Instantaneous Rotational Frequency Research Articles

Rolling Bearings Fault Diagnosis under Variable Speed Conditions Based on Multitime‐Frequency Ridge Extraction

The vibration signals of rolling bearings under variable speed conditions exhibit time‐varying, nonstationary, and nonlinear characteristics, making it challenging to extract fault features. A fast path optimization‐based multitime‐frequency ridge extraction (FPOMRE) method is proposed in this paper to address the problem. First, the time‐frequency representation of the fault signal is obtained through the adaptive short‐time Fourier transform (STFT), and then the FPOMRE algorithm is used to extract additional ridges below the lowest time‐frequency ridge multiple times. The frequency ratio of each ridge to the additional ridge, the average value of the ratio, and the variance of the ratio are calculated, respectively, and finally, the matching degree between the average ratio and the fault feature coefficient is compared with the diagnosis of the bearing fault. The simulation and experimental results show that the FPOMRE method can effectively diagnose the bearing fault under constant speed and variable speed conditions, and the extraction accuracy of time‐frequency ridges under constant speed is higher than that under variable speed, and the extraction accuracy under high speed is higher than that under low speed. The relative error of extracting rotation frequency under constant speed and variable speed is 0.5% and 1.3%, respectively. In addition, comparing the FPOMRE with the maximum amplitude method (MAM), both FPOMRE and MAM can effectively extract the instantaneous shaft rotation frequency at a constant speed. However, under variable speed conditions, the ridge extracted by FPOMRE has a higher accuracy and lower average processing time than that extracted by MAM.

A Tacholess Order Tracking Method Based on Spectral Amplitude Modulation for Variable Speed Bearing Fault Diagnosis

Rolling bearings often work under nonstationary conditions, which will lead to spectrum smearing phenomenon and the failure of traditional diagnostic methods. Order tracking is a common technique to alleviate this problem by resampling the time-domain signal using the instantaneous rotation frequency. For tacholess applications, it is of great importance to obtain accurate rotation speeds. In this paper, a tacholess order tracking method based on spectral amplitude modulation (SAM) is proposed. Based on the fact that SAM can separate frequency components with different levels of energy, an appropriate weight is selected to highlight the instantaneous frequency, which is then extracted from the time-frequency representation by peak search algorithm. Moreover, entropy is employed to evaluate the valuable information in the time-frequency representation in order to select an optimal weight adaptively. A simulated signal and experimental signals of bearing outer and inner ring faults demonstrate that the proposed method can extract instantaneous rotation frequency and detect nonstationary bearing fault characteristics effectively.

Research on the Instantaneous Slip Ratio Monitoring of Rolling Bearing Based on Variable Reluctance Sensor

The bearing slip ratio is an effective index for bearing slip evaluation, and is also of great significance for bearing design and condition monitoring. In this paper, we proposed a slip monitoring method based on the variable reluctance sensor (VRS). A magnetoelectric probe is directly aligned with the outer ring to obtain the instantaneous rotation frequency of the inner ring, cage rotation frequency, and rolling element revolution frequency. In addition, a method based on frequency-shifting filter and spectrum correction (FSF-SC) is used to evaluate the instantaneous slip ratio, and a test rig was developed to verify the effectiveness of the proposed method under different speeds, workloads, and oil levels. Experimental results show that the slip ratio measured by the proposed method is very close to that measured by ultrasonic, and the deviation is only 0.02%. Moreover, the proposed instantaneous slip ratio is useful for bearing condition monitoring.

Bearing multi-fault diagnosis with iterative generalized demodulation guided by enhanced rotational frequency matching under time-varying speed conditions

The rotational frequency (RF) is an important information for multi-fault features detection of rolling bearing under varying speed conditions. In the traditional methods, such as the computed order analysis (COA) and the time–frequency analysis (TFA), the RF should be measured using an encoder or extracted by a complex algorithm, which bring challenge to bearing fault diagnosis. In order to address this issue, a novel iterative generalized demodulation (IGD) based method guided by the instantaneous fault characteristic frequency (IFCF) extraction and enhanced instantaneous rotational frequency (IRF) matching is proposed in this paper. Specifically, the resonance frequency band excited by bearing fault is first obtained by the band-pass filter, and its envelope time–frequency​ representation (TFR) is calculated using the Hilbert transform and the short-time Fourier transform (STFT). Second, the IFCF is extracted using the harmonic summation-based peak search algorithm from the envelope TFR. Third, the time-varying RF ridge is transformed into a line paralleling to the time axis using the IGD with the phase function (PF). The PF is calculated by the IFCF function and fault characteristic coefficient (FCC). Lastly, the iterative generalized demodulation spectrum (IGDS) is obtained using the fast Fourier transform (FFT) for identifying fault type corresponding to the extracted IFCF. Based on obtained fault type and FCC ratios, new PFs and frequency points (FPs) are calculated for detecting other faults. Both simulated and experimental results validate that multi-fault features of rolling bearing under time-varying rotational speeds can be effectively identified without RF measurement and extraction.

A parameter-adaptive ACMD method based on particle swarm optimization algorithm for rolling bearing fault diagnosis under variable speed

Fault diagnosis for rolling bearing under variable speed is always a challenging topic since the vibration signal has time-varying characteristics. To overcome this difficulty, a novel method is exploited based on particle swarm optimization (PSO) and adaptive chirp mode decomposition (ACMD), named parameter-adaptive ACMD. Firstly, fast spectral kurtosis algorithm is used to get the resonance band signal. Then, the parameter-adaptive ACMD method decomposes the envelope signal to obtain the time-frequency spectrum. Next, the proposed method of fault diagnosis uses peak search algorithm to estimate instantaneous rotational frequency from the time-frequency graph processed. Finally, the measured rotational frequency is used as the phase function in the resampling process to get the order spectrum and fault characteristic order (FCO). Simulation and actual signals were analyzed and the results indicate that the proposed method can identify fault type with variable speed and has high application value.

A TEO‐based modified Laplacian of Gaussian filter to detect faults in rolling element bearing for variable rotational speed machine

Conventional signal processing techniques for fault detection are usually aimed at constant speed conditions. Nowadays, order tracking, especially tacho-less order tracking is regarded as a valuable tool for extracting fault features under variable rotational speed conditions. Therefore, a teager energy operator (TEO)-based modified Laplacian of Gaussian filter is proposed to enhance the fault detection behaviour of tacho-less order tracking. This method consists of four steps: First, multi-synchrosqueezing transform is employed to estimate and then extract the instantaneous rotation frequency of a shaft. Second, based on the extracted curve, the non-stationary domain signal is converted into a quasi-stationary domain by the re-sampling technique. Third, the obtained quasi-stationary domain signal is de-noise by the novel method. Finally, fault characteristic orders are extracted via envelope order spectrum analysis method and the performance is significantly improved. The results of numerical simulation and experimental investigations are performed to validate the superiority of the novel method for fault extraction under variable rotational speed conditions.

Estimation of the instantaneous rotational frequency of gear transmission with large speed variations using short-time angular resampling and ridge-enhancing techniques

Instantaneous rotational frequency (IRF) is essential for most of the gear fault diagnosis methods used for nonstationary conditions. However, estimating the IRF under large speed-variation conditions is challenging. In this work, we developed a short time angular resampling and ridge-enhancing (STAR-RE) method for estimating the IRF of a gear transmission system of large speed variations. The method is integrated with a rigorous approach for eliminating ridge smearing based on angular resampling and a straightforward scheme for enhancing the target ridge by merging the ridges of several gear meshing harmonics. Data collected from a laboratory gearbox and an in-field wind turbine gearbox operated at variable speed regimes were used to test the accuracy and effectiveness of the STAR-RE method. The results indicated that the STAR-RE method achieved more accurate IRFs than the benchmark methods.

Multiple Time-Frequency Curve Classification for Tacho-Less and Resampling-Less Compound Bearing Fault Detection Under Time-Varying Speed Conditions

Rotating machinery usually runs at time-varying speeds, which often brings failures, especially compound faults. However, currently, there is still a lack of satisfactory coupled fault diagnosis methods at variable speeds. Several methods have been developed, which try to eliminate rotational speed influences by converting nonstationary fault characteristic frequencies into stationary ones. But it brings new problems of computational efficiency and accuracy, and requires an auxiliary tachometer to measure the rotational speed. Time-frequency analysis can solve the problems, and extract time-frequency curves for fault diagnosis through time-frequency representation (TFR). However, for signals containing multiple faults at variable speeds, uninterested components will be generated when extracting multiple time-frequency curve (MTFC), which makes it difficult for compound fault diagnosis. To solve these problems, a novel MTFC classification method for tacho-less and resampling-less compound bearing fault detection under time-varying speed conditions is proposed in this paper. Here the TFR characteristics of a multi-fault vibration signal is analyzed, and a compound fault diagnosis strategy based on MTFC classification is developed. Firstly, MTFC is extracted by a local peak search method. Then, according to the relationship between instantaneous shaft rotational frequency, instantaneous fault characteristics frequency and their respective harmonics, two classification criteria for curves are proposed and MTFC is classified into interested curves and uninterested curves. Finally, the average ratios between interested curves are matched to theoretical fault characteristic coefficients to determine the fault type. Case studies on rolling bearing experiments verified the effectiveness and superiority of the proposed method.

A combined kernel-based fuzzy C-means clustering and spectral centroid for instantaneous frequency estimation

An improved instantaneous frequency estimation algorithm for rotating machines based on a kernel-based fuzzy C-means clustering (KFCM) algorithm used in association with a spectral centroid algorithm is proposed in this study. The clustering algorithm is used first to discriminate the time-frequency points from the sources of the reference axis and other points. The discrete time-frequency points related to the instantaneous rotation frequency of the reference axis are then located based on the values of the time-frequency matrix elements; on the basis of these elements, the instantaneous rotation frequency is then estimated using a spectral centroid algorithm. It is demonstrated that this method effectively reduces the effects of interference and noise while achieving higher estimation precision. To validate the proposed method, numerical simulations of multi-component signals and crossover signals are performed. The results of these simulations indicate that the method can realize instantaneous frequency estimation with high precision, even when the numerical responses are contaminated by Gaussian white noise. In addition, when this method is used to analyze the vibration signal of rotating machinery in the situation of a run-up procedure, remarkable speed estimation results are obtained.

An Adaptive Cross-Validation Thresholding De-Noising Algorithm for Fault Diagnosis of Rolling Element Bearings Under Variable and Transients Conditions

Developing a proper approach for extracting the fault characteristics of rolling element bearings is a significant behavior in variable operation conditions. Generally, without a tachometer, the tacholess order tracking method is well established by instantaneous rotation frequency curve estimation using time-frequency analysis. However, the fault features are often masked in strong background noises. An adaptive cross-validation thresholding de-noising algorithm is employed to improve the performance of the traditional envelope order spectrum method. First, the general linear chirplet transform is applied to estimate the instantaneous rotation frequency curve. Second, with the help of the instantaneous rotation frequency curve, the raw non-stationary signal is transferred to an angular domain by the re-sampling technique. Third, the adaptive cross-validation thresholding de-noising algorithm is employed to purify the angular domain signal to improve the fault feature extraction performance of the envelope order spectrum method. Comparisons using numerical simulations and experimental investigations of bearing faults under variable speed conditions are given to show the superiority of the present method.

Bearing fault diagnosis method with unknown variable speed based on multi-curve extraction and selection

Aiming at the bearing fault diagnosis with unknown time-varying speed, a fault diagnosis method of variable speed bearing based on multi-curve extraction and selection, Vold-Kalman filter (VKF) and generalized demodulation (GD) is proposed. The designed filter decomposes the original vibration signal into signal components carrying speed information and bearing fault information respectively. Fast path optimization algorithm is used to extract time-frequency curves iteratively from the time-frequency representation (TFR) of components. The high-precision instantaneous shaft rotation frequency (ISRF) curve is selected and calculated based on the proposed selection index, and then the demodulation phase function is constructed. Using VKF and GD, the bearing fault pulse harmonics are extracted and demodulated. Finally, by establishing reconstructed demodulation spectrum, bearing fault state identification and classification can be achieved. The experimental and analysis results confirm that the proposed method can successfully extract and demodulate fault pulse harmonics without tachometer. Different fault states can be accurately identified, and the effects of interference components and extraction errors can be avoided. Further analysis demonstrates that it can also identify weak bearings fault. Thus, the proposed method is an effective fault diagnosis technology for unknown time-varying speed bearing.

MOVEMENT OF A RELATIVISTIC ELECTRON IN CROSSED ELECTRIC AND MAGNETIC FIELDS

It is shown that the ideas about the electron motion into electric and magnetic crossed fields existing in the modern literature are incorrected. For relativistic velocity case the electron motion can't be imagined by the superposition of the drift with velocity u = E0/B0 and the rotation with frequency  = eB0/m (e, m is charge and mass of relativistic electron). In the relativistic case these motions are not separated, moreover the instantaneous rotation frequency  of relativistic electron is unstable for fixed coordinate system (E0, B0):  depends on m, which varies considerably during rotation.

A time–frequency-based maximum correlated kurtosis deconvolution approach for detecting bearing faults under variable speed conditions

Rolling bearing vibration signals induced by local faults are highly correlated with structural dynamics. This leads to a great deal of research about signal processing for rolling bearing condition monitoring and fault detection. However, the common diagnostic approaches, which were proposed to manage vibration signals with constant speeds, are unavailable for variable speed cases. Tacholess order tracking is a powerful method to break the limitations of conventional methods while avoiding trouble with tachometer installation and reducing measurement costs. Time–frequency analysis methods are used to estimate the instantaneous rotation frequency (IRF) from vibration signals directly. However, it is difficult to extract the IRF accurately due to the strong nonstationary properties of the signal. Therefore, parameterized time–frequency transform (PTFT) methods are proposed to solve this problem. Polynomial chirplet transform (PCT) is one of the PTFTs that can produce an excellent time–frequency representation with a polynomial kernel function. In this paper, the PCT is employed to estimate the IRF of rolling bearings from the vibration signals. On this basis, a maximum correlated kurtosis deconvolution-based envelope order spectrum is applied to detect the bearing fault characteristic order. The efficiency of the proposed method is certified by numerical signal and rolling bearing vibration data. The diagnostic results indicate that the new fault detection algorithm is superior for rolling bearing fault diagnosis under varying speed conditions.

Gearbox fault diagnosis via generalized velocity synchronous Fourier transform and order analysis

Gearboxes have essential roles in many types of industrial equipment. Fault detection for gearboxes is important yet extremely difficult because volatile working conditions lead to nonstationary vibration signals. Order tracking is a classic and effective technique for nonstationary vibration analysis and fault diagnosis of rotating machinery. Many order tracking methods that do not require a tachometer have been proposed, such as methods based on re-sampling. However, most are complex and often introduce interpolation errors. To avoid such difficulties, a simple yet effective method is proposed in this paper. This method employs the generalized demodulation approach to extract a component with a frequency proportional to the instantaneous shaft rotational frequency from the vibration signal. Then, demodulating the extracted component recovers the instantaneous shaft rotational phase. With such information the order spectrum can be directly obtained via a velocity synchronous discrete Fourier transform. Finally, the fault can be diagnosed by order spectrum analysis. The effectiveness of this method is validated with both simulated and lab experimental vibration signals of a gearbox under time-varying rotational speed conditions.

Ball Screw Fault Detection and Location Based on Outlier and Instantaneous Rotational Frequency Estimation

Ball screw, as a crucial component, is widely used in various rotating machines. Its health condition significantly influences the efficiency and position precision of rotating machines. Therefore, it is important to accurately detect faults and estimate fault location in a ball screw system to make sure that the ball screw system runs safely and effectively. However, there are few research studies concerning the topic. The aim of this paper is to fill the gap. In this paper, we propose a method to automatically detect and locate faults in a ball screw system. The proposed method mainly consists of two steps: fault time estimation and instantaneous rotational frequency extraction. In the first step, a statistics‐based outlier detection method is proposed to involve the fault information mixing in vibration signals and estimate the fault time. In the second step, a parameterized time‐frequency analysis method is utilized to extract the instantaneous rotational frequency of the ball screw system. Once the fault time and instantaneous rotational frequency are estimated, the fault location in a ball screw system is calculated through an integral operation. In order to verify the effectiveness of the proposed method, two fault location experiments under the constant and varying speed conditions are conducted in a ball screw failure simulation testbed. The results demonstrate that the proposed method is able to accurately detect the faults in a ball screw system and estimate the fault location within an error of 22%.

Multiple time-frequency curve extraction Matlab code and its application to automatic bearing fault diagnosis under time-varying speed conditions

Vibration signal analysis is an important technique for bearing fault diagnosis. For bearings operating under constant rotational speed, faults can be diagnosed in the frequency domain since each type of fault has a specific Fault Characteristic Frequency (FCF), which is proportional to the shaft rotational speed. However, bearings often operate under time-varying speed conditions. Additionally, the measurement of the time-varying rotational speed requires instruments, such as tachometers, which leads to extra cost and installation. With the development of time-frequency analysis, the time-varying FCFs manifest as curves in the Time-Frequency Representation (TFR). It has been shown that extracting multiple time-frequency curves from the TFR and then identifying the Instantaneous Fault Characteristic Frequency (IFCF) and Instantaneous Shaft Rotational Frequency (ISRF), bearing faults can be automatically diagnosed under time-varying speed conditions without using tachometers. However, the existing method used to identify the IFCF and the ISRF may lead to inaccurate results. In this study, the complete MATLAB© codes and a more reliable approach to use Multiple Time-Frequency Curve Extraction (MTFCE) for automatic bearing fault diagnosis under time-varying speed conditions are presented.•A Multiple time-frequency curve extraction (MTFCE) Matlab code is presented to extract multiple curves from the TFR.•Custom Matlab code for automatic bearing fault diagnosis under time-varying speed conditions without using tachometer data via the MTFCE is given and explained.•A new parameter, the allowable variance of the curve-to-curve ratio, is proposed to identify the IFCF and ISRF more reliably.

A method for tachometer-free and resampling-free bearing fault diagnostics under time-varying speed conditions

Bearings often operate under time-varying rotational speed conditions. Processing the signal in the time-frequency domain and extracting the Instantaneous Fault Characteristic Frequency (IFCF) and the Instantaneous Shaft Rotational Frequency (ISRF) are important approaches for bearing fault diagnosis under time-varying speed conditions without signal resampling and without using a tachometer. However, there are two problems: (1) the collected bearing signal is often contaminated by random noise and interferences transmitted from other components, which affects the accuracy of the extracted IFCF, and (2) the ISRF cannot always be found in the Time-Frequency Representation (TFR) of the extracted bearing fault transients, which impacts the accuracy of fault identification. Therefore, a new tachometer-free and resampling-free method is proposed for bearing fault diagnosis under time-varying speed conditions which consists of three main steps: (1) bearing fault signature extraction via Oscillatory Behavior-based Signal Decomposition (OBSD) to suppress the influence of random noise and interferences, (2) IFCF and ISRF estimation via applying a IFCF&ISRF search algorithm to the TFR of the decomposed signal, and (3) automatic bearing fault identification based on the average curve-to-curve ratios of the searched IFCF and ISRF. The IFCF&ISRF search algorithm is proposed based on the analysis of the frequency characteristics of bearing vibration signals. The algorithm can estimate the ISFR even if it is not present in the extracted bearing fault signature. The effectiveness of the proposed method is validated using both simulated signals and experimental data.

Bearing fault diagnosis under unknown time-varying rotational speed conditions via multiple time-frequency curve extraction

Under normal operating conditions, bearings often run under time-varying rotational speed conditions. Under such circumstances, the bearing vibrational signal is non-stationary, which renders ineffective the techniques used for bearing fault diagnosis under constant running conditions. One of the conventional methods of bearing fault diagnosis under time-varying speed conditions is resampling the non-stationary signal to a stationary signal via order tracking with the measured variable speed. With the resampled signal, the methods available for constant condition cases are thus applicable. However, the accuracy of the order tracking is often inadequate and the time-varying speed is sometimes not measurable. Thus, resampling-free methods are of interest for bearing fault diagnosis under time-varying rotational speed for use without tachometers. With the development of time-frequency analysis, the time-varying fault character manifests as curves in the time-frequency domain. By extracting the Instantaneous Fault Characteristic Frequency (IFCF) from the Time-Frequency Representation (TFR) and converting the IFCF, its harmonics, and the Instantaneous Shaft Rotational Frequency (ISRF) into straight lines, the bearing fault can be detected and diagnosed without resampling. However, so far, the extraction of the IFCF for bearing fault diagnosis is mostly based on the assumption that at each moment the IFCF has the highest amplitude in the TFR, which is not always true. Hence, a more reliable T-F curve extraction approach should be investigated. Moreover, if the T-F curves including the IFCF, its harmonic, and the ISRF can be all extracted from the TFR directly, no extra processing is needed for fault diagnosis. Therefore, this paper proposes an algorithm for multiple T-F curve extraction from the TFR based on a fast path optimization which is more reliable for T-F curve extraction. Then, a new procedure for bearing fault diagnosis under unknown time-varying speed conditions is developed based on the proposed algorithm and a new fault diagnosis strategy. The average curve-to-curve ratios are utilized to describe the relationship of the extracted curves and fault diagnosis can then be achieved by comparing the ratios to the fault characteristic coefficients. The effectiveness of the proposed method is validated by simulated and experimental signals.

Condition monitoring of rotating machinery under varying operating conditions based on Cyclo-Non-Stationary Indicators and a multi-order probabilistic approach for Instantaneous Angular Speed tracking

During the last decades, an emerging interest has been reported on modelling rotating machinery signals as cyclostationary. Several tools, such as the Spectral Correlation Density (SCD) and the Cyclic Modulation Spectrum (CMS) have been proposed, assuming constant or almost constant rotating speed. To overcome this limitation, generalizations of SCD and CMS have been proposed that display cyclic Order versus Frequency. On the other hand the estimation of the instantaneous angular speed of the rotating machine is a crucial issue that has to be tackled before any analysis or diagnosis of the machine operating under non-stationary conditions. This operation is generally carried out thanks to a tachometer, or an angle encoder, but in some cases this information can be missing for different reasons. The goal of this paper is to combine a novel approach for the analysis of cyclo-non-stationary signals based on the generalization of the indicators of cyclostationarity in order to cover the speed varying conditions, introducing a new speed-dependent angle averaging operator, with a new speed estimation method, that extracts from a vibration signal the most probable instantaneous rotation frequency without focusing on one specific order based on the a priori knowledge of the kinematics of the machine. The effectiveness of the ensemble method is evaluated on an acceleration signal captured at the wind turbine gearbox operating under speed varying conditions.

Rolling element bearing instantaneous rotational frequency estimation based on EMD soft-thresholding denoising and instantaneous fault characteristic frequency

The accurate estimation of the rolling element bearing instantaneous rotational frequency (IRF) is the key capability of the order tracking method based on time-frequency analysis. The rolling element bearing IRF can be accurately estimated according to the instantaneous fault characteristic frequency(IFCF). However, in an environment with a low signal-to-noise ratio (SNR), e.g., an incipient fault or function at a low speed, the signal contains strong background noise that seriously affects the effectiveness of the aforementioned method. An algorithm of signal preprocessing based on empirical mode decomposition (EMD) and wavelet shrinkage was proposed in this work. Compared with EMD denoising by the cross-correlation coefficient and kurtosis(CCK) criterion, the method of EMD soft-thresholding (ST) denoising can ensure the integrity of the signal, improve the SNR, and highlight fault features. The effectiveness of the algorithm for rolling element bearing IRF estimation by EMD ST denoising and the IFCF was validated by both simulated and experimental bearing vibration signals at a low SNR.