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Ridge and Phase Identification in the Frequency Analysis of Transient Signals by Harmonic Wavelets
Abstract
It is difficult to generate high-definition time-frequency maps for rapidly changing transient signals. New details of the theory of harmonic wavelet analysis are described which provide the basis for computational algorithms designed to improve map definition. Features of these algorithms include the use of ridge identification and phase gradient as diagnostic features.
... 23,[34][35][36] Among those researches, the harmonic wavelet (HW) introduced by Newland 27,37 was widely used due to its simple structure and property of non-overlapping Fourier transforms. [38][39][40][41] It is noted that the various members of the HW family, including generalized harmonic wavelet (GHW) and filtered harmonic wavelet (FHW), were introduced to analyze the EPSD of nonstationary process by Spanos. 25 However, most researches have focused on the EPSD analysis of extreme winds. ...
... The FHW is an improved version of the GHW, which enhances the time resolution in the wavelet mean square map with a given frequency resolution by filtering the mother wavelet using a Hanning function in the frequency domain. 38 Specifically, where g (m,n),k is the GHW coefficient and g F (m,n),k is the FHW coefficient. ...
Many long-span bridges are located at typhoon prone regions. With the continuous increase of the bridge span, the typhoon-induced buffeting becomes more and more prominent. In this study, based on the structural health monitoring system installed in the Sutong Bridge, the recorded buffeting responses of the main girder during typhoons Damrey and Haikui were analyzed. The run test method demonstrated that the recorded acceleration responses can be regarded as zero-mean non-stationary random processes. Hence, to capture the energy distribution of the recorded data in the time-frequency domain, the evolutionary power spectral density (EPSD) estimation was conducted using efficient generalized harmonic wavelet (GHW) and filtered harmonic wavelet (FHW), respectively. Compared with the GHW, narrower wavelet bandwidth is required by the FHW to yield a compromise between the time and frequency resolution. For the FHW-based method, the power spectral density amplitudes of the averaging EPSDs are slightly larger for certain major frequency components than those obtained by the Pwelch method. Results show that the non-stationary features of the buffeting of long-span bridges during Typhoon events should be considered. This study can also provide references for non-stationary buffeting analysis of other long-span bridges during extreme wind events.
... where m, n, and k are taken to be positive integer numbers in what follows and However, it is also noted that in all the cited works of Newland (1993;1994a;1994b;1999a and1999b) T o is taken equal to unity. Alternatively, dimensionless time t←t/T o and dimensionless frequency ω←ωT o can be considered . ...
... the thus obtained wavelet coefficients is spaced T o /N seconds from the next. For a total N΄ number of different scales assumed (i.e. for j= 1,…, N΄; see also Fig. 3.8), the above algorithm generates NxN΄ number of harmonic wavelet coefficients: a significantly redundant transformation of the original signal which can be used for representing signals on the time-frequency plane and for structural damage detection purposes (see e.g.Newland 1999a;Newland 1999b;Spanos et al., 2007a).Alternatively, if only the non-zero values of the sequence defined by Equation(3.28) are considered(Figure 2b), an (n j -m j )-point inverse DFT yields a sequence of (n jm j ) wavelet coefficients of scale (m j ,n j ), given by the equation ...
Wavelet-Based Response Spectrum Compatible Synthesis of Accelerograms and Statistical Linearization Based Analysis of the Peak Response of Inelastic Systems
... Going further |V g f (ω, t)| ≈ K k=1 a k (t)|ĝ(ω − φ k (t))|, so that each mode f k is associated with a TF ridge corresponding roughly to the (t, φ k (t)) curve provided |ĝ| attains its maximum at 0. The detection of ridges and their use in mode reconstruction has been pioneered in [13] and subsequently developed in a number of works (e.g. [14] or [15]). ...
... There exist many different ways of computing the ridges associated with the TF representation given by the STFT [14,15]. The ridge detector proposed here is based on the properties of RV. ...
This paper discusses methods for the adaptive reconstruction of the modes of multicomponent AM–FM signals by their time–frequency (TF) representation derived from their short-time Fourier transform (STFT). The STFT of an AM–FM component or mode spreads the information relative to that mode in the TF plane around curves commonly called ridges. An alternative view is to consider a mode as a particular TF domain termed a basin of attraction. Here we discuss two new approaches to mode reconstruction. The first determines the ridge associated with a mode by considering the location where the direction of the reassignment vector sharply changes, the technique used to determine the basin of attraction being directly derived from that used for ridge extraction. A second uses the fact that the STFT of a signal is fully characterized by its zeros (and then the particular distribution of these zeros for Gaussian noise) to deduce an algorithm to compute the mode domains. For both techniques, mode reconstruction is then carried out by simply integrating the information inside these basins of attraction or domains.
... This is not the case for typical wavelet families (e.g., Meyer and Daubechies families) whose frequency content at each scale is implicitly defined by means of a single scalar (i.e., the scaling parameter). An example of four neighbouring scales as part of a basis with constant-width "frequency bins" is shown in Figure 5 Newland (1994) and Newland (1999) for the efficient computation of non-redundant as well as for redundant HWT on the FD. A custom-made implementation of Newland's FFT-based algorithm is used to compute non-constant Q HWT considered in section 5. ...
... Additionally, the considered signals are also processed by means of a harmonic wavelet basis of 128 adjacent non-overlapping "frequency bins" (scales) of constant width equal to 3.91Hz spanning the range of 0-500 Hz on the frequency axis. The non-constant Q HWT analysis is carried out by means of a custom-made code implementing the FFT-based algorithm described by Newland (1994 and1999). ...
A novel numerical study is undertaken to assess the influence of the frequency domain (FD) attributes of wavelet analysis filter banks for vibration-based structural damage detection and localization using the relative wavelet entropy (RWE): a damage-sensitive index derived by wavelet transforming linear response acceleration signals from a healthy/reference and a damaged state of a given structure subject to broadband excitations. Four different judicially defined energy-preserving wavelet analysis filter banks are employed to compute the RWE pertaining to two benchmark structures via algorithms which can efficiently run on wireless sensors for decentralized structural health monitoring. It is shown that filter banks of compactly supported in the FD wavelet bases (e.g., Meyer wavelets and harmonic wavelets) perform significantly better than the commonly used in the literature dyadic Haar discrete wavelet transform filter banks since they achieve enhanced frequency selectivity among scales (i.e., minimum overlapping of the frequency bands corresponding to adjacent scales) and, therefore, reduce energy leakage and facilitate the interpretation of numerical results in terms of scale/frequency dependent contributors to the RWE. Moreover, it is demonstrated that dyadic DWT filter banks with large constant Q values (i.e., ratio of effective frequency over effective bandwidth) are better qualified to capture damage information associated with high frequencies. Finally, it is concluded that wavelet analysis filter banks achieving nonconstant Q analysis are most effective for RWE-based stationary damage detection as they are not limited by the dyadic DWT discretization and can target the structural natural frequencies in cases these are a priori known.
... Still, for stationary damage detection, poor time-localization attributes is of secondary importance. From a computational viewpoint, robust fast Fourier transform (FFT)-based algorithms have been proposed by Newland (1994) and Newland (1999) for the efficient computation of nonredundant as well as for redundant HWT on the FD. A custom-made implementation of Newland's FFT-based algorithm is used to compute non-constant Q HWT considered in section 5. ...
... The effective bandwidth (accounting only for the main lobes of the FAS for the Daubechies wavelets) and the characteristic frequency at which the wavelet FAS is maximized for the first 10 DWT analysis levels of the above three filter banks are reported inTable 2. To facilitate the interpretation of the results presented in the following section, the FAS of the D20, D2, and Meyer wavelets corresponding to the 4 analysis scales indicated by bold fonts in Additionally, the considered signals are also processed by means of a harmonic wavelet basis of 128 adjacent non-overlapping " frequency bins " (scales) of constant width equal to 3.90625Hz spanning the range of 0-500 Hz on the frequency axis. The non-constant Q HWT analysis is carried out by means of a custom-made code implementing the FFT-based algorithm described by Newland (1994 and 1999). Next, the relative wavelet energy in (6) is computed from the wavelet coefficients of the response acceleration signals (healthy and damaged states) at each scale of the 4 different wavelet filter banks. ...
... The continuous wavelet transform has been proved to have a high level of efficiency in accurate information and short processing time. The wavelet transform has dominant advantages in signal filtering and other merits such as time-frequency characteristics, which make it possible to parameter identification [3][4][5]. ...
... 3 x(t) = L aie-(;wo;t sin( wait+ </>i) · i=l (4.1) In order to ident ify "natural" frequency wai of each signal-component from the map in Fig. 2, a numerical algorithm is developed to seek local maxima of t he three-dimensional surface [4]. The positions of the local maxima reveal t he corresponding natural frequencies in frequency axis (see section 3.1). ...
The identification of damping in multi-degree-of-freedom vibration systems is a well-known problem and appears to be of crucial interest. Compared to an estimation of the stiffness and mass, the damping coefficient or, alternatively, damping ratio is the most difficult quantity to determine. In this paper, the continuous wavelet transform based on the Mor let-wavelet function is used to identify the modal damping ratios of multi-degree-of-freedom vibration systems. A new wavelet-based method for the damping identification frommeasured free responses is presented. The proposed method was also tested by experiments on a steel beam.
... Onétudie, dans le cadre de la thèse, trois ondelettes mères complexes représentatives : la première est l'ondelette mère très connue : l'ondelette de Morlet, utilisée par plusieurs auteurs par exemple Torrésani, Carmona et al. [118,25], Staszewski [110]... ; la deuxième est l'ondelette de Cauchy, utilisée par Argoul et al. [12,13] et par plusieurs auteurs dans la mécanique quantique [88] et la dernière est l'ondelette harmonique proposée par Newland [81,82,83] et prise par Tang pour traiter les signaux avec une décroissance exponentielle [115]. ...
... Cette propriété est intéressanteà propos de la séparation des composantes de fréquences très voisines. Newland [83] propose de fenêtrer le spectre de l'ondelette afin d'améliorer la localisation temporelle, toutefois, ce procédé est plus compliqué que la transformation en ondelettes proprement dite. La dernière condition est facilement vérifiée par l'ondelette de Cauchy. ...
The application of the continuous wavelet analysis is proposed for the dynamical monitoring of structures from transient responses under unknown excitations (shock, ambient ...). The numerical tool of the continuous wavelet transform is performed. It is based on the mother wavelet (Morlet or Cauchy) and on the quality factor Q characterizing the mother wavelet. This tool allows extracting instantaneous amplitudes and frequencies within a signal. A domain of the time-frequency plane where the edge effect is negligible, and bounds for Q are determined. The processing of the frequency modulated real signal by the proposed tool facilitates the modal identification of structures (linear and non-linear) and allows an improvement of the impact-echo method. An adapted procedure for every application is detailed. Results obtained from numerical and real tests show the efficiency of the method.
... In this context, the family of generalized harmonic wavelets introduced by Newland [14,15] , features the appealing property of nonoverlapping Fourier transforms, which renders the corresponding harmonic wavelet transform an exceptional tool in cases in which enhanced resolution in the frequency domain is important. To increase the time resolution as well, filtered harmonic wavelets were employed later by the same author (Newland, [16] ). Spanos et al. [13] adopt the filtered harmonic wavelet to model single component accelerograms of four large earthquake events in order to determine the nonlinear response of a high steel building. ...
A numerical procedure, based on an evolutionary optimization algorithm, has been proposed by the authorsfor the simultaneous generation of the three components of the seismic ground acceleration. The methodology allowsthe determination of a train of seismic waves modeled by three different waveforms, for the generation of groundseismic acceleration components. The parameters of each wave, i.e., amplitude, frequency, duration, arrival time anddirection, are determined using an evolutionary optimization algorithm. Although no theoretical justification is knownby the authors for the generation, at the seismic source, of specific initial waveforms, both in case of fracture or ofsliding with friction, waveform acceleration components that satisfy the condition of zero final velocity should inprinciple be preferred. The latter is a physical restriction that is automatically satisfied by anti-symmetrical functions,thus eliminating the need to correct the baseline of simulated accelerograms. The error of fit of simulated accelerogramsgenerated by three different waveforms proposed in the literature was herein determined by comparison with actualseismic records. On that basis, estimations of the expected error of the evolutionary optimization algorithm inengineering applications are presented.
... However, if the frequency axis and the contour parallel to the time axis appear simultaneously and if the shape of the contours is independently circular or square and/or rectangular, it is mostly generated in the direction of the frequency axis, and additional testing may be needed to check the shape of contours again. This phenomenon was also observed by Song and Cho [8] and also in [17]. If the shape of the contours is circular or square/rectangular again, it is confirmed that a debonded state was generated. ...
The impact echo technique is one of the most useful non-destructive test methods for determining the thickness of concrete or detecting possible cracks or cavities in the internal parts of a concrete structure without damaging the surface. Many types of unstable conditions in railway tracks, including various modes of irregularities, may occur when cavities are generated directly under a concrete slab track or when a slight open space is made under a loose sleeper. In this study, we developed a nondestructive testing (NDT) system for detecting abnormalities in concrete tracks and performed 3D numerical simulations using the ABAQUS finite element analysis (FEA) program to investigate the impact echo response from a concrete track slab with different sizes of cavities. Sections of concrete slab were simulated as solid body masses under the railway tracks with gaps in the bodies themselves or with cavities existing between the track concrete layer (TCL) and the hydraulically stabilized base (HSB). We investigated the locations and depths of the cavities and gaps in the model concrete slab using the acoustic impact echo response based on the frequency response of the elastic waves generated in the slab. In addition, a Short-time Fourier Transform (STFT) and a wavelet technique were adopted for a time frequency analysis. Our study demonstrated that the impact echo technique developed in this study by FEA and NDT can measure and confirm the location and depth of cavities in concrete slabs.
... The mathematical basis of wavelet transform is Fourier transform, which is better than Fourier transform for abrupt signal use. However, wavelet transform also has a shortcoming, i.e., the wavelet base needs to be artificially chosen [11][12][13]. Hilbert-Huang transform is a new time-frequency analysis method for non-stationary signals, which takes the transient frequency as the basic quantity and the inherent mode signal as the basic signal. HHT is composed of EMD and Hilbert transform. ...
Large deep foundation pits are usually in a complex environment, so their surface deformation tends to show a stable rising trend with a small range of fluctuation, which brings certain difficulty to the prediction work. Therefore, in this study we proposed a nonlinear autoregressive exogenous (NARX) prediction method based on empirical wavelet transform (EWT) pretreatment is proposed for this feature. Firstly, EWT is used to conduct adaptive decomposition of the measured deformation data and extract the modal signal components with characteristic differences. Secondly, the main components affecting the deformation of the foundation pit are analyzed as a part of the external input. Then, we established a NARX prediction model for different components. Finally, all predicted values are superpositioned to obtain a total value, and the result is compared with the predicted results of the nonlinear autoregressive (NAR) model, empirical mode decomposition-nonlinear autoregressive (EMD-NAR) model, EWT-NAR model, NARX model, EMD-NARX model and EWT-NARX model. The results showed that, compared with the EWT-NAR and EWT-NARX models, the EWT-NARX model reduced the mean square error of KD25 by 91.35%, indicating that the feature of introducing external input makes NARX more suitable for combining with the EWT method. Meanwhile, compared with the EMD-NAR and EWT-NAR models, the introduction of the NARX model reduced the mean square error of KD25 by 78.58% and 95.71%, indicating that EWT had better modal decomposition capability than EMD.
... The amplitude of the reflection is a function of the angle of incidence and is maximum if this angle is 90 (normal incidence). For normal incidence the reflection coefficient, R, is given by the following [7]: ...
Generally, destructive methods such as coring are used for the condition assessment of an existing concrete slab track structure. Although these methods may produce valid data about the corresponding concrete section, they are more expensive and time consuming.
More important than these, destructive methods may damage the structure being investigated and these points usually become focal points for further deterioration. By these reasons, only a few samples can be collected from a structure and these results in a poor representation of the complete structure.
The impact-echo technique is one of the most useful non-destructive test methods that may be used on concrete for thickness determination or for inspection of possible cracks or cavity in the internal parts of a concrete structure without damaging the surface. It has been noticed that reliable results can be obtained quickly (Aktaş, Can Baran,2007).
Many types of unstable conditions of railway tracks including various modes of irregularities can be happened when cavities are generated exactly underneath a concrete slab track or a slight open space is made under a loose sleeper. The cavities can be generated due to various reasons such as settlements of badly compacted soil or poor soil or radical change of track stiffness in transition zones near bridge abutment. The loose sleeper installed in the concrete slab can also produce very undesirable problems including noise, vibration of track and unexpected response between railway car and track structure. However, there are currently no useful and effective detecting methods for investigating the cavity localise underneath the railway concrete slab track and the loose sleeper embedded in the slab. In this study, a comparative study is performed for detecting the cavity and checking the condition of the loose sleeper by adapting conventional impact echo test and acoustic impact echo test on the concrete slab surface. It is confirmed that these two NDT methods can be discriminately adaptive techniques for detecting the unsettled abnormalities in a concrete slab track.
Numerical methods such as finite-difference time-domain (FDTD) and finite element method (FEM) have been exceptionally used to simulate stress wave propagation in solid medium (Chang & Randall, 1988; Ham & Bathe, 2012). These methods have been successfully applied to study the impact-echo response of concrete plates (Sansalone, Carino, & Hsu, 1987; Abramo, 2011).
In this study impact of cavity underneath a concrete track slab was determined by new automatic measurement system configuration which can detected where there are abnormalities under concrete track slab, several model concrete slabs were constructed in a backyard test area and at DongYang University. Numerical simulations using finite element method (FEM) built using ABAQUS 3Dhave also been used to study the impact-echo response for detection of underneath cavity a concrete track slab by considering different size of cavities. The slabs were simulated as railway track slab with gaps in the body itself or cavities between the concrete slab(TCL) and (HSB) layered on soil foundation. The locations and depths of the cavities and gaps in the model concrete slab were investigated by the acoustic impact echo response based on the frequency response of the elastic waves generated in the slab. In this study, a Short-time Fourier Transform (STFT) technique was adapted for time frequency analysis. In-detail procedures about performing STFT to get wavelet forms and Auto Spectrum Density were explained by (Cho et al. 2017)
Keywords: Impact echo Test,Finite element analysis,FFT,Wavelet.
... Thus, they can analyze any signal in the frequency domain by selecting a desired band of frequency to represent in the time-frequency scale. They have been used in fault detection, transient detection, and power quality disturbances in vibration signal analysis [11], [16], [17]. Jordan et al. [6] presented the initial results of intermittent clutter removal from RWP data using classical HWT. ...
Boundary layer radar (L-band) wind profilers frequently encounter a significant problem arising from the contamination of intermittent clutter, produced by seasonal and nocturnal migrating birds, which often yields erroneous wind velocity and boundary layer information. Classical harmonic wavelet transforms (HWTs) are inadequate in removing the transient clutter contamination under certain conditions, particularly when the clutter is significant. We implemented an adaptive complex harmonic discrete wavelet transform with an advanced statistical method to overcome the shortcomings of the classical wavelet method. This algorithm effectively eliminates the bird contamination even where the classical method fails. Finally, a multiple peak-picking (MPP) algorithm was added to select true atmospheric signals and estimate accurate moments. The obtained wind velocity measurements were compared with those derived using the conventional method and validated with global positioning system radiosonde winds. The comparison shows that the proposed method is more effective than the conventional one.
... Substantial methods were proposed to circumvent this problem, among which the ones based on using the properties of the reassignment vector (RV) to estimate TF signatures of MCSs have recently gained a renewed interest [57,59,58]. In contrast to the conventional methods which work directly with the ridge detection for AM-FM modes [100,83,101], the RV-based ones consider these modes as particular TF regions, called basins of attractions (BAs) subsequently used for mode reconstruction. However, the latter still fail to assess the TF signatures associated with a noisy Dirac impulse. ...
Many physical signals including audio (music, speech), medical data (electrocardiogram (ECG), phonocardiogram (PCG)), marine mammals or gravitational-waves can be accurately modeled as a superposition of amplitude and frequency-modulated waves (AM-FM modes), called multicomponent signals (MCSs). Time-frequency (TF) analysis plays a central role in characterizing such signals and in that framework, numerous methods have been proposed over the last decade. However, these methods suffer from an intrinsic limitation known as the uncertainty principle. In this regard, reassignment method (RM) was developed with the purpose of sharpening TF representations (TFRs) given respectively by the short-time Fourier transform (STFT) or the continuous wavelet transform (CWT). Unfortunately, it did not allow for mode reconstruction, in opposition to its recent variant known as synchrosqueezing transforms (SST). Nevertheless, many critical problems associated with the latter still remain to be addressed such as the weak frequency modulation condition, the mode retrieval of an MCS from its downsampled STFT or the TF signature estimation of irregular and discontinuous signals. This dissertation mainly deals with such problems in order to provide more powerful and accurate invertible TF methods for analyzing MCSs. This dissertation gives six valuable contributions. The first one introduces a second-order extension of wavelet-based SST along with a discussion on its theoretical analysis and practical implementation. The second one puts forward a generalization of existing STFT-based synchrosqueezing techniques known as the high-order STFT-based SST (FSSTn) that enables to better handle a wide range of MCSs. The third one proposes a new technique established on the second-order STFT-based SST (FSST2) and demodulation procedure, called demodulation-FSST2-based technique (DSST2), enabling a better performance of mode reconstruction. The fourth contribution is that of a novel approach allowing for the retrieval of modes of an MCS from its downsampled STFT. The fifth one presents an improved method developed in the reassignment framework, called adaptive contour representation computation (ACRC), for an efficient estimation of TF signatures of a larger class of MCSs. The last contribution is that of a joint analysis of ACRC with non-negative matrix factorization (NMF) to enable an effective denoising of phonocardiogram (PCG) signals.
... The harmonic wavelet, constructed by Cambridge University Newland in 1993, is a complex wavelet with clear expression that has strict box-shaped spectrum and good filtering properties [4,5]. It is very sensitive to the changes of the amplitude of vibration signal, from which the weak signals can be extracted [6]. Our preliminary work [7] shows that the HWPT could achieve the extraction of frictional vibration signals. ...
Running-in wear experiments were conducted on a spherical-on-disk tester. The vibration signals collected in the experiments were detected by a combination of harmonic wavelet packet transform (HWPT) and cross-correlation analysis (CCA) methods. Experimental results show that the friction vibration signals detected in tangential and normal directions have the characteristics of no time delay and strong correlation. Their root-mean-square (RMS) values gradually reduce and enter a steady-state of fluctuation with the experiments time, which are consistent with the variation of friction coefficient and reflect the change of wear states from the running-in wear to the stable wear. Therefore, the detection of friction vibration can be realized by a combination of HWPT and CCA methods.
... (4) requires the calculation of the wavelet coefficient by means of Eq. (3) based on an available ensemble of process realizations. From a practical point of view it is worth noting that the GHWT can be numerically determined by utilizing the Fast Fourier Transform (FFT), which offers significant computational advantages [34]. ...
A methodology is proposed for efficient processing of sea wave field data via compressive sensing (CS), and joint time-frequency analysis via harmonic wavelets (HWs) based evolutionary power spectrum (EPS) estimation. In this regard, it is possible to record and store relatively long wave data sequences, whereas the commonly adopted in-practice assumption of stationary data is abandoned. Currently, most wave records are measured by buoys, which acquire data for a time interval representative of stationary time series. Next, following a Fourier transform processing, only few spectral parameters are stored. Thus, detailed information about localized-in-time phenomena are completely lost. Herein, it is shown that CS can be adopted for efficiently compressing and reconstructing wave data, while retaining localized information. For this purpose, CS is used in conjunction with a HW basis for processing long time series. In particular, storage capacity demands are drastically decreased as only the HW coefficients need to be saved. These are determined from a randomly-sampled record by invoking a L1/2 norm minimization procedure. The resulting reconstructed record, being longer than conventional wave time series, can no longer be regarded as stationary; thus, a HW based EPS estimate is employed for describing the joint time-frequency features of the record. Finally, the reliability of the methodology is assessed by analyzing wave field data measured at the Natural Ocean Engineering Laboratory (NOEL) of Reggio Calabria. Specifically, comparisons between original and reconstructed records demonstrate a satisfactory agreement regarding the time-histories, and the estimated EPS and relevant statistical quantities, even for up to 60% missing/removed data.
... on RM [12], [13]. The type of modes sought conditions of the technique for TF signatures estimation: some approaches concentrate on ridge detection for AM/FM modes [14], [15], while some others, using the properties of the reassignment vector (RV), can handle a wider class of TF signatures, the constraints on the modes being less stringent [16] [17]- [19]. In this latter case, the estimated TF signatures are then used to define basins of attraction (BAs) for the modes enabling their reconstruction. ...
This letter addresses the problem of the detection and estimation of the modes of a multicomponent signal using the reassignment framework. More precisely, we propose a new algorithm to estimate the ridges representing the time-frequency (TF) signatures of the modes based on the local orientation of the reassignment vector (RV), and use them to define the so-called “basins of attraction” enabling modes' retrieval. Compared with previous approaches, this new technique not only enable reconstruction of AM/FM modes but also Dirac impulses, which is of great interest in many practical situations. Numerical experiments conducted with both synthetic and real data illustrate the effectiveness of the technique.
... The delay rate in time domain is influenced by discontinuity point in frequency domain, therefbre smoothing the discontinuity points in frequency domain can reform time clomain characteristic of HW. There are several means to construct window function[6] ...
In this paper two kind of rotating stall of blade row is tested on a 4-73No8D centrifugal fan in lab. In order to extract feature of rotating stall of 4-73 fan series, orthogonal harmonic wavelet and the reformed non-orthogonal HW, the delay rate of which in time domain is improved by author, are introduced to analyze time-frequency characteristic of different stall. The time-frequency feature of two kind of stall is obtained and the effectiveness and advantage of harmonic wavelet applying to analyze the unstable flow of centrifugal fans is proved.
... However, should finer frequency or time resolutions over specific frequency bands be required, the basis matrix can be altered accordingly. The harmonic wavelet basis components may be generated efficiently via the Inverse Fast Fourier Transform (IFFT) as shown in [38]. However, a single harmonic wavelet must be shifted n À m times in the time domain to form an orthogonal basis. ...
... Fast Fourier Transform (FFT) did not show good result when it was applied to that type of complex vibrations. Nonadaptive time-frequency methods do not provide with a meaningful interpretation either [50][51][52]. Consequently, a novel methodology for automated fault diagnosis in rotating machinery is needed to avoid loss of time in planning and carrying out unnecessary operation and maintenance (O&M) tasks, reduce machinery downtime, increase reliability, and reduce the cost of energy (COE). ...
Reliable monitoring for the early fault diagnosis of gearbox faults is of great concern for the wind industry. This paper presents a novel approach for health condition monitoring (CM) and fault diagnosis in wind turbine gearboxes using vibration analysis. This methodology is based on a machine learning algorithm that generates a baseline for the identification of deviations from the normal operation conditions of the turbine and the intrinsic characteristic-scale decomposition (ICD) method for fault type recognition. Outliers picked up during the baseline stage are decomposed by the ICD method to obtain the product components which reveal the fault information. The new methodology proposed for gear and bearing defect identification was validated by laboratory and field trials, comparing well with the methods reviewed in the literature.
... Spanos and Kougioumtzoglou, 2012;Spanos and Failla, 2005) ( ) 2 ( , ), ( , ) [ ] ( , ) , (25) is required to compute the wavelet transform. This is realised by taking the inverse discrete Fourier transform of equation (24), having the effect of wrapping the tails of the wavelet outside the recorded time history back in on themselves (Newland, 1994(Newland, , 1999, yielding a periodic function over T 0 , i.e. Note that near the beginning and the end of the time history the discrete wavelet transform introduces edge effects, i.e. the wavelet power is likely to 'leak' around to the other side. ...
The issue of quantifying the uncertainty in stochastic process power spectrum estimates based on realisations with missing data is addressed. In this regard, relying on relatively relaxed assumptions for the missing data, utilizing fundamental concepts from probability theory, and resorting to Fourier and harmonic wavelets based representations of stationary and non-stationary stochastic processes, respectively, a closed-form expression is derived for the probability density function (PDF) of the power spectrum value corresponding to a specific frequency. The significance of the derived PDF relates to cases where incomplete process realisations are available for power spectrum estimation applications. In this setting, standard power spectrum estimation
techniques subject to missing data typically provide with a deterministic estimate for the power spectrum. Thus, no information is provided concerning the uncertainty in the estimates. Numerical examples herein demonstrate the large extent to which any given single estimate may be unrepresentative of the target spectrum.
... Then, the time-domain expression of the harmonic wavelet is obtained as: (1) The frequency spectrum of harmonic wavelet is shown in Fig. 3. It can be seen that the frequency spectrum of harmonic wavelet has the characteristics of the box and excellent compact support features [2]. ...
The working state of rolling bearing has an important influence on the operation of trains, directly related to the safety of
train passengers. Therefore, it has great significance to conduct train bearing fault diagnosis. In this paper, based on
harmonic wavelet envelope, a method for the fault diagnosis of railway bearings is proposed. First of all, the harmonic
wavelet packet was used to translate vibration signal into time-scale representation. Then the decomposed signal was
demodulated. Finally, through the analysis of the envelope spectrum, the bearing fault feature frequency was extracted. In
order to verify the validity of diagnosis method, outer race fault bearing and ball fault bearing were tested. The test
results show that the diagnosis method is effective and practical.
... Regarding the determination of the wavelet coefficients, a computationally efficient algorithm, which takes advantage of the fast Fourier transform (FFT) scheme, is employed (e.g. [44]). ...
A harmonic wavelets based approximate analytical technique for determining the response evolutionary power spectrum of linear and non-linear (time-variant) oscillators endowed with fractional derivative elements is developed. Specifically, time- and frequency-dependent harmonic wavelets based frequency response functions are defined based on the localization properties of harmonic wavelets. This leads to a closed form harmonic wavelets based excitation-response relationship which can be viewed as a natural generalization of the celebrated Wiener–Khinchin spectral relationship of the linear stationary random vibration theory to account for fully non-stationary in time and frequency stochastic processes. Further, relying on the orthogonality properties of harmonic wavelets an extension via statistical linearization of the excitation-response relationship for the case of non-linear systems is developed. This involves the novel concept of determining optimal equivalent linear elements which are both time- and frequency-dependent. Several linear and non-linear oscillators with fractional derivative elements are studied as numerical examples. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the technique.
... Not only is the Morlet wavelet used with the ridgeskeleton method, but Newland (1999) showed a ridgeskeleton extraction scheme with harmonic wavelets. Argoul and Erlicher (2005) used the Cauchy wavelet with the ridge-skeleton method and applied it in the frequency domain. ...
This survey presents the broad range of research on using wavelets in the analysis and design of dynamic systems. Though wavelets have been used with all types of systems, the major focus of this survey is mechanical and electromechanical systems in addition to their controls. However, the techniques presented can be applied to any category of dynamic systems such as economic, biological, and social systems. Wavelets can be classified into three different types: orthogonal, biorthogonal, and pseudo, all of which are employed in dynamic systems engineering. Wavelets-based methods for dynamic systems applications can be divided into vibrations analysis and systems and control analysis. Wavelets applications in vibrations extend to oscillatory response solutions and vibrations-based systems identification. Also, their applications in systems and control extend to time–frequency representation and modeling, nonlinear systems linearization and model reduction, and control design and control law computation. There are serious efforts within systems and control theory to establish time–frequency and wavelets-based Frequency Response Functions (FRFs) parallel to the Fourier-based FRFs, which will pave the road for time-varying FRFs. Moreover, the natural similarity of wavelets to the representation of neural networks allows them to slip into neural-networks-based and fuzzy-neural-networks-based controllers. Additionally, wavelets have been considered for applications in feedforward and feedback control loops for computation, analysis, and synthesis of control laws
... In 1997, Liu and Ling (1997) introduced ball bearing fault diagnosis using a WPT method; the wavelet packet coefficients were used to classify bearing faults. In 1999, Newland (1999) introduced the WPT in the engineering field, with several calculation methods and examples of his application in vibratory signal analysis. In 2002, Zheng et al. (2002) published a gear fault diagnosis method based on a continuous wavelet transform and proposed a new concept of time-averaged wavelet spectrum for reducing the enormous operand. ...
The gearbox is an important component in industrial drives, providing safe and reliable operation for industrial production. Wavelet packet transform (WPT) analysis was used to extract fault features in the vibration signals generated by a gearbox. The extracted features from the WPT were used as input in a rough set (RS) for attribute reduction and then combined with a genetic algorithm to obtain global optimal attribute reduction results. The fault features gained after the attribute reductions were used to generate decision rules. The unknown gear status signal attributes were used as input to match the generated decision rules for fault diagnosis purposes. Gearbox vibration signals contain a significant amount of gear status information; a WPT has an acute portion-locked ability to extract attribute information from the vibration signals. However, WPT frequency aliasing would lead to the generation of spurious frequency components, affecting gear fault diagnosis. In this paper, we introduce an improved WPT to eliminate frequency aliasing, thus improving the accuracy of fault diagnosis. This paper studies the use of wavelet packet for feature extraction and the RS for classification; the results demonstrate that this method can accurately and reliably detect failure modes in a gearbox.
... In 1994, Newland researches on their properties and applications, and coins the term harmonic wavelet. Harmonic wavelets are used for ridge and phase identification in signals [54]. The results showed that the cracks found reduced the rotor speed. ...
The wavelets ψ s,τ (t) generated from the same mother wavelet function ψ(t) have different scale s and location τ, but the same shape. Scale factors are always s>0. The wavelets are dilated when the scale s>1 and contracted when s<1. Thus, the changing of the value s can cover different ranges of frequencies. Large values for the parameter s correspond to lower frequencies ranges or a large scale for ψ s,τ (t). Small values of s correspond to lower frequencies ranges or very small scales.
Functions in the time domain can be represented as a linear combination of all frequency components present in a signal, where the coefficients are the amount of energy provided by each frequency component to the original signal. The main decomposition is associated with a 4 (main or mother wavelet) that usually has the highest energy, though it is not always necessarily the case. It has a similar pattern to the original signal. The first (d 4 ), second (d 3 ), third (d 2 ) and fourth (d 1 ) transformed signals have decreasing energy rates, being s the original signal. Usually a 4 is the low frequency component of the original signal while d i is the high frequency component, having d 1 the biggest value.
It has been observed that for the outer ends of the engine and the generator, the appearance of a pronounced peak amplitude at the natural frequency or 2X (vibration) was associated to the maximum energy values for the main signal, the most suitable with the original, and minimum values for decomposed signals d 1 and d 2 (sound). In contrast, the results obtained close to the coupling did not follow a clear trend as the results were conditioned by the type of experiment. The numerical values of each peak were also taken into account in the establishment of the pattern recognitions, being different for each experiment. The same conclusion was reached for the energy values. Different models and results were expected because the objective was not to find similar patterns between different experiments, and the tests were never performed under identical conditions. The objective was to have different vibration patterns and their associated sound models in order to create a catalogue of possible scenarios for predictive maintenance in the mechanisms. Thus, it is possible to extend the range of possibilities to relate the result of an acoustic signal with the frequency domain using the Fast Fourier Transform.
... For instance, the well-known Shannon wavelets correspond to the imaginary parts of Equation (48), for m,n=1,2; 2,4; 4,8;…. Harmonic wavelets are used in many mechanics applications such as acoustics, vibration monitoring, and damage detection (Newland, 1994b(Newland, , 1994c(Newland, , 1999Butler, 1998, 1999). ...
... Regarding the estimation of nonstationary stochastic process power spectra, a time/frequency localized wavelet basis is used to model the process based on a framework developed in [18]. In this case, the chosen basis is comprised of generalized harmonic wavelets [19,20], (2) which have a box-shaped frequency spectrum making them ideal for representing non-stationary harmonic processes. ...
A compressive sensing (CS) based approach is developed in conjunction with an adaptive basis reweighting procedure for stochastic process power spectrum estimation. In particular, the problem of sampling gaps in stochastic process records, occurring for reasons such as sensor failures, data corruption, and bandwidth limitations, is addressed. Specifically, due to the fact that many stochastic process records such as wind, sea wave and earthquake excitations can be represented with relative sparsity in the frequency domain, a CS framework can be applied for power spectrum estimation. To this aim, an ensemble of stochastic process realizations is often assumed to be available. Relying on this attribute an adaptive data mining procedure is introduced to modify harmonic basis coefficients, vastly improving on standard CS reconstructions. The procedure is shown to perform well with stationary and non-stationary processes even with up to 75% missing data. Several numerical examples demonstrate the effectiveness of the approach when applied to noisy, gappy signals.
... Dans l'équipe Dynamiqueà Navier, trois ondelettes mères complexes ontété choisies pour le traitement des signaux modulés en temps et en fréquence. La première est l'ondelette mère de Morlet [1], la deuxième est l'ondelette de Cauchy [3] et la troisième est l'ondelette harmonique proposée par Newland [65]. ...
Modification of modal parameters is considered the main tool for the evaluation of characteristic changes of a non stationary dynamic system. Therefore, our first interest is to obtain these modal parameters from vibration measures using identification methods. Three methods are discussed here: Proper Orthogonal Decomposition (POD), Singular Value Decomposition (SVD) and Smooth Orthogonal Decomposition (SOD). Then, in order to evaluate the mass changes in non stationary systems, three steps are proposed: instant localization of mass changes (step 1), determination of geometrical location of the mass changes (step 2) and quantification of mass changes (step 3). The Wavelet transform (WT), considered to be a time-frequency analysis, is indented in step 1. In step 2, three methods for the detection of the position of the mass changes are developed. Finally, the relative variation of the natural frequencies of the system is used to evaluate the relative variation of the mass in step 3. The efficiency of these methods is verified by numerical tests. Moreover a building experimental model, instrumented with accelerometers, is studied in the case of after-shock vibrations. These experimental tests permit to validate the methods proposed in this thesis
... In addition, the ridge and skeleton of wavelet transform have shown excellent capability in tracing the instantaneous characteristics of the system dynamic responses such as the timedependent amplitude and the amplitude-dependent frequency of the free response of both the linear and nonlinear systems. This distinct capability allows the WT method to play important roles in the parameter identifications for both linear and nonlinear systems [20][21][22][23][24][25][26][27][28][29][30][31][32][33]. A comprehensive review about the application of the WT method in mechanical signal analysis can be found in Ref. [34]. ...
The present work carries out a comprehensive investigation into the border distortion deficiency for the conventional scalogram and the reassigned scalogram. The reasons for this deficiency are analyzed, and a simple way is suggested to determine the border distortion ranges. New methods are proposed to reduce the border distortions in both the scalograms. The practical meanings of the border distortion improvement method are demonstrated by applying both the scalograms with and without border distortion improvements to estimate the modal parameters for a 2-DOF linear system. The estimation results indicate that, in the presence of noise, which is inevitable in practice, for the mode with weak amplitude, the reassigned scalogram can perform better than the conventional scalogram in estimating the modal parameters. In addition, for the mode with short effective duration, its frequency components at the border distortion ranges are of great importance for the modal parameter estimation purpose, and the border distortion improvement method can greatly increase the estimation accuracy.
... For instance, the well-known Shannon wavelets correspond to the imaginary parts of Equation (48), for m,n=1,2; 2,4; 4,8;…. Harmonic wavelets are used in many mechanics applications such as acoustics, vibration monitoring, and damage detection (Newland, 1994b(Newland, , 1994c(Newland, , 1999Butler, 1998, 1999). ...
In stochastic dynamics, ensuring the structural reliability of buildings and structures is of paramount importance, especially when subjected to environmental loads such as wind or earthquakes. To adequately address these loads and the uncertainties associated with them, it is often necessary to utilise advanced load models, frequently expressed using a power spectral density (PSD) function. The construction of these load models becomes challenging when only limited data is available and meaningful statistics cannot be reliably derived. To address this issue, safety bounds are commonly used in load models to account for uncertainties. Many PSD functions, such as the Clough-Penzien model, are described by parameters with a physical background and can therefore reflect the real case. The aim of this work is to expand these physical parameters in order to account for uncertainties. For this purpose, bootstrapping is used to derive more reliable statistics. By introducing a scaling parameter that allows for flexibility, bounds of the data set can be derived. Consequently, suitable PSD models are fitted to the derived bounds. The PSD function is thus represented by intervals for its physical properties instead of relying on discrete values. When applying such a bounded load model to a structure, advanced interval propagation schemes can be utilised to bound the failure probability.
n order to study the failure law of rockburst phenomenon which happened frequently at rock mass with joints in deep fault zone of XinJiang water delivery tunnel. The field prediction and prevention of rockburst are guided, so that the construction process can be carried out safely and efficiently. The rockburst test of intact rock specimens and jointed rock samples was carried out by the modified true-triaxial apparatus (MTTA). Firstly, the high-speed photography and the acoustic emission (AE) system were applied to record the experimental process. The crack propagation and failure process were recorded and the acoustic emission signals of this process were analyzed. At the same time, the acoustic emission signals during crack propagation and failure are analyzed by wavelet analysis and fast Fourier transform (FFT). Then, the numerical simulation software named PFC was adopted to simulate the whole process of crack propagation in the rockburst of two kinds of specimens. And the simulation result is consistent with the experimental result. Some conclusions were drawn as follows: for calcareous sandstone with uniaxial compressive strength of 135.7 MPa, the rockburst of complete rock sample occurred at 129.6 MPa, and the joint rockburst occurred at 82.42 MPa. The more sudden time of joint rockburst was less than 0.1 s. During experiments, the main frequency band of complete sample is 62.5–250 MHz, while the main frequency band of jointed rockburst is lower and more concentrated at 62.5–125 MHz. The normalized energy proportion at the time of rockburst is focused at frequency part (62.5–250 MHz), and both types of rockburst energy exceed 80%, this signal can be used to judge the stage of stress adjustment in rocks. Therefore, it is considered that the existence of joint plane reduces the critical strength of rockburst and makes rockburst occur in a lower stress state. The distribution of the main frequency band can judge the extent of rock damage and help to predict rockburst disaster.
The purpose of this paper is to study the capabilities of the impulse response method in length and flaw detecting for concrete piles and provide a suggested method to find small-sizeflaws in piles. In this work, wavelet transform is used to decompose the recorded time domain signal into a series of levels. These levels are narrowband, so the mix of different dominant bandwidths can be avoided. In this study, the impulse response method is used to analyze the signal obtained from the wavelet transform to improve the judgment of the flaw signal so as to detect the flaw location. This study provides a new wayof thinking in non-destructive testing detection. The results show that the length of a pile is easy to be detected in the traditional reflection time or frequency domain method. However, the small flaws within pile are difficult to be found using these methods. The proposed approach in this paper is able to greatly improve the results of small-size flaw detection within pilesby reducing the effects of anynoise and clarifyingthe signal in the frequency domains.
The subject of this study is the vertical mass-spring-like oscillation of a pendant droplet and its resonant detachment, which was experimentally observed in the process of laser droplet generation from a metal wire. The process was characterized by various time series, which were generated from a sequence of infrared intensity images of the process. Following a visual inspection of pendant droplet images and an analysis of a wavelet based time-frequency map of the droplet’s vertical displacement time series, the pendant droplet’s oscillation is described by a time-variable mass-spring system. Based on the characteristics of the time-frequency map, the resonant nature of the pendant droplet detachment was demonstrated. Additionally, an algebraic expression was formulated, which can be used to predict the detached droplet’s diameter as a function of the laser pulse frequency.
Six wind turbines were blown to the ground by the wind gust during the attack of Typhoon Soudelor in August 2015. Survey using unmanned aerial vehicle, UAV, found the collapsed wind turbines had been broken at the lower section of the supporting towers. The dynamic behavior of wind turbine systems is thus in need of attention. The vibration of rotor blades and supporting towers of two wind turbine systems have been measured remotely using IBIS, a microwave interferometer. However the frequency of the rotor blade can be analyzed only if the microwave measurements are taken as the wind turbine is parked and secured. Time-frequency analyses such as continuous wavelet transform and reassigned spectrograms are applied to the displacement signals obtained. A frequency of 0.44Hz exists in both turbines B and C at various operating conditions. Possible links between dynamic characteristics and structural integrity of wind turbine –tower systems is discussed.
In order to overcome the defect that the frequency domain was divided by the dyadic mode in the traditional harmonic wavelet packet, an improved harmonic wavelet packet that could divide the frequency domain more finely was proposed. In order to overcome the defect that the parameters of band-pass filter were selected by the experience of the user in the conventional envelope analysis and enhance the performance of kurtogram, by combining the improved harmonic wavelet packet with kurtogram, the improved harmonic wavelet packet kurtogram was given, based on which a method to diagnose the mechanical fault was presented. The simulation and fault signal analysis show the method overcomes the defect in the conventional envelope analysis and enhances the performance of kurtogram.
In order to investigate the vibration characteristics of the reciprocating compressors for the offshore platform, the vibration from the reciprocating compressor is analyzed by applying the harmonic wavelet packet transform. The energy distribution characteristics of vibration signal under the different frequencies are discussed. Results show that the vibration energy of the reciprocating compressor is mainly concentrated in the low frequencies of 25 Hz and 50 Hz in the horizontal and vertical directions, and the vibration energy in other frequencies is small and smooth. In the axial direction, the vibration energy of the reciprocating compressors extends to the medium-high frequency, and the large energy appears in the 225 Hz. Therefore, the harmonic wavelet packet transform can be used to research the vibration characteristics of the reciprocating compressor.
Vibration signal processing method has been an active research topic all the time, and the equipment vibration monitoring and fault diagnosis are crucial. Though the vibration signal processing methods developed fast in recent years, they still need to be improved and optimized. Some typical approaches referring to recent literatures are classified and summarized in this paper. The developments, features and applications are presented and discussed for amplitude domain analysis, Fourier transform, correlation analysis in traditional methods, and Wigner-Ville distribution, spectral analysis, wavelet analysis, blind source separation, Hilbert-Huang transform, higher order statistics analysis in modern methods. Finally, we make a conclusion for this paper and an overview is made to guide the future development in this field.
In this paper, a characteristic signal detection method based on the parameter-tuned resonance of a second-order linear system is proposed. The second-order linear system is taken as a signal detection model, and its resonance characteristics are used to obtain the characteristic curve for describing the change law between the maximal value of system response and the natural frequency. According to the maximum value of the curve, we can identify the characteristic signals from the noise background. Simulation analysis and the experiment of rotor fault diagnosis indicate that the principle of the presented method is simple, and that it may provide a practical method for signal detection and processing.
The signal characteristics of the coherent heterodyne Brillouin optical time-domain reflectometry (BOTDR), as well as the impacts of the signal envelope on the spatial resolution and Brillouin frequency shift (BFS) measurement accuracy of the BOTDR are analyzed, respectively. A general algorithm model for the digital envelope detection of an amplitude-modulated (AM) signal and the features of an ideal digital envelope detection algorithm are summarized, respectively. According to the zero phase shift ideal bandpass filtering features and passband designing flexibility of the generalized harmonic wavelet (GHW), a digital BOTDR envelope detection technique based on the generalized harmonic wavelet transform (GHWT) is proposed, and the envelope demodulation scheme is designed. The parameters optimization and experiments are made. The results show that, compared with the existing techniques, an undistorted signal envelope with a higher signal-to-noise ratio can be obtained by the proposed method, resulting in an undegraded spatial resolution and a higher BFS measurement accuracy for the BOTDR system. The experimental results are also analyzed according to the frequency domain characteristics of the algorithms.
Existing system identification methods are mostly dealing with local changes that are time-invariant. However, the process of local change in a real structure caused by damage is mostly time-varying, which can be modeled with a change in the physical parameters such as structural stiffness and damping. An identification algorithm for a time-varying beam system is proposed based on wavelet analysis. The response signal is firstly decomposed using the Daubechies wavelet scaling function. The governing differential equation of a structure is then transformed into a set of linear equations based on the orthogonality property of the scaling functions in a wavelet space. Finally, the proposed algorithm is illustrated with studies on a cantilever beam structure. The precision of identification with respect to different wavelet scales is investigated and discussed. Numerical results demonstrate that the proposed method can identify smoothly, periodically and abruptly time-varying physical parameters with excellent accuracy.
By introducing modified coefficients of spectral magnitudes, an improved harmonic wavelet packet transformation was proposed. With the improved harmonic wavelet packet transformation, the spectral magnitudes of subbands of a signal kept from varying after signals decomposition. With, the improved harmonic wavelet packet transformation, the amplitude and frequency of fault modulation signals could be extracted accurately and it became easy to diagnose mechanial faults. The simulation and test results for bearing fault diagnosis showed that the harmonic wavelet packet possesses a perfect band-pass filtering feature and an excellent ability to detect weak signals, the improved harmonic wavelet packet transformation really has good capabilities mentioned above and is valuable for engineering application.
Gear as an important transmission component in the production of modern industrial, used in all areas of production and life, its stable and reliable work has great social significance. In this paper, gear fault diagnosis based on wavelet packet for fault feature extraction has been proposed for gear fault detection and diagnosis. First, this paper analyzes the variations of gear fault vibration signal, using time-domain and frequency-domain sign attributes to characterize these gear vibration signal and then extract fault sign attributes by using wavelet packet. This paper introduces a kind of new method for wavelet denoise, eliminating the problem of wavelet de-noise decompose level and de-noise threshold value selection, at the same time analysis a kind of wavelet packet transform method, eliminating the frequency and frequency band confusion, reducing the error in fault sign attribute extraction. At last, using fault simulation platform to simulate different conditions and different gear fault vibration signals. The results demonstrate that this method can accurately and reliably detect failure modes in a gearbox.
A new approach of gear fault diagnosis based on Gaborlet Transform A tlas is presented. Gaborlet Transform A tlas is extended from wavelet transform. It is a linear transformation of time-frequency scale three-dimensional space and a combination of wavelet transform and Gabor transform. It has powerful analysis capabilities for the non-stationary signals. The signal spectral estimation based on this method has the wavelet transform advantage of a high frequency resolution. Besides, it is not limited to the width of signal frequency range. The needed scale parameter can be chosen freely and the spectral estimated value is accurate and efficient. The experimental results indicate that the proposed method can get better results than the classic local power spectrum estimation in gear fault diagnosis. It highlights the gear sideband structure. So, it is suitable for local fault diagnosis. 1553-9105/
The discrete wavelet transform (DWT) has attracted a rising interest in recent years to monitor the condition of rotating electrical machines in transient regime, because it can reveal the time–frequency behavior of the current’s components associated to fault conditions. Nevertheless, the implementation of the wavelet transform (WT), especially on embedded or low-power devices, faces practical problems, such as the election of the mother wavelet, the tuning of its parameters, the coordination between the sampling frequency and the levels of the transform, and the construction of the bank of wavelet filters, with highly different bandwidths that constitute the core of the DWT. In this paper, a diagnostic system using the harmonic WT is proposed, which can alleviate these practical problems because it is built using a single fast Fourier transform of one phase’s current. The harmonic wavelet was conceived to perform musical analysis, hence its name, and it has spread into many fields, but, to the best of the authors’ knowledge, it has not been applied before to perform fault diagnosis of rotating electrical machines in transient regime using the stator current. The simplicity and performance of the proposed approach are assessed by comparison with other types of WTs, and it has been validated with the experimental diagnosis of a 3.15-MW induction motor with broken bars.
Monitoring drill wear is a major topic in automated manufacturing operations. This paper presents an effective drill wear feature identification scheme based on robust clustering techniques. Three types of drill wear (namely; chisel wear, chipped edge and flank wear) are artificially induced on the drill point. The drill cutting edge wear related features are extracted experimentally by processing the force signals from a three-axis piezoelectric load cell in both the time and frequency domains. Techniques based on the short time Fourier transform (STFT), wavelet transform (WT) and statistical parameters are utilized for feature extraction. The sensitivity of the proposed method is tested under different cutting feed and speed conditions. The computational study is conducted using the features extracted from three dimensional vibration and cutting force signals. The type of drill wear and related variations in the cutting forces are identified using robust clustering methods. The objective is to isolate regions in the feature space, each region corresponding to one of the drill wear types. Results show that power spectral density data clusters better than data obtained using wavelet coefficients. The clustering results can be used to design classifiers for real time monitoring of wear conditions while drilling.
An efficient method of transforming a discrete Fourier transform (DFT) into a constant Q transform, where Q is the ratio of center frequency to bandwidth, has been devised. This method involves the calculation of kernels that are then applied to each subsequent DFT. Only a few multiples are involved in the calculation of each component of the constant Q transform, so this transformation adds a small amount to the computation. In effect, this method makes it possible to take full advantage of the computational efficiency of the fast Fourier transform (FFT). Graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin.
A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the `empirical mode decomposition' method with which any complicated data set can be decomposed into a finite and often small number of 'intrinsic mode functions' that admit well-behaved Hilbert transforms. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and non-stationary processes. With the Hilbert transform, the 'instrinic mode functions' yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert spectrum. In this method, the main conceptual innovations are the introduction of `intrinsic mode functions' based on local properties of the signal, which make the instantaneous frequency meaningful; and th
The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24-oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventional DFT that yields a constant spacing between frequency components. In addition to advantages for resolution, representation with a constant pattern has the advantage that note identification ("note identification" rather than the term "pitch tracking," which is widely used in the signal processing community, is being used since the editor has correctly pointed out that "pitch" should be reserved for a perceptual contest), instrument recognition, and signal separation can be done elegantly by a straightforward pattern recognition algorithm.
Signal decomposition by time-frequency and time-scale mapping is an essential element of most diagnostic signal analysis. Is the wavelet method of decomposition any better than the short-time Fourier transform and Wigner-Ville methods? This paper explores the effectiveness of wavelets for diagnostic signal analysis. The author has found that harmonic wavelets are particularly suitable because of their simple structure in the frequency domain, but it is still difficult to produce high-definition time-frequency maps. New details of the theory of harmonic wavelet analysis are described which provide the basis for computational algorithms designed to improve map definition.
Signal decomposition by time-frequency and time-scale mapping is an essential element of most diagnostic signal analysis. Is the wavelet method of decomposition any better than the short-time Fourier transform and Wigner-Ville methods? This paper explores the effectiveness of wavelets for diagnostic signal analysis. The author has found that harmonic wavelets are particularly suitable because of their simple structure in the frequency domain, but it is still difficult to produce high-definition time-frequency maps. New details of the theory of harmonic wavelet analysis are described which provide the basis for computational algorithms designed to improve map definition.
New details of the theory of harmonic wavelets are described and provide the basis for computational algorithms designed to compute high-definition time-frequency maps. Examples of the computation of phase using the complex harmonic wavelet and methods of signal segmentation based on amplitude and phase are described.
If one direction of (three-dimensional) space is singled out, it makes sense to formulate the description of embedded curves and surfaces in a frame that is adapted both to the embedded manifold and to the special direction, rather than a frame based upon the curvature landscape. Such a case occurs often in computer vision, where the image plane plays a role that differs essentially from the direction of view. The classical case is that of geomorphology, where the vertical is the singled out dimension. In computer vision the `ridges' and `(water-)courses' are recognized as important entities and attempts have been made to make the intuitive notions precise. These attempts repeat the unfortunate misunderstandings that marked the course of the late 19th century struggle to define the `Talweg' (equals `valley path' or `(water-)course'). We elucidate the problems and their solution via novel examples.
Wavelet maps provide a graphical picture of the frequency composition of a vibration signal. This paper, which is Part 2 of a pair, describes their construction and properties. In the case of harmonic wavelets, there are close similarities between wavelet maps and sonograms. A range of practical examples illustrate how the wavelet method may be applied to vibration analysis and some of its advantages.
The concept of a harmonic wavelet is generalized to describe a family of
mixed wavelets with the structure wm,n(x) = {exp (in2π
x)-exp (im2π x)}/i(n - m) 2π x. It is shown that this family
provides a complete set of orthogonal basis functions for signal
analysis. By choosing the (real) numbers m and n (not necessarily
integers) appropriately, wavelets whose frequency content ascends
according to the musical scale can be generated. These musical wavelets
provide greater frequency discrimination than is possible with harmonic
wavelets whose frequency interval is always an octave. An example of the
wavelet analysis of music illustrates possible applications.
A new harmonic wavelet is suggested. Unlike wavelets generated by
discrete dilation equations, whose shape cannot be expressed in
functional form, harmonic wavelets have the simple structure w(x) =
{exp(i4π x)-exp(i2π x)}/i2π x. This function w(x) is
concentrated locally around x = 0, and is orthogonal to its own unit
translations and octave dilations. Its frequency spectrum is confined
exactly to an octave band so that it is compact in the frequency domain
(rather than in the x domain). An efficient implementation of a discrete
transform using this wavelet is based on the fast Fourier transform
(FFT). Fourier coefficients are processed in octave bands to generate
wavelet coefficients by an orthogonal transformation which is
implemented by the FFT. The same process works backwards for the inverse
transform.
Wavelets provide a new tool for the analysis of vibration records. They allow the changing spectral composition of a nonstationary signal to be measured and presented in the form of a time-frequency map. The purpose of this paper, which is Part 1 of a pair, is to introduce and review the theory of orthogonal wavelets and their application to signal analysis. It includes the theory of dilation wavelets, which have been developed over a period of about ten years, and of harmonic wavelets which have been proposed recently by the author. Part II is about presenting the results on wavelet maps and gives a selection of examples. The papers will interest those who work in the field of vibration measurement and analysis and who are in positions where it is necessary to understand and interpret vibration data.
Preface. 1. Introduction. 2. Mathematical Preliminaries. 3. Ridges in Euclidean Geometry. 4. Ridges in Riemannian Geometry. 5. Ridges of Functions Defined on Manifolds. 6. Applications to Image and Data Analysis. 7. Implementation Issues. Bibliography. Index.