Characteristic Frequency Ratio Research Articles

FPGA implementation of an improved envelope detection approach for bearing fault diagnosis

This study presents an enhanced envelope detection technique implemented on a field-programmable gate array (FPGA) to diagnose bearing faults in rotating machinery. Bearing faults frequently result in machinery breakdowns, incurring substantial downtime and maintenance expenses. In our approach, we employ the Teager energy operator (TEO) to extract the vibration signal envelope. Subsequently, we subject the envelope signal to the fast Fourier transform (FFT) to generate the envelope spectrum of the vibration signal. Finally, we further refine the envelope spectrum using TEO for a second time, resulting in a pronounced fault peak that facilitates early fault detection. We evaluate the effectiveness and performance of the proposed method using two distinct types of bearing vibration signals, one being simulated and the other measured. Our findings reveal that the suggested approach outperforms traditional envelope detection methods, leading to a substantially enhanced fault diagnosis capability. For instance, when we assess the characteristic frequency ratio (FCFR) for faults in the inner and outer rings of the bearing using the proposed method, we observe that the FCFR values are significantly elevated, ranging from 160 to 330% higher compared to the analysis performed by the TEO and HT methods. Consequently, this indicates that the proposed approach has the ability to detect faults at an earlier stage than other methods. Furthermore, the FPGA-based implementation makes it suitable for critical industrial applications where rapid fault detection is essential to prevent catastrophic failures.

Statistical Tool Size Study for Computer-Controlled Optical Surfacing

Over the past few decades, computer-controlled optical surfacing (CCOS) systems have become more deterministic. A target surface profile can be predictably achieved with a combination of tools of different sizes. However, deciding the optimal set of tool sizes that will achieve the target residual error in the shortest run time is difficult, and no general guidance has been proposed in the literature. In this paper, we present a computer-assisted study on choosing the proper tool size for a given surface error map. First, we propose that the characteristic frequency ratio (CFR) can be used as a general measure of the correction capability of a tool over a surface map. Second, the performance of different CFRs is quantitatively studied with a computer simulation by applying them to guide the tool size selection for polishing a large number of randomly generated surface maps with similar initial spatial frequencies and root mean square errors. Finally, we find that CFR = 0.75 achieves the most stable trade-off between the total run time and the number of iterations and thus can be used as a general criterion in tool size selection for CCOS processes. To the best of our knowledge, the CFR is the first criterion that ties tool size selection to overall efficiency.

An Adaptive Optimized TVF-EMD Based on a Sparsity-Impact Measure Index for Bearing Incipient Fault Diagnosis

Bearing incipient fault diagnosis is very crucial to the timely condition-based maintenance (CBM), but it is also relatively difficult because of the faint feature and environmental interference. In this article, an adaptive optimized time-varying filtering-based empirical mode decomposition (TVF-EMD) (AO-TVF-EMD) is put forward to extract more explicit and abundant incipient fault features by optimizing the parameter combination in conventional TVF-EMD. First, a novel integrated indicator termed sparsity-impact measure index (SIMI) constructed from dispersion entropy (DE) and envelope characteristic frequency ratio (ECFR) is designed as the parameter selection criterion. This index fully guarantees the optimal sparsity and impact properties of the obtained modes simultaneously, considering both time- and frequency-domain characteristics. Simulation verified that SIMI can not only sensitively characterize the variation of fault feature but also is robust to the environmental disturbance. Subsequently, we employ the Salp swarm algorithm (SSA) to realize the accurate search of parameter combinations where the minimum SIMI is defined as the fitness function. Moreover, comparative analysis in two runs to failure tests of bearings demonstrate that AO-TVF-EMD is more effective to extract incipient fault features than the maximum weighted Kurtosis (MWK) optimized TVF-EMD, the conventional TVF-EMD with fixed parameter, and some other advanced signal processing methods, which highlights the superiority of the proposed method.

Resonance-based bandwidth Fourier decomposition method for gearbox fault diagnosis

This paper aims to investigate the response mechanism of faulty gears, and further proposes a resonance-based bandwidth Fourier decomposition (RBBFD) method for gearbox fault identification. To reveal the vibration patterns of the gear-pair system, a dynamic model with time-varying stiffness is established to obtain the transient response excited by local damage. The analysis results demonstrate that the impulse response of a faulty gear is essentially carried by the natural frequency instead of the widely recognized meshing frequency and its harmonies. After that, the RBBFD method is developed based on the dynamic characteristics. In the proposed method, the resonance frequencies of the gear system are estimated according to the maximum point distribution of the Fourier spectrum, and then the Fourier spectrum bandwidth optimization algorithm is introduced to decompose the vibration signal into mono-components centered on the estimated natural frequencies. A new criterion termed the characteristic frequency ratio is utilized to measure the fault-related degree of the decomposed modes, and the fault type is identified by Hilbert demodulation. The effectiveness of RBBFD is verified through two diagnostic cases. The comparison with several state-of-art works validates the superiority of the proposed method in gearbox fault diagnosis.

Extended Slip-Weakening Model and Inference of Rupture Velocity

The slip-weakening model developed by Ohnaka and Yamashita is extended over the breakdown zone by equating the scaling relationships for the breakdown zone and the whole rupture area. For the extension, the study uses the relationship between rupture velocity and radiation efficiency, which was derived in the theory of linear elastic fracture mechanics, and the definition of fmax given in the specific barrier model proposed by Papageorgiou and Aki. The results clearly show that the extended scaling relationship is governed by the ratio of rupture velocity to S wave velocity, and the velocity ratio can be determined by the ratio of characteristic frequencies of a Fourier amplitude spectrum, which are corner frequency, fc, and source-controlled cut-off frequency, fmax, or vice versa. The derived relationship is tested by using the characteristic frequencies extracted from previous studies of more than 130 shallow crustal events (focal depth less than 25 km, MW 3.0~7.5) that occurred in Japan. Under the assumption of a dynamic similarity, the rupture velocity estimated from fmax/fc and the modified integral timescale give quite similar scale-dependence of the rupture area to that given by Kanamori and Anderson. Also, the results for large earthquakes show good agreement to the values from a kinematic inversion in previous studies. The test results also indicate the unavailability of the spectral self-similarity proposed by Aki because of the scale-dependent rupture velocity and the rupture velocity-dependent fmax/fc; however, the results do support the local similarity asserted by Ohnaka. It is also remarkable that the relationship between the rupture velocity and fmax/fc is quite similar to Kolmogorov’s hypothesis on a similarity in the theory of isotropic turbulence.

Determination of Individual Parameters for the Model Mapping of a Boundary Curve in the Phase Diagram of a Stratifying Liquid Mixture

The properties of a model based on the analogy between binary stratifying liquid mixtures and coupled self-oscillating systems is studied. In this paper, an expression representing a curve mapping in the phase diagram of a binary system is derived. The mapping of a boundary curve is presented in the form of the dependence between the ratio of characteristic frequencies assigned to pure components and the mixture concentration. In the framework of this model, the temperature dependence of the squared ratio of frequencies $$({\nu}_{a}/{\nu}_{b})^{2}$$ is also determined. One important aspect of this model is that the temperature-dependent characteristic frequencies $$\nu_{a}$$ and $$\nu_{b}$$ reflect the properties of pure components (instead of solutions). The problem is posed of investigating the physical aspects of the model proposed in this paper in more detail. In the present study, it has been shown that the use of Pade approximants makes it possible to determine the character of the temperature dependence of each characteristic frequency separately on the basis of the temperature dependence derived by linking to experimental data for the ratio $$({\nu}_{a}/{\nu}_{b})^{2}$$ .

An Automatic Bearing Fault Diagnosis Method Based on Characteristics Frequency Ratio.

Bearing is a key component of satellite inertia actuators such as moment wheel assemblies (MWAs) and control moment gyros (CMGs), and its operating state is directly related to the performance and service life of satellites. However, because of the complexity of the vibration frequency components of satellite bearing assemblies and the small loading, normal running bearings normally present similar fault characteristics in long-term ground life experiments, which makes it difficult to judge the bearing fault status. This paper proposes an automatic fault diagnosis method for bearings based on a presented indicator called the characteristic frequency ratio. First, the vibration signals of various MWAs were picked up by the bearing vibration test. Then, the improved ensemble empirical mode decomposition (EEMD) method was introduced to demodulate the envelope of the bearing signals, and the fault characteristic frequencies of the vibration signals were acquired. Based on this, the characteristic frequency ratio for fault identification was defined, and a method for determining the threshold of fault judgment was further proposed. Finally, an automatic diagnosis process was proposed and verified by using different bearing fault data. The results show that the presented method is feasible and effective for automatic monitoring and diagnosis of bearing faults.

An Improved Empirical Mode Decomposition Based on Adaptive Weighted Rational Quartic Spline for Rolling Bearing Fault Diagnosis

As a powerful time-frequency signal analysis technique, empirical mode decomposition (EMD) has been commonly applied in fault diagnosis. However, the cubic spline-curve often causes outstanding over and undershoot problem, which significantly limits the performance of conventional EMD. To address this problem, an improved EMD (I-EMD) based on adaptive weighted rational quartic spline is proposed. Firstly, the original cubic spline interpolation in conventional EMD is replaced with the weighted rational quartic spline interpolation (WRQSI) which has two adjustable shape control parameters. Secondly, a novel parameter selection criterion termed envelope characteristic frequency ratio (ECFR) is designed to guide the construction of the optimal local envelope. And simulation analysis proved that ECFR is not only sensitive to the characteristic information but also robust to the weak noise and sudden impulses. Subsequently, the optimal shape control parameter can be searched by grasshopper optimization algorithm (GOA) using the maximum ECFR as the objective function. Then the sensitive modes selected via weighted Kurtosis index are employed for further Hilbert envelope spectrum analysis. Finally, two case studies on rolling bearing fault diagnosis are constructed to verify the rationality and effectiveness of the I-EMD method. The results show that I-EMD method can evidently solve the over and undershoot problem and restrain the mode mixing phenomenon. Moreover, I-EMD also performs better fault feature extraction ability under the same conditions compared with EMD, VMD and CEEMDAN. So it is expected that I-EMD will serve as a potential improvement for signal processing, fault feature extraction and fault diagnosis.

Bandwidth Fourier decomposition and its application in incipient fault identification of rolling bearings

The incipient fault identification of rolling bearings is of great significance in avoiding the occurrence of malignant accidents in rotating machinery. However, at early stages the fault-related features are weak and easily contaminated by environmental noise, making them difficult to identify by traditional methods. Hence, in this paper, a new optimized Fourier spectrum decomposition method, termed bandwidth Fourier decomposition (BFD), is proposed for early fault detection in rolling bearings. Firstly, in the BFD method, the vibration signal is adaptively decomposed into sparse narrow-band sub-signals in the frequency domain through bandwidth optimization. In order to improve the performance of spectrum decomposition, a new bandwidth estimation method and an improved variable initialization strategy are proposed on the basis of spectral energy distribution. Then, the obtained sub-signals are converted into time-domain bandwidth mode functions (BMFs) by inverse Fourier transform. After that, the fault characteristic frequency ratio (FCFR) is introduced to select the effective component from the decomposition results. Finally, the bearing faults are identified by matching the envelope spectrum with the defect frequency of the theoretical calculation. To verify the validity of the proposed method, simulation and experimental analysis are carried out in this paper. Preliminary results indicate that the proposed BFD can effectively enhance the recognition of incipient faults in rolling bearings. The superiority of the proposed BFD is also demonstrated by comparing it with ensemble empirical mode decomposition (EEMD), variational mode decomposition (VMD) and an improved kurtogram method.

A Hybrid Approach for Weak Fault Feature Extraction of Gearbox

A novel hybrid fault diagnosis method based on ensemble empirical mode decomposition and weighted adaptive multi-scale morphological analysis (WAMMA) is proposed to detect the early damage of gearboxes. In this method, we propose a characteristic frequency ratio (CFR) method to determine the weighted coefficient for each scale of AMMA. First, multiple scales are obtained using the AMMA method. Second, the weighted coefficient of each scale in the AMMA method is calculated using the CFR. Third, the final results can be obtained by multiplying the weighted coefficients and filtering results with all scales. Since the performance of each scale of AMMA is evaluated using the CFR, the demodulation ability can be effectively improved. However, the WAMMA is easily disturbed by heavy noise when extracting early fault feature directly. A method combined EEMD with the WAMMA is proposed. The effectiveness of the proposed method has been verified using two experimental vibration signals of gearboxes. The results demonstrate that the proposed method has a superior performance in the extraction of weak fault characteristics of gearboxes in comparison with the WAMMA and EEMD-AMMA methods.

A Coarse TF Ridge-Guided Multi-Band Feature Extraction Method for Bearing Fault Diagnosis Under Varying Speed Conditions

Currently, most of the vibration signal analysis methods for bearing fault diagnosis under varying speed conditions are based on the resampling technology with shaft rotational frequency (SRF). However, the SRF obtained by a fixed tachometer or time-frequency (TF) ridge detection reduces measuring flexibility or introduces errors inevitably. In this paper, a multi-band feature extraction method using coarse TF ridge-guided variational nonlinear chirp mode decomposition (VNCMD) is proposed for bearing fault diagnosis under varying speed conditions. Specifically, the proposed method is conducted as follows. First, the low-frequency component (LFC) and resonance component are extracted by the low-pass filtering and the fast kurtogram method, respectively, to alleviate the noise interference. Second, the coarse TF ridges are identified by a tractable ridge estimation method that is based on the TF representation for preliminary selection of the initial instantaneous frequency. Third, the coarse TF ridge-guided VNCMD is constructed to track the SRF and instantaneous fault characteristic frequency (IFCF) from the envelope signals of the LFC and the resonance component, respectively. Finally, the characteristic frequency ratio is computed on the basis of the values of SRF and IFCF to determine the fault type of ball bearing without resampling. The simulation studies and experimental verifications confirm that the proposed method can accurately locate bearing defect types and outperforms some existing methods.

A fault diagnosis method for planetary gearboxes under non-stationary working conditions using improved Vold-Kalman filter and multi-scale sample entropy

This paper presents a novel signal processing scheme by combining an improved Vold-Kalman filter and the multi-scale sample entropy (IVKF-MSSE) for planetary gearboxes under non-stationary working conditions. In this scheme, we propose a method based on the characteristic frequency ratio (CFR) to select the VKF bandwidth. First, a CFR is adopted to select a VKF bandwidth with the largest CFR value as the optimal VKF bandwidth. Second, IVKF is used to extract fault-induced information under time-varying speed conditions. Because an optimal bandwidth is used in VKF, the feature extraction capability of VKF is enhanced. Then, the MSSE is applied to extract gearbox fault features. After that, the Laplacian score (LS) approach is introduced to refine the fault features by sorting the scale factors. At the end, the selected features are fed into the least square support vector machine (LSSVM) for effective fault pattern identification. Simulation and experimental vibration signals are employed to evaluate the effectiveness of the proposed method. Results show that the proposed method outperforms the auto-regressive AR-MSSE, VKF-MSSE and EEMD-MSSE in identifying fault types of planetary gearboxes.

Asteroseismic and orbital analysis of the triple star system HD 188753 observed by Kepler

Context. The NASA Kepler space telescope has detected solar-like oscillations in several hundreds of single stars, thereby providing a way to determine precise stellar parameters using asteroseismology. Aims. In this work, we aim to derive the fundamental parameters of a close triple star system, HD 188753, for which asteroseismic and astrometric observations allow independent measurements of stellar masses. Methods. We used six months of Kepler photometry available for HD 188753 to detect the oscillation envelopes of the two brightest stars. For each star, we extracted the individual mode frequencies by fitting the power spectrum using a maximum likelihood estimation approach. We then derived initial guesses of the stellar masses and ages based on two seismic parameters and on a characteristic frequency ratio, and modelled the two components independently with the stellar evolution code CESTAM. In addition, we derived the masses of the three stars by applying a Bayesian analysis to the position and radial-velocity measurements of the system. Results. Based on stellar modelling, the mean common age of the system is 10.8 ± 0.2 Gyr and the masses of the two seismic components are MA = 0.99 ± 0.01 M⊙ and MBa = 0.86 ± 0.01 M⊙. From the mass ratio of the close pair, MBb/MBa = 0.767 ± 0.006, the mass of the faintest star is MBb = 0.66 ± 0.01 M⊙ and the total seismic mass of the system is then Msyst = 2.51 ± 0.02 M⊙. This value agrees perfectly with the total mass derived from our orbital analysis, Msyst = 2.51−0.18+0.20 M⊙, and leads to the best current estimate of the parallax for the system, π = 21.9 ± 0.2 mas. In addition, the minimal relative inclination between the inner and outer orbits is 10.9° ± 1.5°, implying that the system does not have a coplanar configuration.

Early fault feature extraction of rolling bearing based on ICD and tunable Q-factor wavelet transform

When a fault occurs on bearings, the measured bearing fault signals contain both high Q-factor oscillation component and low Q-factor periodic impact component. TQWT is the improvement of the traditional single Q-factor wavelet transform, which is very suitable for separating the low Q-factor component from the high Q-factor component. However, the accuracy of its decomposition heavily depended on the selection of Q-factors. There is no reported simple but effective method to select the Q-factors with enough accuracy. This study aims to develop a strategy to diagnostic the early fault of rolling bearings. In this paper, a characteristic frequency ratio (CFR) is used to optimize Q-factors of TQWT (OTQWT). However, directly application of OTQWT is difficult to extract fault signatures at early stage due to the weak fault symptoms and strong noise. A strategy of combination of intrinsic characteristic-scale decomposition (ICD) and TQWT is proposed. ICD owns significant advantages on computation efficiency and alleviation of mode mixing. The effectiveness of the proposed strategy is tested with both simulated and experimental vibration signals. Meanwhile, comparisons are conducted between the proposed method and other methods like: envelope demodulation and EEMD-TQWT. Results show that the proposed method has superior performance in extracting fault features of defective bearings at an early stage.

Impedance Spectra of Energy-Storage Electrodes Obtained with Commercial Three-Electrode Cells: Some Sources of Measurement Artefacts

ABSTRACT The information derived from electrochemical ac impedance spectroscopy (EIS) measurements on Li-ion batteries is important for the non-destructive diagnostics of the battery state of health, as well as for identifying the best cell materials for automotive electrification and characterizing their electrochemical performance. The present work aims to identify and determine some of the sources of systematic errors in EIS measurements on small (“coin” type) cells. The data reveal perfect capacitive behavior of both electrodes at low frequencies and kinetic limitation at higher frequencies. Artefacts due to a small misalignment between the working electrode (WE) and the counter electrode (CE) in symmetric cells, or to an unfavorable ratio of characteristic frequencies for the WE and CE in mixed cells, are negligible only in aqueous electrolyte solutions of high conductivity. In contrast, measurement artefacts seriously complicate impedance measurements in non-aqueous, low conductivity solutions, as is the case for non-aqueous electrolyte solutions used in Li-ion batteries and supercapacitors. Our study shows that the necessary condition for a good CE in 3-electrode impedance measurements, namely, a small impedance (relative to the WE) at low frequencies is not sufficient to ensure the absence of high-frequency artefacts, since it must also have a small impedance at high frequencies. When this is not the case (a common practical situation), a severe contamination of the measured WE impedance by the large, high-frequency CE impedance will occur during 3-electrode impedance measurements. The report ends with recommendations regarding best practices for electrochemical ac impedance measurements.

Non-Langevin high-temperature magnetization of nanoparticles in a weak magnetic field

Experimental evidence and theoretical substantiation are presented for the asymptotic behavior of high-temperature magnetization of an ensemble of nanoparticles in a weak magnetic field, which was predicted earlier and which differs qualitatively from the “Langevin” limit for ideal superparamagnetic particles. It is shown that the physical reason for the new asymptotic behavior is the temperature-independent “positive” tilt of the uniform magnetization vector at local energy minima in the direction of the field; this asymptotic behavior is associated with the nonstandard thermodynamics of single-domain particles, which depends on the ratio of characteristic frequencies of regular precession and random diffusion of this vector. An alternative approach is proposed for describing the magnetic dynamics of an ensemble of nanoparticles in a magnetic field, and the precession orbits of the magnetization vector are considered as stochastic states of each particle, whereas each state is characterized by the trajectory-averaged value of magnetization.

NONLINEAR ADIABATIC DYNAMICS OF SMALL FERROMAGNETIC PARTICLES

The purpose of this work is to present an analytical description of dynamics of small ferromagnetic particles with uniaxial anisotropy and the slowly varying magnetic field applied at an arbitrary angle to the anisotropy axis. Considerable attention is given to the nonlinear aspects of adiabatic dynamics. Theoretical analysis based on the consideration of the Landau–Lifshitz–Gilbert equation employs an asymptotic expansion similar to the famous semi-classical WKBJ solution of quantum mechanics equations. The small parameter of the expansion is the ratio of characteristic frequency of the applied magnetic field to the precession frequency. The nonlinear equation describing the slow dynamics of small ferromagnetic particles is derived. The applicability conditions of the developed theory are presented. The nonlinear corrections to the precessional frequency are derived. It is shown that in numerical simulation using the Landau–Lifshitz–Gilbert equation it is not possible to neglect precessional terms at long simulation times due to the nonlinear dynamic correction to a position of the orbit center.

Adiabatic dynamics of small ferromagnetic particles

The purpose of this work is to present an analytical description of dynamics of small ferromagnetic particles (SFP) with uniaxial anisotropy energy and slowly varying magnetic field applied at an arbitrary angle to it. Theoretical analysis based on the consideration of the Landau–Lifshits–Gilbert (LLG) equation employs an asymptotic expansion similar to the famous semiclassical WKBJ solution of quantum mechanics equations. The small parameter of the expansion is the ratio of characteristic frequency of the applied magnetic field to the precession frequency. The equation describing slow dynamics of SFP is derived. Different cases of slow variation of magnitude and direction of magnetic field are considered. The formal solution of linearized equations is obtained and exact solutions are presented in the cases of magnetic field parallel and perpendicular to the anisotropy axis.

Spectral Representation Theory for Dielectric Behavior of Nonspherical Cell Suspensions**The project supported by the Research Grants Council of the Hong Kong SAR Government

Recent experiments revealed that the dielectric dispersion spectrum of fission yeast cells in a suspension was mainly composed of two sub-dispersions. The low-frequency sub-dispersion depended on the cell length, while the high-frequency one was independent of it. The cell shape effect was simulated by an ellipsoidal cell model but the comparison between theory and experiment was far from being satisfactory. Prompted by the discrepancy, we proposed the use of spectral representation to analyze more realistic cell models. We adopted a shell-spheroidal model to analyze the effects of the cell membrane. It is found that the dielectric property of the cell membrane has only a minor effect on the dispersion magnitude ratio and the characteristic frequency ratio. We further included the effect of rotation of dipole induced by an external electric field, and solved the dipole-rotation spheroidal model in the spectral representation. Good agreement between theory and experiment has been obtained.

Coherent oscillations and incoherent tunneling in a one-dimensional asymmetric double-well potential.

For a model one-dimensional asymmetric double-well potential we calculated the so-called survival probability (i.e., the probability for a particle initially localized in one well to remain there). We use a semiclassical (WKB) solution of the Schrödinger equation. It is shown that behavior essentially depends on transition probability, and on a dimensionless parameter Lambda that is a ratio of characteristic frequencies for low-energy nonlinear in-well oscillations and interwell tunneling. For the potential describing a finite motion (double-well) one has always a regular behavior. For Lambda<<1, there are well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However, for Lambda>>1 there are no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case, one may not restrict himself to only resonance pair levels. The number of levels perturbed by tunneling grows proportionally to square root of [Lambda] (in other words, instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of Lambda one has a crossover between both limiting cases, namely, the exponential decay with subsequent long period recurrent behavior.