Traditional Time-frequency Analysis Methods Research Articles

Synchronous decomposition match-reassigning transform and its application in planetary gearbox fault diagnosis

Abstract Under time-varying operating conditions, the instantaneous frequency (IF) of the vibration signal of the planetary gearbox exhibits non-stationary time-varying closely spaced characteristics as well as non-proportional and non-synchronous characteristics, making it difficult for traditional time-frequency analysis (TFA) methods to accurately identify its fault features and obtain accurate time-frequency representations (TFR). To address this challenge, this paper proposes a TFA method based on Synchronous Decomposition Match-Reassigning Transform (SDMRT). Firstly, the successive variational mode decomposition (SVMD) is used to adaptively separate non-synchronous frequency components in the original vibration signal. Then, the chirp rate (CR) describing the signal harmonic structure is used to synchronously match each frequency component, obtaining the TFR corresponding to each component. Next, all TFRs are merged to obtain a complete TFA result. Finally, Finally, the energy spread is re-aggregated to the ridge of the IF using the reassignment principle, resulting in a new frequency estimation operator called the Synchronous Decomposition Match-Reassigning Operator (SDMRO). SDMRT can adaptively separate non-proportional and non-synchronous IFs and achieve precise matching of time-varying closely spaced IFs, thereby completing the TFR of the vibration signal generated by the planetary gearbox. By analyzing simulated signals and measured vibration signals from planetary gearboxes, the method proposed in this paper is significantly superior to other TFA methods, thereby validating the effectiveness and superiority of SDMRT.

Robustness evaluation of acceleration-based early rub detection methodologies with real fluid-induced noise

The early detection of rotor-casing rub allows to prevent permanent damage and even catastrophic failure of rotating machines, saving maintenance costs and decreasing downtime. Very often, aeroderivative gas turbines can be monitored only with accelerometers on the casing, which provide signals that contain both stochastic and deterministic noise that masks the vibration measurements. Although real vibration data from turbomachines present a background noise that is neither white nor Gaussian, the rule of thumb is to evaluate the robustness of fault detection under white signal noise, which may lead to estimation errors of such robustness. Here we evaluate and compare the performance of several signal processing methodologies for very early rub detection on accelerometer signals under fluid-induced signal noise due to turbulent fluid excitation. The independence between the fluid-induced vibrations and the rub and unbalance vibration was proved, and experimental data of fluid-induced noise were added as signal noise to clean rub signals. The robustness and sensitivity of rub detection under fluid-induced noise were calculated and compared. This study proves the superiority of reassigned time–frequency analysis methods such as the Wavelet Synchrosqueezed Transform for very early rub detection as compared with traditional time–frequency analysis methods and with the Fourier Transform. To the authors’ knowledge, this is the first time that non-Gaussian fluid-induced signal noise is used to evaluate fault detection performance.

A Micro-Motion Parameters Estimation Method for Multi-Rotor Targets without a Prior

Multi-rotor aircraft have the advantages of a simple structure, low cost, and flexible operation in the unmanned aerial vehicle (UAV) family, and have developed rapidly in recent years. Radar surveillance and classification of the growing number of multi-rotor aircraft has become a challenging problem due to their low-slow-small (LSS) characteristics. Estimation of the blade number is an important step in distinguishing LSS targets. However, most of the current research on micro-motion parameters estimation has focused on the analysis of rotational frequency, length, and the initial phase of blades with a prior of blade number, affecting its ability to identify LSS targets. In this article, a micro-motion parameters estimation method for multi-rotor targets without a prior is proposed. On the basis of estimating the flashing frequency of the blades, a validation function is constructed through spectral analysis to judge the number of blades, and then the rotational frequency is estimated. The blade length is calculated by estimating the maximum Doppler shift. Moreover, the variational mode decomposition (VMD)-based atomic scaling orthogonal matching pursuit (AS-OMP) method is jointly applied to estimate the blade length when suffering from the low PRF and insufficient SNR conditions. Extensive experiments on the simulated and measured data demonstrate that the proposed method outperforms robust micro-motion parameter estimation capability in low PRF and insufficient SNR conditions compared to the traditional time-frequency analysis methods.

Local maximum synchrosqueezing reassigning chirplet transform and its application to gearbox fault diagnosis

Abstract Aiming at the problems of poor time-frequency aggregation and severe noise interference when traditional time-frequency analysis methods deal with complex multi-component signals, this paper proposes a new time-frequency analysis method – local maximum synchrosqueezing reassigning chirplet transform (LMSRCT). The core idea of the method is to introduce the principle of general linear chirplet transform (GLCT) into synchro-reassigning transform (SRT), followed by reassigning the results of local maximum synchrosqueezing chirplet transform (LMSSCT), and then introducing the innovative synchro-reassigning operator (SRO). This results to a novel three-step method for extracting instantaneous frequency, which ultimately yields the final time-frequency representation. This method ensures the integrity of each instantaneous frequency, solves the problem of energy divergence problems, and improves aggregation. Simulation and experimental results show that the LMSRCT can provide better characterization results compared to other methods and can effectively solve the different situations that occur between the instantaneous frequencies of complex signals. The proposed method can effectively estimate transmission speed, which achieves fault diagnosis under tacholess conditions, and the order analysis results are more accurate and reliable.

Decoupling and Parameter Extraction Methods for Conical Micro-Motion Object Based on FMCW Lidar.

Micro-Doppler time-frequency analysis has been regarded as an important parameter extraction method for conical micro-motion objects. However, the micro-Doppler effect caused by micro-motion can modulate the frequency of lidar echo, leading to coupling between structure and micro-motion parameters. Therefore, it is difficult to extract parameters for micro-motion cones. We propose a new method for parameter extraction by combining the range profile of a micro-motion cone and the micro-Doppler time-frequency spectrum. This method can effectively decouple and accurately extract the structure and the micro-motion parameters of cones. Compared with traditional time-frequency analysis methods, the accuracy of parameter extraction is higher, and the information is richer. Firstly, the range profile of the micro-motion cone was obtained by using an FMCW (Frequency Modulated Continuous Wave) lidar based on simulation. Secondly, quantitative analysis was conducted on the edge features of the range profile and the micro-Doppler time-frequency spectrum. Finally, the parameters of the micro-motion cone were extracted based on the proposed decoupling parameter extraction method. The results show that our method can effectively extract the cone height, the base radius, the precession angle, the spin frequency, and the gravity center height within the range of a lidar LOS (line of sight) angle from 20° to 65°. The average absolute percentage error can reach below 10%. The method proposed in this paper not only enriches the detection information regarding micro-motion cones, but also improves the accuracy of parameter extraction and establishes a foundation for classification and recognition. It provides a new technical approach for laser micro-Doppler detection in accurate recognition.

Application of sparse S transform network with knowledge distillation in seismic attenuation delineation

Time-frequency analysis is a successfully used tool for analyzing the local features of seismic data. However, it suffers from several inevitable limitations, such as the restricted time-frequency resolution, the difficulty in selecting parameters, and the low computational efficiency. Inspired by deep learning, we suggest a deep learning-based workflow for seismic time-frequency analysis. The sparse S transform network (SSTNet) is first built to map the relationship between synthetic traces and sparse S transform spectra, which can be easily pre-trained by using synthetic traces and training labels. Next, we introduce knowledge distillation (KD) based transfer learning to re-train SSTNet by using a field data set without training labels, which is named the sparse S transform network with knowledge distillation (KD-SSTNet). In this way, we can effectively calculate the sparse time-frequency spectra of field data and avoid the use of field training labels. To test the availability of the suggested KD-SSTNet, we apply it to field data to estimate seismic attenuation for reservoir characterization and make detailed comparisons with the traditional time-frequency analysis methods.

Multiple synchro-tuning chirplet transform

Time-frequency analysis (TFA) is an effective tool for characterizing non-stationary signals. Due to the limited time-frequency resolution capability, traditional TFA methods cannot achieve the optimal time-frequency representation (TFR). Based on the post-processing technique, a novel TFA method named multiple synchro-tuning chirplet transform (MSTCT) is proposed to obtain high-resolution and robust TFR for non-stationary signals. The MSTCT first estimates the spectral energy, instantaneous frequency (IF), and chirp rate (CR) of non-stationary signals with high time-frequency resolution, enabling precise energy localization on the time-frequency plane. Then, a multiple reassignment procedure is proposed to concentrate the blurry time-frequency energy and yield a better TFR. Moreover, benefiting from the multiple squeezing mechanism in the frequency axis, the MSTCT holds the potential to reconstruct the signal with high accuracy. Both the simulation results and the real-world experiment using the bat echolocation data demonstrate that the proposed MSTCT algorithm achieves superior performance in the time-frequency resolution, noise immunity, and reconstruction capability compared to the state-of-art linear TFA methods.

Sinusoidal-assisted synchrosqueezing transform: Algorithms and biomedical applications

Intermittent oscillation signals (IOSs) exist widely in biomedical systems. They are always contaminated by various randomnesses like noise and artifacts. Compared with traditional time–frequency analysis (TFA) methods, adaptive TFA methods show significant advantages in separating IOSs. However, these methods are implemented by assuming that the input signal is in a weakly noisy environment and have narrow-band limitation for extracting IOSs. Besides, it is difficult for these methods to alleviate the noise in the gap between time where IOSs happen. Therefore, in this paper, a new method called the sinusoidal-assisted synchrosqueezing transform (SASST) is proposed for separating IOSs in strongly noisy environments. The SASST method locates the IOSs with the aid of a sinusoidal signal. Such a sinusoidal signal fills the energy blank in the gap between time where IOSs happen. This enables the ridge detection approach, which is the necessary tool for mode retrieval of SST-based methods, to focus on the assisted sinusoidal signal rather than noise. Thus, the background noise in the gap is suppressed. Moreover, this sinusoidal-assisted framework can be introduced into any SST-based TFA method. Depending on the characteristics of different biomedical signals, different SST-based methods are chosen to achieve better separation. As a result, SASST is capable of extracting both narrow- and wide-band IOSs in strongly noisy environments. The effectiveness and advantages of the proposed algorithm are verified by both numerical simulations and real-world cases. Even in a strongly noisy environment of −10 dB, SASST achieves output SNR of 12.66 dB and 10.53 dB in narrow- and wide-band simulation, respectively.

Chaotic Characterization of Milling Vibration Information

In view of the strong nonlinearity of the signals in the gestation period of milling chatter, and the problem that the traditional time-frequency analysis methods cannot reveal the weak characteristics of the gestation period of chatter well, a chaotic characteristic analysis method of milling vibration information is proposed. The milling force signals of stable milling, chatter gestation and chatter outbreak states are collected through variable working condition milling force measurement experiments, and the chaotic phase space reconstruction method is used to obtain the attractor images of milling force signals in different vibration states. The experiments show that the attractor features in the chatter gestation period are more significant than the traditional time-frequency features, and the chaotic attractor images can better reveal the weak features in the chatter gestation period.

PN Codes Estimation of Binary Phase Shift Keying Signal Based on Sparse Recovery for Radar Jammer.

Parameter estimation is extremely important for a radar jammer. With binary phase shift keying (BPSK) signals widely applied in radar systems, estimating the parameters of BPSK signals has attracted increasing attention. However, the BPSK signal is difficult to be processed by traditional time frequency analysis methods due to its phase jumping and abrupt discontinuity features which makes it difficult to extract PN (PN) codes of the BPSK signal. To solve this problem, a two-step PN codes estimation method based on sparse recovery is introduced in this paper. The proposed method first pretreats the BPSK signal by estimating its center frequency and converting it to zero intermediate frequency (ZIF). The pretreatment transforms phase jumps of the BPSK signal into the level jumps of the ZIF signal. By nonconvex sparsity promoting regularization, the level jumps of the ZIF signal are extracted through an iterative algorithm. Its effectiveness is verified by numeric simulations and semiphysical tests. The corresponding results demonstrate that the proposed method is able to estimate PN codes from the BPSK signal in serious electromagnetic environments.

Generalized Synchrosqueezing Transform: Algorithm and Applications

Time–frequency analysis (TFA) is a vital method to deal with nonstationary signals. However, the traditional TFA methods cannot produce a concentrated time–frequency representation (TFR) for the nonstationary signal with strongly time-varying instantaneous frequency (IF). In this work, a new TFA method named generalized synchrosqueezing transform is proposed to deal with the nonstationary signal with complicated IF. First, the explicit formula of the high-order IF estimation is calculated by a new computational framework. Next, the generalized synchrosqueezing operator (GSSO) is constructed by iterative operations. After that, the time–frequency energy is rearranged according to the constructed GSSO to obtain a more concentrated TFR. Finally, the efficient algorithm implementation is introduced to show the feasibility of the proposed method. Numerical data are used to demonstrate the effectiveness and superiority of the proposed method in improving the concentration and accuracy of the TFR of the nonstationary signal with complex IF. Experimental validation proves that the proposed method is effective in analyzing real-world signals and fault diagnosis.

Vibration Diagnosis and Optimization of Industrial Robot Based on TPA and EMD Methods

This paper proposed method that combined transmission path analysis (TPA) and empirical mode decomposition (EMD) envelope analysis to solve the vibration problem of an industrial robot. Firstly, the deconvolution filter time-domain TPA method is proposed to trace the source along with the time variation. Secondly, the TPA method positioned the main source of robotic vibration under typically different working conditions. Thirdly, independent vibration testing of the Rotate Vector (RV) reducer is conducted under different loads and speeds, which are key components of an industrial robot. The method of EMD and Hilbert envelope was used to extract the fault feature of the RV reducer. Finally, the structural problems of the RV reducer were summarized. The vibration performance of industrial robots was improved through the RV reducer optimization. From the whole industrial robot to the local RV Reducer and then to the internal microstructure of the reducer, the source of defect information is traced accurately. Experimental results showed that the TPA and EMD hybrid methods were more accurate and efficient than traditional time-frequency analysis methods to solve industrial robot vibration problems.

Local maximum synchrosqueezes form scaling-basis chirplet transform.

In recent years, time-frequency analysis (TFA) methods have received widespread attention and undergone rapid development. However, traditional TFA methods cannot achieve the desired effect when dealing with nonstationary signals. Therefore, this study proposes a new TFA method called the local maximum synchrosqueezing scaling-basis chirplet transform (LMSBCT), which is a further improvement of the scaling-basis chirplet transform (SBCT) with energy rearrangement in frequency and can be viewed as a good combination of SBCT and local maximum synchrosqueezing transform. A better concentration in terms of the time-frequency energy and a more accurate instantaneous frequency trajectory can be achieved using LMSBCT. The time-frequency distribution of strong frequency-modulated signals and multicomponent signals can be handled well, even for signals with close signal frequencies and low signal-to-noise ratios. Numerical simulations and real experiments were conducted to prove the superiority of the proposed method over traditional methods.

A novel complex network community clustering method for fault diagnosis

The complex network, as a method for the analysis of nonlinear and non-stationary signals, overcomes the shortcomings of traditional time-frequency analysis methods and proves its effectiveness in mechanical fault diagnosis. Community clustering, a type of complex network, has made great progress in recent years. However, the existing community clustering algorithms have disadvantages in that they lack significant global extreme value and huge search spaces. Therefore, a Fast Newman algorithm based on reliability judgment is proposed. Starting from the community structure characteristics of the complex network, with the fault sample as a network node, the relationship between the samples as a connected edge and a complex network model of fault data is established. Clusters in troubleshooting are transformed into community structure discovery in the network. Firstly, the initial division of the community is obtained by measuring the distance between the samples. Then, the modularity index of the network is used as a standard function of the community division, and the bottom-up community merger is performed. The local edge density index is used for reliability determination before each combination to achieve global optimization, and the network block structure is the most obvious. Finally, with all the data merged into one community, the optimal division of the community structure is obtained, while accurate community clustering and fault diagnosis is realized. The benchmark graphs for testing community detection (Lancichinetti–Fortunato–Radicchi benchmark standard test network, LFR) and different fault data of rolling bearings under multiple operating conditions are applied to verify the effectiveness of this method; the results prove that the modified Fast Newman algorithm has better clustering effects and a higher accuracy rate than the original algorithm. Compared with K-means clustering and fuzzy clustering, the modified Fast Newman algorithm achieves higher performance in fault diagnosis of rolling bearings under multiple operating conditions.

Scaling operator demodulation spectrum-based planetary gearbox fault diagnosis method under variable speed conditions

It is a challenging task to detect planetary gearbox fault type under time-varying speeds. Firstly, under time-varying speeds, the fault-induced frequency harmonics of the planetary gearbox are time-varying and close-spaced, and the fault feature is difficult to detect from the spectrum or traditional time-frequency analysis (TFA) results. Secondly, the interference component in the vibration signal resulting from the complicated gearbox configuration is another obstacle for fault detection. As such, the scaling operator demodulation spectrum (SODS) technique with high readability is developed for planetary gearbox fault diagnosis under variable speed conditions. The developed technique consists of two terms: scaling demodulation operator term for transforming all time-varying fault-related frequency harmonics; and spectral selection criterion term for obtaining fault-related frequency harmonics and removing the other smeared frequency components. With the proposed technique, the time-varying and close-spaced fault-induced frequency harmonics can be quantitatively characterized by the spectral peaks. The performance of the SODS-based planetary gearbox fault diagnosis technique is validated by a simulated signal and two experimental signals collected from different test rigs. Furthermore, comparison results with the traditional TFA methods show that the proposed technique has much better ability to identify the fault-related frequency harmonics of the planetary gearbox.

Spatial transformation of general sampling-aliasing frequency region for rotating-blade parameter identification with emphasis on single-probe blade tip-timing measurement

Blade-health monitoring is intensely required for turbomachinery because of the high failure risk of rotating blades. Blade-Tip Timing (BTT) is considered as the most promising technique for operational blade-vibration monitoring, which obtains the parameters that characterize the blade condition from recorded signals. However, its application is hindered by severe undersampling and stringent probe layouts. An inappropriate probe layout can make most of the existing methods invalid or inaccurate. Additionally, a general conflict arises between the allowed and required layouts because of arrangement restrictions. For the sake of economy and safety, parameter identification based on fewer probes has been preferred by users. In this work, a spatial-transformation-based method for parameter identification is proposed based on a single-probe BTT measurement. To present the general Sampling-Aliasing Frequency (SAFE) map definition, the traditional time–frequency analysis methods are extended to a time-sampling frequency. Then, a SAFE map is projected onto a parameter space using spatial transformation to extract the slope and intercept parameters, which can be physically interpreted as an engine order and a natural frequency using coordinate transformation. Finally, the effectiveness and robustness of the proposed method are verified by simulations and experiments under uniformly and nonuniformly variable speed conditions.

Enhanced order spectrum analysis based on iterative adaptive crucial mode decomposition for planetary gearbox fault diagnosis under large speed variations

Variable speed conditions are omnipresent in the operation of rotating machinery. Tacholess order tracking (TOT) technique based on time-frequency representation (TFR) has been attracting much attention in recent years, but the challenges of low time-frequency resolution, parameter selection and computational consumption limit its practical applications. To this end, this paper presents a novel time-frequency analysis (TFA) method named iterative adaptive crucial mode decomposition (IACMD), which accurately extracts the time-varying component in the signal through the instantaneous frequency refinement strategy and the adaptive bandwidth parameter update rule. Then, an enhanced order spectrum analysis scheme based on IACMD is further proposed, which highlights the fault characteristic order by eliminating the amplitude modulation effect and is completely independent of the tachometers. Both simulation analysis and test-rig experiments have demonstrated the effectiveness of the proposed strategy in fault diagnosis of planetary gearboxes under various large speed variations. Compared with traditional TFA methods, the results show that the IACMD provides higher accuracy and efficiency even in the case of large time-varying and strong noise.

Parameterized Instantaneous Frequency Estimation Method for Vibration Signal with Nonlinear Frequency Modulation

The vibration signal from the rotatory machinery condition monitoring under time-varying speed is usually amplitude-modulated (AM) and frequency-modulated (FM). It is important to efficiently and accurately estimate the instantaneous frequency (IF) of the vibration signal. In this paper, a novel parameterized IF estimation method is proposed. The method employs a high-order polynomial function to approximate the nonlinear IF and subsequently constructs overdetermined systems of linear equations by calculating the Fourier transform of the derivative of the signal. The IF can be estimated by using least squares estimation to solve the equations. The proposed method has high computational efficiency because it can obtain the estimation of IF over a period of time simultaneously; it differs from traditional time-frequency analysis methods that need to calculate the IF at each point in the time axis. It is demonstrated that the proposed method is not only particularly powerful for the nonlinear FM mono-component signal but also applicable to the multi-component signal constructed by multiple harmonics. The numerical simulation validates the effectiveness of the proposed method, and the experiment’s results show that the method is suitable for the IF estimation of the vibration signal from the varying-speed rotor system.

An automatic real-time cable modal frequency identification and tracking algorithm by combining recursive band-pass filter and recursive Hilbert transform

Cables are essential elements in bridge engineering. The cable force is usually determined by the vibration method using the cable modal frequency. This paper proposes a real-time online identification method of cable modal frequency based on band-pass filter and Hilbert transform. A recursive form of band-pass filtering (RBF) is employed so that the original signal can be filtered for the target frequency band to obtain a single modal signal of the target order of the cable. The recursive Hilbert transform (RHT) is then used to obtain a real-time resolved signal for real-time frequency identification, allowing for real-time identification and tracking of the time-varying modal frequencies at the cable's target order. This method is more efficient than traditional time–frequency analysis methods. According to numerical simulations and actual measurement data. The method is noise-resistant and does not necessitate priori information of the order. The method allows online identification and tracking of time-varying frequency of interest at a specific order. The procedure can be applied to the real-time identification of cable incident frequencies in an online monitoring system, and can also provide a basis for further real-time identification of time-varying cable forces.

Research on Wheel-Rail Local Impact Identification Based on Axle Box Acceleration

Abnormal vibration signals of tramcar are mostly nonstationary and nonlinear signals. This study applied the Hilbert–Huang transform (HHT) to analyze abnormal vibration of the tramcar, aiming to overcome the limitations of some traditional time-frequency analysis methods, such as Fourier transform, in dealing with nonstationary and nonlinear signals. Additionally, to address mode aliasing produced during empirical mode decomposition (EMD) used in classical HHT, this study proposed to first use complete EMD with adaptive noise for the decomposition of original vibration data, then eliminate the trend-term components with the calculated correlation coefficients, and finally perform denoising on high-frequency noisy components using the wavelet threshold method. After weighted reconstruction using denoised high-frequency components and low-frequency information components, data processing was finally optimized via HHT. Taking a tramcar as an example, the Hilbert spectra of the vertical acceleration of axle box were plotted via HHT, and the time-instantaneous, frequency-instantaneous energy 3D relations were obtained for the location of impact points. Further, the vibration characteristics were analyzed and quality indexes were calculated in combination with the marginal spectra so as to judge the reasons for abnormal vibration and failure modes of the tramcar. The results revealed that the proposed method was feasible and effective in vibration feature extraction and vibration impact analysis for tramcars.