Generalized Demodulation Transform for Bearing Fault Diagnosis Under Nonstationary Conditions and Gear Noise Interferences
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Abstract
It is a challenging issue to detect bearing fault under nonstationary conditions and gear noise interferences. Meanwhile, the application of the traditional methods is limited by their deficiencies in the aspect of computational accuracy and efficiency, or dependence on the tachometer. Hence, a new fault diagnosis strategy is proposed to remove gear interferences and spectrum smearing phenomenon without the tachometer and angular resampling technique. In this method, the instantaneous dominant meshing multiple (IDMM) is firstly extracted from the time-frequency representation (TFR) of the raw signal, which can be used to calculate the phase functions (PF) and the frequency points (FP). Next, the resonance frequency band excited by the faulty bearing is obtained by the band-pass filter. Furthermore, based on the PFs, the generalized demodulation transform (GDT) is applied to the envelope of the filtered signal. Finally, the target bearing is diagnosed by matching the peaks in the spectra of demodulated signals with the theoretical FPs. The analysis results of simulated and experimental signal demonstrate that the proposed method is an effective and reliable tool for bearing fault diagnosis without the tachometer and the angular resampling.
Keywords
Bearing fault diagnosis Generalized demodulation transform Nonstationary conditions Gear noise1 Introduction
Bearing failure is one of the most common causes for rotating machinery breakdowns and accidents, and condition monitoring and fault detection of the bearing has been a subject of extensive research in recent years [1, 2, 3]. With efforts of the researchers, bearing fault information can be extracted from vibration, sound, current, acoustic emission and temperature signals. In particular, vibration signal analysis techniques, such as the time-domain statistics [4], the spectrum [5] and the time-frequency analysis [6], are the preferred approaches due to their satisfactory performance.
To tackle this issue, the most common methods are combining the computed order analysis (COA) and the gear noise elimination-based algorithms. Borghesani et al. [9] proposed an algorithm which includes eliminating influences of speed fluctuation with the angular resampling, removing gear interferences by the denoising method, applying the reverse order tracking and the angular resampling to the denoised signal, and detecting fault by the envelope analysis. This method is complicated and critically depends on the tachometer which is not always available. Focused on the above defects, Wang et al. [10] proposed an adaptive noise cancellation-based method. In this method, the IDMM trend is extracted from the TFR of the raw signal as a reference to remove the gear noise, then with the help of instantaneous fault characteristic frequency extracted from the envelope TFR of the denoised signal, the COA is applied to diagnose fault bearing. In addition, a bearing fault diagnosis method based on the empirical mode decomposition and the COA is proposed in Ref. [11]. Compared with the method proposed by Borghesani et al., the latter two methods can effectively remove the limitation of the tachometer. However, all the above methods rely on the COA.
The COA is an effective method to transform the nonstationary signal (in time domain) into the stationary one (in angle domain) for eliminating the influences of speed fluctuation with help of rotating speed or angular position measured by additional sensor [12, 13, 14], even the instantaneous rotating frequency can be extracted or estimated [15, 16], the deficiencies in aspect of computational accuracy and efficiency cannot be ignored. Saavedra et al. [17] pointed out that the computational errors are caused by calculation of resampled times and estimation of resampled amplitude. Cheng et al. [18] pointed out that the deformation of the envelope signal causes a variation in the angular interval between the peaks of the impulses, which will directly affect the accuracy of envelope analysis results. Furthermore, lots of equations calculation for determining the angular positions is time-consuming. Lastly, bearing fault type is determined by the ratio between the fault characteristic order and the rational frequency order, which isn’t obvious in some case. The unobvious rational frequency order brings challenges to locate fault.
The GDT is developed by Olhede et al. [19] for speech-type signal processing. This method can transform an arbitrary time-varying frequency ridge into a line paralleling to the time axis, and make it possible to separate any interested time-frequency ridge into a mono-component from other components by conventional filters. For separating the overlapped multi-components, an improved GDT is proposed in Ref. [20]. In this improved method, a varying window function is used to improve the time–frequency concentration of the generalized short time Fourier transform. Due to the characteristics of the GDT, some researchers applied it or its improved versions to process the nonstationary signals, especially for processing amplitude modulation (AM)—frequency modulation (FM) signal in recent years [21, 22]. A joint envelope and frequency order spectrum analysis method based on iterative GDT is proposed by Feng et al. [23] for diagnosing planetary gearbox under nonstationary conditions, where the iterative GDT can decompose all the constituent components into mono-components of constant frequency.
According to the above analysis, a satisfactory bearing fault diagnosis strategy should be able to eliminate the influences of gear noise and rotating speed fluctuation without the COA, the tachometer and the complicated denoising algorithm. As such, based on the GDT, the bearing high frequency resonance phenomenon and the gear meshing characteristic, a new bearing fault diagnosis method under nonstationary conditions and gear noise interferences is developed. In the method, the prominent time-frequency ridge named IDMM is firstly extracted from the TFR of raw signal, and the extracted IDMM can be used to calculate the PFs and FPs. Secondly, the band-pass filter is used to separate faulty bearing vibration and gearing meshing vibration due to their different distributed levels in the TFR, and based on the AM-FM features of bearing vibration signal [24, 25] the envelope of the filtered signal including FCF is obtained by the Hilbert transform. Lastly, the GDT is applied to the envelope signal for transforming the time-varying FCF and its harmonics into straight lines paralleling to the time axis, in this case, the peaks represented FCF and its harmonics can be easily captured in envelope spectrum.
The rest of this paper is structured as follows. In Section 2, theoretical basis of the proposed method is presented. Simulated signal is constructed for validating the proposed method in Section 3. In Section 4, the proposed method is further examined by different faulty bearing signals. The conclusions are presented in Section 5.
2 Theoretical Background
2.1 Gear Meshing Vibration Characteristic
2.2 High Frequency Resonance and AM-FM Characteristic of Faulty Bearing Signal
Meanwhile, the localized fault on a bearing will excites a mechanical structural resonance in operating condition, and the FCF of the bearing will be modulated by the resonance band. Therefore, faulty bearing vibration signals often exhibit AM-FM phenomenon. In general, the resonance band excited by bearing fault is located at a frequency band far from the gear meshing vibration as shown in Figure 3. The region of gear meshing vibration is around from 0 Hz to 700 Hz, and the faulty bearing vibration has high level from 5500 Hz to 10000 Hz. As such, high frequency resonance band that including bearing fault information can be separated by the band-pass filter.
Based on the above analysis, it can be concluded that the FCF and its harmonics have amplitude advantage in the envelope of high frequency resonance signal. Under time-varying speed conditions, the time-frequency ridges in envelope TFR will change with time, which is the main reason for the spectrum smearing. In traditional methods, angular resampling is used to transform the time-varying ridges into lines paralleling to the angular axis, and the fault characteristic order and rotational frequency order can be captured in the envelope order spectrum, then the bearing fault type is determined. Inevitably, if the rotational frequency order is not obvious, the bearing fault cannot be determined. Most importantly, other irrelevant components are also transformed which will bring interferences for detecting the fault characteristic order and its harmonics. Inspired by angular resampling technique, if an algorithm can be used to reset the interested time-varying frequency ridge only, and new parameter is used to quantize the FCF and define fault type, the faulty bearing can be diagnosed.
2.3 Generalized Demodulation Transform
Namely, if the function of the interested time-frequency curve is calculated, the time-varying frequency ridge can be transformed into linear path paralleling to the time axis.
- 1.
Obtain the equation of the target time-frequency trajectory;
- 2.
Calculate its PF based on Eq. (13);
- 3.
Based on the PF, the GDT is applied to the analytic signal x(t), and a new signal \( d(t) = x(t)e^{{ - 2\pi js_{0} (t)}} \) is obtained;
- 4.
An analytic signal z(t) = d(t) + jH(t) is obtained to avoid negative frequencies, and it is the final demodulated result.
An arbitrary mono-component signal x(t) = sin[(25t − 31)^{2}t + 1257t] is constructed to verify the GDT, whose instantaneous frequency function f(t) = 298t^{2} − 493t + 353 is calculated.
- 1.
A time-varying component can be converted into a line paralleling to the time axis, and the energy of the original signal would be concentrated on the constant frequency, which is equal to the frequency starting point of the original time-frequency path.
- 2.
The energy concentration degree of the time-frequency trajectory in the TFR of the demodulated signal is improved.
2.4 Overall Framework of the Proposed Method
- 1.
Calculated the TFR of the raw signal, from which the IDMM is extracted by the peak search algorithm;
- 2.
Envelope of resonance frequency signal is obtained via band-pass filter and Hilbert transform, where the filtering parameters are estimated from the TFR of the raw signal;
- 3.
The PFs and FPs corresponding to the different fault types are calculated by the IDMM function and mechanical structure parameters;
- 4.
The GDT is applied to envelope signal with different PFs, and then the spectra of demodulated signals are obtained by the fast Fourier transform (FFT);
- 5.
Fault diagnosis by locating the peak in the spectra with the FPs;
- 6.
The envelope signal is demodulated again with PFs for further determining the fault type of the target bearing, where the PFs correspond to the second and third harmonics of the captured peak in Step (5).
3 Simulation Verifications
Parameters of simulated model
Parameter | Value |
---|---|
Length of the signal t (s) | 4 |
Sampling rate f_{s} (Hz) | 20000 |
FCC | 3 |
Structural damping characteristic β | 750 |
Bearing resonance frequency w (Hz) | 5000 |
Number of teeth z | 18 |
Slippage coefficient u | 0.01 |
Rotating frequency ratio c | 1.2 |
From Figure 6(b), the high frequency resonance band excited by bearing fault also can be easily found, and it distributes from around 4000 Hz to 6000 Hz. This resonance band has an obvious difference with the gear meshing vibration. Hence, it can be obtained via the band-pass filter algorithm, and the filtering parameters are estimated from Figure 6(b).
Applied the GDT to the envelope signal (Figure 6(d)) with the s_{0}(t), and the spectrum of the demodulated signal is presented in Figure 6(e). It can be seen that a peak whose value is 44.4 Hz is clearly identified, and approximately equal to the theoretical FP f_{p} = 45 Hz, which corresponds to the preset fault type. In addition, its harmonics are also located in the corresponding spectra of demodulated signals as shown in Figure 6(g) and (h), respectively. The demodulated signals are calculated by the 2s_{0}(t) and 3s_{0}(t), respectively.
In comparison, an arbitrary FCC (2.3) is selected. A new PF s_{1}(t) = 2.875t^{2} is calculated, and the FP is 34.5 Hz. Based on the new PF, the spectrum is obtained as shown in Figure 6(f), in which there isn’t obvious peak corresponding to the FP f_{p1} = 34.5 Hz.
The above analysis results indicate that the GDT-based bearing fault diagnosis method is capable to detect bearing fault type under nonstationary conditions and gear noise interferences.
4 Experimental Verifications
Parameters of bearing and gearbox
Parameter | Value |
---|---|
Bearing type | ER-10k |
Rolling element diameter d (mm) | 7.94 |
Pitch diameter D (mm) | 33.5 |
Number of rolling elements n_{b} | 8 |
Contact angle α | 0 |
FCC of the outer race F_{o} | 3.052 |
FCC of the inner race F_{i} | 4.95 |
FCC of the rolling element F_{b} | 1.99 |
Number of pinion teeth z_{g} | 18 |
Transmission ratio of belts i_{b} | 2.56 |
Transmission ratio of gear box i_{g} | 1.5 |
Serial number of the IDMM c_{d} | 3 |
4.1 Case 1: Target Bearing with Outer Race Fault
In this subsection, outer race fault bearing signal is measured from the test rig to verify the proposed algorithm. In this test, the rotational frequency changes from 23.25 Hz to 31.1 Hz at first and then goes down to 25.61 Hz in 4.5 s. The sampling rate is 48000 samples/s.
PFs and FPs under outer race fault
Fault type | PF | FP |
---|---|---|
Outer race | S_{o}(t) = F_{o} × C_{s} | f_{po}(t) = F_{o} × C_{f} = 70.91 Hz |
Inner race | S_{i}(t) = F_{i} × C_{s} | f_{pi}(t) = F_{i} × C_{f} = 115.09 Hz |
Ball | S_{b}(t) = F_{b} × C_{s} | f_{pb}(t) = F_{b} × C_{f} = 46.27 Hz |
It can be seen from Figure 9(b), the high frequency resonance band excited by bearing fault distributes from around 7000 Hz to 10000 Hz, which has obvious difference with the gear meshing vibration. As a result, the resonance band can be extracted by the band-pass filter with parameters 7000 Hz and 10000 Hz.
Then, the GDT is applied to the envelope signal with the PFs as shown in Table 3. The spectra of the demodulated signals are presented in Figure 9(f), (g) and (h), respectively. From the figures, only one spectrum obtained by the s_{o}(t) has the obvious peak, whose value is approximately equal to the f_{po}. Furthermore, its harmonics are also detected in the corresponding spectra of the demodulated signals as shown in Figure 9(i) and (j), respectively. The analysis results reveal that the target bearing has an outer race fault.
4.2 Case 2: Target Bearing with Inner Race Fault
PFs and FPs under inner race fault
Fault type | PF | FP |
---|---|---|
Outer race | S_{o}(t) = F_{o} × C_{s} | f_{po}(t) = F_{o} × C_{f} = 94.13 Hz |
Inner race | S_{i}(t) = F_{i} × C_{s} | f_{pi}(t) = F_{i} × C_{f} = 152.77 Hz |
Ball | S_{b}(t) = F_{b} × C_{s} | f_{pb}(t) = F_{b} × C_{f} = 61.41 Hz |
Figure 11(e) shows the envelope of the filtered signal. Figure 11(f), (g) and (h) display the spectra of the demodulated signals obtained with the s_{i}(t), s_{o}(t) and s_{b}(t), respectively. In the spectra, only one spectrum has obvious peak at its corresponding FP f_{pi} = 152.77 Hz. Furthermore, the envelope signal is demodulated with the 2s_{i}(t) and 3s_{i}(t), respectively. The spectra are presented in Figure 11(g) and (h), respectively, which reveal the obvious peak near its corresponding FP 2f_{pi}(t) and 3f_{pi}(t), respectively. This finding confirms the inner race fault.
5 Conclusions
- 1.
The IDMM trend is prominent in the TFR of the raw signal based on the gear meshing vibration characteristic, hence it can be extracted for estimating the PFs and FPs corresponding to different fault types. As a result, the proposed method eliminates the dependence on the tachometer;
- 2.
Based on the high frequency resonance phenomenon, the band-pass filter algorithm can be used to remove the gear meshing vibration without complicated preferences selection and calculation;
- 3.
Due to the AM-FM characteristic, the FCF and its harmonics will appear as time-frequency ridges in the envelope TFR of the resonance signal, and the GDT can be used to transform the time-varying frequency ridges into lines paralleling to the time axis. As a result, the spectrum smearing caused by rotational speed fluctuation is removed without the angular resampling. Furthermore, only the interested time-frequency ridges are mapped, which eliminates the interferences of other spectral lines;
- 4.
The bearing fault type is determined by the calculated PFs and FPs, hence the proposed method is free from capturing the peak corresponding to the rotational frequency information in the envelope spectrum or envelope order spectrum.
Notes
Authors’ contributions
WC was in charge of the whole trial; DZ wrote the manuscript; JL and ZH assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
Dezun Zhao, born in 1990, is currently a postdoctoral fellow at Department of Mechanical Engineering, Tsinghua University, China. His research interests include fault diagnosis of rotating machinery and signal processing.
Jianyong Li, born in 1962, is currently a professor at School of Mechanical Electronic and Control Engineering, Beijing Jiaotong University, China. His research interests include advanced manufacturing processes and systems.
Weidong Cheng, born in 1967, is a professor at School of Mechanical Electronic and Control Engineering, Beijing Jiaotong University, China. His research interests include fault diagnosis of rotating machinery and signal processing.
Zhiyang He, born in 1991, is currently a PhD candidate at School of Mechanical Electronic and Control Engineering, Beijing Jiaotong University, China.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Natural Science Foundation of China (Grant Nos. 51335006 and 51605244).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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