A TEO‐based modified Laplacian of Gaussian filter to detect faults in rolling element bearing for variable rotational speed machine

Funding information National Natural Science Foundation of China, Grant/Award Numbers: U1909217, U1709208; Zhejiang Special Support Program for High-level Personnel Recruitment of China, Grant/Award Number: 2018R52034; Wenzhou Key Innovation Project for Science and Technology of China, Grant/Award Numbers: 2018ZG023, ZG2019018 Abstract Conventional signal processing techniques for fault detection are usually aimed at constant speed conditions. Nowadays, order tracking, especially tacho-less order tracking is regarded as a valuable tool for extracting fault features under variable rotational speed conditions. Therefore, a teager energy operator (TEO)-based modified Laplacian of Gaussian filter is proposed to enhance the fault detection behaviour of tacho-less order tracking. This method consists of four steps: First, multi-synchrosqueezing transform is employed to estimate and then extract the instantaneous rotation frequency of a shaft. Second, based on the extracted curve, the non-stationary domain signal is converted into a quasi-stationary domain by the re-sampling technique. Third, the obtained quasi-stationary domain signal is de-noise by the novel method. Finally, fault characteristic orders are extracted via envelope order spectrum analysis method and the performance is significantly improved. The results of numerical simulation and experimental investigations are performed to validate the superiority of the novel method for fault extraction under variable rotational speed conditions.


INTRODUCTION
It is known that bearings are a core part installed in modern machinery, which is wildly applied in rotating machinery, for example, transmissions, planetary gearboxes and motor device.
Failures of the rolling-element bearing may cause a shutdown of a machine system, even accidents and disasters. As a consequence, it is necessary to provide advanced techniques to accurately detect the faults and then reduce the probability of occurring accidents [1][2][3][4][5].
In recent decades, some fault feature detection techniques have been developed and further advanced, such as acoustic emission technology [6], temperature analysis [7], wear debris [8] and vibration analysis [9]. Vibration analysis is considered a powerful tool for faulty bearing detection in industrial applications. Once the surface of bearing with localised defects is in operation and the broken area is knocked by rollers, the acquisition data would appear as a series of transient impulses, which can be considered a fault characteristic. Generally, in industrial applications, multiple complicated frequency components compose vibration signal, which includes machine elements and noises [10]. Fast Fourier transform (FFT) is considered an important method in condition monitoring, and it can apparently present the biggest energy components in the spectrum [11]. However, the local faults usually introduce sideband components that may cause aliasing in the spectrum. On the other hand, rotating machinery is always set in harsh working environments, and thus the collected vibration signals contain heavy background noise [12]. The fault features are masked by the unwanted components, and in recent years, it has been a challenge to accurately recognise the faults.
In particular, rolling-element bearing usually operates in variable operation conditions, thus the vibration signal is always non-stationary and its natural properties are no longer invariable [13]. Especially, the signal has wide frequency bands, and the fault features may be hidden with unwanted components.
On the basis of the above-mentioned conditions, researchers developed effective signal processing methods to avoid the variable speed influence.
Tacho-less order tracking (TLOT) is wildly used due to its low cost and convenience, and the most important procedure of the method is to accurately estimate as well as extract the instantaneous rotation frequency (IRF) of vibration signal without referring tachometer information [14]. Time-frequency (TF) analysis is an essential approach to exhibit the IRF curve, which can trace the speed variation of a shaft, for instance, short-time Fourier transform (STFT), wavelet transform (WT) and Cohen and affine class distribution based on Wigner-Ville distribution (WVD) [15][16][17][18]. Nevertheless, STFT and WT are restricted by the principle of Heisenberg uncertainty; therefore, we cannot acquire a good TF resolution [19]. When analysing the multicomponent signal with WVD, this may introduce cross-term interferences [20]. Yu et al. developed a method called multisynchrosqueezing transform (MSST), which has better energy concentration and stronger robustness to noise interferences for the estimation of vibration signal under non-stationary conditions [21].
On the basis of IRF curve, a constant angle was made to realise re-sampling processing for eliminating the smear effect caused by the variable speed. Unfortunately, according to the envelope order spectrum manifestation results, some interesting features are easily masked. Wang et al. had introduced a hybrid method to process the vibration signal at the variable speed [22]. Yang et al. combined order tracking technique with local mean decomposition to extract fault components [23]. The aforementioned methods need to input some related a priori knowledge of the tested elements to accurately detect the bearing faults. A method was proposed by improving the envelope order spectrum analysis method to achieve faulty bearing detection [24]. This method need not preset a priori parameters; however, the computational complexity is increased. Laplacian of Gaussian (LOG) filter is usually applied in images de-noising. Laplacian operator is often applied for edge detection and the tested signal is smoothed by Gaussian mask [25]. The edge of an image can be considered a sudden change, and it is similar to the transient impulse in a faulty vibration signal. Saad et al. proposed a modified method to pick the P-arrival time clearly [26]. This filter method need not input any prior knowledge. However, it weakens the characterisation ability in rotating machinery faults.
With the purpose to extract fault components of the non-stationary signal, this study investigates a novel method named teager energy operator (TEO)-based modified LOG (MLOG; TEO-MLOG) filter. Based on the extracted IRF curve, obtained by MSST method, the time-varying signal is converted into an angular domain signal with re-sampling method. TEO is introduced to pre-process the re-sampled signal for strengthening the impact component [27] using MLOG filter to pick the transient component and then recognising fault characteristic orders (FCOs) from the envelope spectrum.
Subsequently, the rest of this study is as follows. Three theories of MLOG filter, TEO and MSST are given in Section 2.
The specifics of TEO-MLOG method is studied in Section 3. In Section 4, the performance of the TEO-MLOG method is testified in numerical simulation case. In Section 5, experimental investigations of two types of fault bearings are investigated to further test the validity of the TEO-MLOG method. Section 6 draws the conclusion.

MLOG filter
Second-order edge detector of LOG filter is used to process an image when the scale of the interesting features are often unknown [28][29]. According to the 2D LOG formula, the 1D Gaussian filter is defined as where is defined with standard deviation and Gaussian index is n. We can obtain the first-order differential of Equation (1) as And the second-order differential of Equation (1) can be shown as Equation (3): The normalised LOG filter is described as The effect of the finite impulse response (FIR) filter for the vibration signal is similar to the LOG filter for vibration signal. In image processing, a good high-pass FIR filter is the taps sum to zero, which can effectively achieve edge extraction. In the same way, the transient impulses of the fault vibration signal can be extracted. However, the LOG filter is sensitive to the background noise. Furthermore, an MLOG filter is proposed, which not only picks the transient impulse components but also smooths the background noise: where K is the order of filter of the MLOG filter.

Teager energy operator
TEO has a powerful ability to measure sudden changes in the vibration signal. It demonstrates the total energy of the collected signal within a specific limited frequency band [30]. Moreover, TEO also amplifies the discontinuous points and sudden amplitude changes into the signal when the soft transitions between points decrease. In continuous-time signal, the definition of TEO is shown as where the y(t ) is collected vibration signal, t is differential processing with respect to time t and [ ] denotes TEO. If the signal is discrete, the new definition of TEO can be easily obtained as where s(n) is a discrete-time signal.

Multi-synchrosqueezing transform
STFT is regarded as a classical TF analysis method; many advanced methods are based on it [31][32][33]. The function of STFT of the signal x(t ) is defined by the below formula: where g( − t ) is the window function along the time axis, X (t, ) represents the correlation between x( ) and g( − t )e − j ( −t ) . According to the Taylor expansion, we can construct a signal model in which the form is written as A( ) = A(t )and ( ) = (t ) + ′ (t )( − t ) at time t , and the formula is given as Combining Equations (9) and (8), we have where the Fourier transform of g(⋅) isg(⋅), the obtained IRF curve by calculating the derivative of X (t, ) iŝ In order to trace the signal features in the time axis, it is necessary to reduce the window length; however, these results have a poor frequency resolution. As a consequence, the difficulty of this method is to choose the suitable parameters to obtain a satisfactory resolution, either time or frequency. By introducing a frequency reassignment operator to squeeze the energy around the rotational frequency component, which is named synchrosqueezing transform (SST) [34], we can formulate it as Summarising the last related studies, the error between the estimated IRF and reality IRF becomes bigger as the signal changes fast, and SST may cause the blurry in TF plane. MSST overcomes this drawback by applying an iterative processing based on SST. Equation (13) describes the procedure: where N is the iteration number; finally, the ideal IRF curve is estimated by MSST.

THE TEO-ENHANCED MLOG TO DETECT FAULTS USING ENVELOPE ORDER SPECTRUM (EOS)
In order to realise the transformation of different domains, the primary stage is to estimate the IRF curve of the shaft. The most irrelevant components have been removed by the re-sampling technique; however, the accurate fault diagnosis of quasi-stationary (angular domain) signal still presents some hindrance. This study introduces the TEO-MLOG filter to purify the polluted signal, which not only smoothes the background noise but also picks transient impulse. Figure 1 presents the detailed flowchart of the TEO-MLOG method. The specific steps are shown as follows: 1. The fault vibration signal is acquired from the machinery fault simulator (MFS) test. 2. The original sampling frequency is decreased at a lower value.
We synchronously keep the rotation frequency information and the obtained vibration signal. In this study, considering the rotation speed range of shaft in the experimental operation, 0-2400 rev/min, the lower value is 200 Hz. 3. To extract the first IRF component, a band-pass filter can be adapted to eliminate multiple-order IRF components. In

NUMERICAL SIMULATION
A numerical simulation of bearing with a localised failure on outer race under variable rotational speed conditions is conducted for testing the behaviour of the TEO-MLOG method. Randall et al. [35] generated a model as follows: and r (t ) = e − t cos(2 f n + ) (15) where T represents the fault transient impulses interval in time series, m denotes the influence of the random slippage of rollers-we assume the range from 0.01 to 0.02-and A m is the amplitude of the mth impulse, denotes damping characteristic, f n is natural frequency and is original phase. Combining Equations (14) and (15), the formula is rewritten as where (t ) is the added white Gaussian noise with signal-tonoise ratio (SNR) of -10.1 dB, T m means the interval time between two impacts. In order to meet the time-varying operation condition, we assume that f r (t ) is the instantaneous frequency and t m represents the mth impulse occurrence time, respectively. K is FCO. Considering the influence of the ball slippage, we obtain the formula of fault occurrence time as In this case, some parameters of the model are given: m = 0.02, A m = × t m , is constant and the rotation frequency f r (t ) = 6 × t + 25, and the other specific parameters are given in Table 1.
The waveform of the simulated signal along the time axis is displayed in Figure 2(a) and then calculated by FFT; the result is shown in Figure 2(b). Non-stationary signal properties are variable with speed change, which lead to the spectrum as smeared; therefore, the conventional signal processing methods are limited in this condition. The vital step of TLOT is to transform the non-stationary to the quasi-stationary domain using the resampling technique.
According to the procedure of the flowchart, the simulated signal is down-sampled and then filtered. We obtain the lower sampling frequency value from Table 1. Next, the parameters of the filter are-the central frequency is 35 Hz and the bandwidth is 32 Hz. Before conducting the re-sampling technique for simulated signal, TF representation (TFR) is calculated via TF analysis method to obtain high-quality IRF curve and extract it. From Figure 2(c), the result of calculation of TFR using MSST method is clearly displayed and its extracted IRF is displayed in Figure 2(d). MSST has the ability to improve the TF resolution. From Figure 2(d), the actual IRF curve is the green line, which is obtained by the collected key-phase signal via a tachometer. The calculated details can be referred to [22]. Subsequently, on that basis, it is essential to change over from the non-stationary to the angular domain. The effect of de-noising of MLOG can

EXPERIMENTAL INVESTIGATIONS
The good performance of the TEO-MLOG method is testified via two faulty rolling-element bearing signals, which are acquired from the MFS test rig of SpectraQuest Co [36,37]. Two groups of the tests are prepared (inner-race and outer-race faults) [22].  (17): Substituting the detail parameters of the tested bearings into Equation (17), the formulas are: The FCO of the inner-race and outer-race bearings are obtained as 4.95 and 3.05, respectively.

Innersrace fault detection case
The rotation speed range of the shaft is set from 0 to 40 Hz and the signal length is 12.8 s. We select 106,600 points as the tested signal. Figures 6(a) and (b) show the waveforms of the nonstationary signal and spectrum. Similarly, Figure 6(c) displays the lower sampling frequency signal with the down-sampled technique. Although some pre-processing steps are employed, no useful components can be detected directly. Either in the nonstationary or the frequency domain, any fault information is overwhelmed in the unwanted components. In order to extract the first rotational frequency curve, the initial step is to decrease the raw sampling frequency to 200 Hz. According to the range of rotation frequency of the shaft, which is from 0 to 40 Hz, the important parameters of filter are determined (central frequency is 28 Hz and bandwidth is 30 Hz). The next step is to obtain TFR of the down-sampled signal. The result calculated by WT is shown in Figure 7(a). Divergent energy of TF results in low-quality IRF curve estimation. Figures 7(b) and (c) present the TFR generated by MSST and the estimated IRF curve by the MSST-based algorithm. It is the same as the numerical simulation; the green line corresponds to actual IRF, which is obtained by the processed collected keyphase signal. Apparently, MSST can provide better TF resolution than WT in analysing the non-stationary signal. As in step 5, the non-stationary signal is transformed to the quasi-stationary domain using the re-sampling technique.
The re-sampling step is accomplished, and the waveform of the quasi-stationary signal along radian axis is shown in Figure 8(a); furthermore, the result of the angular domain signal is calculated by the envelope order spectrum analysis and is shown in Figure 8(b). High amplitude values 1 and 2 can be found, which belongs to shaft orders. De-noising by MLOG to acquire purified signal and through envelope order spectrum analysis to obtain features are presented in Figure 8(c); the acquired result is the same with Figure 8(b). Figure 8(d) shows the comparison order spectrum result obtained by TEO-MLOG-filtered signal, 2, 4.97, 9.94, 14.89 are the shaft orders, which approach the computational ideal FCO of the inner-race fault and its harmonic orders 1 × FCO, 2 × FCO, 3 × FCO. It is obvious that TEO-MLOG can effectively improve the behaviour of envelope order spectrum analysis

Outer-race fault detection case
During this part, the experimental data length is 12.8 s. To raise the computation efficiency, we select 106,600 points to testify the good behaviour of the TEO-MLOG method, the raw signal length is 4.16 s and the speed range is the same as the innerrace fault detection case. Therefore, we set the central frequency as 28 Hz with a bandwidth frequency of 30 Hz as the parameters of the band-pass filter. From the waveform of the nonstationary signal, the amplitudes become bigger cause by speed run-up in Figure 9(a). Spectrum is shown in Figure 9(b), which is processed by FFT. Unfortunately, the fault information is hidden due to the shaft speed variation. The experimental data is down-sampled, and then the first rotation shaft frequency component is obtained using the band-pass filter. Figure 9(c) shows the former result. The tested signal is decimated at 1/128 times of 25,600 Hz. After that, the obtained filtered experimental data is applied to accomplish the IRF curve extraction by TF analysis methods. Figure 10(a) exhibits the TFR of the processed signal, which is obtained by WT. From Figure 10(a), there is a coarser TF resolution by WT. The result of MSST is shown in Figure 10(b), which can provide high energy concentration in 2D plane. The extracted IRF is shown in Figure 10(c), where green line represent the real IRF curve. By comparing the results of Figures 10(a) and (b), we can obtain the more precise IRF curve with the MSST method.
The experimental data is re-sampled from non-into quasistationary via the re-sampling technique. Figures 11(a) and (b) exhibit the transformed signal and its corresponding order spec-   Figure 11(c) exhibits the order spectrum result of the filtered signal by MLOG; similarly, the peaks of shaft orders can be simply recognised. TEO-MLOG method is introduced to pick fault transient impulse component, after calculating the processed sig-

CONCLUSION
This study proposes an approach named TEO-MLOG-based TLOT for rolling-element bearing fault diagnosis under variable rotational speed conditions. MSST is able to improve TF energy to extract the corresponding vibration signal IRF curve. On this basis, to confirm a constant angle to realise re-sampling processing for eliminating the smear effect caused by the variable speed, TEO-MLOG is further employed to purify the angular domain signal from heavy background noise and unrelated components. Subsequently, the FCO is detected from the order spectrum and recognises the faulty type of rolling bearing. To address the advantages of the aforementioned methods, the comparison investigations are presented by both numerical simulation and experimental investigations. Comparison results show that TEO-MLOG method has superior fault component detection, much easier than those using MLOG. It indicates that TEO-MLOG method for bearing fault diagnosis when operating in variable rotational speed conditions is available.