Generalized demodulation with tunable E-Factor for rolling bearing diagnosis under time-varying rotational speed

A time-varying instantaneous frequency fault features extraction method of rolling bearing under variable speed

Synchrosqueezing transform (SST) benefits from an instantaneous frequency (IF) estimator in the time-frequency domain, providing an energy-concentrated time-frequency representation to describe the time-varying frequency of seismic signals. To enhance the concentration performance of SST, this paper theoretically proposes a statistical SST (SSST) by constructing a spectrum-weighted IF estimator in the short-time Fourier transform (STFT) domain. In this SSST, a linear chirp signal is introduced to better capture its chirp rate, and then the quadratic STFT spectra are weighted by a window-related weighting function. On this basis, two equations related to the IF and chirp rate are constructed to derive the spectrum-weighted IF estimator. Finally, the STFT coefficients are squeezed to the estimated IF trajectories by a frequency fixed-point iterative algorithm, thereby providing a more concentrated time-frequency representation for multi-component signals than existing advanced methods, while enabling to retrieve each component. Two synthetic examples and one field seismic data on thin interbeds are utilized to demonstrate the effectiveness of the proposed SSST and show its ability to highlight the time-varying frequency features of seismic signals, which is a promising seismic data analysis tool, such as characterizing thin interbed thickness variations.

Time-frequency (TF) analysis (TFA) to nonstationary signals can reveal the nonlinearly changing instantaneous frequencies (IFs) of signals; it is, therefore, widely used for rotating machinery fault diagnosis under time-varying speed conditions. However, the traditional TFA methods may only reveal the outline of IFs due to the extra TF diffusion caused by limited TF resolution, hindering the fault diagnosis of rotating machinery. To enhance the readability of the time-frequency representation (TFR) of the signals with IF trajectories linearly time-varying, linear chirplet transform has been proved effective. To effectively tackle the signal with nonlinearly changing IFs, a string of chirp-rates is preferred, where the final TFR is obtained by superposition of each corresponding sub-TFR at each TF point. However, the extra cross-term interferences resulted from a string of chirp-rates cannot be neglected on TFRs. Aiming at alleviating the cross-terms from the TFR, matching and reassignment based TF enhancement strategy is proposed, where only the appropriate chirp-rate and its corresponding TFR slice is retained at each time instant. The appropriate chirp-rate is adaptively selected by the index—spectral kurtosis. To further increase the readability of the resulting TFR, a reassignment technique synchrosqueezing transform is integrated with the proposed matching strategy. By iteratively employing reassignments, TFR with the enhanced energy and sharp IF ridges can be generated. The effectiveness of the proposed method is validated by both simulated and experimental analyses. It is shown that the proposed method is effective in processing time-varying signals and can provide more accurate IF estimation, which paves the way for rotating machinery fault diagnosis under nonstationary speed conditions.

Slope synchronous chirplet transform and its application to tacho-less order tracking of rotating machineries

The use of linear time-frequency (TF) representa-tions as instantaneous frequency (IF) estimators arises in various fundamental disciplines, mainly thanks to their simplicity and immunity against interfering cross-terms. In this paper, we derive the optimal window width for IF estimation of noisy multicomponent signals based on a family of linear TF transforms that use Gaussian windowing functions and the Fourier oscillatory kernel. Closed-form formulas concerning the estimation bias and the variance are presented, which, thanks to their generality, describe the statistical performance of IF estimators based on transforms with fixed, time-adaptive, frequency-adaptive, or time-frequency adaptive windows. The optimal window is dependent on the unknown first derivative of the IF; therefore, we employ a low-complexity procedure to infer the derivative and optimize the width accordingly. Two adaptive and fully automated TF representations (TFRs) are developed; the first employs a time-adaptive window that minimizes the sum of the mean-squared errors (MSEs) of the IF estimates at each time instant, while in the second TFR, the window is adaptive over time and frequency, minimizing the estimation MSE at each location in the TF domain. Examples using synthetic and real-world signals demonstrate that the proposed algorithms may outperform many popular state-of-the-art techniques, including those that are signal-adaptive, in terms of IF estimation.

Accurate extraction of time-varying instantaneous frequency (IF) is crucial for fault diagnosis in rotating machinery operating under time-varying speed conditions, as real-world signals often exhibit nonlinearity and complexity. Variational nonlinear chirp mode decomposition (VNCMD) demonstrates significant capability and adaptability in extracting time-varying IFs by converting broadband signals into narrowband signals through a demodulation technique. However, its accuracy is limited by the fact that it only enforces smoothness on the demodulated signals, and it does not consider the inherent proportional relationships among IFs in rotating machine vibration signals. These limitations hinder VNCMD from accurately extracting IFs and result in poor noise robustness, a strong dependence on the initial IFs, and difficulties in handling nonlinear chirp signals (NCSs) with closely spaced IFs. To overcome these limitations, this article proposes hybrid constraints embedding VNCMD (HCVNCMD), which places further constraints on demodulated signals and IFs. Specifically, a sparse constraint is first applied to the demodulated signals. This improves their estimation accuracy and reduces errors in the subsequent IF estimates. Second, a low-rank constraint is applied to the IFs, effectively capturing their proportional relationships and optimizing inaccurate initial IFs to achieve the true IFs. The hybrid constraints embedded in HCVNCMD significantly enhance noise robustness and improve IF estimation accuracy, enabling effective processing of NCSs with closely spaced IFs. Both simulation and experimental cases demonstrate that the HCVNCMD outperforms existing time–frequency analysis (TFAs) methods in accurately estimating the IFs of NCSs. Furthermore, it successfully identifies fault features in gear and bearing under time-varying speed conditions, confirming its effectiveness and practicability.

Time-varying demodulation analysis for rolling bearing fault diagnosis under variable speed conditions

Polynomial Wigner Ville distribution (PWVD) is considered as a tool for recognition of signals with an unknown frequency modulation law. This time-frequency distribution (TF) with multi-linear kernel is an optimal form for signals with a polynomial frequency modulation law. Instantaneous frequency (IF) is the important property of non-stationary signals, i.e. signals whose spectral contents vary with time. Then the recognition of unknown signals can be formulated as the problem of estimation of its instantaneous frequency. The studies of estimation of IF for noiseless signals can be found in literature. The noise embedded in a signal has large influence on time-frequency distribution and can change the true value of the IF. Time-frequency distribution strongly depends on the kind of noise. In this paper signals embedded in additive gaussian and impulse noise have been considered. Estimation of the instantaneous fiequency in impulsive noise background is different than in gaussian noise and requires robust forms of time-frequency distributions. In this paper the application of the robust form of the Wigner-Ville distribution (WVD_r) and the robust form of the polynomial Wigner-Ville distribution (PWVD_r) for IF estimation is presented. Maxima of these distributions are assumed as the estimates of true values of IF. The theoretical background of WVD_r and PWVD_r distributions is introduced and several numerical experiments of IF estimation are presented.

Chirplet transform (CT) is effective in characterization of instantaneous frequency (IF) for monocomponent linear-frequency-modulated signal. However, the CT is not suitable to analyze multicomponent signal with nonlinear-frequency-modulated component. In this paper, a time-frequency fusion technique based on polynomial CT (PCT) (TFPCT) is proposed to characterize the time-frequency structure of such signals. The TFPCT relies on the fact that the PCT is able to concentrate the energy closely along the IF of the monocomponent signal in time-frequency distribution (TFD). For multicomponent signal, the TFPCT first estimates the proper coefficients with respect to individual component and, second, produces a series of the TFD using the PCT. Each TFD has better energy concentration along the IF of one component. Then, in order to reduce the interference of unwanted component and preserve the component of interest, each TFD is filtered and grouped as an image. At last, the TFPCT combines these TFDs to be an eventual fused TFD, which has the energy concentrating closely along the IF of all components. Comparison with several conventional TFD methods on both numerical multicomponent signal and bat echolocation signal validates the potential and the effectiveness of the proposed method.

Conventional bearing fault diagnosis methods require specialized instruments to acquire signals that can reflect the health condition of the bearing. For instance, an accelerometer is used to acquire vibration signals, whereas an encoder is used to measure motor shaft speed. This study proposes a new method for simplifying the instruments for motor bearing fault diagnosis. Specifically, a video clip recording of a running bearing system is captured using a cellphone that is equipped with a camera and a microphone. The recorded video is subsequently analyzed to obtain the instantaneous frequency of rotation (IFR). The instantaneous fault characteristic frequency (IFCF) of the defective bearing is obtained by analyzing the sound signal that is recorded by the microphone. The fault characteristic order is calculated by dividing IFCF by IFR to identify the fault type of the bearing. The effectiveness and robustness of the proposed method are verified by a series of experiments. This study provides a simple, flexible, and effective solution for motor bearing fault diagnosis. Given that the signals are gathered using an affordable and accessible cellphone, the proposed method is proven suitable for diagnosing the health conditions of bearing systems that are located in remote areas where specialized instruments are unavailable or limited.

Self-paced learning for instantaneous frequencies estimation in heavy noise environments

Under diverse conditions, the vibration signals of complex rotating machinery exhibit non-stationary behavior, multi-component characteristics, closely spaced frequencies, and non-proportionality, posing challenges to conventional time-frequency analysis (TFA) methods. These limitations hinder accurate instantaneous frequency (IF) estimation and time-frequency representation (TFR) construction, directly impacting machinery fault diagnosis. As such, we propose the Local Entropy Selection Scaling-Extracting Chirplet Transform (LESSECT), which optimizes entropy-based chirp rate (CR) selection to match non-proportional fundamental frequencies. By adaptively selecting multiple CRs at the same time center, LESSECT enhances TFR resolution and energy concentration, leading accurate IF identification. Experimental validation on bat echolocation, bearing fault, and planetary gearbox signals shows its superior performance in resolving non-proportional, closely spaced IFs. This significantly improves state estimation and enhances machinery diagnostics, contributing to greater system reliability.

A time-frequency distribution (TFD) signal processor, developed at the Applied Physics Laboratory, is currently under evaluation using simulated signals and actual laser vibration sensor (LVS) data that we collected on various ship targets. Preliminary results for one instantaneous frequency (IF) estimator implementation, the smoothed cross Wigner-Ville Distribution (XWVD), indicate 8 to 10 dB demodulation (CNR) advantage compared to a digital FM limiter-discriminator. A second approach, using the unsmoothed XWVD TFD, demonstrated a 3-5 dB advantage. Regarding spectral estimation, we are investigating performance of our reduced interference distribution (RID) implementation through comparison with the short-time Fourier transform (STFT). From the LVS data processed, indications are that a significant increase in spectral and temporal resolution exists using our RID approach. Our processor also provided improved detectability over the STFT for transient signals and short-lived sinusoids. Significant correlation between accepted acoustic lines and LVS-derived vibration lines are indicated. Details are presented that describe our signal simulation, the LVS measurements, and signal processing implementations along with assumptions based on measured speckle-induced amplitude modulation.

Traditional signal postprocessing methods suffer from the repeated assignment problem (RAP), which can result in inaccurate instantaneous frequency (IF) estimation and signal recovery. In this article, to solve this problem, a novel time-frequency (TF) analysis (TFA) method called the synchro-reassigning transform (SRT) is proposed. This method aims to obtain the ideal TF representation (TFR) by utilizing derivatives of the constructed amplitude function and a three-step selection rule to adaptively extract the TF coefficients on the IF trajectories in the TF plane, and reassigning these TF coefficients into a new TFR. In this way, SRT can eliminate the RAP, thus obtaining an approximately ideal TFR that helps to realize more accurate IF estimation and signal reconstruction. Furthermore, SRT shows satisfactory performance on signals with high nonlinear IFs or relatively close IFs, even under strong noise conditions. Two simulated signal and three real-life signals were used to demonstrate the performance of SRT.

Joint extraction-based multivariate chirp mode decomposition and its application to fault diagnosis of rotating machinery under variable speed conditions