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Signals with multiple components and fast-varying instantaneous frequencies reduce the readability of the time-frequency representations obtained by traditional synchrosqueezing transforms due to time-frequency blurring. We discussed a vertical synchrosqueezing transform, which is a second-order synchrosqueezing transform based on the short-time Fourier transform and compared it to the traditional short-time Fourier transform, synchrosqueezing transform, and another form of the second-order synchrosqueezing transform, the oblique synchrosqueezing transform. The quality of the time-frequency representation and the accuracy of mode reconstruction were compared through simulations and experiments. Results reveal that the second-order frequency estimator of the vertical synchrosqueezing transform could obtain accurate estimates of the instantaneous frequency and achieve highly energy-concentrated time-frequency representations for multicomponent and fast-varying signals. We also explored the application of statistical feature parameters of time-frequency image textures for the early fault diagnosis of roller bearings under fast-varying working conditions, both with and without noise. Experiments showed that there was no direct positive correlation between the resolution of the time-frequency images and the accuracy of fault diagnosis. However, the early fault diagnosis of roller bearings based on statistical texture features of high-resolution images obtained by the vertical synchrosqueezing transform was shown to have high accuracy and strong robustness to noise, thus meeting the demand for intelligent fault diagnosis.

Fault diagnosis of rotating machinery has long been studied, and many effective fault diagnosis methods have been proposed. A nonideal environment, with varying operating conditions, time-varying loads, fast-varying speeds, and random transient phenomena, results in extraordinarily complex vibration patterns in rotating machinery [

As mentioned above, nonstationary operating conditions cause the components of vibration signals to show fast variations of both the IF and instantaneous amplitude (IA). Time-frequency analysis (TFA) is a powerful method to obtain insights into the time-frequency (TF) structures of nonstationary signals. However, traditional TFA methods cannot meet the needs of nonstationary signal analysis [

The change of speed has a considerable impact on the vibration signals through modulation, and a small change will give rise to significant frequency aliasing. When the speed changes significantly, the IF changes faster, the vibration signal becomes more nonstationary, and the TF spectrogram becomes more blurred. Much research has been conducted on the theory and application of the SST, most importantly in the analysis of multicomponent nonstationary signals. Oberlin et al. [

The second-order SST, high-order SST, multi-SST, and MSST have made great breakthroughs in the acquisition of highly concentrated TFR, and TF images obtained from nonstationary and multicomponent signals have high sharpness and good legibility. However, the robustness to noise has not been verified in the research of high-order SSTs, which is important for equipment fault diagnosis, because noise is unavoidable in real engineering. Multi-SST methods have been validated by numerical simulations, but experimental verification has been carried out at constant speeds, and the performance depends on the iteration number

For machine fault diagnosis, many methods, such as the multifeature entropy distance method, have been proposed. Most TFA-related methods rely on observations of an experienced technician to detect fault frequency features based on the frequency distributions in TF images, that is, manual diagnosis. This depends on the quality of the TFR, especially under time-varying operating conditions. This is the reason that scholars are committed to improving the IF and IA estimation accuracy and obtaining highly concentrated TF ridges, which can reduce the recognition difficulty for human eyes. For intelligent fault diagnosis, although high-order SSTs and the MSST greatly increase the concentration of TFR, there has been no verification that high-resolution TFR can improve the accuracy of intelligent diagnosis, because it depends on the extraction of fault characteristics. Therefore, in addition to the acquisition of high-quality TF images, this paper aims to determine whether highly concentrated TF images result in high accuracy of intelligent diagnosis using the textures of images with the same recognition algorithm, especially under fast-varying operating conditions and complex frequency structures. Based on a comprehensive comparison and our previous research [

The remainder of this paper is organized as follows. Section

Multicomponent and fast-varying signals are usually composed of several single-component fast-varying signals overlapping in the time domain. The mathematical model of a multicomponent mixed signal containing

The Fourier transform of the function

The representation

Thus, we can obtain the definitions of the so-called reassignment operators for the STFT reassignment method.

Frequency reassignment operator:

Group delay reassignment operator:

Assuming that the window

If the signal is analytic (i.e.,

As an alternative TF reassignment method, the STFT-based SST has three main steps. The first is to calculate the STFT representation

Knowing the time-varying phase function

A previous study showed that the STFT-based SST can also obtain reasonable accuracy with slowly time-varying frequency-modulated signals [

The hypothesis of the first-order SST is suitable for low-modulation signals; that is, for any time

Motivated by this heuristic, a second-order difference using the phase of the STFT was proposed by Oberlin et al. [

The synchrosqueezed Fourier transform “squeezes” the TF spectrum by reassigning the amplitude from

The reason the term “vertical” is used is related to the oblique synchrosqueezing transform (OSST), which is introduced in Section

This significantly enhances the sharpness of the TFR, which leads to a more accurate and robust estimate of the local frequency of the signal, as illustrated below. This gives a sharpened energy distribution on the phase space [

The SST only reallocates the energy of signals in the frequency direction, while ignoring the time direction [

The strategy of the FSST and VSST to deal with perturbations is to integrate the coefficients near the ridge to compensate for the error caused by the estimated IF. However, the OSST simultaneously moves the coefficients in the time and frequency directions and therefore cannot achieve regularized treatment.

A multicomponent and fast-varying test signal is represented by

The three components of the test signal are illustrated in Figure

Three components of multicomponent and fast-varying test signal and the reconstruction error of each component based on short-time Fourier transform (STFT-) based synchrosqueezing transform (FSST).

Accuracy of mode retrieval expressed in SNR.

Mode components | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

SNR/dB | 82.39 | 6.99 | 2.95 |

Figure

Time-frequency rearrangements (TFRs) obtained by (a) STFT, (b) synchrosqueezing transform (SST), (c) oblique SST (OSST), and (d) vertical SST (VSST).

Local enlarged images of high-frequency bands marked by yellow rectangle in Figure

Local enlarged images of crossed region marked by red rectangle in Figure

To facilitate the comparison of the quality of each TFR, we measured the amount of information contained in the maximum coefficient amplitude in each time step and adopted a cumulative normalized energy [

Noise is inevitable in scientific research and practical engineering. Thus, robustness analysis should be conducted to support the bearing fault experimental study presented below. Two groups of numerical simulation analyses were carried out, one with added Gaussian white noise (noise level SNR = 0 dB) and one without. RM is a general method that is suitable for sharp TF representations; it has been proven to be able to accurately represent multicomponent signals [

Cumulative normalized energy as a function of the number of coefficients for numerical simulation tests: (a) noise-free and (b) with noise (SNR = 0 dB).

Since the signal contained modulation and noise, it can be deduced from Figure

Because the relationship between the output SNR and

Quality of reconstruction as a function of (d): (a) noise-free and (b) with noise.

Based on the above numerical analysis of the test signal

Therefore, the VSST was applied for roller bearing fault signal processing with multiple components and fast-varying conditions to study whether this method can minimize FM interference, which cannot be avoided in practical engineering. This is equivalent to eliminating the false energy in the monitoring signals as much as possible and obtaining high-resolution TFR sensitivity to faults. Whether TF images obtained by this method can accurately capture bearing fault features for fault diagnosis also warrants further study, as it will have important engineering value for the early prediction and diagnosis of equipment faults caused by bearings.

We carried out a practical implementation study to demonstrate the effectiveness of the VSST for the fault diagnosis of roller bearings at fast-varying speeds and its effect on improving the clarity of the TFRs for signals with strong FM. The experimental platform is shown in Figure

Schematic diagram of experimental platform.

The left column in Figure

TFRs under different fault states: (a) fault-free, (b) ball fault, (c) inner race fault, (d) outer race fault, and (e) combined fault of inner and outer rings. Left column, from top to bottom: TFRs of STFT, SST, OSST, and VSST. Partial enlarged images of each representation are displayed in the right column.

Figures

Normalized energy as a function of the number of coefficients for experimental signals: (a) fault-free, (b) ball fault, (c) inner ring fault, (d) outer ring fault, and (e) combined fault of inner and outer rings.

We examined the relationship between SNR and

Output signal-to-noise ratio (SNR) for original signal (top) and original signal with added noise (bottom) under different working conditions: (a) fault-free, (b) inner race fault, (c) outer race fault, (d) ball fault, and (e) combined fault of inner and outer rings.

The results showed that the reconstruction accuracy of the STFT was far lower than that of the other two, and it did not meet the signal reconstruction accuracy requirements. While the reconstruction effect of the SST was worse than that of the VSST in the inner ring fault state, the reconstruction effects of the VSST and SST in other states were basically similar, except that the effect of the SST was slightly lower. This was mainly because the fast-varying speed scheme in this experiment was linear acceleration, so the vibration signals contained linear AM and linear FM. The analysis results show that when

The TFRs obtained in Section

Haralick et al. [

This experiment also adopted the vibration data in five states collected by the above-mentioned roller bearing experimental platform with a fast-varying speed. Forty samples were randomly selected for each state. The colored TF images transformed using the STFT, SST, OSST, and VSST were converted to gray TF images, and 200 gray images were obtained for each TF transformation method, for a total of 800 gray images between the four methods. The GLCM of each gray image was calculated, and 24 SFPs were separately extracted to generate a feature matrix with dimensions of 200 × 24 for each TF transform method. The distributions of the SFPs for the five states are shown for the STFT, SST, OSST, and VSST in Figures

Distribution of 24 statistical feature parameters (SFPs) obtained by (a) STFT, (b) SST, (c) OSST, and (d) VSST.

The sensitivity of the SFPs obtained by the four TF transform methods to the different states of the bearing was comprehensively evaluated from the aspects of intraclass aggregation and interclass separability. Figure

From the viewpoint of interclass separability, the distribution of the SFPs calculated by the OSST was the worst. In the normal and inner ring fault states, the distribution area of the maximum probability in four directions obtained by the VSST, SST, and SFTF had the same distribution interval, that is, they completely overlapped. The eigenvalue distribution interval of the other three states obtained by the VSST was slightly improved by the SST, and in particular, it was much better than that of the STFT. Thus, the performance ranking of the maximum probability of the four TF transformation methods was VSST > SST > STFT > OSST. Using the same judgment method, the intraclass discrimination of the other five SFPs in the four directions was compared. The order when sorting by the entropy and the contrast was still VSST > SST > STFT > OSST. However, for the correlation, energy, and homogeneity, they were in the order of VSST > STFT > SST > OSST.

Roller bearing fault diagnosis in fast-varying speed conditions relies heavily on capturing the time-varying fault features. From the comparison of intraclass aggregation and interclass separability, the six SFPs obtained by the VSST had the best sensitivity to different states of the bearing; thus, these parameters can be used for fault identification and diagnosis.

From the direct observation of the time-frequency images and the quality of the TFR under different fault states presented in Sections

We have shown that more concentrated TF images do not necessarily provide high-quality SPFs, and for intelligent fault diagnosis approaches, feature extraction is key. We may need to conduct further research on whether we can extract characteristics that are conducive to intelligent fault diagnosis from highly concentrated TF images using methods such as the high-order SST, multi-SST, and MSST.

Based on the above simulation and experimental data analysis, we compared the bearing fault diagnosis performances based on the texture features of TF images obtained by different methods. The flowchart in Figure

Flowchart of bearing fault diagnosis based on gray level cooccurrence matrix (GLCM) from grayed TFRs.

Forty vibration data samples under each of the four fault states and the fault-free state were obtained from the test platform, and each of the 40 samples was divided into 20 samples for training and 20 samples for testing. Each testing dataset contained 100 samples, and there were 20 samples for each fault state. As shown in Figure

Comparison of accuracy of state recognition.

Method | STFT | SST | OSST | VSST | ||||
---|---|---|---|---|---|---|---|---|

Noise | Without | With | Without | With | Without | With | Without | With |

Accuracy (%) | 97.19 | 84.84 | 94.40 | 51.46 | 90 | 80.64 | 99.2 | 91.33 |

Table

We compared the performances of the vertical and oblique second-order synchrosqueezing transforms for analyzing strongly modulated and multicomponent signals with a fast-varying IF to the traditional STFT, SST, and RM. Based on the simulations, the second frequency estimator of the VSST and OSST could estimate the IF more accurately. The accuracy of the IF estimation was also guaranteed to obtain more energy-concentrated TFRs than the STFT and SST, and the sharpness of the TF images obtained by the OSST was slightly higher than that obtained by the VSST. The VSST allowed for better mode reconstruction than the original SST, while the OSST did not, and it showed strong robustness to noise in the reconstruction. Experimental verification of the quality of the TFR and the accuracy of the mode reconstruction was conducted on a roller bearing experimental platform. As a postprocessing method, the VSST method produced more energy-concentrated TFRs than some nonreassignment and reassignment methods.

We experimentally studied practical applications for roller bearing fault diagnosis under fast-varying speed conditions. A comprehensive comparison of the STFT, SST, OSST, and VSST was conducted from the perspective of intraclass aggregation and interclass separability. Six SFPs in four directions were verified to be more sensitive at detecting fast-varying fault characteristics from gray TF images when the VSST was applied. An experimental study validated the VSST as a powerful tool for condition monitoring and bearing fault diagnosis under a fast-varying speed, and we concluded that TF images with high sharpness do not necessarily lead to high accuracy of fault diagnosis. The results clearly showed that the VSST method can effectively characterize the time-varying fault features by using texture features of TF images for fast-varying signals. Thus, the VSST method can meet the requirements of intelligent fault diagnosis.

Further work should be devoted to examining the influence of reassignment operators at high frequencies, especially in frequency-intersection regions when frequency components cross. This paper only studied the second-order SST based on the STFT. Further study is needed to introduce the second-order SST to the CWT, S-transform, and WPT, to understand the influence of different versions of the TFA on multicomponent and fast-varying signals, and to determine the application effect on intelligent fault diagnosis. The comparative analysis of high-order SSTs and the MSST can also be introduced in future studies.

Data are available upon request.

The authors declare that they have no conflicts of interest.

The authors are grateful for the financial support of this study. This research was funded by the Fundamental Research Funds for the Central Universities (Grant no. 2021YQJD14), the National Natural Science Foundation of China (Grant no. U1361127), the Yue Qi Distinguished Scholar Project of China University of Mining and Technology (Beijing, Grant no. 800015Z1145), and the National Key Research and Development Program of China (Grant no. 2016YFC0600900).