Time-varying Frequency Components Research Articles

Rotating Machinery Fault Diagnosis under Time–Varying Speed Conditions Based on Adaptive Identification of Order Structure

Rotating machinery fault diagnosis is of key significance for ensuring safe and efficient operation of various industrial equipment. However, under nonstationary operating conditions, the fault–induced characteristic frequencies are often time–varying. Conventional Fourier spectrum analysis is not suitable for revealing time–varying details, and nonstationary fault feature extraction methods are still in desperate need. Order spectrum can reveal the rotational–speed–related time–varying frequency components as spectral peaks in order domain, thus facilitating fault feature extraction under time–varying speed conditions. However, the speed–unrelated frequency components are still nonstationary after angular–domain resampling, thus causing wide–band features and interferences in the order spectrum. To overcome such a drawback, this work proposes a rotating machinery fault diagnosis method based on adaptive separation of time–varying components and order feature extraction. Firstly, the rotational speed is estimated by the multi–order probabilistic approach (MOPA), thus eliminating the inconvenience of installing measurement equipment. Secondly, adaptive separation of the time–varying frequency component is achieved through time–varying filtering and surrogate test. It effectively eliminates interference from irrelevant components and noise. Finally, a high–resolution order spectrum is constructed based on the average amplitude envelope of each mono–component. It does not involve Fourier transform or angular–domain resampling, thus avoiding spectral leakage and resampling errors. By identifying the fault–related spectral peaks in the constructed order spectrum, accurate fault diagnosis can be achieved. The Rényi entropy values of the proposed order spectrum are significantly lower than those of the traditional order spectrum. This result verifies the effective energy concentration and high resolution of the proposed order spectrum. The results of both numerical simulation and lab experiments confirm the effectiveness of the proposed method in accurately presenting the time–varying frequency components for rotating machinery diagnosing faults.

Fault diagnosis of offshore wind turbines based on component separable synchroextracting transform

The fault diagnosis of wind turbines is crucial for wind power generation. However, the violent variation in wind power and directions of offshore wind turbines often results in high nonstationary vibration signals, which poses a challenge for effective fault recognition. Time-frequency analysis (TFA) is a common method to reveal the time-varying frequency components caused by faults. In this paper, we propose a component separable synchroextracting transform (CSSET) by exploring the unique modulation characteristics of wind turbine vibration signals. For the method, a fundamental frequency is firstly estimated by a time-frequency ridge optimization method, and the potential fault characteristic frequency (FCF) set is constructed according to the modulation characteristics of the wind turbine vibration signal. Next, the instantaneous amplitudes (IAs) of these components in the set are estimated by a time-varying bandpass filter. Then, the instantaneous frequency (IF) of the reconstructed single component is extracted by the synchronous extraction operator (SEO), and a high-precision time-frequency representation (TFR) is obtained. The proposed method takes full advantage of the physical modulation characteristics of vibration signals and the frequency components which are most relevant to fault information are only preserved. The effectiveness of the proposed method is verified by analyzing simulation and experimental signals.

Time-varying filtering for nonstationary signal analysis of rotating machinery: Principle and applications

Rotating machinery vibration signals are composed of multiple time-varying frequency components under nonstationary conditions, and they usually overlap in the frequency domain. Effective extraction of these time-varying components is vital for rotating machinery fault diagnosis. This paper proposes a time-varying filtering (TVF) method for extraction of time-varying components. Its upper and lower cutoff frequencies are time-varying and independent. The TVF uses locally designed bandpass filters to achieve local filtering in the time domain, and exploits filtering with phase correction (FPC) and local filter lengths to ensure the phase and amplitude consistency of filtered signals. It cannot only extract time-varying mono-components but also separate time-varying frequency bands from highly nonstationary signals, for further time–frequency analysis: (1) fine time–frequency representations based on mono-components extraction, and (2) time-varying demodulation of time-varying frequency band of interests. The proposed method is illustrated and validated through numerical simulation, and signal analyses of typical rotating machines including a hydraulic turbine rotor, an aircraft engine rolling bearing, a multistage planetary gearbox, and a fixed shaft gearbox.

Multivariate time-varying complex signal processing framework and its application in rotating machinery rotor-bearing system

Affected by multi-field coupling factors, the vibration response of rotating machinery similar to the hydro-generator unit often exhibits strong time-varying frequency components, which makes rotor fault detection more challenging. The fusion analysis of the vibration signals of multiple bearing sections of the rotor has been proved to be a very effective method for rotor vibration fault diagnosis. However, how to more accurately and synchronously extract the instantaneous features of rotor non-stationary vibration signals associated with multiple sections has been unresolved. To this end, a framework for multivariate time-varying complex signal decomposition of the rotor-bearing system (RBS) is proposed, namely multivariate complex nonlinear chirp mode decomposition. First, the decomposition of multivariate time-varying complex signals is realized by two-stage processing. Second, instantaneous orbit features (IOFs) are obtained through the proposed framework. Finally, a three-dimensional instantaneous orbit map reflecting the time-varying process is constructed through the IOFs. The framework not only realizes the decomposition of the multi-channel time-varying complex signals of the rotor but also simultaneously extracts the instantaneous features of the multi-channel signals. In addition, it also realizes the description of the instantaneous vibration state of the RBS in the non-stationary process (such as startup and shutdown). Simulation experiments show that the framework is superior to other multi-channel signal processing methods in processing time-varying complex signals. The results based on field-measured signals show that the framework can guide the real-time analysis of the signals generated by rotating machinery, which improves the intuition of condition monitoring.

Time-varying characteristics analysis of vehicle-bridge interaction system based on modified S-transform reassignment technique

The health monitoring of bridge under moving vehicles demands a high-efficient time-varying characteristic analysis method to extract the frequency fluctuations caused by vehicle-bridge dynamic interaction effect. Characteristics identification of structures becomes a challenging task especially when time-varying vibration modes are involved. For this purpose, an instantaneous frequency identification technique based on modified S-transform reassignment is presented in this paper. Firstly, the general approximation between structural instantaneous frequency and S-transform is deduced. The Gaussian window is employed in S-transform and a novel first-order frequency function with two parameters is added into the Gaussian window function to improve the time–frequency resolution. The added function makes the Gaussian window scalable-sliding and the optimal values of the parameters are selected based on time–frequency concentration criterion. Considering the Rényi entropy as the measurement of criterion, the information performance extracted from S-transform spectrogram is evaluated. Then the time–frequency reassignment technique is applied to improve the readability of S-transform spectrogram to assure the accuracy of the extracted instantaneous characteristics, which enhances the time–frequency concentration efficiently. The proposed method is verified by numerical simulation, the results obtained by the spectrograms derived from short-time Fourier transform, wavelet transform, modified S-transform and reassigned modified S-transform are compared. Afterwards, the time-varying frequency components related to the vehicle-bridge interaction are extracted from the response signal. Numerical simulations involving different parameter analysis and laboratory experiments are conducted to validate the flexibility of the technique in vehicle-bridge system. The effect of vehicle weight, speed and road surface roughness on the identified results are discussed. Both numerical and experimental results have demonstrated that a higher time–frequency concentration in the spectrogram can be observed of the proposed modified S-transform reassignment method than other time–frequency methods, and subsequently a higher accuracy of instantaneous frequency identification is achieved which provides a good tool for time-varying characteristic analysis in bridge health monitoring.

A Methodology to Handle Spectral Smearing in Gearboxes Using Adaptive Mode Decomposition and Dynamic Time Warping

Tacho-less order tracking is an effective tool for fault detection in gearboxes operating under speed fluctuations. This technique's performance depends on extracting instantaneous shaft speed information from a complex multicomponent gearbox signal, which is first preprocessed by a bandpass filter. However, spectral overlap resulting from the time-varying frequency components makes it challenging to isolate the shaft speed harmonic mono-component using this approach. In order to overcome such issues, an adaptive tacho-less order-tracking (OT) method combining the variational mode decomposition (VMD) and the fast dynamic time warping (FDTW) is proposed in this article. The proposed method first decomposes the measured gearbox vibration signal using VMD to estimate the instantaneous shaft speed profile to construct the shaft vibration signal. The gearbox vibration signal is then resampled based on the optimal warping path obtained by FDTW, which performs an “elastic” stretching and compression along the time axis of the extracted shaft vibration signal with respect to a sinusoidal reference signal with a constant shaft rotational frequency. Finally, the presence of gear fault is detected by constructing the order spectrum of the resampled vibration signal. The effectiveness of the proposed algorithm is demonstrated using both simulation analysis and experimental validation using measurements from two different wind-turbine gearboxes collected under test and field conditions, respectively.

Research on singular spectrum decomposition and its application to rotor failure detection

As an important part of rotating machinery, a healthy rotor is critical to ensuring optimal working conditions of the entire system. Considering that the vibration signal of rotor consists of different frequency components when the failure arises, a novel rotor failure detection method based on singular spectrum decomposition (SSD) is presented. The original vibration signal is adaptively decomposed into a number of singular spectrum components (SSCs) by the SSD method. Then, energy separation algorithm (ESA) is adopted to demodulate each singular spectrum component. Finally, the SSD-ESA time-frequency spectrum can be obtained and the fault features contained in the SSD-ESA time-frequency spectrum can be identified to determine the fault types. The effectiveness of SSD for harmonic separation was assessed through tones separation analyses, the results show that SSD is able to separate more harmonic pairs of different amplitude ratios than empirical mode decomposition (EMD). Furthermore, three simulations of multi-component signals were designed to investigate the use of SSD for signal decomposition. The SSD method was then applied to detect signatures caused by rotor oil film whirl in experimental signals and compared to both EMD and ensemble EMD (EEMD). The simulated analysis results reflect that SSD shows superiority to EMD and EEMD in inhibiting mode mixing and extracting the time-varying frequency components. The experimental analysis results demonstrate that the SSD based rotor failure detection method is an alternative method under both constant and variable speed conditions.

A novel method for speed recovery from vibration signal under highly non-stationary conditions

Many engineering structures operate under non-stationary conditions resulting in signals containing time-varying frequency components. Analysis of such signals in time or frequency domain naturally yields smeared results. Typically implemented solution to this problem is the angle and order domain analysis. However, in some scenarios, acquisition of speed signal may be not available, due to practical limitations. If the speed fluctuations are limited, or recorded vibrational signal is free from environmental and measurement disturbances, it is possible to extract the speed signal from raw time waveforms. State-of-the-art methods include e.g. semi-automatic tracking of the ridges of the Short-Time Fourier Transform for speed signal reconstruction. However, these methods can fail if the time-varying frequency components cross with each other. Moreover, it is very common that complicated structures operating in non-stationary conditions are controlled by sophisticated electronic systems, which can introduce constant frequency noise to the measuring equipment. This can lead to some interference of these time-invariant and time-variant frequency components. The paper presents a simple and easy way to use a tool for hand selection and tracking of desired components, thus making complex time-frequency filtration easy. Finally, instantaneous frequency is calculated with use of Centre Moments of Frequency to extract the speed signal. The presented method shows how a particular tool developed within 3-dimensional image processing branch could be implemented in analysis of 2-dimensional signals characterized by time-frequency varying components.

Time–frequency representation using IEVDHM–HT with application to classification of epileptic EEG signals

Time–frequency representation (TFR) is useful for non-stationary signal analysis as it provides information about the time-varying frequency components. This study proposes a novel TFR based on the improved eigenvalue decomposition of Hankel matrix and Hilbert transform (IEVDHM–HT). In the proposed method, first the authors decompose non-stationary signals using the IEVDHM with suitably defined criterion for eigenvalue selection, requirement of number of iterations, and new component merging criteria. Furthermore, the HT is applied on extracted components in order to obtain the TFR of non-stationary signals. The performance of proposed TFR has been evaluated on synthetic signals in clean and white noise environment with different signal-to-noise ratios. The proposed method gives good performance in terms of Renyi entropy measure in comparison with other existing methods. Application of the proposed TFR is also shown for the classification of epileptic seizure electroencephalogram (EEG) signals. The least-square support vector machine (LS-SVM) with radial basis function kernel is used for classification of seizure and seizure-free EEG signals obtained from the publicly available database by the University of Bonn, Germany. The proposed method has achieved classification accuracy 100% for the studied EEG database.

A Method for Estimation of Motor Rotational Speed from STFT Spectrogram at a Non-stationary Conditions

Vibration signals acquired from rotational machines may be measured without a tacho signal for different reasons - either the speed sensor cannot be mounted due to construction reasons, speed was not measured, or the tacho signal is corrupted or not accurate enough in case of higher speeds. This paper shows STFT spectrogram based approach to determine speed profile from multiple time-varying frequency components that can be used for instance for direct processing by a Vold-Kalman filter (VKF). The method functionality is demonstrated on a simulated signal as well as on a vibration signal measured using an accelerometer on electric motor during the run-up conditions, while the rotational speed was not measured. The only known information is maximum rotational speed, presence of run-up in the signal and the motor geometry. Conducted experiment showed that despite the simple approach that resides in utilization of only one method that results directly the speed profile, the method can handle larger rotational speed changes while having enough resolution needed for VKF processing.

Online frequency estimation with applications to engine and generator sets

Frequency and spectral analysis based on the discrete Fourier transform is a fundamental task in signal processing and machine diagnostics. This paper aims at presenting computationally efficient methods for real-time estimation of stationary and time-varying frequency components in signals. A brief survey of the sliding time window discrete Fourier transform and Goertzel filter is presented, and two filter banks consisting of: (i) sliding time window Goertzel filters (ii) infinite impulse response narrow bandpass filters are proposed for estimating instantaneous frequencies. The proposed methods show excellent results on both simulation studies and on a case study using angular speed data measurements of the crankshaft of a marine diesel engine-generator set.

Joint envelope and frequency order spectrum analysis based on iterative generalized demodulation for planetary gearbox fault diagnosis under nonstationary conditions

Planetary gearbox vibration signals under nonstationary conditions are characterized by time-varying nature and complex multi-components, making it very difficult to extract features for fault diagnosis. Order spectrum analysis is one of the effective approaches for nonstationary signal analysis of rotating machinery. The main idea of order analysis is to map the time-varying frequency components into constant ones. Inspired by this idea, we propose a new order spectrum analysis method to exploit the unique property of iterative generalized demodulation in converting arbitrary instantaneous frequency trajectories of multi-component signals into constant frequency lines on the time–frequency plane. This new method is completely algorithm-based and tachometer/encoder-free, thus easy to implement. It does not involve equi-angular resampling commonly required by most order tracking methods and is hence free from the decimation and/or interpolation error. The proposed order analysis method can eliminate the time-variation effect of frequency and thus can effectively reveal the harmonic order constituents of nonstationary multi-component signals. However, the planetary gearbox vibration signals also lead to complex sideband orders. As such, we further propose to analyze the order spectrum of amplitude envelope. This will eliminate the complex sideband orders in the order spectrum of original signals, leading to a substantially simplified and more reliable gear characteristic frequency identification process. Nevertheless, the gear and/or planet carrier rotating frequency orders, which are irrelevant to gear fault, may still exist. To avoid possible misleading results due to such frequency orders, we also propose to analyze the order spectrum of instantaneous frequency. Theoretically, the peaks present in frequency order spectrum directly correspond to the gear characteristic frequency orders, which can be used to extract gear fault signature more explicitly. The proposed method has been illustrated via analysis of the simulated signals of planetary gearboxes, and validated using lab experimental planetary gearbox datasets, both under variable speed conditions. The analysis results have shown that the method is effective in extracting both distributed and localized gear faults under nonstationary conditions.

Time–frequency analysis based on Vold-Kalman filter and higher order energy separation for fault diagnosis of wind turbine planetary gearbox under nonstationary conditions

Planetary gearbox fault diagnosis under nonstationary conditions is important for many engineering applications in general and for wind turbines in particular because of their time-varying operating conditions. This paper focuses on the identification of time-varying characteristic frequencies from complex nonstationary vibration signals for fault diagnosis of wind turbines under nonstationary conditions. We propose a time–frequency analysis method based on the Vold-Kalman filter and higher order energy separation (HOES) to extract fault symptoms. The Vold-Kalman filter is improved such that it is encoders/tachometers-free. It can decompose an arbitrarily complex signal into mono-components without resorting to speed inputs, thus satisfying the mono-component requirement by the HOES algorithm. The HOES is then used to accurately estimate the instantaneous frequency because of its high adaptability to local signal changes. The derived time–frequency distribution features fine resolution without cross-term interferences and thus facilitates extracting time-varying frequency components from highly complex and nonstationary signals. The method is illustrated and validated by analyzing simulated and experimental signals of a planetary gearbox in a wind turbine test rig under nonstationary running conditions. The results have shown that the method is effective in detecting both distributed (wear on every tooth) and localized (chipping on one tooth) gear faults.

A time–frequency approach for generation of synthetic time-histories of earthquake signals

Seismic signals are non-stationary in nature, generated by earthquake and some other sources like explosive, landslides etc. In this paper, pattern of time varying frequency components of real-time recorded data of seismic signal are investigated by computing the mean of Gabor transform. In most of the cases, it is not possible to use recorded data of strong ground motion at a given site. It is therefore essential to analyze synthetic time-history which gives the prediction of strong ground motion in the regions where actual records can’t be used. In this work, the problem has been shown in synthetic time-histories where time–frequency variation is not controlled and suggested Gabor transformation based approach for generation of site specific synthetic time-history having control on time–frequency variation. Synthetic time-histories have been generated at five different sites by computing the average of Gabor transform of eight records for different earthquakes at each site. It is also shown that shape of response spectrum calculated from the generated synthetic time-history at each site is quite comparable to shape of mean response spectrum of recorded time-histories. Synthetic time-histories of a realistic time–frequency variation will be very useful for the researchers who wish to perform the nonlinear analysis of structures for a particular site.

Recursive Parametric Frequency/Spectrum Estimation for Nonstationary Signals With Impulsive Components Using Variable Forgetting Factor

This paper proposes a general and computationally efficient parametric model-based framework for recursive frequency/spectrum estimation and feature detection of nonstationary signals, which may contain different extents of nonstationarities and impulsive components. The estimation of time-varying frequency or spectrum is formulated as a time-varying linear model identification problem, where the spectral information is estimated from the model coefficients. We then employ a QR-decomposition-based recursive least M-estimate (QRRLM) algorithm for recursive estimation of the time-varying model coefficients in impulsive environment using M-estimation. New variable forgetting factor (VFF) schemes are developed to improve the tracking performance of the QRRLM method in nonstationary environment and we use theoretical derivation and simulations to prove that the proposed VFF schemes can approach the optimal VFF selection. The resultant VFF-QRRLM algorithm is able to restrain and isolate impulsive components whereas it is able to handle different extents of spectral variations. Simulation results show that the proposed VFF-QRRLM algorithm is more robust and accurate than conventional recursive least squares-based methods in estimating both time-varying narrowband frequency components and broadband spectral components with impulsive components. Potential applications of the proposed method can be found in power quality monitoring, online fault detection and speech analysis.

Using Time Deformation to Filter Nonstationary Time Series with Multiple Time-Frequency Structures

For nonstationary time series consisting of multiple time-varying frequency (TVF) components where the frequency of components overlaps in time, classical linear filters fail to extract components. TheG-filter based on time deformation has been developed to extract components of multicomponentG-stationary processes. In this paper, we explore the wide application of theG-filter for filtering different types of nonstationary processes with multiple time-frequency structure. Simulation examples illustrate that theG-filter can be applied to filter a broad range of multicomponent nonstationary process where TVF components may in fact overlap in time.

G-Filtering Nonstationary Time Series

The classical linear filter can successfully filter the components from a time series for which the frequency content does not change with time, and those nonstationary time series with time-varying frequency (TVF) components that do not overlap. However, for many types of nonstationary time series, the TVF components often overlap in time. In such a situation, the classical linear filtering method fails to extract components from the original process. In this paper, we introduce and theoretically develop the G-filter based on a time-deformation technique. Simulation examples and a real bat echolocation example illustrate that the G-filter can successfully filter a G-stationary process whose TVF components overlap with time.

Synchronization of business cycles in Poland and the euro zone – the wavelet domain approach

In the paper time-scale (wavelet) analysis is suggested as a tool for examining business cycle synchronization. Wavelet analysis enables to examine stochastic processes simultaneously in the time and frequency domains. Due to this the approach makes it possible to investigate time varying frequency components of economic processes. Among such components one can distinguish cycles of length over one year up to twelve years, which are known as business cycles (see Burns and Mitchell, 1946). In the paper this approach is illustrated with an examination of business cycle synchronization between Poland and its main trading partners in the euro zone: Germany, France, Italy and the Netherlands. The empirical analysis confirms the stylized facts on the Polish business cycle: its length, similarity with the German cycle and the lead-lag relations with national cycles in the euro zone. Besides, verification of the OCA endogenity hypothesis has been undertaken, but the results obtained are ambiguous.

Time–frequency representation of musical rhythm by continuous wavelets

A method is described that exhaustively represents the periodicities created by a musical rhythm. The continuous wavelet transform is used to decompose an interval representation of a musical rhythm into a hierarchy of short-term frequencies. This reveals the temporal relationships between events over multiple time-scales, including metrical structure and expressive timing. The analytical method is demonstrated on a number of typical rhythmic examples. It is shown to make explicit periodicities in musical rhythm that correspond to cognitively salient ‘rhythmic strata’ such as the tactus. Rubato, including accelerandos and ritardandos, are represented as temporal modulations of single rhythmic figures, instead of timing noise. These time varying frequency components are termed ridges in the time–frequency plane. The continuous wavelet transform is a general invertible transform and does not exclusively represent rhythmic signals alone. This clarifies the distinction between what perceptual mechanisms a pulse tracker must model, compared to what information any pulse induction process is capable of revealing directly from the signal representation of the rhythm. A pulse tracker is consequently modelled as a selection process, choosing the most salient time–frequency ridges to use as the tactus. This set of selected ridges is then used to compute an accompaniment rhythm by inverting the wavelet transform of a modified magnitude and original phase back to the time domain.

A fault detection tool using analysis from an autoregressive model pole trajectory

A new scheme is proposed that combines autoregressive (AR) modelling techniques and pole-related spectral decomposition for the study of incipient single-point bearing defects for a vibration-based condition monitoring system. Vibration signals obtained from the ball bearings from the high vacuum (HV) and low vacuum (LV) ends of a dry vacuum pump run in normal and faulty conditions are modelled as time-variant AR series. The appearance of spurious peaks in the frequency domain of the vibration signatures translates to the onset of defects in the rolling elements. As the extent of the defects worsens, the amplitudes of the characteristic defect frequencies’ spectral peaks increase. This can be seen as the AR poles moving closer to the unit circle as the severity of the defects increase. The number of poles equals the AR model order. Although not all of the poles are of interest to the user. It is only the poles that have angular frequencies close to the characteristic bearing defect frequencies that are termed the ‘critical poles’ and are tracked for quantification of the main spectral peaks. The time-varying distance, power and frequency components can be monitored by tracking the movement of critical poles. To test the efficacy of the scheme, the proposed method was applied to increasing frame sizes of vibration data captured from a pump in the laboratory. It was found that a sample size of 4000 samples per frame was sufficient for almost perfect detection and classification when the AR poles’ distance from the centre of unit circle was used as the fault indicator. The power of the migratory poles was an alternative perfect classifier, which can be used as a fault indicator. The analysis has been validated with actual data obtained from the pump. The proposed method has interesting potential applications in condition monitoring, diagnostic and prognostic-related systems.