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Real Order Derivatives and Spectral Norms in Fault Detection
Abstract
Fault diagnosis can utilise signal processing in various ways: informative time domain signals are separated or transformed from the source signals and used in feature extraction to find useful fault indicators. Frequency domain analyses focus on identifying interesting signal components. This research integrates different solutions together: differentiation is used for modifying the source signal and feature extraction is based on the generalised spectral norms which have been studied by the authors before. The concept of these norms is intended to give a method of spectral analysis with less human labour and, potentially, even entirely without it. We discuss the possibility of utilising the spectral norms along with real order derivatives. The solution was tested for bearing and misalignment faults induced in two test grids. The potential challenges of selecting parameters and interpreting results are discussed. Results show that when used with a properly selected set of indices, the presented method can be an effective tool for condition monitoring. Moreover, this technique provides an opportunity for automated analytics also for frequency domain analyses.KeywordsFault detectionGeneralised spectral normReal order derivative
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NASA’s Juno mission, involving a pioneering spacecraft the size of a basketball court, has been instrumental in observing Jupiter’s atmosphere and surface from orbit since it reached the intended orbit. Over its first decade of operation, Juno has provided unprecedented insights into the solar system’s origins through advanced remote sensing and technological innovations. This study focuses on change detection in terms of Juno’s trajectory, leveraging cutting-edge data computing techniques to analyze its orbital dynamics. Utilizing 3D position and velocity time series data from NASA, spanning 11 years and 5 months (August 2011 to January 2023), with 5.5 million samples at 1 min accuracy, we examine the spacecraft’s trajectory modifications. The instantaneous average acceleration, jerk, and snap are computed as approximations of the first, second, and third derivatives of velocity, respectively. The Hilbert transform is employed to visualize the spectral properties of Juno’s non-stationary 3D movement, enabling the detection of extreme events caused by varying forces. Two unsupervised machine learning algorithms, DBSCAN and OPTICS, are applied to cluster the sampling events in two 3D state spaces: (velocity, acceleration, jerk) and (acceleration, jerk, snap). Our results demonstrate that the OPTICS algorithm outperformed DBSCAN in terms of the outlier detection accuracy across all three operational phases (OP1, OP2, and OP3), achieving accuracies of 99.3%, 99.1%, and 98.9%, respectively. In contrast, DBSCAN yielded accuracies of 98.8%, 98.2%, and 97.4%. These findings highlight OPTICS as a more effective method for identifying outliers in elliptical orbit data, albeit with higher computational resource requirements and longer processing times. This study underscores the significance of advanced machine learning techniques in enhancing our understanding of complex orbital dynamics and their implications for planetary exploration.
Spectral analysis is a very common tool in vibration monitoring. While useful in machine diagnostics, spectral analysis can be rather time consuming. Tasks that require an extensive amount of time are often considered too expensive, especially in modern industry. To achieve an automatic monitoring system of some kind, different features, such as the root mean square (RMS) or the peak value of signals, are often monitored. These are special cases of generalised norms, which can be effective tools to determine whether a signal has shown some kind of change and how notable the change is when compared to a previously measured signal, for example. However, no spectral information is obtained in this way and the question regarding the frequency in the signal at which the change has occurred remains unanswered. Generalised spectral norms are a frequency-domain application of these norms and, as presented in this paper, provide a suitable way to perform automatic monitoring regarding the spectral information, for example.
Automatic fault detection with condition indices enables reliable condition monitoring to be combined with process control. Useful information on different faults can be obtained by selecting suitable features. Generalised norms can be defined by the order of derivation, the order of the moment and sample time. These norms have the same dimensions as the corresponding signals. The nonlinear scaling used in the linguistic equation approach extends the idea of dimensionless indices to nonlinear systems. Condition indices are obtained from short samples by means of the scaled values and linear equations. Indices, which are used in the same way as the process measurements in process control, detect differences between normal and faulty conditions and provide an indication of the severity of the faults. The generalised norms represent the norms from the minimum to the maximum in a smoothly increasing way. The new nonlinear scaling methodology based on generalised norms and skewness provides good results for the automatic generation of condition indices. Additional sensitivity is achieved for the values which differ only slightly from the centre values. For cavitation, the new approach provides four levels from cavitation-free to clear cavitation. For the supporting rolls of a lime kiln, it provides an efficient indication of faulty situations. Sensitivity is also improved for small fluctuations. All the supporting rolls can be analysed using the same approach throughout the data set. The results of both the applications are consistent with the vibration severity criteria: good, usable, still acceptable, and not acceptable. Warning and alarm limits can be defined and fault types can also be identified with fuzzy set systems and specialised condition indices.
Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Rapid changes in acceleration become emphasised upon the derivation of the signal x(2). The aim is to detect faults at an early stage by using the absolute values of the dynamic part of the signals. Generalised norms ‖τMαp‖p can be defined by the order of derivation (α), the order of the norm (p) and sample time (τ), where α and p are real numbers. These norms have the same dimensions as the corresponding signals. A need for maintenance is indicated for lime kilns, which are essential parts of the chemical recovery cycle in a pulp mill. These large machines with very slow rotational speeds must run at different production capacities and speeds. The generalised norms provide an efficient indication of faulty situations in the supporting rolls. Surface damage and alignment problems are clearly detected and in the present system also identified with two norms of different order obtained from the signals x(4). An early indication of friction increase is also achieved. The data set covers the following cases: (1) surface problems, (2) good conditions after grinding, (3) misalignment, (4) stronger misalignment, (5) very good conditions after repair work, and (6) very good conditions one year later. Maintenance was done for one of the supporting rolls. All the rolls can be analysed using the same features throughout the data set. All the faults, friction and minor fluctuations were validated by listening to the recorded acceleration signals and analysing time domain signals and frequency spectra with an oscilloscope and a real time analyser.
Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. The signals x(3) and x(4) are very suitable for the condition monitoring of slowly rotating bearings since rapid changes in acceleration become emphasised upon the differentiation of the signal x(2). Real order derivatives x(α) provide additional possibilities, e.g. in a bearing fault case the sensitivities of some features have been found to reach a maximum when α = 4.75. In earlier cavitation indicators, the rms values were combined with either kurtosis or peak values. Generalised moments τMαp can be defined by the order of derivation (α), the order of the moment (p) and sample time (τ). The moment normalised by standard deviation can be used as kurtosis in the model-based analysis.
This paper introduces a new norm ‖τMαp‖p = (τMαp)1/p = [1/NΣi=1N|xi(α)|p]1/p, where the orders α and p are real numbers. The number of signal values s N =τ N where Ns is the number of samples per second. The new norm has the same dimensions as the corresponding signals x(α ) . The cavitation of a Kaplan water turbine was analysed in the power range 1.5…59.4 MW based on measurements collected with sampling frequency 12800 Hz. The order p was compared in the range from 0.25 to 8 with a step 0.25, and a total of 11 sample times were used: τ = 1, 2,…, 6, 8, 10, 20, 30, and 40 seconds. An optimum order p was detected for each sample time τ. The relative max(‖3^M_4^2.75‖) compared to a cavitation-free case is alone a good indicator for cavitation: strong cavitation and cavitation-free cases are clearly detected, and the power ranges for shortterm cavitation are only slightly wider than in the previously developed knowledge-based cavitation index. Short sample times and relatively small requirements for the frequency ranges make this approach feasible for on-line analysis and power control. The weighted sums of features on a different order of derivation form fault-specific measurement indices MIT, and several indices MIT define the health index SOL. The sensitivity increases with the order of derivation to some limit of α. For faults causing impacts, high MIT values for lower orders of derivation indicate an increase in the severity of the fault.
This paper examines a derivate x(α), the order of which is α, where α is an arbitrary real number. It is shown with practical examples that x(α) can be used in condition monitoring and that without it we can only obtain an erratic picture of condition in certain cases.
A combined signal processing and feature extraction approach can be based on generalised moments and norms, which are defined by the order of derivation (α), the order of the moment (p) and sample time (τ), where α and p are real numbers. The analysis can be further improved by taking into account nonlinear effects by monotonously increasing scaling functions. Several features can be combined in condition indices and, in some cases, only one feature is needed if the orders p and α are chosen properly. The aim of the application analysis is to test new features and whether their sensitivity is sufficient to detect faults from the absolute values of the dynamic part of the signals x(α) at an early stage when the faults are still very small. For faults causing impacts, the sensitivity of the features is clearly improved by higher and real order derivation, and low order norms can be used if the order of derivation is sufficient. Correspondingly, subharmonic vibrations can be amplified by integration. A limit of sensitivity is reached on a certain order α, and on each order α an optimal sensitivity is chosen by the order p. Both the orders can be chosen fairly flexibly from the optimal area. Sample time, which connects the features to the control application, is process-specific. The analysis methods allow the use of lower frequency ranges, and a multisensor approach can also be used. The approach is well suited for rotating process equipment with speeds ranging from very slow to very fast. In this article, the types of slow rotating equipment are a lime kiln, a washer and the scraper of a digester, all from pulp mills. A centrifuge and a turbo compressor are examples of very fast rotating machines. Features can also be combined in stress indices, which react to harmful process conditions. For example, strong cavitation, cavitation-free cases and even short-term cavitation are clearly detected in a Kaplan water turbine. Several features, also from different frequency ranges, can be combined in condition and stress indices.
A new methodology for bearing defect detection is described in this article. The technology involves the analysis of peak values (PeakVue™) in a band limited signal. For bearings, it focuses on the analysis of stress waves which are set up in metal-to-metal impacting, scuffing, abrasive wear, etc. Unlike demodulation, PeakVue does not employ any low pass filter, i.e., there are no implicit assumptions about a low frequency occurrence modulating a high frequency carrier. In theory, as well as practice, the PeakVue analysis procedure is capable of detecting bearing defects on very slow machinery (<1 RPM) as well as on intermediate to fast machinery. It is demonstrated that PeakVue analysis is comparable to demodulation analysis on intermediate speed machinery and significantly superior to demodulation on slow speed machinery.
Advanced signal processing methods combined with automatic fault detection enable reliable condition monitoring for long periods of continuous operation. Rapid changes in acceleration become emphasised upon derivation of the signal x(2). Higher order derivatives, especially x(4), work very well in the whole range from slowly to very fast rotating rolling bearings. Real order derivatives x(α) provide additional possibilities. If α < 0, we can also talk about the fractional integration of displacement. It has interesting application potential for faults that occur in frequencies below the rotating frequency. The aim of the analysis is to introduce new features whose sensitivity is sufficient to detect faults from the absolute values of the dynamic part of the signals x(α) at an early stage, ie when the faults are still very small. Generalised moments τMαp and norms, ‖τMαp‖p = (τMαp)1/p = [1/NΣi=1N|xi(α)|p]1/p, can be defined by the order of derivation (α), the order of the moment (p) and sample time (τ), where α and p are real numbers. The number of signal values N = τNs, where Ns is the number of signal values taken in a second. The norm ‖τMαp‖p has the same dimensions as the corresponding signals x(α). Both the order of derivation and the order of the moment improve sensitivity to impacts. There are many alternative ways to normalise the moments: absolute mean deviation is an easy solution and normalisation by standard deviation generalises the moments to standardised variables. Dimensionless vibration indices obtained using different orders α and p can be combined in measurement indices. The analysis can be further improved by taking into account non-linear effects by monotonously increasing the scaling functions. Several features can be combined in condition indices and, in some cases, only one feature is needed if the orders p and α are chosen properly.
Dependent Component Analysis(DCA) as an extension of Independent Component Analysis(ICA) for Blind Source Separation(BSS) has more applications than ICA and received more and more attentions during the last several years in the study of signal processing and neural networks. After a general and detailed definition of the DCA model is given, the separateness and uniqueness of the DCA model have been discussed in theory. Then, the state-of-art DCA algorithms are overviewed, these methods include multidimensional ICA, variance dependent BSS, subband decomposition ICA, maximum non-Gaussianity method, Wold decomposition method and time-frequency method are constructed for the BSS problem in theories and some simulations of these algorithms are also exhibited for different applications.
Feature selection techniques have become an apparent need in many bioinformatics applications. In addition to the large pool of techniques that have already been developed in the machine learning and data mining fields, specific applications in bioinformatics have led to a wealth of newly proposed techniques.
In this article, we make the interested reader aware of the possibilities of feature selection, providing a basic taxonomy of feature selection techniques, and discussing their use, variety and potential in a number of both common as well as upcoming bioinformatics applications.
Contact:yvan.saeys@psb.ugent.be
Supplementary information:http://bioinformatics.psb.ugent.be/supplementary_data/yvsae/fsreview
This study is focused on the application of automated techniques on low-speed bearing diagnostics. The diagnosis in low-speed conditions is hampered by the long periods between defect-related impulses and the high level of noise relative to the magnitude of the impulses. To detect a localised defect in such conditions, a new approach that uses vibration signals and information on the bearing defect frequencies is proposed. At first, the vibration signal is filtered in a specific frequency range to enable the detection of the impulses hidden in the signal. The filtered signal is then segmented into short time windows, the length of which are selected based on the bearing defect frequencies. Statistical time domain features are calculated from these windows to amplify and compress the impulses inflicted by the defect. Then, a criterion based on the autocorrelation values of specific time lags is calculated. An exhaustive search procedure is used to determine the frequency band for signal filtering and to select the statistical feature, which together maximises the proposed criterion. The highest value of the criterion is finally compared with the corresponding value from the baseline condition to detect the localised defect. The proposed technique is demonstrated on simulated signals, and validated based on the vibration signals from laboratory tests with undamaged, slightly damaged and severely damaged rolling elements in a rolling element bearing. Different conditions with shaft speeds from 20 to 80 rpm were studied in the laboratory tests. The proposed technique was compared with automated envelope spectrum diagnosis approaches based on the peak ratio and peak-to-median indicators and the fast kurtogram. The results reveal that the criterion based on autocorrelation gave defect indications associated with the correct type of defect in various circumstances while the tested envelope spectrum approaches were prone to induce an incorrect conclusion. Moreover, the results indicate that the approach could be used successfully on signals with a length that includes relatively few defect periods or impulses. The approach requires a high sampling rate relative to the defect frequencies, which may limit its suitability for the higher shaft speeds.
The shock pulse method provides a completely new approach with regard to both theory and application for the measurement of wear in rolling-type bearings. The method measures indirectly the velocity of a mechanical impact caused by a bearing irregularity and allows a quantative measurement of bearing condition in operational machinery without influence from vibration, noise, temperature, or other external factors. The equipment in its simplest form consists of a portable instrument with a hand-held, probe-type transducer. Adapter studs permanently mounted in the bearing housing are also available for more detailed evaluations or for permanent installations with continuous monitoring.
In condition monitoring, detecting changes in measured signals is often desired. Another crucial piece of information is the frequency, which shows the change. In industrial applications today, it is quite common to monitor some features, for example RMS value, and visually analyse the frequency spectrum for information about the frequency content. The RMS is, in fact, a special case of the lp norms, which have proven to be quite a useful way to monitor changes in signals. While effective for detecting changes, the lp norms calculated from the time domain signal fail to answer the question of which frequency range the changes are in. Calculating spectral norms is a way to perform both of these tasks simultaneously and effectively.
In the last thirty years there have been many developments in the use of vibration measurement and analysis for monitoring the condition of rotating machinery while in operation. These have been in all three areas of interest, namely fault detection, diagnosis and prognosis. Of these areas, diagnosis and prognosis still require an expert to determine what analyses to perform and to interpret the results. Currently much effort is being put into automating fault diagnosis and prognosis. Major economic benefits come from being able to predict with reasonable certainty how much longer a machine can safely operate (often a matter of several months from when incipient faults are first detected). This article discusses the different requirements for detecting and diagnosing faults, outlining a robust procedure for the former, and then goes on to discuss a large number of signal processing techniques that have been proposed for diagnosing both the type and severity of the faults once detected. Change in the severity can of course be used for prognostic purposes. Most procedures are illustrated using actual signals from case histories. Part 2 of this article will appear in the May 2004 issue of S&V.
The use of enveloping or demodulation techniques in machinery analysis instrumentation allows a user to detect fault signals in rotating element bearings during their operating history. This article describes the technique and provides many examples which compare the results of various measurements on the same bearings.
This chapter provides an overview of history of source separation applications, presents how to apply independent component analysis, and describes a few (blind or semiblind) source separation applications. A “fully blind” approach to source separation would assume strictly no prior knowledge about either source properties or type of mixture. The source separation problem cannot be solved in such conditions. Therefore, the most classical source separation approaches, which are called “blind methods,” are based on generic priors. In order to successfully apply blind source separation (BSS) methods to practical problems, one first has to define the considered relationship between the observations and the sources, which is obtained by modeling the physics of the system when this is possible, and to check if the assumed source properties are realistic. When the number of free unknown parameters in independent component analysis (ICA) is too high, as compared to the available sample size, the ICA model is likely to overfit or overlearn the data. Solutions to the overfitting problem include, in addition to the acquisition of more data, a reduction of the dimensions of the data. Such a process can be performed during whitening, which typically precedes most ICA algorithms.
The aim of this work is to evaluate, in a theoretical sense, the current application of bispectrum on rolling element bearings diagnosis. A mathematical model of the vibration generated by defective rolling element bearings is used and substituted into bispectrum formulas. This work demonstrated that using this statistical tool in order to detect a local fault on rolling element bearings is not effective, contrasting with practical results achieved in other papers. In that sense, some arguments concerning such a contradiction are exposed and proved.
This chapter introduces the reader to the various aspects of feature extraction covered in this book. Section 1 reviews definitions
and notations and proposes a unified view of the feature extraction problem. Section 2 is an overview of the methods and results
presented in the book, emphasizing novel contributions. Section 3 provides the reader with an entry point in the field of
feature extraction by showing small revealing examples and describing simple but effective algorithms. Finally, Section 4
introduces a more theoretical formalism and points to directions of research and open problems.
In this paper we propose a new statistical method for process monitoring that uses independent component analysis (ICA). ICA is a recently developed method in which the goal is to decompose observed data into linear combinations of statistically independent components and . Such a representation has been shown to capture the essential structure of the data in many applications, including signal separation and feature extraction. The basic idea of our approach is to use ICA to extract the essential independent components that drive a process and to combine them with process monitoring techniques. I2, Ie2 and SPE charts are proposed as on-line monitoring charts and contribution plots of these statistical quantities are also considered for fault identification. The proposed monitoring method was applied to fault detection and identification in both a simple multivariate process and the simulation benchmark of the biological wastewater treatment process, which is characterized by a variety of fault sources with non-Gaussian characteristics. The simulation results clearly show the power and advantages of ICA monitoring in comparison to PCA monitoring.
Vibration monitoring of sealess pumps using spike energy
- Le Bleu
- J J Xu
Schwingungsdiagnostische Beurteilung von Maschinen und Anlagen, 3, überarbeitete
- U Klein