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The authors have declared that no competing interests exist.

Stockwell transform(ST) time-frequency representation(ST-TFR) is a time frequency analysis method which combines short time Fourier transform with wavelet transform, and ST time frequency filtering(ST-TFF) method which takes advantage of time-frequency localized spectra can separate the signals from Gaussian noise. The ST-TFR and ST-TFF methods are used to analyze the fault signals, which is reasonable and effective in general Gaussian noise cases. However, it is proved that the mechanical bearing fault signal belongs to Alpha(

The fault signal received by the sensors is non-stationary when the rotating machinery bearing break down, and time-frequency representation is a effective method to analyze the non-stationary signal[

The mentioned methods in [

In

Similarly, the time frequency filtering methods in [

In this paper, the improved S transform time-frequency representation and S transform time-frequency filtering methods based on fractional lower statistical moment are proposed for machine fault diagnosis. The paper is structured in the following manner.

The signal received by the vibration sensors is a non-stationary mixture when the rolling machinery bearing breaks down, which includes the fault signal, the other vibration signals, and the noises, etc. The experimental signals are selected from the case western reserve university data center in this paper[

The time waveforms of the normal signal received by DE and FE accelerometers are shown in

parameters | |||||
---|---|---|---|---|---|

S transform of a non-stationary signal can be defined as:

The mechanical fault signal containing ^{<p>}(^{p−1}⋅^{<p>}(^{p−1}⋅^{*}(

The ST method is proposed based on short-time Fourier transform(STFT) and continuous wavelet transform (CWT). We assume that the Eq (^{−j2πft}, then, the Eq (

After FLOST in the Eq (

When

Fourier transform of

The traditional ST-TFR method fails under

According to the Eq (

_{1} ≤ _{2}, _{1} ≤ _{2}) as time-frequency passed domain, and the other regions are regarded as noises.

After

We assume that a time-time function _{1}(_{1}(_{1}(

According to the calculation process in section 4.2, we multiply weight function

With _{s} = 1/_{0}, _{s}/_{0}. Then, the discrete FLOST form in the Eq (^{<p>}(

According to fractional lower order Fourier transform in the Eq (

Step 1:Computing

Step 2: Selecting appropriate weight function

Step 3: Computing IFLOST with substituting

Step 4: Performing inverse operation of

We design the following experiments to test the proposed FLOST method, FLOST-TFF algorithm, the existing ST method and ST-TFF algorithm. The test signals _{1}(_{2}(_{1}(_{2}(_{3}(_{4}(_{1} = 80, _{2} = 180, _{10} {^{2}]/^{α}}.

In this simulation, we select _{1}(_{1}(

In this simulation, _{2}(_{2}(_{2}(_{2}(_{2}(_{2}(

In this simulation, we select _{2}(_{5} and _{6} are two original LFM signals,

Let _{5} and _{6} in different

In this simulation, the experimental signals adopt the out race bearing fault in BA, DE and FE from

This paper proves that bearing fault signals belong to

(MAT)

(MAT)

(MAT)

(MAT)

This work is financially supported by Natural Science Foundation of China (61261046, 61362038), the Natural Science Foundation of Jiangxi Province China(20151BAB207013), the Research Foundation of health department of Jiangxi Province China(20175561), science and technology project of jiujiang university China(2016KJ001, 2016KJ002, 2016KJ003).