# Fault Feature Extraction Method of Ball Screw Based on Singular Value Decomposition, CEEMDAN and 1.5DTES

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## Abstract

**:**

## 1. Introduction

## 2. Introduction of the Basic Theory

#### 2.1. Singular Value Decomposition

_{1}≥ σ

_{2}≥ … ≥ σ

_{r}. If the source signal X is composed of a useful signal and noise, the singular value σ

_{i}of matrix H can reflect the concentration of signal and noise energy. The first few large singular values will mainly reflect the useful part of the signal, while the smaller singular values will mainly reflect the noise. The noise in the signal can be removed by setting the singular values that reflect the noise to zero. By adding and averaging the items corresponding to the reconstructed matrix, the denoised signal can be restored [13,14].

#### 2.2. CEEMDAN

_{0}refers to the amplitude. The new signal is decomposed using the EMD algorithm to obtain the first-order intrinsic mode component IMF

_{1}(Equation (3)) and the first residual ${\mathrm{r}}_{1}\left(\mathrm{t}\right)$ (Equation (4)):

_{2}(Equation (5)), and the second residual, ${\mathrm{r}}_{2}\left(\mathrm{t}\right)$ (Equation (6)):

_{k}. The IMF

_{(k+1)}obtained by EMD decomposition is:

#### 2.3. Teager Energy Operator

#### 2.4. The 1.5-Dimensional Spectrum

_{3x}(τ,τ) of the third-order cumulant R

_{3x}(τ

_{1},τ

_{2}) of the stationary random signal x(t) can be defined as:

_{3x}(τ,τ) of the third-order cumulant of the stationary random signal x(t) is defined as a 1.5-dimensional spectrum R(ω), as shown in Equation (11):

## 3. Fault Diagnosis Process

_{XY}is the correlation coefficient between each component and the original signal, E is the expectation, and D is the variance. The kurtosis value of the original signal is taken as X and the kurtosis value of the IMF component is taken as Y, which is substituted into Equation (13) to obtain the size of the correlation coefficient.

## 4. Simulation Signal Analysis

#### Ball Screw Simulation Signal Fault Diagnosis

_{0}is the displacement constant, A

_{0}= 0.3; f

_{r}is the frequency of the motor, f

_{r}= 30; C is the attenuation coefficient, C = 700; f

_{n}is the natural frequency, f

_{n}= 3 kHz; h(t) is the impulse function; A

_{i}is the amplitude of the impact signal; n(t) is the added Gaussian white noise, with a value of 4 dB; x(t) is the fault simulation signal after adding Gaussian white noise; t is the period of characteristic frequency, where the fault characteristic frequency $\mathrm{f}=1/\mathrm{T}=90\mathrm{Hz}$.

## 5. Measured Signal Analysis

_{r}; if the pitting defect occurs in the nut raceway, the corresponding fault characteristic frequency is f

_{o}; if the pitting defect appears in the screw raceway, the fault characteristic frequency is f

_{i}[20], and the expression is as follows:

_{b}is the ball diameter, d

_{0}is the nominal diameter of the screw, α is the contact angle of the ball with the nut and the screw, and ω

_{s}is the derivative of the angular displacement generated by the screw rotation. Where ${\gamma}^{\prime}={d}_{b}/2{r}_{m}$, rm is the vertical distance from the ball center to the screw center.

## 6. Conclusions

- (1)
- In this paper, a fault diagnosis method for ball screw faults based on the SVD-CEEMDAN-1.5DTES algorithm is proposed, which can effectively extract the fault characteristics of a ball screw in a strong background noise environment and accurately diagnose the fault in a ball screw.
- (2)
- The SVD algorithm can effectively suppress noise interference. The application of the 1.5-dimensional envelope spectrum offers more advantages than the traditional envelope spectrum. It can accurately extract the weak fault information generated by the ball screw under the interference of strong noise, and can effectively suppress the background noise. The diagnosis results, which are based on the SVD-CEEMDAN-1.5DTES algorithm, are better than those with the SVD-CEEMDAN Teager algorithm.
- (3)
- Within a strong background noise environment, the fault diagnosis results for the ball screw based on the SVD-CEEMDAN denoising algorithm are better than those based on the WRN-CEEMDAN algorithm and SVD-WOA-VMD algorithm. In future research, this method will be further optimized.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, Y.; Lc, Y.; Ge, M. Complementary ensemble adaptive local iterative filtering and its application to rolling bearing fault diagnosis. IEEE Access
**2021**, 9, 47275–47293. [Google Scholar] [CrossRef] - Xiangnan, L.; Xuezhi, Z.; Wenbin, S. Feature extraction of rolling bearing fault impact in strong background noise vibration signal. J. Vib. Eng.
**2021**, 34, 202–210. [Google Scholar] - Wang, Z.; Li, Y.; Dong, L.; Li, Y.; Du, W. A Rul Prediction of Bearing Using Fusion Network through Feature Cross Weighting. Meas. Sci. Technol.
**2023**, 34, 105908. [Google Scholar] [CrossRef] - Shi, Y.; Zhang, Q. Bearing Fault Diagnosis Based on Wavelet Denoising and XGBoost Fusion Feature Selection. Available online: http://kns.cnki.net/kcms/detail/41.1148.TH.20220421.1700.002.html (accessed on 4 October 2023).
- Wu, Y.; Wang, J.; Xu, X.; Jiang, Z. Application of FastICA combined noise reduction method based on wavelet analysis in fault diagnosis of rolling bearings. China Mech. Eng.
**2017**, 28, 2183–2188+2197. [Google Scholar] - Cai, K.; Wang, L.; Xu, Z. Feature extraction and analysis of weak bearing faults based on SVD and VMD. Comb. Mach. Tool Autom. Mach. Technol.
**2022**, 578, 70–73+78. [Google Scholar] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Math. Phys. Eng. Sci.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Torres, M.E.; Colominas, M.A.; Schlotthauer, G.; Flandrin, P. A complete ensemble empirical mode decomposition with adaptive noise. In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, Czech Republic, 22–27 May 2011; IEEE: New York, NY, USA, 2011; pp. 4144–4147. [Google Scholar]
- Wang, Z.; Zhao, W.; Li, Y.; Dong, L.; Wang, J.; Du, W.; Jiang, X. Adaptive staged RUL prediction of rolling bearing. Measurement
**2023**, 222, 113478. [Google Scholar] [CrossRef] - Liu, L.; Wei, Y.; Song, X.; Zhang, L. Fault Diagnosis of Wind Turbine Bearings Based on CEEMDAN-GWO-KELM. Energies
**2022**, 16, 48. [Google Scholar] [CrossRef] - Yan, H.; Zhou, H.; Wang, Y. Combining the synchrosqueezing generalized S-transform of variational mode decomposition with the Teager–Kaiser energy operator to calculate the attenuation gradient for identifying oil and gas reservoirs. Acta Geophys.
**2022**, 71, 795–812. [Google Scholar] [CrossRef] - Tang, G.; Xue, G.; Wang, X.; Ding, A. Rolling bearing fault diagnosis based on multivariate variational mode decomposition and 1.5-dimensional spectrum. Bearings
**2022**, 12, 74–82. [Google Scholar] - Dai, L.; Cao, W.; Yi, S.; Wang, L. Damage identification of concrete structures based on WPT-SVD and GA-BPNN. J. Zhejiang Univ.
**2023**, 57, 100–110+132. [Google Scholar] - Qi, L.; Shen, Z.; Guo, Q.; Wang, Y.; Mykola, K. Chirp Rates Estimation for Multiple LFM Signals by DPT-SVD. Circuits Syst. Signal Process.
**2022**, 42, 2804–2827. [Google Scholar] [CrossRef] - He, Z.; Li, J.; Liu, S.; Qin, Z. CEEMD-VMD combined with parameter optimization SVM for roller bearing fault diagnosis. Mech. Sci. Technol. 1–7.
- Kaiser, J.F. Some useful properties of Teager’s energy operators. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Minneapolis, MN, USA, 27–30 April 1993; Volume 3, pp. 149–152. [Google Scholar]
- Wang, H.; Shan, C.; Meng, J.; Chen, G.; Wu, L. A composite fault diagnosis method for rolling bearings based on Autogram resonance demodulation and 1.5-dimensional spectrum. J. Vib. Eng.
**2022**, 35, 1541–1551. [Google Scholar] - Zhu, Y.; Guan, L. Gearbox fault diagnosis based on fast spectral kurtosis and 1.5-dimensional spectrum. Baogang Technol.
**2020**, 46, 67–69. [Google Scholar] - Zhou, Z. Machine Learning; Tsinghua University Press: Beijing, China, 2016. [Google Scholar]
- Nie, C.; Zhou, C.; Liu, D.; Feng, H.; Wang, Z.; Ou, Y. Application of HHT-SVM in fatigue pitting failure diagnosis of ball screw pair. Comb. Mach. Tool Autom. Mach. Technol.
**2020**, 12, 80–84+89. [Google Scholar] - Ren, L.; Zhen, L.; Zhao, Y.; Dong, Q.; Zhang, Y. Rolling bearing fault diagnosis under strong background noise environment based on SSA-VMD-MCKD. Vib. Shock.
**2023**, 42, 217–226. [Google Scholar]

**Figure 3.**Simulation signal diagram. (

**a**) Simulation signal time domain diagram; (

**b**) The time domain diagram of the simulated signal after adding noise; (

**c**) The frequency domain diagram of the simulated signal after adding noise.

**Figure 10.**The experimental fault signal diagram. (

**a**) Ball screw nut fault time domain diagram; (

**b**) Ball screw nut fault frequency domain diagram.

Name | Screw Diameter | Ball Diameter | Contact Angle | Screw Lead Angle |
---|---|---|---|---|

unit | d_{0}/mm | d_{b}/mm | α/° | λ/° |

numerical value | 40 | 5.953 | 45 | 4.55 |

IMF | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

WRN-CEEMDAN | 0.9761 | 0.2690 | 0.0572 | 0.0071 | 0.0028 |

SVD-CEEMDAN | 0.9860 | 0.3396 | 0.0511 | 0.1790 | 0.0373 |

SVD-WOA-VMD | 0.5598 | 0.4701 | 0.5004 | 0.4768 | 0.3937 |

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**MDPI and ACS Style**

Wu, Q.; Niu, J.; Wang, X.
Fault Feature Extraction Method of Ball Screw Based on Singular Value Decomposition, CEEMDAN and 1.5DTES. *Actuators* **2023**, *12*, 416.
https://doi.org/10.3390/act12110416

**AMA Style**

Wu Q, Niu J, Wang X.
Fault Feature Extraction Method of Ball Screw Based on Singular Value Decomposition, CEEMDAN and 1.5DTES. *Actuators*. 2023; 12(11):416.
https://doi.org/10.3390/act12110416

**Chicago/Turabian Style**

Wu, Qin, Jun Niu, and Xinglian Wang.
2023. "Fault Feature Extraction Method of Ball Screw Based on Singular Value Decomposition, CEEMDAN and 1.5DTES" *Actuators* 12, no. 11: 416.
https://doi.org/10.3390/act12110416