# Frequency Loss and Recovery in Rolling Bearing Fault Detection

- 67 Downloads

**Part of the following topical collections:**

## Abstract

Rolling element bearings are key components of mechanical equipment. The bearing fault characteristics are affected by the interaction in the vibration signals. The low harmonics of the bearing characteristic frequencies cannot be usually observed in the Fourier spectrum. The frequency loss in the bearing vibration signal is presented through two independent experiments in this paper. The existence of frequency loss phenomenon in the low frequencies, side band frequencies and resonant frequencies and revealed. It is demonstrated that the lost frequencies are actually suppressed by the internal action in the bearing fault signal rather than the external interference. The amplitude and distribution of the spectrum are changed due to the interaction of the bearing fault signal. The interaction mechanism of bearing fault signal is revealed through theoretical and practical analysis. Based on mathematical morphology, a new method is provided to recover the lost frequencies. The multi-resonant response signal of the defective bearing are decomposed into low frequency and high frequency response, and the lost frequencies are recovered by the combination morphological filter (CMF). The effectiveness of the proposed method is validated on simulated and experimental data.

## Keywords

Rolling element bearing Signal processing Frequency loss Fault detection Morphological filter## 1 Introduction

Bearing defects are a common cause of machine breakdown. It is crucial to accurately diagnose the existence of rolling bearing fault. Vibration analysis is extensively employed rolling bearing fault detection. Many analysis methods have been proposed in the time domain [1, 2] and the frequency domain, such as demodulation algorithm [3, 4], cyclostationary analysis [5]. Time-frequency analysis techniques wavelet-transform (WT) [6, 7], EMD and LMD [8], for instance, are devoted to process bearing vibration signals as well.

When a rolling element moves over a damaged surface, it generates an impulsive force and excites resonances in the bearing and machine. The period of the impulses or characteristic frequency can be calculated, according to the rotating velocity, position of faults and bearing dimensions [9]. However, the bearing fault signals are almost always masked by background noise [10] and the defect frequency identification from direct vibration signals becomes difficult. Randall [11] illustrated that the low harmonics of the bearing characteristic frequencies are almost invariably strongly masked by other vibration components. It is difficult to identify the bearing fault in the spectrum using conventional FFT methods.

In the current literature, although the low harmonics of the bearing characteristic frequencies [11] and the shaft-speed frequency [12] can sometimes be found in raw spectrum, it is widely recognized that low frequency components of bearing fault signal cannot be usually observed in the spectrum. Here, we define the phenomenon that the low frequency components cannot be observed in the FFT spectrum as frequency loss. The low frequency components include bearing characteristic defect frequencies, rotating frequency and their harmonics. As for the causes of the frequency loss, the explanation in published literatures is summarized (i) masked by the strong background noise [13], (ii) masked by the interference vibrations from other machine elements [11, 14], (iii) may not exist at all in the measured signal sometimes [15].

However, frequency loss can be observed when the bearing is running under the condition of low background noise and no other vibration components existing. Moreover, if the lost frequencies are masked by background noise, the low frequency band in the spectrum must present the feature of noise, but sometimes there is no such noise feature when the frequency loss occurs. In addition, it is noticed in the gear/bearing model study that the gearmesh frequency is modulated at both the shaft speed and the bearing characteristic frequency [16]. Obviously, if the lost frequencies are masked by other vibration components, features of the external vibration components and the action between the characteristic frequency and the external vibration should be found. It does not mean that the low frequency components cannot be observed. Furthermore, frequency loss is also observed in the sideband frequencies and the resonant frequencies in the paper. The features of the spectrum are used in the calculation of spectral kurtosis [14], correlated kurtosis [17] and entropy [18, 19] for the central frequency and band width selection in the bearing fault detection. Obviously, the existence of frequency loss will affect the calculation results of these indicators. Study on the root cause of the frequency loss in bearing fault signal is helpful to obtain a better understanding of the bearing fault feature and its detection.

This paper discusses the interaction of the bearing vibration signal. It is demonstrated that the phenomenon of the frequency loss is generated by the internal vibrations rather than the external interference. The interaction mechanism of the bearing signal is revealed through theoretical and practical analysis. A new method based on morphological filter (MF) is proposed to recover the lost frequencies in bearing fault signal.

The paper is organized as follows. The phenomenon of frequency loss in the bearing vibration signal is presented through experiments in Section 2. The interaction mechanism of the bearing signal is explained in Section 3. A novel method is presented to recover the lost frequencies based on morphological filter. And the proposed technique is evaluated using the simulation and experimental signals in Section 4. Some remarks and conclusions are drawn in Section 5.

## 2 Frequency Loss in the Bearing Vibration Signal

In this section, the phenomenon of frequency loss in the bearing vibration signal is presented through two independent experiments. It is demonstrated that the frequency loss can be observed even the bearing is running under the condition of low background noise and no other vibration components existing.

Characteristic frequencies

Bearing type | BPFO (Hz) | BPFI (Hz) |
---|---|---|

N205 | 116.0 (at 1440 r/min) | 172.0 (at 1440 r/min) |

6205 | 107.7 (at 1797 r/min) | 159.9 (at 1772 r/min) |

### 2.1 Experiment 1: Vibration Signal of Bearing with Severe Defect

### 2.2 Experiment 2: Vibration Signal of Bearing with Slight Defect

The second experimental data of rolling bearing are collected from the CWRU Bearing Data Centre Website [20]. In this experiment, single point faults are seeded to the test bearing with different fault diameters. More detail about the experiment condition can be found in the Website. The measurement is performed with sampling frequency of 12 kHz. Vibration data of fault size 0.177 mm (0.07 in) in outer race and inner race are selected in the paper. The 0.177 mm is the smallest fault size in this experiment. The defect is classified as slight.

The phenomenon of frequency loss is observed in the two independent experiments. Comparing the two experiments, some common features can be concluded: (1) the experiments are carried out in the laboratory, where has low background noise. (2) Only the test rigs are running, no external vibration interference. It is demonstrated that background noise or external interference cannot be the root cause of the frequency loss. Furthermore, various fault degree leads to contrary result in the experiments. All of the low frequency components are lost in the severe bearing fault test (N205) and the low frequency components are partly lost in the slight bearing fault test (6205). The cause of frequency loss must be the result of internal action in the bearing fault signal.

## 3 Interaction Mechanism of the Bearing Signal

*B*denotes the amplitude of the impulsive response, \(f_{n}\) is the excited natural frequency, \(\phi_{n}\) is the initial phase angle, \(\xi\) is the attenuation factor, for a single degree of freedom (SDOF), \(\xi = \alpha 2\uppi{f_{n}}\), where \(\alpha\) denotes the relative damping ratio.

From the above experimental cases, multiple resonant frequency bands are found in the spectrum, i.e., frequency bands about 1500 Hz and 2800 Hz in Figure 2(b), 2800 Hz and 4000 Hz in Figure 2(c), 1200 Hz, 2700 Hz and 3600 Hz in Figure 3. Therefore, the actual bearing fault signal is the cumulating of multiple impulse responses with different resonant frequencies. The transmission path of different resonant responses to the transducer varies to different components. So the collected vibration signal is multi-resonant responses with time difference. Two impulse responses are also reported in the vibration signal of a spalled bearing when the rolling element entry into and exit from the spall [21].

*FT*[·] denotes the Fourier transform.

Thus, \(Y(f)\) has a discrete spectrum, and the corresponding phase is the sum of \(\phi (f)\) and \(\phi_{n}\).

Superposition conditions

No. | Response 2 | |
---|---|---|

Initial phase | Start position | |

Condition 1 | 0° | 0 |

Condition 2 | −180° | 0 |

Condition 3 | 0° | Point 32 |

Condition 4 | −180° | Point 32 |

Theoretical calculation using Eq. (6) indicates that the amplitude of the superposed response in the spectrum is the vector sum of that of each response. From Eq. (5), the phase of the two exponential decay functions approach − 90° at the low frequency band. When the initial phase \(\phi_{n}\) is set to zero (condition 1), the final phase of the two responses is almost same (approximately − 90°). Therefore, the phase difference of the two responses at the low frequency band is almost 0°, and the amplitude of the superposed response is the summation of that of the each response. Similarly, when the initial phase of the second response is set to − 180° (condition 2), the phase difference of the two responses is − 180°, and the amplitude of the two responses is cancelled out each other. So the frequency loss is observed under the superposition condition two.

The second response starts at 32 point, which amounts to a quarter of the period, the phase delays of the first four harmonics are calculated as \(- {\uppi \mathord{\left/ {\vphantom {\uppi 2}} \right. \kern-0pt} 2}\), \(- \pi\), \({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2}\), 0. Due to the time-shifting of the second response, a total − 180° phase difference is produced between the two responses at 2 N (Figure 11(b)) under condition three and at 4 N (Figure 12(b)) under condition 4.

According to the theoretical calculation and simulation, the interaction mechanism of the defective bearing signal is summarized. When rolling elements strike a local fault on the inner or outer race a shock is introduced that excites high frequency resonances of the whole structure between the bearing and the transducer. The collected vibration signal of the defective bearing consists of multi-resonant response. The final amplitude of the harmonics in the spectrum is the vector sum of that of each response. The different responses can cancel out each other when the phase difference between the responses is equal to − 180°. The interaction of the multi-resonant response leads to frequency loss and distortion in the Fourier transform spectrum. Therefore frequency loss can be found in the low frequency harmonic, the sideband frequency and the resonant frequency.

## 4 Recovery the Lost Frequencies Based on Mathematical Filter

The root cause of frequency loss in the defective bearing signal is the interaction of the multi-resonant response. Therefore the key point to recover the lost frequencies depends on the separation of the different resonant response.

Mathematical morphology (MM) [22, 23] is a kind of nonlinear analysis method which was first used in image analysis. Unlike the traditional frequency domain filter, the morphological filter (MF) decomposes the signal into several physical components according to the signal geometric characteristics. The morphological filters have been adopted for fault feature extraction of bearing vibration signal [24, 25, 26].

### 4.1 Morphological Filter in Brief

*g*:

### 4.2 Property Morphological Filter

Based on our previous analysis of working mechanism of mathematical morphological operators, the CMF is employed to recover the lost frequencies in the defective bearing signal. More details of the properties of different morphological operators, frequency response, selection principle of SE length can be found in Ref. [27]. The CMF presents low pass filter property. The high frequency component can be extracted by subtracting the CMF output from the original signal. Therefore, the multiresonant response can be decomposed into two parts after the CMF processing, low frequency and high frequency response. Once the multiresonant response is separated, the effect of the interaction is eliminated and the lost frequencies are visible in the spectrum.

### 4.3 Recovery the Lost Frequencies in Bearing Fault Signal Based on CMF

## 5 Conclusions

This paper focuses on the interaction of the bearing vibration signal. It is presented that the phenomenon of frequency loss is generated by the internal vibrations rather than the external interfering. Multi-resonant response is excited when rolling elements strike a local fault. The harmonics in the Fourier transform spectrum of the collected vibration signal is the vector sum of that of each response. The interaction of the multi-resonant response leads to frequency loss and distortion in the spectrum. Frequency loss can be found in the low frequency harmonic, the sideband frequency and the resonant frequency. Theoretical and practical analysis demonstrates the existence of the frequency loss in bearing signal. Since frequency loss can occur in side band frequency and the resonant frequency, the interaction of the bearing fault signal should be considered in the frequency band selection based demodulation method. Based on morphological filter, a new method is provided to recover the lost frequencies. Compared with the traditional digital filter, the property of CMF in impulsive type signal processing is analyzed. The simulation and experiment results show that the morphological filter can effectively separate the interaction of bearing signal and enhance the bearing fault feature.

## Notes

### Authors’ Contributions

AH was in charge of the whole trial; AH, LX and SX wrote the manuscript; SX assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

Aijun Hu, born in 1971, is currently a professor at *Department of Mechanical Engineering, North China Electric Power University, China.* He received his PhD degree from *North China Electric Power University, China,* in 2008. His research interests are signal process, condition monitoring and fault diagnosis.

Ling Xiang, born in 1971, is currently a professor at *Department of Mechanical Engineering, North China Electric Power University, China*. She received her PhD degree from *North China Electric Power University, China*. Her research interests include, condition monitoring and fault diagnosis.

Sha Xu, born in 1989, is currently a master candidate at *Department of Mechanical Engineering, North China Electric Power University, China*.

Jianfeng Lin, born in 1991, he received his master degree from *North China Electric Power University, China,* in 2018.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Natural Science Foundation of China (Grant Nos. 51675178, 51475164).

## References

- [1]W R Thomas, A B Francisco, M V Flávio. Heterogeneous feature models and feature selection applied to bearing fault diagnosis.
*IEEE Transactions on Industrial Electronics*, 2015, 62(1): 637–646.CrossRefGoogle Scholar - [2]J Yu. Local and nonlocal preserving projection for bearing defect classification and performance assessment.
*IEEE Transactions on Industrial Electronics*, 2012, 59(5): 2363–2376.CrossRefGoogle Scholar - [3]L Barbini, M Eltabach, A J Hillis, et al. Amplitude-cyclic frequency decomposition of vibration signals for bearing fault diagnosis based on phase editing.
*Mechanical Systems and Signal Processing*, 2018, 103: 76–88.CrossRefGoogle Scholar - [4]D Zhao, J Li, W Cheng, et al. Compound faults detection of rolling element bearing based on the generalized demodulation algorithm under time-varying rotational speed.
*Journal of Sound and Vibration*, 2016, 378: 109–123.CrossRefGoogle Scholar - [5]I Antoniadis, G Glossiotis. Cyclostationary analysis of rolling-element bearing vibration signals.
*Journal of Sound and Vibration*, 2001, 248(5): 829–845.CrossRefGoogle Scholar - [6]J Yuan, Y Wang, Y Peng, et al. Weak fault detection and health degradation monitoring using customized standard multiwavelets.
*Mechanical Systems and Signal Processing*, 2017, 94: 384–399.CrossRefGoogle Scholar - [7]Jinglong Chen, Jun Pan, Zipeng Li, et al. Generator bearing fault diagnosis for wind turbine via empirical wavelet transform using measured vibration signals.
*Renewable Energy*, 2016, 89: 80–92.CrossRefGoogle Scholar - [8]L Wang, Z Liu, Q Miao, et al. Time-frequency analysis based on ensemble local mean decomposition and fast kurtogram for rotating machinery fault diagnosis.
*Mechanical Systems and Signal Processing*, 2018, 103: 60–75.CrossRefGoogle Scholar - [9]C Mishra, A K Samantaray, G Chakraborty. Ball bearing defect models: A study of simulated and experimental fault signatures.
*Journal of Sound and Vibration*, 2017, 400: 86–112.CrossRefGoogle Scholar - [10]K Sidra, J K Dutt, N Tandon. Extracting rolling element bearing faults from noisy vibration signal using Kalman filter.
*ASME J. Vibration and Acoustics*, 2014, 136: 031008.CrossRefGoogle Scholar - [11]R B Randall, J Antoni. Rolling element bearing diagnostics—A tutorial.
*Mechanical Systems and Signal Processing*, 2011, 25(2): 485–520.CrossRefGoogle Scholar - [12]M C Pan, W C Tsao. Using appropriate IMFs for envelope analysis in multiple fault diagnosis of ball bearings.
*International Journal of Mechanical Sciences*, 2013, 69: 114–124.CrossRefGoogle Scholar - [13]A Siddique, G S Yadava, B Singh. A review of stator fault monitoring techniques of induction motors.
*IEEE Trans. Energy Convers*, 2005, 20(1): 106–114.CrossRefGoogle Scholar - [14]Y X Wang, M Liang. An adaptive SK technique and its application for fault detection of rolling element bearings.
*Mechanical Systems and Signal Processing*, 2011, 25(5): 1750–1764.CrossRefGoogle Scholar - [15]J Zarei, M A Tajeddini, H R Karimi. Vibration analysis for bearing fault detection and classification using an intelligent filter.
*Mechatronic*, 2014, 24(2): 151–157.CrossRefGoogle Scholar - [16]N Sawalhi, R B Randall. Simulating gear and bearing interactions in the presence of faults_ Part II_ Simulation of the vibrations produced by extended bearing faults.
*Mechanical Systems and Signal Processing*, 2008, 22(8): 1952–1966.CrossRefGoogle Scholar - [17]Yonghao Miao, Ming Zhao, Jing Lin, et al. Application of an improved maximum correlated kurtosis deconvolution method for fault diagnosis of rolling element bearings,
*Mechanical Systems and Signal Processing*, 2017, 92: 173–195.CrossRefGoogle Scholar - [18]The infogram: Entropic evidence of the signature of repetitive transients.
*Mechanical Systems and Signal Processing*, 2016, 74: 73–94.CrossRefGoogle Scholar - [19]Zhipeng Feng, Haoqun Ma, Ming J Zuo. Spectral negentropy based sidebands and demodulation analysis for planet bearing fault diagnosis.
*Journal of Sound and Vibration*, 2017, 410: 124–150.CrossRefGoogle Scholar - [20]Bearing Data Center, Case Western Reserve Univ., Cleveland, OH. [Online]. Available: http://www.eecs.case.edu/laboratory/bearing.
- [21]N Sawalhi, R B Randall. Vibration response of spalled rolling element bearings: Observations, simulations and signal processing techniques to track the spall size.
*Mechanical Systems and Signal Processing*, 2011, 25(3): 846–870.CrossRefGoogle Scholar - [22]P Maragos, R W Schafer. Morphological fi1ters -Part I: Their set theoretic analysis and relation to linear shift invariant filters.
*IEEE Transactions on Acoustics, Speech and Signal Processing*, 1987, 35(8): 1153–1169.MathSciNetCrossRefGoogle Scholar - [23]P Maragos, R W Schafer. Morphological fi1ters -Part II: Their relation to median, order-statistic, and stack filters.
*IEEE Transactions on Acoustics, Speech and Signal Processing*, 1987, 35(8): 1170–1184.MathSciNetCrossRefGoogle Scholar - [24]J Lv, J Yu. Average combination difference morphological filters for fault feature extraction of bearing.
*Mechanical Systems and Signal Processing*, 2018, 100: 827–845.CrossRefGoogle Scholar - [25]Y Li, X Liang, J Lin, et al. Train axle bearing fault detection using a feature selection scheme based multi-scale morphological filter.
*Mechanical Systems and Signal Processing*, 2018, 101: 435–448.CrossRefGoogle Scholar - [26]D Yu, M Wang, X Cheng. A method for the compound fault diagnosis of gearboxes based on morphological component analysis.
*Measurement*, 2016, 91: 519–531.CrossRefGoogle Scholar - [27]A J Hu, L Xiang. Selection principle of mathematical morphological operators in vibration signal processing.
*Journal of Vibration and Control*, 2016, 22(14): 3157–3168.MathSciNetCrossRefGoogle Scholar - [28]Antoni. Fast computation of the kurtogram for the detection of transient faults.
*Mechanical Systems and Signal Processing*, 2007, 21(1): 108–124.CrossRefGoogle Scholar - [29]A Garcia-Perez, R J Romero-Troncoso, E Cabal-Yepez, et al. The application of high-resolution spectral analysis for identifying multiple combined faults in induction motors.
*IEEE Transactions on Industrial Electronics*, 2011, 58(5): 2002–2010.CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.