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Synchrosqueezing transform (SST) is a high resolution time frequency representation technology for nonstationary signal analysis. The short time Fourier transform-based synchrosqueezing transform (FSST) and the S transform-based synchrosqueezing transform (SSST) time frequency methods are effective tools for bearing fault signal analysis. The fault signals belong to a non-Gaussian and nonstationary alpha (

Synchrosqueezing transform is a new time frequency analysis technology for the nonstationary signals. Its principle is to calculate time frequency distribution of the signal, then squeeze the frequency of the signal in time frequency domain, and rearrange its time frequency energy, so as to improve time frequency resolution greatly. Synchrosqueezing transform mainly includes continuous wavelet transform-based synchrosqueezing transform [

Daubechies et al. firstly gave synchrosqueezing transform concept based on the continuous wavelet transform and proposed a continuous wavelet transform-based synchrosqueezing transform (WSST) time frequency representation method and its inverse method. The method squeezes the time frequency energy of continuous wavelet transform in a certain frequency range to nearby instantaneous frequency of the signal, and the time frequency resolution was improved effectively [

Recently, it is verified that probability density function (PDF) of the mechanical bearing fault signals has an obvious trail, which is a nonstationary and non-Gaussian distribution and belongs to

In this paper, the improved FLOFSST and FLOSSST time frequency representation technologies based on fractional lower order statistics and synchrosqueezing transform are presented for the bearing fault diagnosis under Gaussian and

Probability density function (PDF) of

The waveform of

PDFs of

The real bearing fault signals data are obtained from the Case Western Reserve University (CWRU) bearing data center [

The waveform of the bearing fault signals. (a) The normal signals in DE and FE; (b) the inner race fault signals in BA, DE, and FE; (c) the ball fault signals in BA, DE, and FE; (d) the outer race fault signals in BA, DE, and FE.

In order to further verify the pulse characteristics of bearing failure signals, we use

The parameters of the bearing fault signals based on

Parameters | |||||
---|---|---|---|---|---|

Normal | DE | 2.000 | −0.2863 | 0.0532 | 0.0121 |

FE | 2.000 | 1.000 | 0.0583 | 0.0236 | |

Inner race fault | BA | 1.7682 | 0.0872 | 0.0590 | 0.0062 |

DE | 1.4195 | 0.0155 | 0.2407 | 0.0175 | |

FE | 1.8350 | 0.0322 | 0.1495 | 0.0291 | |

Ball fault | BA | 1.9790 | 0.0592 | 0.0293 | 0.0055 |

DE | 1.8697 | 0.1215 | 0.0772 | 0.0193 | |

FE | 1.998 | −0.0371 | 0.0674 | 0.0321 | |

Outer race fault | BA | 1.6077 | −0.1731 | 0.0530 | 0.0012 |

DE | 1.1096 | 0.0433 | 0.1341 | 0.0367 | |

FE | 1.5435 | −0.0169 | 0.0968 | 0.0296 |

PDFs of the inner race fault, ball fault, and outer race fault signals are shown in Figures

PDFs of the normal and bearing fault signals. (a), (b) PDFs of the normal and inner race fault signals in DE, FE, and BA; (c), (d) PDFs of the normal and ball fault signals in DE, FE, and BA; (e), (f) PDFs of the normal and outer race fault signals in DE, FE, and BA.

Short time Fourier transform (STFT) of the fault machinery vibration signal contaminated by

Letting

Letting

Substituting (

Letting

Substituting (

Fourier transform of

The instantaneous frequency (IF) formula of

After synchrosqueezing the frequency in (

The FLOFSST can “squeeze” a frequency interval to a frequency point in the time frequency domain; therefore, the process can greatly improve the time frequency resolution.

A multicomponent signal can be expressed as

FLOSTFT is just as linear as STFT; then

The instantaneous frequency (IF) calculation method of

The corresponding instantaneous frequency calculation method of

By substituting (

According to the definition of inverse STFT-based synchrosqueezing transform in [

Step 1: compute

Step 2: compute instantaneous frequency

Step 3: solve

Step 4: solve the discrete values

Step 5: compute

In this section, we design the following experiments to compare the proposed FLOFSST method with the existing STFT, FLOSTFT, and FSST methods. The simulation signal

Let

The waveforms in time domain. (a) The signal

Time frequency representations of the signal

Time frequency representations of the signal

In order to compare the effectiveness of the IFSST and IFLOFSST methods, letting

The STFT, FLOSTFT, FSST, and FLOFSST time frequency methods in Figure

The STFT and FSST are unsuitable for

Figure

Figure

The fault machinery vibration signal contaminated by the noise may be given by

Equation (

Let

The right side of (

Letting

By substituting (

Fourier transform of

The instantaneous frequency of

By substituting (

For the calculation of IF and FLOSSST of a multicomponent signal,

FLOST is just as linear as ST; then

The IF calculation method of

The corresponding IF calculation method of

By substituting (

Multiplying

Let

For the real signal

In the piecewise constant approximation corresponding to the binning in

FLOST of the signal

Multiplying

Substituting (

We can reconstruct the signal

Step 1: compute

Step 2: solve

Step 3: compute instantaneous frequency

Step 4: solve

Step 5: compute the discrete values

Step 6: solve

In this section,

Time frequency representations of the signal

Time frequency representations of the signal

Letting

The waveforms of signal reconstruction under

Figure

The reconstructed signal

As a result, the SSST time frequency method and the ISSST signal reconstruction method are only suitable to analyze and reconstruct the signals under Gaussian noise environment, but the improved FLOSSST and IFLOSSST methods can work in Gaussian and

In this simulation, the experiment signal adopts the bearing outer race fault signal (DE) in Section

The time frequency representations of the outer race fault signal

Figures

In order to further prove the advantages of the improved FLOFSST and FLOSSST methods,

The time frequency representations of the outer race fault signal under

The data used to support the findings of this study are provided in the Supplementary Materials.

The authors declare that they have no conflicts of interest.

This work was financially supported by the Natural Science Foundation of China (61962029), the Natural Science Foundation of Jiangxi Province, China (20192BAB207002), the Science and Technology Project of Provincial Education Department of Jiangxi (GJJ170954), and the Science and Technology Project of Jiujiang University, China (2014SKYB009).

This section contains the original experimental data of this paper.