critical in the efficiency of the de-noising task. The first foundations in wavelet-based de-noising were set by (Donoho, 1995). Let *x(t)* be the discrete signal acquired during condition monitoring. The signal series consists of impulses and noise. x(t) can alternatively be expressed as *x(t)=p(t)+n(t),* where *p(t)* indicates the impulses to be determined, whereas *n(t)* indicates equally distributed and independent Gaussian noise with mean zero and standard deviation r. In principle, the wavelet threshold de-noising procedure has the following steps:


The second step is probably the most critical and has quite an impact upon the effectiveness of the procedure. There are plenty of thresholding techniques and many different thresholds proposed in the literature. Hard thresholding sets any coefficient less than or equal to the threshold to zero.

$$c\_{jk} = \begin{cases} 0, & c\_{jk} < t \\ c\_{jk}, & c\_{jk} \ge t \end{cases} \tag{50}$$

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 289

2. At each resolution level j, group the noisy wavelet coefficients into disjoint blocks bij of length *L0=log(n)/2*; then extend each block bij by an amount of max(1,L0/2) in each

3. Within each block bij, each noisy wavelet coefficient is processed via "NeighCoeff"

In Fig. 10 various de-noising algorithms were applied on an AE signal from a bearing with seeded defect. In a) the original signal is depicted. In b) the method of spectral kurtosis (Randall and Antoni, 2011) is utilized. Spectral kurtosis is not a wavelet-based technique and relies on the location of the frequency band where kurtosis is maximized and then the band-pass filtering of the signal in the resulted band. In figure c) the DTCWT wavelet transform is applied in combination with "NeighCoeff" thresholding whilst in d) a parametric procedure was used by the authors to determine the optimum parameters of DWT (wavelet type, number of levels, threshold type, soft or hard application of threshold) that maximize the kurtosis and crest factor of the signal. DTCWT- and DWT-based de-



0

0.05

0.1

direction to form overlapping larger blocks Bij of length *L=L0+2L1*

noising proved the most efficient in terms of the resulting signal kurtosis.

Fig. 10. Effect of various de-noising schemes on an AE signal from defective bearing a) original signal b) de-noised signal via spectral kurtosis technique c) de-noised signal via

<sup>0</sup> 0.005 0.01 0.015 0.02 0.025 0.03 0.035 -0.1

SK-denoised Kurtosis=34.6826

Time(sec)

DWT-denoised Kurtosis=84.8574

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Time(sec)

Fault symptoms of running gearboxes must be detected as early as possible to avoid serious accidents. An efficient monitoring plan is needed for any industry because it can optimize the resources management and improve the plant economy, by reducing unnecessary costs

DTCWT d) de-noised signal via DWT

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Time(sec)

Time(sec)

DTCWT-denoised Kurtosis=38.8058

Original signal Kurtosis=8.8508

**4.2 Gearboxes** 



4. Calculate the de-noised signal using inverse wavelet domain

shrinkage rule

Hard thresholding is the simplest approach but tends to miss useful parts of the signal. In soft thresholding, the threshold is subtracted from any coefficient that is greater than it.

$$c\_{jk} = \text{sign}\{c\_{jk}\} \cdot (|c\_{jk}| - t) \tag{51}$$

*t* is universal threshold ݐൌߪή ඥʹ ή ݈݃ܰ , σ is the standard deviation of the noise and N is the number of data samples in the measured signal. The true value of the noise standard deviation σ is, generally, unknown. It is often estimated by σ = MAD/0.6745, where MAD refers to the median absolute value of the finest scale wavelet coefficients. The combination of the soft thresholding policy and universal threshold is also referred to as "VisuShrink". It ensures a noise-free reconstruction but often the threshold is set too high. (Donoho and Jonestone, 1994) introduced the "minimax" threshold an enhancement of the universal threshold. The "minimax" threshold level can be much lower than the universal threshold level when it comes to small-to-moderate sample sizes. "SureShrink" or "rigsure" approach relies on the minimization of Stein's unbiased estimator of risk (Donoho and Jonestone, 1995). When the wavelet representation is not very sparse, it yields better results. The universal threshold and "minimax" threshold are more effective when it comes to detecting sparse impulses. All the above methods assume that the noise properties are known, which is rarely the case in industrial applications. The maximum likelihood estimation de-noising method is suitable for non-Gaussian noise. A specific threshold rule, which is based on the maximum likelihood estimation method, incorporates a priori information on the impulse probability density function. The probability density function of the impulse to be identified must be known in advance though. The so-called ''sparse code shrinkage'' method, proposed by (Hyvarinen, 1999), can be utilized for wavelet coefficients shrinkage.

The DTCWT can give a substantial performance enhancement to the conventional DWTbased noise reduction methodologies due to its interesting properties of near shiftinvariance and reduced frequency aliasing. (Wang et al., 2010) proposed a scheme based on "NeighCoeff" scheme (Cai and Silverman, 2001). "NeighCoeff" uses lower threshold than "VisuShrink" and outperforms all other shrinkage methods. The de-noising using DTCWT and "NeighCoeff" shrinkage is implemented in the following stages:

1. Transform the data x into the wavelet domain via DTCWT (or any other wavelet transform in general)


In Fig. 10 various de-noising algorithms were applied on an AE signal from a bearing with seeded defect. In a) the original signal is depicted. In b) the method of spectral kurtosis (Randall and Antoni, 2011) is utilized. Spectral kurtosis is not a wavelet-based technique and relies on the location of the frequency band where kurtosis is maximized and then the band-pass filtering of the signal in the resulted band. In figure c) the DTCWT wavelet transform is applied in combination with "NeighCoeff" thresholding whilst in d) a parametric procedure was used by the authors to determine the optimum parameters of DWT (wavelet type, number of levels, threshold type, soft or hard application of threshold) that maximize the kurtosis and crest factor of the signal. DTCWT- and DWT-based denoising proved the most efficient in terms of the resulting signal kurtosis.

Fig. 10. Effect of various de-noising schemes on an AE signal from defective bearing a) original signal b) de-noised signal via spectral kurtosis technique c) de-noised signal via DTCWT d) de-noised signal via DWT

#### **4.2 Gearboxes**

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

1. Transform the signal *x(t)* to the time-scale plane by means of a wavelet transform. The

2. Assess the threshold *t* and, in accordance with the established rules, shrink the wavelet

3. Use the shrunken coefficients to carry out the inverse wavelet transform. The series

The second step is probably the most critical and has quite an impact upon the effectiveness of the procedure. There are plenty of thresholding techniques and many different thresholds proposed in the literature. Hard thresholding sets any coefficient less than or equal to the

ܿ ൌ ൜Ͳǡ ܿ ൏ ݐ

Hard thresholding is the simplest approach but tends to miss useful parts of the signal. In soft thresholding, the threshold is subtracted from any coefficient that is greater than it.

*t* is universal threshold ݐൌߪή ඥʹ ή ݈݃ܰ , σ is the standard deviation of the noise and N is the number of data samples in the measured signal. The true value of the noise standard deviation σ is, generally, unknown. It is often estimated by σ = MAD/0.6745, where MAD refers to the median absolute value of the finest scale wavelet coefficients. The combination of the soft thresholding policy and universal threshold is also referred to as "VisuShrink". It ensures a noise-free reconstruction but often the threshold is set too high. (Donoho and Jonestone, 1994) introduced the "minimax" threshold an enhancement of the universal threshold. The "minimax" threshold level can be much lower than the universal threshold level when it comes to small-to-moderate sample sizes. "SureShrink" or "rigsure" approach relies on the minimization of Stein's unbiased estimator of risk (Donoho and Jonestone, 1995). When the wavelet representation is not very sparse, it yields better results. The universal threshold and "minimax" threshold are more effective when it comes to detecting sparse impulses. All the above methods assume that the noise properties are known, which is rarely the case in industrial applications. The maximum likelihood estimation de-noising method is suitable for non-Gaussian noise. A specific threshold rule, which is based on the maximum likelihood estimation method, incorporates a priori information on the impulse probability density function. The probability density function of the impulse to be identified must be known in advance though. The so-called ''sparse code shrinkage'' method,

proposed by (Hyvarinen, 1999), can be utilized for wavelet coefficients shrinkage.

and "NeighCoeff" shrinkage is implemented in the following stages:

transform in general)

The DTCWT can give a substantial performance enhancement to the conventional DWTbased noise reduction methodologies due to its interesting properties of near shiftinvariance and reduced frequency aliasing. (Wang et al., 2010) proposed a scheme based on "NeighCoeff" scheme (Cai and Silverman, 2001). "NeighCoeff" uses lower threshold than "VisuShrink" and outperforms all other shrinkage methods. The de-noising using DTCWT

1. Transform the data x into the wavelet domain via DTCWT (or any other wavelet

(50) ݐ ܿǡܿ

ܿ ൌ ݏ݅݃݊൫ܿ൯ ή ሺหܿห െ ݐሻ (51)

wavelet coefficients on various scales are obtained.

recovered is the estimation of impulse p(t).

coefficients.

threshold to zero.

Fault symptoms of running gearboxes must be detected as early as possible to avoid serious accidents. An efficient monitoring plan is needed for any industry because it can optimize the resources management and improve the plant economy, by reducing unnecessary costs and increasing the level of safety. A great percentage of breakdowns in industrial processes as well as in rotorcraft transportation (helicopters etc) are caused by gearbox related failures. Fault symptoms usually begin from early stages, rather long before a destructive failure making the use of effective condition monitoring schemes very attractive. Many highquality investigations can be found in the recent literature.

(YanPing et al., 2006) explored the statistical characteristics of the continuous wavelet transform scalogram of vibration signals from rotating machinery. Two features, wavelet grey moment (WGM) and first-order wavelet grey moment vector (WGMV), were proposed for condition monitoring of rotating machinery. Wavelet grey moments are defined as:

$$\mathbf{g}\_{k} = \frac{1}{m \times n} \boldsymbol{\Sigma}\_{l=1}^{\mathbf{m}} \boldsymbol{\Sigma}\_{l=1}^{\mathbf{n}} \mathbf{c}\_{l]}^{\mathbf{k}} \sqrt{(\mathbf{i} - 1)^{2} + (\mathbf{j} - 1)^{2}} \tag{52}$$

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 291

classification system. The modulus maxima distribution was utilized as the input observation sequence of the system. An adaptive algorithm was proposed and validated by three sets of real gearbox vibration data to classify two conditions: normal and failure. In addition, in condition classification (stage 2), three HMM models were set up to classify three different machine conditions, namely, adjacent tooth failure, distributed tooth failure and normal condition. The validation results showed an excellent performance of the

(Saravanan et al., 2008) investigated the effectiveness of wavelet-based features for fault diagnosis in a bevel gearbox using support vector machines (SVM) and proximal support vector machines (PSVM). The statistical feature vectors from Morlet wavelet coefficients resulted after CWT at sixty-four scales, were classified using the J48 algorithm and the predominant features were fed as input for training and testing SVM and PSVM. The coefficients of Morlet wavelet were used for feature extraction from the time domain vibration signals. Various statistical features like kurtosis, standard deviation, maximum value, etc. calculated from the wavelet coefficients formed the feature sets. It was concluded that PSVM has an edge over SVM in the classification efficiency of various fault conditions. (Li et al., 2008) presented a new signal-adapted lifting scheme for rotating machinery fault diagnosis, which allows the construction of a wavelet directly from the statistics of a given signal. The prediction operator based on genetic algorithms was designed to maximize the kurtosis of detail signal produced by the lifting scheme, and the update operator was designed to minimize a reconstruction error. The signal-adapted lifting scheme was applied to analyze bearing and gearbox vibration signals. The conventional diagnosis techniques and nonadaptive lifting scheme were also used to analyze the same signals for comparison. The results demonstrated that the signal-adapted lifting scheme was more effective in extracting inherent fault features from complex vibration signals. (Kar and Mohanty, 2008) conducted an experimental investigation of fault diagnosis in a multistage gearbox under transient loads. The signals studied were vibration measurements, recorded from an accelerometer fitted at the tail-end bearing of the gearbox as well as the current transients monitored at the induction motor. Three defective cases and three transient load conditions were investigated. DWT (with "db8") and a corrected multi-resolution Fourier transform (MFT) were applied to process the vibration and current transients. A statistical feature extraction technique was proposed in search of a trend in detection of defects. A condition monitoring scheme is devised that can facilitate in monitoring vibration and current transients in the gearbox with simultaneous presence of transient loads and defects. (Jafarizadeh et al., 2008) suggested a new noise canceling method, based on time-averaging method for asynchronous input, and CWT with complex Morlet wavelet. The complex Morlet wavelet depends on non-fixed parameters. For the feature extraction from time-domain vibration signals, the optimum values of the Morlet wavelet parameters should be estimated. Wavelet entropy was used towards this optimization. Then CWT was applied and 3-D scalograms were utilized for damage detection. The proposed method was successfully implemented on a simulated signal and real test rig of

(Loutas et al., 2009) reported on the condition monitoring of a lab-scale, single stage, gearbox with cracked gears using different non-destructive inspection methodologies and the processing of the acquired waveforms with advanced signal processing techniques is the aim of the present work. Acoustic emission (AE) and vibration measurements were utilized for this purpose. Emphasis was given on the signal processing of the acquired vibration and

proposed classification system.

a Yahama motorcycle gearbox.

Where cij is the element of matrix [C]mxn, �(� − 1)� + (j − 1)� is the Euclidean distance between element cij and c11, that is corresponding to the geometry length between the point (i,j) and reference point (1,1) in the scalogram. In (Fan and Zuo, 2006) a new fault detection method that combines Hilbert transform and wavelet packet transform was proposed. The wavelet packet node energy method is used as feature. WPT at the 4th decomposition level using "db10" wavelet was utilized. Their results showed that the proposed method is effective to extract modulating signal and help to detect the early gear fault.

(Sanz et al., 2007) proposed a method which combines the capability of DWT to treat transient vibration signals with the ability of auto-associative neural networks (AANNs) for feature extraction. "db6" and 3 levels of decomposition were chosen for real application vibration data from a pump rotor gearset. The detail coefficient vectors of the DWT were taken as input parameters of the AANN. An advantage of the proposed method is that DWT is performed directly on the raw vibration signals not on time-synchronous averaged signals. (Rafiee et al., 2007) presented a new procedure which experimentally recognized gears and bearings faults of a typical gearbox system using a multi-layer perceptron ANN. The feature vector was populated by the standard deviation of wavelet packet coefficients after WPT on the recorded vibration signals. "db4" wavelet and 4 levels of decomposition were used. The gear conditions were considered to be normal gearbox, slight- and mediumworn, broken-teeth gears faults and a general bearing fault. (He et al., 2007) proposed a novel non-linear feature extraction scheme from the time-domain features with wavelet packet preprocessing and frequency-domain features of the vibration signals using the kernel principal component analysis (KPCA) to characterize various gearbox conditions. Experimental analysis on a fatigue test of an automobile transmission gearbox have shown that the KPCA features outperformed PCA features in terms of clustering capability, and both the two KPCA-based subspace methods can be effectively applied to gearbox condition monitoring. The time-domain statistical features with wavelet packet preprocessing and frequency-domain statistical features proved more effective than the conventional timedomain features without WPT preprocessing for extracting the KPCA features. (Li et al., 2007) used the Haar wavelet CWT (HCWT) to diagnose three types of machine faults. To assess its effectiveness, the diagnosis information obtained by HCWT is compared with that by Morlet wavelet CWT (MCWT), which is more popular in machine diagnosis. Their results demonstrate that Haar wavelet is also a feasible wavelet in machine fault diagnosis and HCWT can provide abundant graphic features for diagnosis than MCWT. (Miao and Makis, 2007) have introduced a new feature extraction approach based on wavelet modulus maxima and proposed a Hidden Markov Model (HMM) based two-stage machine condition

and increasing the level of safety. A great percentage of breakdowns in industrial processes as well as in rotorcraft transportation (helicopters etc) are caused by gearbox related failures. Fault symptoms usually begin from early stages, rather long before a destructive failure making the use of effective condition monitoring schemes very attractive. Many high-

(YanPing et al., 2006) explored the statistical characteristics of the continuous wavelet transform scalogram of vibration signals from rotating machinery. Two features, wavelet grey moment (WGM) and first-order wavelet grey moment vector (WGMV), were proposed for condition monitoring of rotating machinery. Wavelet grey moments are defined as:

Where cij is the element of matrix [C]mxn, �(� − 1)� + (j − 1)� is the Euclidean distance between element cij and c11, that is corresponding to the geometry length between the point (i,j) and reference point (1,1) in the scalogram. In (Fan and Zuo, 2006) a new fault detection method that combines Hilbert transform and wavelet packet transform was proposed. The wavelet packet node energy method is used as feature. WPT at the 4th decomposition level using "db10" wavelet was utilized. Their results showed that the proposed method is

(Sanz et al., 2007) proposed a method which combines the capability of DWT to treat transient vibration signals with the ability of auto-associative neural networks (AANNs) for feature extraction. "db6" and 3 levels of decomposition were chosen for real application vibration data from a pump rotor gearset. The detail coefficient vectors of the DWT were taken as input parameters of the AANN. An advantage of the proposed method is that DWT is performed directly on the raw vibration signals not on time-synchronous averaged signals. (Rafiee et al., 2007) presented a new procedure which experimentally recognized gears and bearings faults of a typical gearbox system using a multi-layer perceptron ANN. The feature vector was populated by the standard deviation of wavelet packet coefficients after WPT on the recorded vibration signals. "db4" wavelet and 4 levels of decomposition were used. The gear conditions were considered to be normal gearbox, slight- and mediumworn, broken-teeth gears faults and a general bearing fault. (He et al., 2007) proposed a novel non-linear feature extraction scheme from the time-domain features with wavelet packet preprocessing and frequency-domain features of the vibration signals using the kernel principal component analysis (KPCA) to characterize various gearbox conditions. Experimental analysis on a fatigue test of an automobile transmission gearbox have shown that the KPCA features outperformed PCA features in terms of clustering capability, and both the two KPCA-based subspace methods can be effectively applied to gearbox condition monitoring. The time-domain statistical features with wavelet packet preprocessing and frequency-domain statistical features proved more effective than the conventional timedomain features without WPT preprocessing for extracting the KPCA features. (Li et al., 2007) used the Haar wavelet CWT (HCWT) to diagnose three types of machine faults. To assess its effectiveness, the diagnosis information obtained by HCWT is compared with that by Morlet wavelet CWT (MCWT), which is more popular in machine diagnosis. Their results demonstrate that Haar wavelet is also a feasible wavelet in machine fault diagnosis and HCWT can provide abundant graphic features for diagnosis than MCWT. (Miao and Makis, 2007) have introduced a new feature extraction approach based on wavelet modulus maxima and proposed a Hidden Markov Model (HMM) based two-stage machine condition

��(i − 1)� + (j − 1) � �

��� (52)

quality investigations can be found in the recent literature.

�� <sup>=</sup> �

��� ∑ ∑ c��

effective to extract modulating signal and help to detect the early gear fault.

��� �

classification system. The modulus maxima distribution was utilized as the input observation sequence of the system. An adaptive algorithm was proposed and validated by three sets of real gearbox vibration data to classify two conditions: normal and failure. In addition, in condition classification (stage 2), three HMM models were set up to classify three different machine conditions, namely, adjacent tooth failure, distributed tooth failure and normal condition. The validation results showed an excellent performance of the proposed classification system.

(Saravanan et al., 2008) investigated the effectiveness of wavelet-based features for fault diagnosis in a bevel gearbox using support vector machines (SVM) and proximal support vector machines (PSVM). The statistical feature vectors from Morlet wavelet coefficients resulted after CWT at sixty-four scales, were classified using the J48 algorithm and the predominant features were fed as input for training and testing SVM and PSVM. The coefficients of Morlet wavelet were used for feature extraction from the time domain vibration signals. Various statistical features like kurtosis, standard deviation, maximum value, etc. calculated from the wavelet coefficients formed the feature sets. It was concluded that PSVM has an edge over SVM in the classification efficiency of various fault conditions.

(Li et al., 2008) presented a new signal-adapted lifting scheme for rotating machinery fault diagnosis, which allows the construction of a wavelet directly from the statistics of a given signal. The prediction operator based on genetic algorithms was designed to maximize the kurtosis of detail signal produced by the lifting scheme, and the update operator was designed to minimize a reconstruction error. The signal-adapted lifting scheme was applied to analyze bearing and gearbox vibration signals. The conventional diagnosis techniques and nonadaptive lifting scheme were also used to analyze the same signals for comparison. The results demonstrated that the signal-adapted lifting scheme was more effective in extracting inherent fault features from complex vibration signals. (Kar and Mohanty, 2008) conducted an experimental investigation of fault diagnosis in a multistage gearbox under transient loads. The signals studied were vibration measurements, recorded from an accelerometer fitted at the tail-end bearing of the gearbox as well as the current transients monitored at the induction motor. Three defective cases and three transient load conditions were investigated. DWT (with "db8") and a corrected multi-resolution Fourier transform (MFT) were applied to process the vibration and current transients. A statistical feature extraction technique was proposed in search of a trend in detection of defects. A condition monitoring scheme is devised that can facilitate in monitoring vibration and current transients in the gearbox with simultaneous presence of transient loads and defects. (Jafarizadeh et al., 2008) suggested a new noise canceling method, based on time-averaging method for asynchronous input, and CWT with complex Morlet wavelet. The complex Morlet wavelet depends on non-fixed parameters. For the feature extraction from time-domain vibration signals, the optimum values of the Morlet wavelet parameters should be estimated. Wavelet entropy was used towards this optimization. Then CWT was applied and 3-D scalograms were utilized for damage detection. The proposed method was successfully implemented on a simulated signal and real test rig of a Yahama motorcycle gearbox.

(Loutas et al., 2009) reported on the condition monitoring of a lab-scale, single stage, gearbox with cracked gears using different non-destructive inspection methodologies and the processing of the acquired waveforms with advanced signal processing techniques is the aim of the present work. Acoustic emission (AE) and vibration measurements were utilized for this purpose. Emphasis was given on the signal processing of the acquired vibration and

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 293

well as the number of neurons in the ANNs hidden layers, resulted in a high-speed, effective two-layer ANN with a small-sized structure. "db11", level 4 and 14 neurons have been selected as the best values for Daubechies order, decomposition level, and the number of nodes in hidden layer, respectively. (Singh and Al Kazzaz, et al., 2009) studied the effect of dry bearing fault on multi-sensor measurements (three line to line voltages, three currents, two vibration signals, four temperatures and one speed signal) in induction machines. Different families of WT have been introduced and implemented with vibration signals covering the dry bearing fault in induction machine. The results of testing various popular types of the WT showed different degree of success in relating the decomposed band with machine condition. It was concluded that the fluctuation in the RMS value of the first and second decomposition level was larger in the case of Mexican hat wavelet and it was thus proposed to investigate the random vibration of all machines in case of dry bearing fault. It was concluded that WT can be used effectively to specify one machine fault at a time, while it cannot treat multiple faults simultaneously. Instead, the combined use of wavelet and Fourier transform proved an effective tool for extracting important information about the machine condition. An intelligent diagnostic methodology for fault gear identification and classification based on vibration signals using DWT and adaptive neuro-fuzzy inference system (ANFIS) is presented in (Wu et al., 2009). After the vibration signal acquisition, 4-level decomposition via the DWT followed resulting in four high frequency details (D1–D4) and one low frequency approximation (A4). Three Daubechies wavelets (db4, db8 and db20) were utilized for the decomposition. The energy distribution of the five subbands was calculated and trained two different ANNs for the successful fault identification. No major differences were observed on the ANNs recognition rates in regard to the different mother wavelets utilized in the DWT. (Wu and Hsu, 2009) described a development of the fault gear identification system using the vibration signal with discrete wavelet transform and fuzzy–logic inference for a gear-set experimental platform. The extraction method of feature vector is based on DWT decomposition followed by level energy calculation. The recognition rate of the classification task using three different Daubechies wavelets ("db4, db8 and db20") coefficients under various working conditions did not show significant discrepancies. The

A diagnostic methodology of artificial defects in a single stage gearbox operating under various load levels and different defect states was proposed by (Loutas et al., 2010) based on vibration recordings as well as advanced signal analysis techniques. Two different waveletbased signal processing methodologies, using the DWT as well as the CWT, were utilized for the analysis of the recorded vibration signals and useful diagnostic information were

DWT was applied with "db10" and 10-level decomposition whilst CWT was applied with Morlet wavelet (bandwidth parameter and wavelet center frequency were set at 1 and 1.5 respectively. Averaging across all scales was utilized instead of time synchronous averaging giving very characteristic scalograms for each artificial defect case. A novel method incorporating customized (i.e., signal-based) multiwavelet lifting schemes with sliding window de-noising was proposed in (Yuan et al., 2010). On the basis of Hermite spline interpolation, various vector prediction and update operators with the desirable properties of biorthogonality, symmetry, short support and vanishing moments are constructed. The minimum entropy principle is recommended to determine the optimal vector prediction and update operators in the lifting scheme, by means of measuring the sparsity. Due to the

fault recognition rates were in general over 96%.

extracted out of them.

acoustic emission signals in order to extract conventional as well as novel parametersfeatures of potential diagnostic value from the monitored waveforms. Wavelet-based parameters-features were proposed utilizing the DWT and "db10" wavelet. The evolution of selected parameters/features versus test time is provided, evaluated and the parameters with the most interesting diagnostic behavior were highlighted. The differences in the parameters evolution of each NDT technique are discussed and the superiority of AE over vibration recordings for the early diagnosis of natural wear in gear systems was concluded. In (Saravanan and Ramachandran, 2009) the coefficients of Morlet wavelet were used for feature extraction. CWT and sixty four scales were chosen to extract the Morlet wavelet coefficients of the vibration signals. A group of statistical features like kurtosis, standard deviation, maximum value, etc., widely used in fault diagnostics, were extracted from the wavelet coefficients of the time domain signals. For the selection of best features, the decision tree using J48 algorithm was used. The selected features were fed as input to SVM for classification. (Xian and Zeng, 2009) developed a new intelligent method for the fault diagnosis of the rotating machinery based on wavelet packet analysis (WPA) and hybrid support vector machines (hybrid SVM). The faulty vibration signals obtained from a gearbox were decomposed by WPA via Dmeyer wavelet. Shannon entropy was calculated from the coefficients at each subspace of the WPA decomposition and formed the feature vectors that trained/tested the hybrid SVM for estimating the fault type. (Belsak and Flasker, 2009) studied the influence of a fatigue gear crack in a single-stage gear unit on the recorded vibrations. They applied the sparse code shrinkage method to de-noise vibration signals from a faulty gearbox. They discriminated between healthy and cracked gear using scalograms of the resulted CWT coefficients. Gabor wavelet was adopted in their work. (Wu and Chan, 2009) utilized the sound emission from a multi-stage gearbox towards gear fault diagnostics. Continuous wavelet transform with Morlet mother wavelet combined with a feature selection of energy spectrum was proposed for analyzing fault signals and feature extraction. Two artificial neural network (ANN) approaches i.e. the probability neural network and conventional back-propagation network were compared in the recognition of six faulty states and one healthy. (Saravanan and Ramachandran, 2009) recorded vibration signals from a spur bevel gearbox in different lubrication, loading and gear state conditions. They used various members of the Daubechies family (db1-db15) for statistical feature extraction. J48 Decision Tree was used for two reasons, feature selection and classification of the faulty signals. (Rafiee and Tse, 2009) processed vibration signals from a gearbox with three different fault conditions (slight-worn, medium-worn, and broken-tooth) of a spur gear. CWT was used with packet decomposition through the scales. After synchronizing the raw vibration signals, the CWT and autocorrelation function were applied to the synchronized signals and generated continuous wavelet coefficients of synchronized vibration signals. They found that a simple sinusoidal summation function can approximate the waveforms generated by autocorrelation of CWC-SVS for normal gearboxes as well as other defective gears with satisfactory performance. The function achieved proper approximation even though the waveforms were different from one condition to another as they possess different frequency contents of vibration signals. (Rafiee et al., 2009) presented an optimized gear fault identification system using genetic algorithms (GAs) to investigate the type of gear failures of a complex gearbox system using artificial neural networks (ANNs). Slightly-worn, medium-worn, and broken-tooth of a spur gear of the gearbox system were selected as the faults types. GAs were exploited to optimize the selection of mother wavelet function (among several members of the Daubechies family), the number of the decomposition levels of the wavelet packet transform (WPT) as

acoustic emission signals in order to extract conventional as well as novel parametersfeatures of potential diagnostic value from the monitored waveforms. Wavelet-based parameters-features were proposed utilizing the DWT and "db10" wavelet. The evolution of selected parameters/features versus test time is provided, evaluated and the parameters with the most interesting diagnostic behavior were highlighted. The differences in the parameters evolution of each NDT technique are discussed and the superiority of AE over vibration recordings for the early diagnosis of natural wear in gear systems was concluded. In (Saravanan and Ramachandran, 2009) the coefficients of Morlet wavelet were used for feature extraction. CWT and sixty four scales were chosen to extract the Morlet wavelet coefficients of the vibration signals. A group of statistical features like kurtosis, standard deviation, maximum value, etc., widely used in fault diagnostics, were extracted from the wavelet coefficients of the time domain signals. For the selection of best features, the decision tree using J48 algorithm was used. The selected features were fed as input to SVM for classification. (Xian and Zeng, 2009) developed a new intelligent method for the fault diagnosis of the rotating machinery based on wavelet packet analysis (WPA) and hybrid support vector machines (hybrid SVM). The faulty vibration signals obtained from a gearbox were decomposed by WPA via Dmeyer wavelet. Shannon entropy was calculated from the coefficients at each subspace of the WPA decomposition and formed the feature vectors that trained/tested the hybrid SVM for estimating the fault type. (Belsak and Flasker, 2009) studied the influence of a fatigue gear crack in a single-stage gear unit on the recorded vibrations. They applied the sparse code shrinkage method to de-noise vibration signals from a faulty gearbox. They discriminated between healthy and cracked gear using scalograms of the resulted CWT coefficients. Gabor wavelet was adopted in their work. (Wu and Chan, 2009) utilized the sound emission from a multi-stage gearbox towards gear fault diagnostics. Continuous wavelet transform with Morlet mother wavelet combined with a feature selection of energy spectrum was proposed for analyzing fault signals and feature extraction. Two artificial neural network (ANN) approaches i.e. the probability neural network and conventional back-propagation network were compared in the recognition of six faulty states and one healthy. (Saravanan and Ramachandran, 2009) recorded vibration signals from a spur bevel gearbox in different lubrication, loading and gear state conditions. They used various members of the Daubechies family (db1-db15) for statistical feature extraction. J48 Decision Tree was used for two reasons, feature selection and classification of the faulty signals. (Rafiee and Tse, 2009) processed vibration signals from a gearbox with three different fault conditions (slight-worn, medium-worn, and broken-tooth) of a spur gear. CWT was used with packet decomposition through the scales. After synchronizing the raw vibration signals, the CWT and autocorrelation function were applied to the synchronized signals and generated continuous wavelet coefficients of synchronized vibration signals. They found that a simple sinusoidal summation function can approximate the waveforms generated by autocorrelation of CWC-SVS for normal gearboxes as well as other defective gears with satisfactory performance. The function achieved proper approximation even though the waveforms were different from one condition to another as they possess different frequency contents of vibration signals. (Rafiee et al., 2009) presented an optimized gear fault identification system using genetic algorithms (GAs) to investigate the type of gear failures of a complex gearbox system using artificial neural networks (ANNs). Slightly-worn, medium-worn, and broken-tooth of a spur gear of the gearbox system were selected as the faults types. GAs were exploited to optimize the selection of mother wavelet function (among several members of the Daubechies family), the number of the decomposition levels of the wavelet packet transform (WPT) as well as the number of neurons in the ANNs hidden layers, resulted in a high-speed, effective two-layer ANN with a small-sized structure. "db11", level 4 and 14 neurons have been selected as the best values for Daubechies order, decomposition level, and the number of nodes in hidden layer, respectively. (Singh and Al Kazzaz, et al., 2009) studied the effect of dry bearing fault on multi-sensor measurements (three line to line voltages, three currents, two vibration signals, four temperatures and one speed signal) in induction machines. Different families of WT have been introduced and implemented with vibration signals covering the dry bearing fault in induction machine. The results of testing various popular types of the WT showed different degree of success in relating the decomposed band with machine condition. It was concluded that the fluctuation in the RMS value of the first and second decomposition level was larger in the case of Mexican hat wavelet and it was thus proposed to investigate the random vibration of all machines in case of dry bearing fault. It was concluded that WT can be used effectively to specify one machine fault at a time, while it cannot treat multiple faults simultaneously. Instead, the combined use of wavelet and Fourier transform proved an effective tool for extracting important information about the machine condition. An intelligent diagnostic methodology for fault gear identification and classification based on vibration signals using DWT and adaptive neuro-fuzzy inference system (ANFIS) is presented in (Wu et al., 2009). After the vibration signal acquisition, 4-level decomposition via the DWT followed resulting in four high frequency details (D1–D4) and one low frequency approximation (A4). Three Daubechies wavelets (db4, db8 and db20) were utilized for the decomposition. The energy distribution of the five subbands was calculated and trained two different ANNs for the successful fault identification. No major differences were observed on the ANNs recognition rates in regard to the different mother wavelets utilized in the DWT. (Wu and Hsu, 2009) described a development of the fault gear identification system using the vibration signal with discrete wavelet transform and fuzzy–logic inference for a gear-set experimental platform. The extraction method of feature vector is based on DWT decomposition followed by level energy calculation. The recognition rate of the classification task using three different Daubechies wavelets ("db4, db8 and db20") coefficients under various working conditions did not show significant discrepancies. The fault recognition rates were in general over 96%.

A diagnostic methodology of artificial defects in a single stage gearbox operating under various load levels and different defect states was proposed by (Loutas et al., 2010) based on vibration recordings as well as advanced signal analysis techniques. Two different waveletbased signal processing methodologies, using the DWT as well as the CWT, were utilized for the analysis of the recorded vibration signals and useful diagnostic information were extracted out of them.

DWT was applied with "db10" and 10-level decomposition whilst CWT was applied with Morlet wavelet (bandwidth parameter and wavelet center frequency were set at 1 and 1.5 respectively. Averaging across all scales was utilized instead of time synchronous averaging giving very characteristic scalograms for each artificial defect case. A novel method incorporating customized (i.e., signal-based) multiwavelet lifting schemes with sliding window de-noising was proposed in (Yuan et al., 2010). On the basis of Hermite spline interpolation, various vector prediction and update operators with the desirable properties of biorthogonality, symmetry, short support and vanishing moments are constructed. The minimum entropy principle is recommended to determine the optimal vector prediction and update operators in the lifting scheme, by means of measuring the sparsity. Due to the

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 295

kurtogram. The results were evaluated through the kurtosis calculated for each signal after the de-noising. NeighCoeff shrinkage scheme was applied in all wavelet-based cases. Denoised results of signals collected from a gearbox with tooth crack showed that the DTCWTbased de-noising approach yielded more promising result than the SGWT- and DWT-based methods, and it can effectively remove the noise and retain valuable information as much as possible. In the case of multiple features detection, diagnosis results of rolling element bearings with combined faults and actual industrial equipment confirmed that the proposed DTCWT-based method is powerful and consistently outperformed the widely used SGWT

(Loutas et al. 2011a) conducted multi-hour tests in healthy gears in a single-stage gearbox. Three on-line monitoring techniques were implemented in the tests. Vibration and acoustic emission recordings in combination with data coming from oil debris monitoring (ODM) of the lubricating oil were utilized in order to assess the condition of the gears. A plethora of parameters/features were extracted from the acquired waveforms via conventional (in time and frequency domain) and non-conventional (wavelet-based) signal processing techniques. DWT was utilized to process vibration and AE signals with "db10" mother wavelet and 10 levels of decomposition. The wavelet levels energy and entropy were used as features. Data fusion was accomplished in the level of integration of the most representative among the extracted features from all three measurement technologies in a single data matrix. Principal component analysis (PCA) was utilized to reduce the dimensionality of the data matrix whereas independent component analysis (ICA) was further applied to identify the independent components among the data and correlate them to different damage modes of the gearbox. (Miao and Makis, 2011) presented an on-line fault classification system with an adaptive model re-estimation algorithm. The machinery condition is identified by selecting the HMM which maximizes the probability of a given observation sequence. The proper selection of the observation sequence is a key step in the development of an HMM-based classification system. In this paper, the classification system is validated using observation sequences based on the wavelet modulus maxima distribution obtained from real vibration signals, which has been proved to be effective in fault detection in previous research. (Li et al., 2011) utilized the Hermitian wavelet to diagnose the gear localized crack fault. The complex Hermitian wavelet is constructed based on the first and the second derivatives of the Gaussian function to detect signal singularities. The Fourier spectrum of Hermitian wavelet is real; therefore, Hermitian wavelet does not affect the phase of a signal in the complex domain. This gives a desirable ability to extract the singularity characteristic of a signal precisely. The proposed method is based on Hermitian wavelet amplitude and phase map of the time-domain vibration signals. Hermitian wavelet amplitude and phase maps

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

and fast kurtogram.

are used to evaluate healthy and cracked gears.

**4.3 Bearings**

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 load.

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

kurtogram. The results were evaluated through the kurtosis calculated for each signal after the de-noising. NeighCoeff shrinkage scheme was applied in all wavelet-based cases. Denoised results of signals collected from a gearbox with tooth crack showed that the DTCWTbased de-noising approach yielded more promising result than the SGWT- and DWT-based methods, and it can effectively remove the noise and retain valuable information as much as

possible. In the case of multiple features detection, diagnosis results of rolling element bearings with combined faults and actual industrial equipment confirmed that the proposed DTCWT-based method is powerful and consistently outperformed the widely used SGWT and fast kurtogram.

(Loutas et al. 2011a) conducted multi-hour tests in healthy gears in a single-stage gearbox. Three on-line monitoring techniques were implemented in the tests. Vibration and acoustic emission recordings in combination with data coming from oil debris monitoring (ODM) of the lubricating oil were utilized in order to assess the condition of the gears. A plethora of parameters/features were extracted from the acquired waveforms via conventional (in time and frequency domain) and non-conventional (wavelet-based) signal processing techniques. DWT was utilized to process vibration and AE signals with "db10" mother wavelet and 10 levels of decomposition. The wavelet levels energy and entropy were used as features. Data fusion was accomplished in the level of integration of the most representative among the extracted features from all three measurement technologies in a single data matrix. Principal component analysis (PCA) was utilized to reduce the dimensionality of the data matrix whereas independent component analysis (ICA) was further applied to identify the independent components among the data and correlate them to different damage modes of the gearbox. (Miao and Makis, 2011) presented an on-line fault classification system with an adaptive model re-estimation algorithm. The machinery condition is identified by selecting the HMM which maximizes the probability of a given observation sequence. The proper selection of the observation sequence is a key step in the development of an HMM-based classification system. In this paper, the classification system is validated using observation sequences based on the wavelet modulus maxima distribution obtained from real vibration signals, which has been proved to be effective in fault detection in previous research. (Li et al., 2011) utilized the Hermitian wavelet to diagnose the gear localized crack fault. The complex Hermitian wavelet is constructed based on the first and the second derivatives of the Gaussian function to detect signal singularities. The Fourier spectrum of Hermitian wavelet is real; therefore, Hermitian wavelet does not affect the phase of a signal in the complex domain. This gives a desirable ability to extract the singularity characteristic of a signal precisely. The proposed method is based on Hermitian wavelet amplitude and phase map of the time-domain vibration signals. Hermitian wavelet amplitude and phase maps are used to evaluate healthy and cracked gears.
