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Complex equipment deterioration refers to a nonlinear multistate deterioration process, where the deterioration curve may not follow a typical shape such as exponential or linear function. A general solution is presented to nonlinear multistate deterioration modeling for deterioration level assessment and remaining useful life prognostics under no-label lifetime data including multi signals. In the solution, a three layer nonlinear multistate deterioration model of complex equipment is established based on hyper-parameter optimization. Hyper-parameter I and M, which determine the first two layers, are optimized by the proposed unsupervised extraction method based on greedy kernel principal components analysis and the improved Mann–Kendall criterion, respectively. As a determinant of the third layer, hyper-parameter N is optimized by the improved Bayesian information criterion to obtain optimized model, when parameters have been estimated under each alternative model structure at different N. The no-label lifetime dataset of turbofan engines are adopted for case study, and more accurate deterioration level and remaining useful life are obtained. The results verify the effectiveness of the presented solution. The study indicates that: nonlinearity makes an important effect on multistate deterioration modeling through hyper-parameters; the solution deals with nonlinearity in a systematic manner by hyper-parameter optimization of three layers.

The gearbox of a wind turbine (WT) has dominant failure rates and highest downtime loss among all WT subsystems. Thus, gearbox health assessment for maintenance cost reduction is of paramount importance. The concurrence of multiple faults in gearbox components is a common phenomenon due to fault induction mechanism. This problem should be considered before planning to replace the components of the WT gearbox. Therefore, the key fault patterns should be reliably identified from noisy observation data for the development of an effective maintenance strategy. However, most of the existing studies focusing on multiple fault diagnosis always suffer from inappropriate division of fault information in order to satisfy various rigorous decomposition principles or statistical assumptions, such as the smooth envelope principle of ensemble empirical mode decomposition and the mutual independence assumption of independent component analysis. Thus, this paper presents a joint subspace learning-based multiple fault detection (JSL-MFD) technique to construct different subspaces adaptively for different fault patterns. Its main advantage is its capability to learn multiple fault subspaces directly from the observation signal itself. It can also sparsely concentrate the feature information into a few dominant subspace coefficients. Furthermore, it can eliminate noise by simply performing coefficient shrinkage operations. Consequently, multiple fault patterns are reliably identified by utilizing the maximum fault information criterion. The superiority of JSL-MFD in multiple fault separation and detection is comprehensively investigated and verified by the analysis of a data set of a 750 kW WT gearbox. Results show that JSL-MFD is superior to a state-of-the-art technique in detecting hidden fault patterns and enhancing detection accuracy.

In this paper, a wavelet-based periodic group-sparse signal denoising approach is proposed for detecting faults in rotary machines. The proposed approach exploits group sparsity in the wavelet domain. For this purpose, a periodicity-induced overlapping group shrinkage technique is utilized to threshold the wavelet coefficients. The wavelet coefficients are obtained by using the tunable Q-factor wavelet transform to decompose the measured vibration signals. The proposed approach is constrained to promote sparsity more strongly than convex regularization for estimating periodic group-sparse signals in noise, while avoiding nonconvex optimization. In addition, this maximally sparse convex approach has the advantage of preserving the oscillatory behavior of the useful fault features. A simulated signal is formulated to verify the performance of the proposed approach in periodic feature extraction. The detection performance of the proposed approach is compared with that of the comparative methods via root mean square error values. Finally, the proposed approach is applied to fault diagnosis of both experimental cases and engineering application. The processed results demonstrate that the proposed feature extraction technique can effectively detect the fault features from heavy background noise.

As one of most critical component of high-speed locomotive, wheel set bearing fault identification has attracted an increasing attention in recent years. However, non-stationary vibration signal with modulation phenomenon and heavy background noise make it difficult to excavate the hidden weak fault feature. Variational Mode Decomposition (VMD), which can decompose the non-stationary signal into couple Intrinsic Mode Functions adaptively and non-recursively, brings a feasible tool. However, heavy background noise seriously affects setting of mode number, which may lead to information loss or over decomposition problem. In this paper, an independence-oriented VMD method via correlation analysis is proposed to adaptively extract weak and compound fault feature of wheel set bearing. To overcome the information loss problem, the appropriate mode number is determined by the criterion of approximate complete reconstruction. Then the similar modes are combined according to the similarity of their envelopes to solve the over decomposition problem. Finally, three applications to wheel set bearing fault of high speed locomotive verify the effectiveness of the proposed method compared with original VMD, EMD and EEMD methods.

It is significant to perform condition monitoring and fault diagnosis on rolling mills in steel-making plant to ensure economic benefit. However, timely fault identification of key parts in a complicated industrial system under operating condition is still a challenging task since acquired condition signals are usually multi-modulated and inevitably mixed with strong noise. Therefore, a new data-driven mono-component identification method is proposed in this paper for diagnostic purpose. First, the modified nonlocal means algorithm (NLmeans) is proposed to reduce noise in vibration signals without destroying its original Fourier spectrum structure. During the modified NLmeans, two modifications are investigated and performed to improve denoising effect. Then, the modified empirical wavelet transform (MEWT) is applied on the de-noised signal to adaptively extract empirical mono-component modes. Finally, the modes are analyzed for mechanical fault identification based on Hilbert transform. The results show that the proposed data-driven method owns superior performance during system operation compared with the MEWT method.

Prognostics of the remaining useful life (RUL) has emerged as a critical technique for ensuring the safety, availability, and efficiency of a complex system. To gain a better prognostic result, degradation information is quite useful because it can reflect the health status of a system. However, due to the lack of accurate information about the plants’ degradation, the prognostic model is usually not well established. To solve this problem, this paper proposes a two-stage strategy that is in the context of data-driven modeling to predict the future health status of a bearing, where the degradation information was estimated by calculating the deviation of multiple statistics of vibration signals of a bearing from a known healthy state. Then, a prediction stage based on an enhanced Kalman filter and an expectation–maximization algorithm were used to estimate the RUL of the bearing adaptively. To verify the effectiveness of the proposed approach, a real-bearing degradation problem was implemented.

The implementation of condition monitoring and fault diagnosis system (CMFDS) on wind turbine is significant to lower the unscheduled breakdown. Generator is one of the most important components in wind turbine, and generator bearing fault identification always draws lots of attention. However, non-stationary vibration signal of weak fault and compound fault with a large amount of background noise makes this task challenging in many cases. So, effective signal processing method is essential in the accurate diagnosis step of CMFDS. As a novel signal processing method, empirical Wavelet Transform (EWT) is used to extract inherent modulation information by decomposing signal into mono-components under an orthogonal basis, which is seen as a powerful tool for mechanical fault diagnosis. Moreover, in order to avoid the inaccurate identification the internal modes caused by the heavy noise, wavelet spatial neighboring coefficient denoising with data-driven threshold is applied to increase Signal to Noise Ratio (SNR) before EWT. The effectiveness of the proposed technique on weak fault and compound fault diagnosis is first validated by two experimental cases. Finally, the proposed method has been applied to identify fault feature of generator bearing on wind turbine in wind farm successfully.

As a significant role in industrial equipment, rotating machinery fault diagnosis (RMFD) always draws lots of attention for guaranteeing product quality and improving economic benefit. But non-stationary vibration signal with a large amount of noise on abnormal condition of weak fault or compound fault in many cases would lead to this task challenging. As one of the most powerful non-stationary signal processing techniques, wavelet transform (WT) has been extensively studied and widely applied in RMFD. Numerous publications about the study and applications of WT for RMFD have been presented to academic journals, technical reports and conference proceedings. Many previous publications admit that WT can be realized by means of inner product principle of signal and wavelet base. This paper verifies the essence on inner product operation of WT by simulation and field experiments. Then the development process of WT based on inner product is concluded and the applications of major developments in RMFD are also summarized. Finally, super wavelet transform as an important prospect of WT based on inner product are presented and discussed. It is expected that this paper can offer an in-depth and comprehensive references for researchers and help them with finding out further research topics.

Fault identification timely of rolling mill drivetrain is significant for guaranteeing product quality and realizing long-term safe operation. So, condition monitoring system of rolling mill drivetrain is designed and developed. However, because compound fault and weak fault feature information is usually sub-merged in heavy background noise, this task still faces challenge. This paper provides a possibility for fault identification of rolling mills drivetrain by proposing customized maximal-overlap multiwavelet denoising method. The effectiveness of wavelet denoising method mainly relies on the appropriate selections of wavelet base, transform strategy and threshold rule. First, in order to realize exact matching and accurate detection of fault feature, customized multiwavelet basis function is constructed via symmetric lifting scheme and then vibration signal is processed by maximal-overlap multiwavelet transform. Next, based on spatial dependency of multiwavelet transform coefficients, spatial neighboring coefficient data-driven group threshold shrinkage strategy is developed for denoising process by choosing the optimal group length and threshold via the minimum of Stein’s Unbiased Risk Estimate. The effectiveness of proposed method is first demonstrated through compound fault identification of reduction gearbox on rolling mill. Then it is applied for weak fault identification of dedusting fan bearing on rolling mill and the results support its feasibility.

A generalized and efficient model for rotating anisotropic rotor-bearing systems is presented in this paper with full considerations of the system’s anisotropy in stiffness, inertia and damping. Based on the 3D finite element model and the model order reduction method, the effects of anisotropy in shaft and bearings on the forced response and whirling of anisotropic rotor-bearing systems are systematically investigated. First, the coefficients of journal bearings are transformed from the fixed frame to the rotating one. Due to the anisotropy in shaft and bearings, the motion is governed by differential equations with periodically time-variant coefficients. Then, a free-interface complex component mode synthesis (CMS) method is employed to generate efficient reduced-order models (ROM) for the periodically time-variant systems. In order to solve the obtained equations, a variant of Hill’s method for systems with multiple harmonic excitations is developed. Four dimensionless parameters are defined to quantify the types and levels of anisotropy of bearings. Finally, the effects of the four types of anisotropy on the forced response and whirl orbits are studied. Numerical results show that the anisotropy of bearings in stiffness splits the sole resonant peak into two isolated ones, but the anisotropy of bearings in damping coefficients mainly affect the response amplitudes. Moreover, the whirl orbits become much more complex when the shaft and bearings are both anisotropic. In addition, the cross-coupling stiffness coefficients of bearings significantly affect the dynamic behaviors of the systems and cannot be neglected, though they are often much smaller than the principle stiffness terms.

The various multi-sensor signal features from a diesel engine constitute a complex high-dimensional dataset. The non-linear dimensionality reduction method, t-distributed stochastic neighbor embedding (t-SNE), provides an effective way to implement data visualization for complex high-dimensional data. However, irrelevant features can deteriorate the performance of data visualization, and thus, should be eliminated a priori. This paper proposes a feature subset score based t-SNE (FSS-t-SNE) data visualization method to deal with the high-dimensional data that are collected from multi-sensor signals. In this method, the optimal feature subset is constructed by a feature subset score criterion. Then the high-dimensional data are visualized in 2-dimension space. According to the UCI dataset test, FSS-t-SNE can effectively improve the classification accuracy. An experiment was performed with a large power marine diesel engine to validate the proposed method for diesel engine malfunction classification. Multi-sensor signals were collected by a cylinder vibration sensor and a cylinder pressure sensor. Compared with other conventional data visualization methods, the proposed method shows good visualization performance and high classification accuracy in multi-malfunction classification of a diesel engine.

As one of the most important dynamic excitation sources, the gear time-varying mesh stiffness is the key parameter of gear system dynamic model. So how to calculate the gear mesh stiffness accurately is of great importance. In this paper, an accumulated integral potential energy method is proposed to calculate the mesh stiffness of helical gears. Compared with the finite element model (FEM) and the ISO standard, the proposed analytical method is verified as its results agree well with the others'. Meanwhile the effects of different helical gears parameters (e.g., helix angle, normal module) on mesh stiffness are studied. The results show that the fluctuation degree of mesh stiffness becomes smaller as helix angle or face width increases, by contrast the influence of normal module on the mesh stiffness shows an opposite trend. Furthermore, the mesh stiffness reduction of helical gears due to the tooth crack is quantified. Finally, a dynamic model of helical gear transmission is established to study the vibration characteristics of helical gears. The results show that the mesh stiffness plays a vital role in controlling the vibration of helical gears and has a significant effect on the vibration characteristic of helical gears with tooth crack.

A novel reduced-order modeling method is presented in this paper for dynamics analysis of rotating impeller–shaft-bearing assembly with cracked impellers. Based on three-dimensional finite element model, the complex component mode synthesis (CMS) method is employed to generate an efficient reduced-order model (ROM) for studying the effects of crack on the global vibration of the rotating assembly. First, a modeling framework for impeller–shaft-bearing systems in rotating frame is presented. Rotational effects, including Coriolis matrix and centrifugal softening, have been taken into account. Then, the governing equation of motion of the damped gyroscopic system is reduced by the complex CMS method. Finally, the obtained ROM is employed to study the effects of crack on assembly׳s vibration. During the steady-state response analysis, external excitations on the impeller due to rotor–stator interactions have been taken into account, which was however neglected in previous investigations on rotordynamics. Numerical results show that the lower-order eigenvalues and the unbalance response of the assembly are not sensitive to the local crack on impeller. Nevertheless, the flexible coupling between impeller and shaft becomes more complex when the air flow-induced excitations are considered. Under EO1 traveling wave excitations, a crack leads to slight changes in the assembly׳s response. In contrast, the effect of crack becomes significant when the assembly is excited by EO2 and higher EO excitations. Moreover, the nonlinear crack breathing effects affect the assembly׳s response obviously. Finally, a potential technique for detecting the crack on impeller during operation is discussed.

Traditional reliability evaluation method generally requires a large amount of previous data and information on historical lifetime. For an individual mechanical device without historical lifetime data, it is difficult to carry out the reliability assessment by using the traditional method. To attempt exploring this difficult problem, support evidence statistics (SES) as an approach to operation reliability assessment is presented in this paper. Moreover, this presented method is also expected to indicate the physical state changes of the individual mechanical device. Since in scientific research, evidence usually goes towards supporting or rejecting a hypothesis. For a running device, evidences derived from the running state information should consistently demonstrate its current sole-running-state within a given short time interval. In practice, due to the interference of environmental noises, these evidences lose the consistency. Accordingly, they can be classified into two classes: the firm evidences and the flimsy evidences. Analogous to the support vector data description (SVDD), these firm evidences which show remarkable consistency can form a support evidence space (SESP) through one-class classification. Suppose that a SESP is obtained by using the evidences accumulated from the normal running state, the device operation reliability at any time of unknown running state can be evaluated through the statistical comparison between the normal SESP and the unknown SESP. This reliability evaluation process is named as SES. The most fundamental distinction between the proposed method and the traditional method lies in different statistical objects. The traditional methods are to analyze lifetime data while the proposed methods are to analyze running state data. Obviously, the evidence feature optimization plays a crucial role in the presented method. The maximum correlation and minimum redundancy (MCMR) method is improved by principal component analysis (PCA) to select evidence features based on vibration signals. Finally, the effectiveness of the presented method is validated through a new experiment of rolling bearing.

Mechanical anomaly is a major failure type of induction motor. It is of great value to detect the resulting fault feature automatically. In this paper, an ensemble super-wavelet transform (ESW) is proposed for investigating vibration features of motor bearing faults. The ESW is put forward based on the combination of tunable Q-factor wavelet transform (TQWT) and Hilbert transform such that fault feature adaptability is enabled. Within ESW, a parametric optimization is performed on the measured signal to obtain a quality TQWT basis that best demonstrate the hidden fault feature. TQWT is introduced as it provides a vast wavelet dictionary with time-frequency localization ability. The parametric optimization is guided according to the maximization of fault feature ratio, which is a new quantitative measure of periodic fault signatures. The fault feature ratio is derived from the digital Hilbert demodulation analysis with an insightful quantitative interpretation. The output of ESW on the measured signal is a selected wavelet scale with indicated fault features. It is verified via numerical simulations that ESW can match the oscillatory behavior of signals without artificially specified. The proposed method is applied to two engineering cases, signals of which were collected from wind turbine and steel temper mill, to verify its effectiveness. The processed results demonstrate that the proposed method is more effective in extracting weak fault features of induction motor bearings compared with Fourier transform, direct Hilbert envelope spectrum, different wavelet transforms and spectral kurtosis.

Satellite communication antennas are key devices of a measurement ship to support voice, data, fax and video integration services. Condition monitoring of mechanical equipment from the vibration measurement data is significant for guaranteeing safe operation and avoiding the unscheduled breakdown. So, condition monitoring system for ship-based satellite communication antennas is designed and developed. Planetary gearboxes play an important role in the transmission train of satellite communication antenna. However, condition monitoring of planetary gearbox still faces challenges due to complexity and weak condition feature. This paper provides a possibility for planetary gearbox condition monitoring by proposing ensemble a multiwavelet analysis method. Benefit from the property on multi-resolution analysis and the multiple wavelet basis functions, multiwavelet has the advantage over characterizing the non-stationary signal. In order to realize the accurate detection of the condition feature and multi-resolution analysis in the whole frequency band, adaptive multiwavelet basis function is constructed via increasing multiplicity and then vibration signal is processed by the ensemble multiwavelet transform. Finally, normalized ensemble multiwavelet transform information entropy is computed to describe the condition of planetary gearbox. The effectiveness of proposed method is first validated through condition monitoring of experimental planetary gearbox. Then this method is used for planetary gearbox condition monitoring of ship-based satellite communication antennas and the results support its feasibility.

This paper proposes rhombic micro-displacement amplifier (RMDA) for piezoelectric actuator (PA). First, the geometric amplification relations are analyzed and linear model is built to analyze the mechanical and electrical properties of this amplifier. Next, the accurate modeling method of amplifier is studied for important application of precise servo control. The classical Preisach model (CPM) is generally implemented using a numerical technique based on the first-order reversal curves (FORCs). The accuracy of CPM mainly depends on the number of FORCs. However, it is generally difficult to achieve enough number of FORCs in practice. So, Support Vector Machine (SVM) is employed in the work to circumvent the deficiency of the CPM. Then the hybrid model, which is based on discrete CPM and SVM is developed to account for hysteresis and dynamic effects. Finally, experimental validation is carried out. The analyzed result shows that this amplifier with the hybrid model is suitable for control application.

An efficient method for nonlinear vibration analysis of mistuned centrifugal impellers with crack damages is presented. The main objective is to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. Firstly, in order to reduce the input information needed for component mode synthesis (CMS), the whole model of an impeller is obtained by rotation transformation based on the finite element model of a sector model. Then, a hybrid-interface method of CMS is employed to generate a reduced-order model (ROM) for the cracked impeller. The degrees of freedom on the crack surfaces are retained in the ROM to simulate the crack breathing effects. A novel approach for computing the inversion of large sparse matrix is proposed to save memory space during model order reduction by partitioning the matrix into many smaller blocks. Moreover, to investigate the effects of mistuning and cracks on the resonant frequencies, the bilinear frequency approximation is used to estimate the resonant frequencies of the mistuned impeller with a crack. Additionally, statistical analysis is performed using the Monte Carlo simulation to study the statistical characteristics of the resonant frequencies versus crack length at different mistuning levels. The results show that the most significant effect of mistuning and cracks on the vibration response is the shift and split of the two resonant frequencies with the same nodal diameters. Finally, potential quantitative indicators for detection of crack of centrifugal impellers are discussed.

When a tooth crack failure occurs, the vibration response characteristics caused by the change of time-varying mesh stiffness play an important role in crack fault diagnosis. In this paper, an improved time-varying mesh stiffness algorithm is presented. A coupled lateral and torsional vibration dynamic model is used to simulate the vibration response of gear-rotor system with tooth crack. The effects of geometric transmission error (GTE), bearing stiffness, and gear mesh stiffness on the dynamic model are analyzed. The simulation results show that the gear dynamic response is periodic impulses due to the periodic sudden change of time varying mesh stiffness. When the cracked tooth comes in contact, the impulse amplitude will increase as a result of reductions of mesh stiffness. Amplitude modulation phenomenon caused by GTE can be found in the simulation signal. The lateral–torsional coupling frequency increases greatly within certain limits and thereafter reaches a constant while the lateral natural frequency nearly remains constant as the gear mesh stiffness increases. Finally, an experiment was conducted on a test bench with 2 mm root crack fault. The results of experiment agree well with those obtained by simulation. The proposed method improves the accuracy of using potential energy method to calculate the time-varying mesh stiffness and expounds the vibration mechanism of gear-rotor system with tooth crack failure.

Rapid expansion of wind turbines has drawn attention to reduce the operation and maintenance costs. Continuous condition monitoring of wind turbines allows for early detection of the generator faults, facilitating a proactive response, minimizing downtime and maximizing productivity. However, the weak features of incipient faults in wind turbines are always immersed in noises of the equipment and the environment. Wavelet denoising is a useful tool for incipient fault detection and its effect mainly depends on the feature separation and the noise elimination. Multiwavelets have two or more multiscaling functions and multiwavelet functions. They possess the properties of orthogonality, symmetry, compact support and high vanishing moments simultaneously. The data-driven block threshold selected the optimal block length and threshold at different decomposition levels by using the minimum Stein’s unbiased risk estimate. A multiwavelet denoising technique with the data-driven block threshold was proposed in this paper. The simulation experiment and the feature detection of a rolling bearing with a slight inner race defect indicated that the proposed method successfully detected the weak features of incipient faults.

Wavelet denoising is an effective approach to extract fault features from strong background noise. It has been widely used in mechanical fault detection and shown excellent performance. However, traditional thresholds are not suitable for nonstationary signal denoising because they set universal thresholds for different wavelet coefficients. Therefore, a data-driven threshold strategy is proposed in this paper. First, the signal is decomposed into different subbands by wavelet transformation. Then a data-driven threshold is derived by estimating the noise power spectral density in different subbands. Since the data-driven threshold is dependent on the noise estimation and adapted to data, it is more robust and accurate for denoising than traditional thresholds. Meanwhile, sliding window method is adopted to set a flexible local threshold. When this method was applied to simulation signal and an inner race fault diagnostic case of dedusting fan bearing, the proposed method has good result and provides valuable advantages over traditional methods in the fault detection of rotating machines.

Mechanical fault diagnosis is important to reduce unscheduled machine downtime and avoid catastrophic accidents. It is significant to extract incipient fault and compound fault features as early as possible, which is a complex and challenging task that requests advanced analytical methods with high reliability, high accuracy and high efficiency. Compound fault features are mutually coupled in dynamic signals from the complex system. Weak features of incipient faults are always submersed in background noises. Multiwavelet transform is a remarkable development of wavelet transform, which uses vector scaling functions and wavelet functions. Multiwavelets possess the property of orthogonality, symmetry, compact support and high vanishing moments simultaneously. These advantages promote the development of multiwavelets and their applications in mechanical fault diagnosis in the past decades. This paper attempts to summarize the recent development of multiwavelet transform and its applications in mechanical fault diagnosis. First, the history of wavelets and multiwavelets is introduced. Second, the necessity and the overview of preprocessing methods for multiwavelets are summarized. Third, the advantages of multiwavelets and improvements of different generation multiwavelets are addressed. Fourth, different algorithms of these multiwavelet transforms and their flow charts are presented. Fifth, engineering applications of multiwavelets in mechanical fault diagnosis are investigated. This review also describes a simulation experiment and three application examples which provide a better understanding of different generation multiwavelets for compound fault detection. Finally, existent problems and prospects of further researches are discussed. It is expected that this review will construct an image of the contributions of different generation multiwavelets and link the current frontiers with engineering applications for readers interested in this field.

In the optimization of deep hole boring processes, machining condition monitoring (MCM) plays an important role for efficient tool change policies, product quality control and lower tool costs. This paper proposes a novel approach to the MCM of deep hole boring on the basis of the pseudo non-dyadic second generation wavelet transform (PNSGWT). This approach is developed via constructing a valuable indicator, i.e., the wavelet energy ratio around the natural frequency of boring bar. Self-excited vibration occurs at the frequency of the most dominant mode of the machine tool structure. Via modeling dynamic cutting process and performing its simulation analysis during deep hole boring, it is found that the vibration amplitudes at the nature frequency of the machine tool rise with the tool wear. The PNSGWT that has relative adjustable dyadic time-frequency partition grids, good time-frequency localizability and exact shift-invariance is used to extract the wavelet energy in the specified frequency band. Accordingly, the MCM of deep hole boring can be implemented by means of normalizing the wavelet energy. Finally, a field experiment on deep hole boring machine tool is conducted, and the result shows that the proposed method is effective in the process of monitoring tool wear and surface finish quality for deep hole boring.

Fault detection from the vibration measurement data of rotating machinery is significant for avoiding serious accidents. However, non-stationary vibration signal with a large amount of noise makes this task challenging. Multiwavelet not only owns the advantage on multi-resolution analysis but also can offer multiple wavelet basis functions. So it has the possibility of detecting various fault features preferably. However, the fixed basis functions which are not related to the given signal may lower the accuracy of fault detection. Moreover, another major intrinsic deficiency of multiwavelet lies in its critically sampled filter-bank, which causes shift-variance and is harmful to extract the feature of periodical impulses. To overcome these deficiencies, a new method called customized redundant multiwavelet (CRM) is constructed via increasing multiplicity (IM). IM is a simple method to design a series of changeable multiwavelet which are available for the subsequent optimization process. By the rule of the envelope spectrum entropy minimum principle, optimal multiwavelet is searched for. Based on the customized multiwavelet filters, the filters of CRM can be calculated by inserting zeros. The proposed method is applied to analyze the simulation, gearbox and rolling element bearing vibration signals. Compared with some other conventional methods, the results demonstrate that the proposed method possesses robust performance in detecting fault features of rotating machinery.

Vibration signal is an important prerequisite for mechanical fault detection. However, early stage defect of rotating machineries is difficult to identify because their incipient energy is interfered with background noises. Multiwavelet is a powerful tool used to conduct non-stationary fault feature extraction. However, the existing predetermined multiwavelet bases are independent of the dynamic response signals. In this paper, a constructing technique of vibration data-driven maximal-overlap adaptive multiwavelet (MOAMW) is proposed for enhancing the extracting performance of fault symptom. It is able to derive an optimal multiwavelet basis that best matches the critical non-stationary and transient fault signatures via genetic algorithm. In this technique, two-scale similarity transform (TST) and symmetric lifting (SymLift) scheme are combined to gain high designing freedom for matching the critical faulty vibration contents in vibration signals based on the maximal fitness objective. TST and SymLift can add modifications to the initial multiwavelet by changing the approximation order and vanishing moment of multiwavelet, respectively. Moreover, the beneficial feature of the MOAWM lies in that the maximal-overlap filterbank structure can enhance the periodic and transient characteristics of the sensor signals and preserve the time and frequency analyzing resolution during the decomposition process. The effectiveness of the proposed technique is validated via a numerical simulation as well as a rolling element bearing with an outer race scrape and a gearbox with rub fault.

Detecting transient vibration signatures is of vital importance for vibration-based condition monitoring and fault detection of the rotating machinery. However, raw mechanical signals collected by vibration sensors are generally mixtures of physical vibrations of the multiple mechanical components installed in the examined machinery. Fault-generated incipient vibration signatures masked by interfering contents are difficult to be identified. The fast kurtogram (FK) is a concise and smart gadget for characterizing these vibration features. The multi-rate filter-bank (MRFB) and the spectral kurtosis (SK) indicator of the FK are less powerful when strong interfering vibration contents exist, especially when the FK are applied to vibration signals of short duration. It is encountered that the impulsive interfering contents not authentically induced by mechanical faults complicate the optimal analyzing process and lead to incorrect choosing of the optimal analysis subband, therefore the original FK may leave out the essential fault signatures. To enhance the analyzing performance of FK for industrial applications, an improved version of fast kurtogram, named as “fast spatial–spectral ensemble kurtosis kurtogram”, is presented. In the proposed technique, discrete quasi-analytic wavelet tight frame (QAWTF) expansion methods are incorporated as the detection filters. The QAWTF, constructed based on dual tree complex wavelet transform, possesses better vibration transient signature extracting ability and enhanced time-frequency localizability compared with conventional wavelet packet transforms (WPTs). Moreover, in the constructed QAWTF, a non-dyadic ensemble wavelet subband generating strategy is put forward to produce extra wavelet subbands that are capable of identifying fault features located in transition-band of WPT. On the other hand, an enhanced signal impulsiveness evaluating indicator, named “spatial–spectral ensemble kurtosis” (SSEK), is put forward and utilized as the quantitative measure to select optimal analyzing parameters. The SSEK indicator is robuster in evaluating the impulsiveness intensity of vibration signals due to its better suppressing ability of Gaussian noise, harmonics and sporadic impulsive shocks. Numerical validations, an experimental test and two engineering applications were used to verify the effectiveness of the proposed technique. The analyzing results of the numerical validations, experimental tests and engineering applications demonstrate that the proposed technique possesses robuster transient vibration content detecting performance in comparison with the original FK and the WPT-based FK method, especially when they are applied to the processing of vibration signals of relative limited duration.

Fault diagnosis of rotating machinery is of great importance to the high quality products and long-term safe operation. However, the useful weak features are usually corrupted by strong background noise, thus increasing the difficulty of the feature extraction. Thereby, a novel denoising method based on the tunable Q-factor wavelet transform (TQWT) using neighboring coefficients is proposed in this article. The emerging TQWT possesses excellent properties compared with the conventional constant-Q wavelet transforms, which can tune Q-factor according to the oscillatory behavior of the signal. Meanwhile, neighboring coefficients denoising is adopted to avoid the overkill of conventional term-by-term thresholding techniques. Because of having the combined advantages of the two methods, the presented denoising method is more practical and effective than other methods. The proposed method is applied to a simulated signal, a rolling element bearing with an outer race defect from antenna transmission chain and a gearbox fault detection case. The processing results demonstrate that the proposed method can successfully identify the fault features, showing that this method is more effective than the conventional wavelet thresholding denoising methods, term-by-term TQWT denoising schemes and spectral kurtosis.

The rotating machinery, as a typical example of large and complex mechanical systems, is prone to diversified sorts of mechanical faults, especially on their rotating components. Although they can be collected via vibration measurements, the critical fault signatures are always masked by overwhelming interfering contents, therefore difficult to be identified. Moreover, owing to the distinguished time-frequency characteristics of the machinery fault signatures, classical dyadic wavelet transforms (DWTs) are not perfect for detecting them in noisy environments. In order to address the deficiencies of DWTs, a pseudo wavelet system (PWS) is proposed based on the filter constructing strategies of wavelet tight frames. The presented PWS is implemented via a specially devised shift-invariant filterbank structure, which generates non-dyadic wavelet subbands as well as dyadic ones. The PWS offers a finer partition of the vibration signal into the frequency-scale plane. In addition, in order to correctly identify the essential transient signatures produced by the faulty mechanical components, a new signal impulsiveness measure, named spatial spectral ensemble kurtosis (SSEK), is put forward. SSEK is used for selecting the optimal analyzing parameters among the decomposed wavelet subbands so that the masked critical fault signatures can be explicitly recognized. The proposed method has been applied to engineering fault diagnosis cases, in which the processing results showed its effectiveness and superiority to some existing methods.

Remaining useful life (RUL) prognostics is a core problem in prognostics and health management (PHM). Accurate RUL prediction is crucial not only to the verification of mission goals but also to failure prevention and maintenance decision in a more effective and efficient manner. However, the substantial nonlinearity is one of most important challenges in deterioration modeling and RUL estimation of nonlinear mechanical system. An interesting contribution is the improvement of RUL prediction accuracy by the use of both greedy kernel principal components analysis (GKPCA) for dimensional reduction to extract feature from multi dimension data set of monitored nonlinear mechanical system and nonhomogeneous hidden semi-Markov model (NHSMM) to model the multistate deterioration process. A case study with the data set from turbofan engines is analyzed using the methodology, and by comparing the prediction accuracy with the previously linear PCA-NHSMM's, the result verifies the effectiveness (closer to actual RUL, earlier tracked health state, smaller boundary width) and efficiency(higher prognostics robustness) of the methodology.

Confusion of frequency spectrum under varying speed condition leads to the efficacy degradation of fault recognition for rolling element bearings. Meanwhile, a widely-used method is transforming non-stationary vibration signals into angular domain based on angular re-sampling to reduce influence of speed fluctuation. However, single domain usually cannot describe health information comprehensively. Thus a multi-domain indexes description method is proposed in this paper to improve fault recognition by multi-domain fusion. First the input indexes of multiple domains including time domain, frequency domain, angular domain and order domain are organized to train SOM neural network. Then an optimized fault cognition model is established based on the SOM input planes. Finally, faults are recognized based on multi-domain indexes model and extra complexity indexes are added to enhance recognition of compound faults. Experimental results show that the proposed method can distinguish different faults effectively and have good practical significance.

Rolling-element bearing vibrations are random cyclostationary. This paper addresses the problem of noise reduction with simultaneous components extraction in vibration signals for faults diagnosis of bearing. The observed vibration signal is modeled as a summation of two components contaminated by noise, and each component composes of repetitive transients. To extract the two components simultaneously, an approach by solving an optimization problem is proposed in this paper. The problem adopts convex sparsity-based regularization scheme for decomposition, and non-convex regularization is used to further promote the sparsity but preserving the global convexity. A synthetic example is presented to illustrate the performance of the proposed approach for repetitive feature extraction. The performance and effectiveness of the proposed method are further demonstrated by applying to compound faults and single fault diagnosis of a locomotive bearing. The results show the proposed approach can effectively extract the features of outer and inner race defects.

Gearbox fault diagnosis is very important for preventing catastrophic accidents. Vibration signals of gearboxes measured by sensors are useful and dependable as they carry key information related to the mechanical faults in gearboxes. Effective signal processing techniques are in necessary demands to extract the fault features contained in the collected gearbox vibration signals. Overcomplete rational dilation discrete wavelet transform (ORDWT) enjoys attractive properties such as better shift-invariance, adjustable time–frequency distributions and flexible wavelet atoms of tunable oscillation in comparison with classical dyadic wavelet transform (DWT). Due to these advantages, ORDWT is presented as a versatile tool that can be adapted to analysis of gearbox fault features of different types, especially in analyzing the non-stationary and transient characteristics of the signals. Aiming to extract the various types of fault features confronted in gearbox fault diagnosis, a fault feature extraction technique based on ORDWT is proposed in this paper. In the routine of the proposed technique, ORDWT is used as the pre-processing decomposition tool, and a corresponding post-processing method is combined with ORDWT to extract the fault feature of a specific type. For extracting periodical impulses in the signal, an impulse matching algorithm is presented. In this algorithm, ORDWT bases of varied time–frequency distributions and varied oscillatory natures are adopted, moreover an improved signal impulsiveness measure derived from kurtosis is developed for choosing optimal ORDWT bases that perfectly match the hidden periodical impulses. For demodulation purpose, an improved instantaneous time–frequency spectrum (ITFS), based on the combination of ORDWT and Hilbert transform, is presented. For signal denoising applications, ORDWT is enhanced by neighboring coefficient shrinkage strategy as well as subband selection step to reveal the buried transient vibration contents. The proposed fault feature extraction technique is applied in a range of engineering applications, and the processing results demonstrate that the ORDWT-based feature extraction technique successfully identifies the incipient fault features in the cases where DWT and empirical mode decomposition method are less effective.

Gears are common power transmission elements and are frequently responsible for transmission failures. Instantaneous time-frequency spectrum (ITFS) resulted from local mean decomposition is applied to the surveillance and early fault diagnosis of a finishing rolling mill in this paper. Results of practical signals demonstrate that ITFS is effective and reliable for the early detection of gear local fault. In addition, a new parameter to evaluate the damage severity of the gearbox is also developed based on the marginal spectrum derived from ITFS. The utility of the new gear fault symptom has been investigated using practical vibration signals. Results show that the new parameter is only sensitive to the changes caused by the deterioration of a monitored unit and insensitive to the influence of the variable non-deterioration factors such as varying speed and loads. This new index may thus find its wide applications for machine prognostics in the near future.

In order to protect the slider and disk from excessive wear, a numerical scheme is proposed for transient dynamics at slider/disk interface with the emphasis on the slider wear mechanism and the surface topology optimization of sub-ambient pressure slider. For each transient flying height and slider attitude, the modified Reynolds lubrication equation is solved by the control volume method to obtain the simultaneous pressure distribution in the slider surface. Then the dynamic flying attitude of the slider subject to the calculated pressure distribution can be solved by the Quasi-Newton iteration method at each time step. On the basis of above work, the optimization of the slider surface topology is implemented, and it’s concluded that slider crown has the most significant impact on the dynamic flying attitude and stability. Finally, a sub-ambient pressure slider is designed, which can effectively avoid the slider wear and severe fault of hard disk drive.

Kurtogram, due to the superiority of detecting and characterizing transients in a signal, has been proved to be a very powerful and practical tool in machinery fault diagnosis. Kurtogram, based on the short time Fourier transform (STFT) or FIR filters, however, limits the accuracy improvement of kurtogram in extracting transient characteristics from a noisy signal and identifying machinery fault. Therefore, more precise filters need to be developed and incorporated into the kurtogram method to overcome its shortcomings and to further enhance its accuracy in discovering characteristics and detecting faults. The filter based on wavelet packet transform (WPT) can filter out noise and precisely match the fault characteristics of noisy signals. By introducing WPT into kurtogram, this paper proposes an improved kurtogram method adopting WPT as the filter of kurtogram to overcome the shortcomings of the original kurtogram. The vibration signals collected from rolling element bearings are used to demonstrate the improved performance of the proposed method compared with the original kurtogram. The results verify the effectiveness of the method in extracting fault characteristics and diagnosing faults of rolling element bearings.

The characteristic signal of a rolling bearing with a defect acts as a series of periodic impulses. These features are usually immersed in heavy noise and then difficult to extract. It is feasible to make the features distinct through wavelet denoising. Scalar wavelet thresholding has been used to extract features. However, scalar wavelet might not extract the feature available due to its limitation in some important properties, and conventional term-by-term thresholding does not consider the effect of neighboring coefficients. Since multiwavelets have been formulated recently and they might offer good properties in signal processing, a novel denoising method — multiwavelet denoising with improved neighboring coefficients (neighboring coefficients dependent on level, DLNeighCoeff for short) — is proposed in this article. The method proposed is applied to a simulated signal and fault diagnosis of locomotive rolling bearings, obtaining performance superior to conventional methods.

Fault diagnosis of rolling mills, especially the main drive gearbox, is of great importance to the high quality products and long-term safe operation. However, the useful fault information is usually submerged in heavy background noise under the severe condition. Thereby, a novel method based on multiwavelet sliding window neighboring coefficient denoising and optimal blind deconvolution is proposed for gearbox fault diagnosis in rolling mills. The emerging multiwavelets can seize the important signal processing properties simultaneously. Owing to the multiple scaling and wavelet basis functions, they have the supreme possibility of matching various features. Due to the periodicity of gearbox signals, sliding window is recommended to conduct local threshold denoising, so as to avoid the “overkill” of conventional universal thresholding techniques. Meanwhile, neighboring coefficient denoising, considering the correlation of the coefficients, is introduced to effectively process the noisy signals in every sliding window. Thus, multiwavelet sliding window neighboring coefficient denoising not only can perform excellent fault extraction, but also accords with the essence of gearbox fault features. On the other hand, optimal blind deconvolution is carried out to highlight the denoised features for operators’ easy identification. The filter length is vital for the effective and meaningful results. Hence, the foremost filter length selection based on the kurtosis is discussed in order to full benefits of this technique. The new method is applied to two gearbox fault diagnostic cases of hot strip finishing mills, compared with multiwavelet and scalar wavelet methods with/without optimal blind deconvolution. The results show that it could enhance the ability of fault detection for the main drive gearboxes.

Among the vibration-based fault diagnosis methods for rolling element bearing, the shock pulse method (SPM) combined with the demodulation method is a useful quantitative technique for estimating bearing running state. However, direct demodulation often misestimates the shock value of characteristic defect frequency. To overcome this disadvantage, the vibration signal should be decomposed before demodulation. Empirical mode decomposition (EMD) can be an alternative for preprocess bearing fault signals. However, the trouble with this method’s application is that it is time-consuming. Therefore, a novel method that can improve the sifting process’s efficiency is proposed, in which only one time of cubic spline fitting is required in each sifting process. As a consequence, the time for EMD analysis can be evidently shortened and the decomposition results simultaneously maintained at a high precision. Simulations and experiments verify that the improved EMD method, combined with SPM and demodulation analysis, is efficient and accurate and can be effectively applied in engineering practice.

Empirical mode decomposition (EMD) is a self-adaptive analysis method for nonlinear and non-stationary signals. It may decompose a complicated signal into a collection of intrinsic mode functions (IMFs) based on the local characteristic time scale of the signal. The EMD method has attracted considerable attention and been widely applied to fault diagnosis of rotating machinery recently. However, it cannot reveal the signal characteristic information accurately because of the problem of mode mixing. To alleviate the mode mixing problem occurring in EMD, ensemble empirical mode decomposition (EEMD) is presented. With EEMD, the components with truly physical meaning can be extracted from the signal. Utilizing the advantage of EEMD, this paper proposes a new EEMD-based method for fault diagnosis of rotating machinery. First, a simulation signal is used to test the performance of the method based on EEMD. Then, the proposed method is applied to rub-impact fault diagnosis of a power generator and early rub-impact fault diagnosis of a heavy oil catalytic cracking machine set. Finally, by comparing its application results with those of the EMD method, the superiority of the proposed method based on EEMD is demonstrated in extracting fault characteristic information of rotating machinery.

When detecting a weak and high-frequency signal submerged in strong noise, the existing large parameter stochastic resonance (LPSR) models need either a high sampling frequency or a large number of sample points. To breach the above limits and raise the usability of LPSR, a novel method named frequency-shifted and re-scaling stochastic resonance (FRSR) is proposed in this paper. By shifting and re-scaling the frequency, FRSR provides a way to alleviate the contradiction between sampling frequency and the number of sample points. The proposed method is verified with simulated signals. The results show that this method is useful in weak fault diagnosis of mechanical systems which involve high feature frequencies. Compared with the existing LPSR models, FRSR just requires a lower sampling frequency, a smaller data length and has higher efficiency. Finally, the proposed method is applied to a milling machine tool fault diagnosis and an outer ring fault feature of the spindle bearing is found successfully. Thus, FRSR is a meaningful try for SR coming into practical use of mechanical fault diagnosis.

Performance degradation or failure of manufacturing equipment will badly influence machining quality. Because of discontinuousness of the milling process, dynamic signals produced in the milling process become non-stationary. This paper indicates that the essence of SGW (second generation wavelet) transform in non-stationary signal processing is the mathematics principle of inner product transform of a dynamic signal with basis functions. Namely, by means of the inner product operation of a signal with basis functions containing scale function and wavelet function, signal decomposition and reconstruction are obtained. Acoustic emission signals generated in the milling processes of a CNC machine were analyzed by using the basis functions of SGW which are oscillation, decay and compact support. The features of end milling cutter breakage have been extracted, and the influences on machining surface quality have been identified effectively, which provide scientific bases for fault diagnosis, error tracing and quality control.

A novel method is proposed for operational modal analysis OMA of linear rotor systems, combined with correction technique of spectrum analysis (CTSA), harmonic wavelet filtering (HWF), random decrement technique (RDT) and Hilbert transform (HT) method. With CTSA method, operational frequency component is reconstructed and subtracted from rotor vibration signal. The residual signal is composed by harmonics of operational frequency and several model responses. The single mode responses can be obtained by HWF and then RDT method is employed to process these responses, respectively, to extract single mode free responses of the rotor systems. Finally, HT method is applied to obtain modal parameters of the rotor systems from these single mode free responses. The experimental results of the proposed method are close to those of the finite element method which indicates that the proposed method is effective in practical applications.

This paper presents a fault diagnosis method of rotating machinery based on a new clustering algorithm using a compensation distance evaluation technique (CDET). A two-stage feature selection and weighting technique is adopted in this algorithm. Feature weights are computed via CDET according to the sensitivity of features and assigned to the corresponding features to indicate their different importance in clustering. Feature weighting highlights the importance of sensitive features and simultaneously weakens the interference of insensitive features. The new clustering algorithm is described and applied to incipient fault and compound fault diagnosis of locomotive roller bearings. The diagnosis result shows the algorithm is able to reliably recognise not only different fault categories and severities but also the compound faults, and demonstrates the superior effectiveness and practicability of the algorithm. Therefore, it is a promising approach to fault diagnosis of rotating machinery.

In this paper, a novel method based on lifting scheme and Mahalanobis distance (MD) is proposed for detection of tool breakage via acoustic emission (AE) signals generated in end milling process. The method consists of three stages. First, by investigating the specialty of AE signals, a biorthogonal wavelet with impact property is constructed using lifting scheme, and wavelet transform is carried out to separate AE components from the original signals. Second, Hilbert transform is adopted to demodulate signal envelope on wavelet coefficients and salient features indicating the tool state (i.e., normal conditions, slight breakage, and serious breakage) are extracted. Finally, tool conditions are identified directly through the recognition of these features by means of MD. Practical application results on a CNC vertical milling machine tool show that the proposed method is accurate for feature extraction and efficient for condition monitoring of cutting tools in end milling process.

This paper presents a novel method for fault diagnosis based on an improved wavelet package transform (IWPT), a distance evaluation technique and the support vector machines (SVMs) ensemble. The 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 to extract salient frequency-band features from 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 ensemble with AdaBoost algorithm to identify the different abnormal cases. The proposed method is applied to the fault diagnosis of rolling element bearings, and testing results show that the SVMs ensemble can reliably separate different fault conditions and identify the severity of incipient faults, which has a better classification performance compared to the single SVMs.

Degradation information of a complex mechanical system reflects the system's health status and is useful to predict the future progression of the fault or anomalous behaviors. This paper proposed a two-stage strategy to predict the future health status of a bearing by utilizing the bearing's degradation information. The first stage was implemented to monitor the bearing's health status until a degradation point was detected. When the bearing begins to degrade, a prediction stage based on Kalman filter was then used to estimate the remaining useful life (RUL) of the bearing. Finally, a real bearing degradation problem was used to verify our method.

Inner product transform principle reveals that the basis functions most relevant or similar to the fault features are pivotal to the meaningful fault detection. Customized multiwavelet methods and practices have continued to improve over the recent years, focused on two-scale similarity transform (TST), lifting transform (LT), and lifting scheme (LS). Due to the respective advantages and disadvantages, a comparative study on the multiwavelet construction methods by TST, symmetric and dissymmetric LT, and LS is discussed in the paper, covering the differences of construction theories, the synthetic analyses of construction strategies, and the comparison of waveform characteristics along with their applicable occasions. Comprehensively utilizing the capabilities of the construction methods, a novel customized multiwavelet library is established for the accurate fault detection. The proposed method is applied to incipient fault detection of rolling bearing for electric locomotive to verify the effectiveness and feasibility.

This paper addresses the detection of periodic transients in vibration signals for detecting faults in rotating machines. For this purpose, we present a method to estimate periodic-group-sparse signals in noise. The method is based on the formulation of a convex optimization problem. A fast iterative algorithm is given for its solution. A simulated signal is formulated to verify the performance of the proposed approach for periodic feature extraction. The detection performance of comparative methods is compared with that of the proposed approach via RMSE values and receiver operating characteristic (ROC) curves. Finally, the proposed approach is applied to compound faults diagnosis of motor bearings. The non-stationary vibration data were acquired from a SpectraQuest's machinery fault simulator. The processed results show the proposed approach can effectively detect and extract the useful features of bearing outer race and inner race defect.

This paper investigated the instability of head-disk interface caused by the voice coil motor (VCM) end crashing the crash stop during the seeking of magnetic head. To make the whole process of that clear, an in situ measurement method based on maximum likelihood estimation and extended Kalman filter for seeking speed at component level was developed first and was then calibrated by a high speed camera. Given a crash between VCM end and crash stop that may be a consequence of the continuous increasing seeking speed, the seeking speed was carefully controlled by using our developed method to find a critical value that may induce vigorous head-disk interface instability. Acoustic emission sensor and laser Doppler vibrometer were used to capture the transient dynamic behaviors of magnetic head when the crash is happening. Damage analysis and mode identification were carried out to reveal the relationship between the damage patterns on disk surface and head dynamics. The results of this study are helpful to optimize the track seeking profile during the HDD operation, as well as the design of components such as head and head arm.

A new hybrid clustering algorithm based on a three-layer feed forward neural network (FFNN), a distribution density function, and a cluster validity index, is presented in this paper. In this algorithm, both feature weighting and sample weighting are considered, and an optimal cluster number is automatically determined by the cluster validity index. Feature weights are learnt via FFNN based on the gradient descent technique, and sample weights are computed by using the distribution density function of a sample. Feature weighting and sample weighting highlight the importance of sensitive features and representative samples, and simultaneously weaken the interference of insensitive features and vague samples. The presented algorithm is described and applied to the incipient fault diagnosis of locomotive roller bearings. The diagnosis result demonstrates the superior effectiveness and practicability of the algorithm, and shows that it is a promising approach to the fault diagnosis of rotating machinery.