Frequency Loss and Recovery in Rolling Bearing Fault Detection

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.

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.

Two different types of bearing, N205 cylindrical roller bearing and SKF 6205 deep groove ball bearing are employed in the experiments. The calculated bearing defect characteristic frequencies [9], ball pass frequency of the outer race (BPFO) and ball pass frequency of the inner race (BPFI) of the two experimental bearings are given in Table 1.

In this section, the interaction of the bearing fault signal is discussed using the simulated signal. The repetitive impact signal from a localized bearing defect can be described by a train of Dirac delta function \(\delta (t)\) with the period between two successive impulses being \(T_{d}\):where \(T_{d}\) is the reciprocal of the characteristic frequency, and \(A_{k}\) is the amplitude of the impacts.

The impulsive response from the bearing faults can be expressed as:where 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.

Ignoring the influence of roller slippage, the vibration signal arising from a localized fault in a bearing can be expressed bywhere the symbol \(\otimes\) denotes convolution.

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].

In the following sections, frequency domain analysis (amplitude and phase) of the simulated signal in Eq. (3) is performed and then a dual-resonant-frequency bearing fault signal is used to discuss the interaction in the bearing fault signal. The single side Fourier transform of Eq. (1) can be expressed aswhere \(f_{d} = {1 \mathord{\left/ {\vphantom {1 {T_{d} }}} \right. \kern-0pt} {T_{d} }}\), FT[·] denotes the Fourier transform.

The frequency domain expression of the impulsive response iswhere the amplitude \(A(f) = \frac{B}{{\sqrt {\xi^{2} + (2\uppi{f})^{2} } }},\) the phase \(\phi (f) = - \arctan \,({{2\uppi{f}} \mathord{\left/ {\vphantom {{2\uppi{f}} \xi }} \right. \kern-0pt} \xi }),\) it is known that \({{\phi (f) \to\uppi} \mathord{\left/ {\vphantom {{\phi (f) \to\uppi} 2}} \right. \kern-0pt} 2}\) when \(f \to + \infty\) and \(\phi (f) \to {{ -\uppi} \mathord{\left/ {\vphantom {{ -\uppi} 2}} \right. \kern-0pt} 2}\) when \(f \to - \infty\).

The final expression of Eq. (3) in frequency domain is

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

Two different resonant responses excited by the same period impact are simulated. The sampling frequency is set to 8192 Hz, \(T_{d}\) is set to 128 sample points, the corresponding characteristic frequency is 64 Hz. The amplitude of the impulse forces and the impulse responses are both set to 1, the attenuation factor \(\xi\) is equal to 600, and total 8192 samples are used for further analysis. The natural frequency is set to 1000 Hz (response one)and 2000 Hz (response two), respectively. Figure 7 and Figure 8 depict the two simulated responses with 0° initial phases. From the spectra, the two signals have same characteristic frequency and the amplitude of the low frequency components (64 Hz and its low order harmonics) is approximately equal. The spectrum lines are symmetrical relative to the central (resonant) frequency.

The parameters of the first response remains constant, the second response is superposed under different conditions. Superposition conditions are listed in Table 2. Superposition results are provided in Figures 9, 10, 11, 12. Compared to the original response (Figure 7 and Figure 8), the superposition results indicated that the spectrum varies greatly due to the different superposition conditions. As Figure 10(b) shown, superposition result under condition 2 illustrates that the characteristic frequency (64 Hz) and its low order harmonics can hardly be found in the spectrum. The frequency loss phenomenon is observed in the simulation when the two responses have an opposite phase. At the same time, the amplitude of the peaks around 1500 Hz is doubles that of the original response. From the superposition result under condition 1, the amplitude of the low frequency components in Figure 9(b) is approximately twice as that of the original response, and the amplitude of the peaks around 1500 Hz is almost equal to its original value.

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.

Figure 11 and Figure 12 are the superposition results when the start position of the second response is right shifted 32 points. Some harmonics, the second order harmonic (2 N) of the characteristic frequency in Figure 11(b) and 4 N in Figure 12(b), for instance, are almost invisible. Other harmonics, including the resonant frequencies and their sideband frequencies, are changed obviously.

The influence of the start position in response two can be easily explained by the time-shifting property of the Fourier transform:where \(X(f)\) is the Fourier transform of \(x(t)\), \(t_{0}\) is the time delay.

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.

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 [24‐26].

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.