A Fault Diagnosis Method of Train Wheelset Rolling Bearing Combined with Improved LMD and FK

Trackside acoustic signals contain intense noise and nonstationary features even after Doppler distortion correction. Information on bearing defects in these signals is either weak or heavily attenuated. Thus, an improved compound interpolation envelope local mean decomposition (ICIE LMD) method combined with a fast kurtogram (FK) is proposed for wheelset bearings. In this methodology, cubic Hermite interpolation and cubic spline interpolation are employed to find the envelope of the extremal points in the ICIE LMD algorithm to improve accuracy and decrease the computing time of the decomposed signal component. An FK is sensitive to the impact signal and extracts the fault impact features efficiently. In the application, the proposed method uses ICIE LMD to decompose the multicomponent signal into several specific single product function (PF) components. The kurtosis index of the PF is calculated to select the component which contains the most fault information. Then, the selected component of PF is filtered by FK. Finally, the squared envelope spectrum is used to obtain the fault frequency and identify the fault location. The advantages of the ICIE LMD method are verified by simulation analysis. In the application, the results show that the proposed method efficiently extracts the fault features and enhances the target characteristics of the sound signals from a trackside microphone array. Furthermore, the influence of rotating frequency on locating the fault is reduced.


Introduction
The wheelset bearing of a high-speed train is an indispensable component that is prone to damage when the train is running.Bearing damages and defects directly affect the stable operation of equipment [1].The defects, such as pitting corrosion and cracks, happen on the contact surface of the rolling bearing quickly, when the bearing is under the alternating contact force for a long time, especially when a train is running at high speed and with the heavy loads [2].Traditional diagnosis methods for high-speed train rolling bearings are mostly done by the vibration signal.Because of the application of the trackside microphone array system in railway inspection, it is important to analyse the acoustic signal of the rolling bearing obtained from the microphone.The difference between the two types of signal is that the vibration signal is collected directly on the surface, whereas the acoustic signal is picked up through the propagation in the air.The acoustic signal becomes weaker, and much more noise and a complicated reflection are introduced, heavily distorting the original signal.Furthermore, collection diversity influences the efficiency and accuracy of the current fault diagnosis method.
The key to fault diagnosis is mainly extracting the fault feature from the rolling bearing signal [1].When a fault happens, the collected signal is a multicomponent modulation signal, whose characteristic is nonstationary and nonlinear.Time-frequency analysis is widely used for fault diagnosis, such as short-time Fourier transformation (STFT), Wigner-Ville distribution, and wavelet transform [3].However, the methods mentioned produce an individual decomposed result when processing a signal, but the limitation is that they are not adaptive [4].For instance, wavelet analysis cannot denoise a weak signal because of the selection problem of the wavelet basis function and decomposition level [5,6].Empirical mode decomposition (EMD) has the problem of mode mixing.When the signal is decomposed into a sequence of intrinsic mode functions (IMFs) by EMD, the characteristic time scales of IMFs vary [7].
Local mean decomposition (LMD) is a new signalprocessing method developed by EMD and proposed by Smith [8].LMD decomposes the nonstationary signal into the product function (PF) of a purely frequency-modulated signal and envelope signal component [9].It can be decomposed adaptively according to the practical scale characteristics of the signal and can solve the mode-mixing problem caused by EMD [10].However, the end effect and mode mixing still exist in LMD decomposition.Xu et al. [11] introduced an energy-matching extension method to solve the end effects, verified by the simulated signal.To weaken the influence of end effects, Du et al. [12] considered the signal a discrete difference, determining the value of extreme points of continuation according to the specific situation.The result of the new method was adapted to the signal of the rolling bearing with the outer raceway and inner raceway improved by mirror extension.Ensemble LMD (ELMD) was proposed by Yang et al. [13], which adds the Gaussian white noise into the original signal and changes the extreme point distribution of the signal.According to the decomposition characteristic of ELMD, a method of K-L divergence used adaptively in selecting the principal PFs was proposed by He and Zhou [14].
The accuracy of PF components is affected by the envelope function of LMD.The moving averaging method applied in the original LMD produces a phase error of functions under the running of the algorithm [15].Hu et al. [16] developed a method to obtain the envelope estimation function by applying cubic spline interpolation to calculate the envelopes of local extreme points.This method improves computation time, but undershoot and overshoot problems of the envelope are observed for signals with strong nonstationary features [17].Spline-based LMD, combined with the second-generation wavelet, is suitable for extracting the fault feature of the weak fault signals [18].In a previous work [19,20], the moving average was replaced by monotonic piecewise cubic Hermite interpolation (MPCHI) in the LMD algorithm, and this increased the accuracy of the PF components that contained the feature of the nonstationary signal.MPCHI can avoid the undershoot and overshoot problems of the envelope with a nonstationary signal.However, the acoustic signal and vibration signal mix several nonstationary and stationary parts.Hence, Zhao et al. [21] proposed a method combined with cubic spline interpolation (CSI) and MPCHI to obtain the envelope of extreme points in the LMD method.The nonstationary coefficient was defined to divide the envelope of the signal into the CSI and MPCHI parts.
The fast kurtogram (FK) was developed based on spectra kurtosis (SK), which is mainly used to calculate the kurtosis value.The kurtosis value is susceptible to transient signals, which reflect the numbers of nonstationary components included in each frequency curve [22].FK is used to optimize the centre frequency and bandwidth of bandpass filters and then extract the pulses.Barszcz and Randall [23] explored some methods usually applied to fault detection and online diagnosis.The result shows that only the SK method can recognise catastrophic gear failure.Zhang and Randall [24] combined genetic algorithms and FK for rolling component bearings.The experiment illustrated the effectiveness of the approach.It was proved to provide better results than resonance demodulation.The FK gives simple processing results very efficiently.When applied individually, the FK can extract the pulse but cannot recognise the fault location because of the rotating frequency influence.
Concerning the problems mentioned, the improved compound interpolation envelope (ICIE) LMD combined with FK is presented to extract the fault feature of the rail-side acoustic bearing signal.First, the original signal is decomposed into several single components by ICIE LMD.Second, the kurtosis value is calculated for each element.Because kurtosis is sensitive to a shock signal, the component with the maximum value of kurtosis is selected for FK analysis, and the square envelope spectrum is used to extract fault feature information.The average peak signal-to-noise ratio (PSNR) of the method is used to show the efficiency of the proposed method for fault diagnosis.The results show that this method can effectively extract and enhance the fault characteristics of the trackside acoustic rolling bearing and weaken the influence of rotating frequency.
The theories of CSI LMD, ICIE LMD, and FK are briefly introduced in Section 2. In Section 3, the simulation analysis of ICIE LMD with artificial data is shown.In Section 4, the proposed method is applied to the fault diagnosis of the real trackside acoustic bearing signal.The conclusions are outlined in Section 5.

Improved LMD Method
2.1.Principle of CSI LMD.LMD adaptively decomposes the signal into a series of PF components, which are the product of a purely frequency-modulated signal and an envelope signal.The frequencies of the decomposed PF components are arranged automatically from high to low.According to the CSI LMD algorithm, the upper and lower extreme envelopes are derived by CSI for two functions, replacing the movingaveraging method in the classical one [25].The algorithm is as follows [25]: Step 1.The local maximum and minimum points of the given signal xðtÞ are acquired, and then the upper envelope e u and the lower envelope e l of extrema are obtained by CSI Step 2. The local mean value m 11 ðtÞ and the envelope function a 11 ðtÞ are derived by the upper envelope e u and the lower envelope e l as Step 3. The local mean value m 11 ðtÞ is separated from the primitive data xðtÞ 2

Journal of Sensors
Then h 11 ðtÞ is divided by a 11 ðtÞ, illustrated as The above steps are repeated, and the envelope function a 12 ðtÞ can be acquired.Ideally, if s 11 ðtÞ is a purely frequency-modulated signal, which means a 12 ðtÞ = 1, the procedure ends.If not, s 11 ðtÞ is regarded as primitive data, and the above step is iterated again until satisfying the above demand.However, in practice, the envelope function a 12 ðtÞ cannot be exactly 1.The threshold value Δ is set.When 1 − Δ ≤ s 1n ðtÞ ≤ 1 + Δ, iteration ends Step 4. The corresponding envelope a 1 ðtÞ is calculated as The first product function PF 1 ðtÞ can be obtained by Thus, u 1 ðtÞ is given by subtracting PF 1 ðtÞ from the original signal xðtÞ.u 1 ðtÞ is regarded as an initiative signal to do all steps until it is monotonic.
After that, xðtÞ is represented as 2.2.Improved Compound Interpolation Envelope Algorithm.
Because of the envelope problem of undershoot and overshoot in the CSI method, the compound interpolation envelope LMD (CIE LMD) was proposed by Zhao et al. [21].It defined the nonstationary coefficient to represent the state of the signal.The threshold of the nonstationary coefficient is chosen to divide the signal into a stationary section and nonstationary section.The envelope is made compound by CSI for the stationary part and MPCHI for the nonstationary part of the signal.However, compared with CSI LMD, the way to calculate the nonstationary coefficient and envelope increases the computing time in CIE LMD.Therefore, the ICIE LMD method applies the CHI method, replacing the MPCHI method, to simplify the algorithm and reduce the central processing unit (CPU) time.The definition of the CHI method is given as follows.
For the interval [a, b], the node series is For the same series x, the interpolation function of MPCHI is described as Equation ( 8) [21].
The parameters in the formula are defined by The first derivative d i is where If the data are monotonic, i.e., Δy ≥ 0 or Δy ≤ 0, The MPCHI method needs to make a judgment on the data for their monotonicity before processing.Moreover, it has a different way to calculate the derivative values compared with CHI.The factors cause CPU time to increase.
Furthermore, the threshold of the nonstationary coefficient is calculated by the arithmetic mean in Equation (12) in ICIE LMD to adapt the different types of signal and obtain better accuracy.
Then, the steps of the ICIE LMD algorithm [21] are as follows: Step 1.The local extrema series n i ði = 1, ⋯, kÞ of the given signal xðtÞ is calculated, and the maximum extrema series and minimum extrema series are dealt with separately 3 Journal of Sensors Step 2. The slope of one of the extrema series Δ i ð1 = 1, ⋯, k − 1Þ is calculated; after that, the nonstationary coefficient W i ði = 1, ⋯, k − 2Þ is calculated by Equation ( 13) Step 3. The threshold value W of the nonstationary coefficient is obtained by Equation (12).The value of nonstationary points is larger than W. Their position is recorded as P i ði = 1, ⋯, mÞ Step 4. The point is nðP i Þ and its previous one is nðP i − 1Þ, and the following one is nðP i + 1Þ, all as interpolation points for CHI; the local CHI interval is If the number of extrema points between two nonstationary points P i and P i+1 is less than five, the two CHI intervals are connected as ½P i − 1, P i+1 + 1; if the number of extrema points between nonstationary point P i and the left or right extrema series is less than three, the initial local CHI interval is ½1, P i + 1 and the end interval is ½P i − 1, k Step 5.For the rest of the stationary points, the CSI envelope is created in the same way as in Step 4 Step 6.If all CHI intervals are empty, the CSI method creates the whole extreme series.Similarly, if all CSI intervals are empty, the envelope is created by the CHI method Step 7. To compound the two types of envelope, the connecting endpoints of the local CHI interval are assigned to CSI Step 8.The above steps are repeated to obtain the envelopes of other extrema series This method is applied to replace the first step of the CSI LMD method presented in Section 2.1, and keeping the remaining procedures, the ICIE LMD is illustrated.The spectral form of the World-Cramer decomposition is defined in the nonstationary state.
The nonstationary signal xðnÞ and the stationary noise bðnÞ are mixed, and their SK is calculated as where dZ x ð f Þ is an orthonormal spectral increment and Hðn, f Þ is a complex envelope of xðnÞ at frequency f and time note ρð f Þ is the noise-to-signal ratio of the mixed signal.
When the value of ρð f Þ is close to zero, no noise exists here.
When Δf is close to the bandwidth of the fault characteristic signal, the function of the frequency f and window length N w reaches the maximum.Therefore, the centre frequency and bandwidth of the bandpass filter can be optimized by the FK algorithm, and the narrowband impact signal can be extracted under a condition of substantial noise.The flowchart of the proposed method applied in fault detection for rolling bearings is shown in Figure 1.

Simulation Analysis
where y 0 , g, and f n are the parameters of the periodic impact signal with an amplitude of 2, damping coefficient of 0.04, and characteristic frequency of 3000 Hz.Time t is 0.000001 s, and the period of the shock signal is 0.005 s per cycle.The simulation signal is shown in Figure 2.
The three algorithms can fit the shape of the original signal in the stationary part.From the magnified nonstationary part of Figure 3, the results of CSI and CIE in this part are overshoot and undershoot.This results in a decrease of the decomposed accuracy of the signal.
To analyse further the accuracy of the extreme envelope, the local mean function is calculated by the two methods, in contrast to the theoretical local mean function y 2 ðtÞ.The absolute value errors and CPU runtime are calculated to evaluate the utility of different methods, as shown in Table 1.It is illustrated that the CPU time of CSI is the shortest while the absolute errors are the largest.The absolute error value of the ICIE method is close to that of the CIE method, but the computational cost of the improved method is less than that of the CIE method.
The selection of the threshold of nonstationary coefficients causes undershoot and overshoot problems during the envelope calculation.In this study, the arithmetic mean of nonstationary coefficients is used as the threshold, and the effects of particular numerical 2 and geometric mean and harmonic mean as the threshold are analysed, as shown in Figure 4.After the calculation, the value of the geometric mean is 1.97, and the harmonic mean is 2.17.The arithmetic mean is 2.45.The arithmetic mean method can adapt to different signal characteristics so that the envelope does not have overshoot and undershoot problems.However, the other methods have this problem which affects decomposition accuracy.
The impact signal and sinusoidal signal compose the simulation signal.In addition, the impact signal is composed of the high-frequency signal and the exponential signal.Thus, the sinusoidal and exponential signals work together to modulate the amplitude of the high-frequency signal.The ICIE, CSI, and CIE LMD methods are used to obtain different time scale components.The ICIE and CSI algorithms get three PF components, while the CIE method gets two PF components as Figure 5 shows.Regarding the three methods, the first PF represents the feature of the impact signal.The second one is obtained from the sinusoidal signal.For the CIE, the decomposition is stopped, while the   2. The PF components from the ICIE LMD method indicate the inherent features of the simulated signal better.However, distortion exists after processing with the CSI method.

Simulated Evaluation under the Artificial Signal of Rolling
Bearing.The real fault signal of a rolling bearing is more complicated than the individual nonstationary signal because of

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Journal of Sensors the noise.The decomposition performances of the three methods are presented by working on the simulation of the rolling bearing fault signal.The simulation uses an impact signal with 0.1 dB white noise, defined as follows: where y 0 is 3, f n is 3000 Hz, and g is 0.1.The sampling frequency is 30000 Hz.The fault frequency is set to 50 Hz, which means a period of 0.02 s per cycle.The simulation fault signal is shown in Figure 6, where the impact signal represents the fault shock.The decomposition results of the three methods are presented in Figure 7.Then, the kurtosis is calculated to get the PF with maximum impact information in each method, as Table 3 shows.The maximum value is selected to calculate the Hilbert envelope spectrum, as in Figure 8. ICIE LMD can extract clear impact information under noise interference.As for CSI and CIE LMD, parts of the impact information are disrupted by noise.Therefore, compared with CSI and CIE, the ICIE method can fit the upper and lower envelopes of the original signal better.Comparing the absolute error value of the local mean function obtained by the methods shows that the precision of the ICIE method is higher than that of the other methods.The decomposed components of ICIE LMD have better feature information about the original signal.Because the noise

Experiment Validation
To demonstrate the validity of the designed algorithm for rolling bearing fault detection, experimental signals collected from a trackside microphone after Doppler effective correction were analysed, and they were outer ring fault signals.Furthermore, the compound algorithm was compared with other methods in the experiment data analysis, illustrating the superiority of the proposed algorithm.The acoustic signal sampling frequency was 45,249 Hz, and the speed of the inner ring was 380 r/min.The specifications of the bearing used in the experiment are presented in Table 4.
Figure 9 shows the time domain waveform of the bearing acoustic signal.Some effects can be observed; it contains intense noise.To locate the fault position, first, the signal is decomposed by ICIE LMD.The first couple of PF components dominated the fault information.Thus, the first three PF components are shown in Figure 10.The kurtosis values of the PF components were calculated and are presented in Table 5.The kurtosis values of the three components were similar, but the value of PF1 was the largest.Thus, PF1 was filtered by a bandpass filter based on FK.The result is shown in Figure 11(a).The colour of the kurtogram indicates the kurtosis value.The darkest colour represents the largest kurtosis value, which contains the most fault information.
The optimized parameter of the bandpass filter was acquired from the kurtogram.Because the largest kurtosis value is 6 and the centre frequency was 2500 Hz, the band frequency chosen was 5414 Hz.After filtering, the orders of fault characteristic frequency were clearly illustrated by the squared envelope spectrum.The rotating frequency was not     Compared with the performance of the proposed method and individual FK and ICIE LMD methods, the PSNR is applied as Equation (19).The PSNR can determine the effect of extracting the fault feature signal.
where V S is the characteristic frequency spectrum line amplitude and V N is the maximum noise spectral line value near the characteristic frequency spectrum line.There, the PSNR * , which is the mean value of five orders of fault frequency amplitude over related noise frequency amplitude, is calculated, as shown in Table 6.The proposed method has the largest value of PSNR, whereas the individual FK method has the smallest one.The reason is that the rotating frequency is useless in fault diagnosis, relative to the fault frequency.The extracted fault characteristic frequency was 44.19 Hz, close to the ideal one.The proposed method was verified with better results compared with other methods.

Conclusion
A method combining ICIE LMD and FK was proposed for railway wheelset rolling bearing fault diagnosis.The method improved the performance of fault feature extraction and had improved fault recognition accuracy.In the simulation, the ICIE envelope performed better than the CIE and CSI envelopes.The CSI method had minimum CPU time with the sacrifice of the accuracy of the envelope.The ICIE method flexibly adapted to a different signal.The arithmetic mean was defined to calculate the threshold of the nonstationary coefficient.The decomposed result of the proposed method obtained a better signal feature.For the simulated fault signal with noise, ICIE LMD was not easily affected by noise, compared with the other methods.The simulation verified that the ICIE LMD method has better accuracy than other methods.
The proposed method and individual methods were all deployed to process the real train wheelset rolling bearing acoustic signal.This involved two steps.First, the signal was decomposed by ICIE LMD, and the PF was selected according to the kurtosis value.Then, the PF was filtered by FK.The squared envelope spectrum extracted the characteristic fault frequency.The PSNR was proposed to identify the performance of those methods.The result demonstrates that the proposed method obtains clearer fault features.

2. 3 .
Fast Kurtogram.In rolling bearing faults, the existence of transient vibration signals is highly correlated with cracks and clearances.SK is a useful transient signal detection technology.Antoni [26] established the spectral kurtosis theory in 2006, defining the spectral kurtosis by STFT and developed the fast kurtogram algorithm.

Figure 1 :Figure 2 :
Figure 1: The flowchart of the proposed method.

3. 1 .
Simulated Evaluation under Nonstationary Signal.To compare the CSI, CIE, and ICIE envelopes, the following simulation signals with strong nonstationary characteristics are constructed:

Figure 4 :
Figure 4: The result of the different nonstationary coefficients getting the envelope.

Figure 7 :
Figure 7: The PF components by three methods.

Figure 10 :
Figure 10: The PFs from the collected signal processed by ICIE LMD.

2 )Figure 9 :
Figure 9: The rail side acoustic signal of rolling bearing.

Figure 11 :
Figure 11: The kurtogram (a) and the squared envelope spectrum (b) from the designed method.

KFigure 12 :
Figure 12: The kurtogram (a) and the squared envelope spectrum (b) from the individual FK method.

Figure 13 :
Figure 13: The envelope spectrum of the original signal (a) and PF1 decomposed by ICIE LMD (b).

Table 1 :
Performance of different LMD methods.

Table 2 :
The correlation coefficients for PF components.

Table 3 :
Kurtosis of the simulation fault signal.

Table 4 :
Parameters of the rolling bearing.

Table 5 :
Kurtosis coefficient of CIE LMD product functions and the original signal.

Table 6 :
Peak signal-to-noise ratio of the different methods.