# A Tacholess Order Tracking Method Based on Inverse Short Time Fourier Transform and Singular Value Decomposition for Bearing Fault Diagnosis

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## Abstract

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## 1. Introduction

## 2. The Fundamental of SVD

**A**, which the diagonal entries of $\Sigma $ are non-negative values in decreasing order of magnitude, and the positive ones are the singular values of

**A**. That is $\Sigma =\left[diag({\sigma}_{1},{\sigma}_{2},\cdots ,{\sigma}_{m}),0\right]\in {R}^{m\times n}$, ${\sigma}_{1}\ge {\sigma}_{2}\ge ,\cdots ,{\sigma}_{m}>0$.

_{i}is the corresponding sub-matrix of the ith sub-signal.

## 3. Relative Instantaneous Frequency Ratio

#### 3.1. Effect of the Relative Frequency Ratio on Phase Extraction Precision

#### 3.2. The Maximum Relative Frequency Ratio for the Maximum Phase Error of Less Than 5%

## 4. The Effect of the Noise on the Phase Extraction Using SVD

## 5. A New Tacholess Order Tracking Method

- Step 1: Short-time Fourier transform

- Step 2: Component selection and filtering

- Step 3: Inverse short-time Fourier transform

- Step 4: SVD decomposition

- Step 5: Phase extraction

- Step 6: Resampling

## 6. Application to Experimental Data

#### 6.1. Case Study 1

#### 6.2. Case Study 2

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The sinusoidal signal and the first five sub-signals decomposed using the different number of rows m.

**Figure 2.**(

**a**) The actual unwrapped phase and the extracted unwrapped phase using the first two sub-signals; (

**b**) the phase error of the unwrapped phase extracted using the sum of the first two sub-signals.

**Figure 3.**The frequency variable sinusoidal signal of the different relative frequency ratio $\sigma $.

**Figure 4.**(

**a**) The actual and the extracted instantaneous frequencies for different relative frequency ratios; (

**b**) the phase error of the extracted unwrapped phase before phase shifting.

**Figure 5.**(

**a**) The actual and the extracted unwrapped phase for different relative frequency ratios before phase shifting; (

**b**) the phase error of the extracted unwrapped phase using the sum of the first two sub-signals before phase shifting.

**Figure 6.**(

**a**) The actual and the extracted unwrapped phase for different relative frequency ratios after phase shifting; (

**b**) the phase error of the extracted unwrapped phase using the sum of the first two sub-signals after phase shifting.

**Figure 8.**(

**a**) The actual and the extracted unwrapped phase for different frequency variation ratios after the two ends are discarded; (

**b**) the phase difference between the actual and the corresponding extracted unwrapped phase of the first sub-signal.

**Figure 9.**The maximum $\sigma $ for the different initial frequency ${f}_{0}$ with the maximum phase error in one cycle under 5%.

**Figure 10.**The maximum phase error of different SNRs for different number of rows of the Hankel matrix.

**Figure 14.**The defect of the tested bearings: (

**a**) outer ring defect on FAG-804989 of case 1; (

**b**) inner ring defect on SKF NU215 of case 2.

**Figure 16.**(

**a**) The shaft rotation speed of case study 1; (

**b**) the filtered spectrum of the selected component.

**Figure 17.**(

**a**) time-domain signal of the filtered spectrum of case study 1; (

**b**) final shaft phase using the time domain signal.

**Figure 21.**(

**a**) The shaft rotation speed of case study 1; (

**b**) the filtered spectrum of the selected component.

**Figure 22.**(

**a**) The obtained time-domain signal of the filtered spectrum of case study 1; (

**b**) the obtained shaft phase using the time domain signal.

Bearing Code | FTF/NX | BSF/NX | BPFO/NX | BPFI/NX |
---|---|---|---|---|

SKF NU215 | 1.40 | 11.04 | 25.16 | 33.69 |

FAG-804989 | 0.47 | 7.43 | 18.40 | 20.60 |

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**MDPI and ACS Style**

Xu, L.; Chatterton, S.; Pennacchi, P.; Liu, C.
A Tacholess Order Tracking Method Based on Inverse Short Time Fourier Transform and Singular Value Decomposition for Bearing Fault Diagnosis. *Sensors* **2020**, *20*, 6924.
https://doi.org/10.3390/s20236924

**AMA Style**

Xu L, Chatterton S, Pennacchi P, Liu C.
A Tacholess Order Tracking Method Based on Inverse Short Time Fourier Transform and Singular Value Decomposition for Bearing Fault Diagnosis. *Sensors*. 2020; 20(23):6924.
https://doi.org/10.3390/s20236924

**Chicago/Turabian Style**

Xu, Lang, Steven Chatterton, Paolo Pennacchi, and Chang Liu.
2020. "A Tacholess Order Tracking Method Based on Inverse Short Time Fourier Transform and Singular Value Decomposition for Bearing Fault Diagnosis" *Sensors* 20, no. 23: 6924.
https://doi.org/10.3390/s20236924