Multispectral Balanced Automatic Fault Diagnosis for Rolling Bearings under Variable Speed Conditions

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Introduction
Rolling bearings are widely used in rotating machines, and their dynamic performance plays a signifcant efect on stability, reliability, and even safety of the machines [1].Automatic diagnosis of mechanical state based on sensor signals has attracted more and more attention in recent years [2].Since most rolling bearings work in harsh variable speed conditions, such as motor bearings and axle-box bearings in high-speed trains, it is necessary to carry out bearing fault diagnosis under variable speed conditions [3,4].Failure of bearings cause abnormal vibration and noise, which means that vibration signals of rolling bearings carry rich information about their states of health [5].Terefore, classical fault diagnosis methods evaluated health of equipment through taking time domain or frequency domain indicators of vibration signals under normal conditions as a reference [6,7].However, characteristics of signals become more complex under variable speed conditions [8].Specifcally, spectral lines become blurred because of the speed changes; thus, it is difcult to make accurate characteristic frequency analysis to diagnose faults automatically.Consequently, automatic fault diagnosis of rolling bearings under variable speed conditions relies on two conditions: (1) eliminating the infuence of variable speeds; (2) fnding an appropriate adaptive flter to complete automatic diagnosis.
In recent years, many researchers proposed methods to eliminate the infuence of variable speed conditions.Signal processing methods based on order tracking are widely used.Te main step of order tracking is resampling signals at a constant angular increment.Bossley et al. [9] proposed the hardware order tracking (HOT), which resampled equiangular signals of vibration signals by photoelectric encoders and key phasor sensors.Encoder signals could be efectively used for health monitoring of rotating machines [10].However, the required encoders and key phasor sensors are expensive, and their control circuits are complex.Tese disadvantages limit its application.Terefore, Kong et al. [11] proposed the calculation order tracking (COT) technology realized by computer.Tis method is more convenient than the previous one because vibration signals and key phasor signals are collected synchronously by acquisition equipment at equal-time intervals.However, it is diffcult to obtain key phasor information in practical applications of this method.Accordingly, Zhao et al. [12] proposed a tacho-less order tracking technique to extract the tachometer information from a low frequency range.Although all methods mentioned above can achieve fault diagnosis under variable speed conditions to a certain degree, they are only applicable for the ofine fault detection.
Aiming at this shortcoming, Wang et al. [4] proposed an online tacho-less order tracking method which solved three problems including ridge detection under heavy background noises, online monocomponent extraction, and adaptive bandpass flter.In this method, the instantaneous frequency of harmonics was obtained based on adaptive short-time Fourier transform (ASTFT) and ridge search.Although satisfactory resampling results can be derived when processing simulated signals, ridge line search may fail when processing experimental data.With the development of neural networks, many scholars have chosen deep learning to eliminate the impact of variable speed conditions.Lu et al. [13] proposed a novel fault diagnosis method based on spectrum alignment (SA) and deep transfer convolution neural network (DTCNN), where the SA algorithm that can convert nonstationary vibration signals into stationary signals.After verifcation, this method has shown good efectiveness and high accuracy in bearing fault diagnosis.However, the premise of this method is that the speed signal is known.In actual operation, the accuracy of the speed information calculated from the impact signals is very poor, which cannot meet the requirements of the SA algorithm.
Furthermore, it is necessary to demodulate signals by fltering to improve signal-to-noise ratio after resampling.Tus, bearing fault diagnosis can be carried out through prominent fault feature frequencies.In recent years, scholars proposed a series of classical demodulation and frequency analysis methods, such as envelope demodulation analysis, spectral kurtosis analysis, wavelet packet demodulation [14,15], empirical mode decomposition [16,17], and stochastic resonance theory [18,19].Te most important problem of fltering is setting parameters of the flter including bandwidth and central frequency.Dwyer [20] proposed frequency domain kurtosis (FDK), which can determine flter parameters to fnd out fault impulse components in signals.However, the accuracy of this method depended on a reasonable length of analysis segment.Another method calculated the kurtosis of signals in time domain [6].Among them, the fast kurtosis (FK) showed the best efectiveness [7].Te calculation speed of FK is faster than FDK, and there are fewer hyperparameters as well.However, it is difcult to fnd out correct flter parameters when the energies of target signals are low and environmental noises are large.Terefore, Barszcz and JabŁoński [21] proposed Protrugram, which was an adaptive method for optimizing flter parameters.In order to extend the conventional diagnosis technique under variable speed conditions, Wang et al. [22] proposed a rotating speed isolation method based on Protrugram.Tis method took four diferent fault frequencies of rolling bearings as flter bandwidths and selected the flter center frequency adaptively.However, after Protrugram with diferent characteristic frequencies, the diference in amplitudes of the obtained spectrums is large.Tis means that the fltering result with the maximum amplitude will be determined as the fault type by mistake.
At the same time, some scholars have proposed the methods based on deep learning to diagnose the fault of bearings.Compared with traditional methods, intelligent diagnosis can automatically capture useful features from the original signals, instead of manually selecting and extracting them [23].Lu et al. [24] exploited an explainable onedimensional convolutional neural network model by combining with the demodulated frequency features of vibration signals and apply it to the fault classifcation of rolling bearings under time-varying speed conditions.Tis method divided the input signals into a training set and a testing set.Ten, it labeled the training set and trained the intelligent diagnostic model.Te accuracy of fault classifcation obtained by testing the intelligent diagnostic model through a test set is very high.However, the step of labeling the training set must be manually completed.Tis means that the above deep learning-based methods cannot achieve true automatic diagnosis.
According to the above description, there are two problems about automatic fault diagnosis for rolling bearings under variable speed conditions.On the one hand, the angle-domain resampling method must avoid broken of ridge line search to eliminate the infuence of variable speed conditions.On the other hand, amplitudes of signals after fltering need to be unifed to make sure that the fltering method can be applied to the automatic diagnosis process.In order to solve these problems, Multispectral Balanced Automatic Fault Diagnosis (MBAFD) is proposed to diagnose faults of rolling bearings under variable speed conditions.Te proposed method consists of two modules: Multispectral Lossless Preprocessing Module and Balanced Envelope Demodulation Module.
When speed changes too much, the energy distribution range of signals expand

Multispectral Balanced Automatic Fault Diagnosis
Te proposed method consists of two modules: Multispectral Lossless Preprocessing Module and Balanced Envelope Demodulation Module.Multispectral Lossless Preprocessing Module eliminates the infuence of variable speed conditions without losing information about the fault.In this module, Multispectral refers to the fusion of spectrum, envelope spectrum, and known inaccurate speed.In the Balanced Envelope Demodulation Module, the Balanced Envelope Spectrum (BES) is constructed to eliminate the diference of amplitudes between fltered results of Protrugram.Te framework of the proposed method is shown in Figure 1.

Multispectral Lossless Preprocessing Module.
Te energy distribution range of rolling bearing vibration signals is too large under variable speed conditions.Tus, using a single spectrum to analyze bearing fault inevitably leads to fault information loss.Specifcally, the original spectrum line is clear when speed is high, and the envelope spectrum line is clear when speed is low in the experiment.Terefore, the information in these two spectrums should be integrated skillfully.In general, it is easy to get inaccurate bearing speed information.Terefore, the inaccurate rotational speed information is used to revise the ftting range in order to avoid descent of precision.Te spectral line processing method of multispectrum fusion is proposed to solve this problem in the frst module.Tis module includes the following four steps.
Te frst step is to use the inaccurate speed information.In this step, the instantaneous frequency and phase are ftted by shock signals as reference information.It is possible to roughly calculate the instantaneous frequency of equipment by counting the number of pulses per second.Te smooth speed curve S[i] is obtained by quartic polynomial interpolation.Phase of cumulative rotation is calculated by equation (1). ( If true fundamental frequency or its harmonics are observed near the given imprecise rotational speed, it is possible to extract it and use it as true instantaneous frequency.If not, the diference between estimated instantaneous frequency and rotational speed measured by sensors cannot be too large.To summarize, the multispectrum is formed by fusion of spectrum, envelope spectrum, and known inaccurate speed. Te second step is extracting the target order from the spectrum and envelope spectrum.In order to get envelope spectrum h(t), target signals are Hilbert transformed to obtain their analytical forms frstly: Tereinto, H(t) represents Hilbert transform.Generalized Fourier transform (GFT) is an efective nonstationary signal processing method, which can map the continuously changing components of frequency to a constant target frequency.As a result, general bandpass fltering methods can be used to extract nonstationary components which is easy to mix and difcult to extract [25,26].When analyzing signals, it is expected to fnd out the frequency and amplitude corresponding to the highest energy in every moment.Terefore, original and envelope signals are demodulated based on GFT, and time-domain signals which contained a single component of target order are extracted.For given time-domain signals, the GFT is shown in the following equation: In equation ( 2), s 0 (t) is a real number function that depends on time.It expresses the evolution process of signals' phase with time.After GFT, the target order of signals is mapped to a component  x f 0 (t) with certain frequency f 0 .Hereinto, f 0 does not change with time.Tus, it is easy to separate it from adjacent orders.Time-domain signals of target order can be obtained by inverse generalized Fourier transform (IGFT) on the extracted x f(0) (t).IGFT is shown in the following equation: Structural Control and Health Monitoring mapped to fxed-frequency signals after GFT.Consequently, it is necessary to fnd a phase function s 0 (t) that satisfes the above conditions.Te function is shown in the following equation: Te above analysis shows that GFTenables the extraction of harmonic components to be implemented under variable speed conditions.Terefore, original signals and envelope signals are analyzed based on the GFT, and diferent frequencies are used to ft the corresponding instantaneous phase.What is more, in this step, the instantaneous frequency ftted by velocity shock information is used to determine the range of the ridge search and removed the outliers of the instantaneous frequency ftted by the ridge.
Te third step is calculating the instantaneous phase.Te target order of time domain obtained in the second step is converted into the angular domain.For single-component time-domain signals x k (t), the phase is calculated by the following equation: In the formula, unwarp[ ] represents a winding operator, which converts x k (t) from the [−π, π] interval to the accumulated phase space.H(x k (t)) represents the Hilbert transform.For the kth harmonic of signals, there is a linear proportional relationship between its instantaneous phase φ k (t) and instantaneous phase φ(t) of the rotational frequency.Terefore, the instantaneous phase of the rotational frequency is obtained according to the following equation:

Multi-spectrum fusion
Filter with this point as central frequency The fault orders are obvious Function II Roughly calculate phase: The sum of corners Roughly calculate frequency: The number of peaks Angle domain signals 4 Structural Control and Health Monitoring instantaneous phase obtained by ftting velocity shock signals and two ridge searches are weighted to obtain a more accurate instantaneous phase estimation result.Fitting formulas are shown in the following equation: Among them, phase ori , phase env , and phase shock are instantaneous phase information obtained from the original signals, envelope spectrum signals, and speed impact signals, respectively.p is an adaptive confdence.Specifcally, p 1 and p 2 are obtained according to the energy-ratio, and p 3 represents the confdence to ft phase shock to the fnal phase estimation result.Tis paper takes p 3 as 0.3.In addition, E ori,nor [x i ] and E env,nor [x i ] are energy values normalized according to maximum values of their respective power spectrums in the calculation process.
Te fourth step is resampling and order tracking signals.Instantaneous phases of signals are obtained from the above steps.It establishes correspondence between angle and time.Time position of each equal-angle sampling point can be obtained from its inverse function as shown in equation (8).Te deconvolution operation ensures monotonicity of phase change.
After obtaining coordinates of time points, the signals can be resampled according to the timing position segments.In order to meet requirements of high accuracy and small computation at the same time, the amplitude of each point is not calculated by interpolation during resampling.In addition, the interval of angular domain resampling satisfes the condition shown in equation ( 9) in order to avoid confusion.Hereinto, f s is the sampling frequency of original signals and min f(t)   is the lowest rotation frequency of the axis [21].
In conclusion, Multispectral Lossless Preprocessing Module constructs an adaptive instantaneous phase estimation and order tracking method.Exact instantaneous frequency and instantaneous phase can be analyzed from vibration signals and inaccurate speed information.As a result, signals can be resampled to the angular domain.Tis module builds a bridge between fxed and variable speed signals in the angular domain.It is a simple and feasible way without wasting accuracy of data.It is necessary to calculate the equivalent rotational fundamental frequency f 0 of the axis after resampling for calculating the fault characteristic frequency.In the above section, instantaneous phase φ(t) of signals is calculated after resampling.It makes sure that phase function is monotonic.Terefore, the sampling frequency f s ′ of new angular domain can be obtained by the following equation:

Balanced Envelope Demodulation
whereinto N indicates the number of sampling points for one actual rotation of axis.Te equivalent fundamental frequency f 0 can be calculated by f s /f s ′ , and f s is the original frequency of sampled signals.values.As a result, clear characteristic frequencies are extracted and the characteristic frequency spectrum peaks and their harmonic components are observed on its spectrum.However, there is a large diference between amplitudes of four results.When quantitative analyzes spectrum characteristics, it is easy to judge the result with large amplitude as the real fault by mistake.Terefore, in order to comprehensively analyze four results, it is necessary to solve the problem of inconsistent scales.Aiming at the above problem, this module proposed the Balanced Envelope Spectrum (BES) method to realize automatic fault analysis by the Protrugram optimized spectrum.BES eliminates the diference of scale between diferent spectrums.It also suppresses noises of the spectral lines and eliminates the ofset.Te calculation steps of BES are as follows: (1) Calculating the squared envelope spectrum of the fltered signals by Protrugram.(2) Scaling amplitudes of spectrums to similar scale intervals.Protrugram performs fltering with a large bandwidth, and the squared envelope spectrum is calculated after fltering to extract the low frequency components well.Terefore, in squared envelope spectrum, the frequency converted part does not include a distinct spectral peak.But it has a certain amount of energy defnitely.Scaling by energy, then the creation of equivalent scaling factor parameters can be obtained in diferent squared envelope spectra of the same signal.Tus, energy peak at observable harmonic is scaled on each squared envelope spectrum.
(3) Subtracting RMS envelope values of the spectrum.
Tis step changes the variance of the spectral lines instead of the characteristics of their Gaussian distribution.It makes the spectral line directly refect how much each spectral peak is more prominent than other regions.Eventually, the fault threshold is calculated by the 5-sigma method [27].
After the above processing, four-squared envelope spectrums have similar scales.Te spectral lines with the highest energy but without an obvious fault characteristic frequency are also suppressed.Te spectral lines with smaller energy but the clearest characteristic frequency obtain a larger amplitude scale.Tus, the spectral peaks of characteristic frequency and harmonic are clearly expressed.Te spectrum is clean, and the spectrum peak is clear.Te efect of BES is as expected which establishes a new balance in the diferent spectrums after fltering the same signals.Te large amplitude makes it easy to recognize outliers [28].It ensures that the fault type of bearing is clearly judged according to the fault characteristics of each spectrum.

Simulation Analysis
In order to verify the performance of the Multispectral Lossless Preprocessing Module under variable speed conditions, simulation signals are constructed by combining the fault mechanism and signal model of rolling bearings, which is shown in Figure 2.
Simulation signals are obtained by the following equation: indicates the sequence of shocks caused by failure.A i is the amplitude of the ith shock, and signals of fault shock in the measured time period include the superposition of the frst i shocks.It is assumed that the outer ring of a rolling bearing is fxed, the inner ring rotates with a shaft, and the failure occurs in the outer ring of the bearing.Bearings generate exponentially decaying shock signals when passing fault position, as shown in (12).Simulated shock signals are shown as Figure 2(a).Te second part of (12) represents the fundamental frequency and harmonics of the system.Failure characteristic frequency is set at 3 times of rotational frequency, and the angular interval between each shock is 120 °.Superposition signals of shock signals and harmonic signals and signals with noise are shown as Figures 2(b) and 2(c).Instantaneous frequency and confdence range is in Figure 2(d).
Te power spectrum is obtained by short-time Fourier transform and square operation of simulated signals.Energy information can be used as fault characterization [29].Te basic, second, and third harmonics are clearly identifed in Figure 3(a).Te energy of the second harmonic is the largest obviously.Te results of GFT on the second harmonic are shown in Figure 3

Experiment Case 1: SQV Dataset
In this section, the SQV variable-speed bearings dataset of Xi'an Jiaotong University [30] is analyzed to verify performance of MBAFD.As is shown in Figure 8      Figure 12(c) shows the envelope spectrums after BES, and the red line means RMS development of each spectrum.In the BES spectrum fltered by BPFO, the fault characteristic frequencies are well observed and the rest of spectrums do not reach the condition to determine any fault.Ten, fault threshold used to determine the spectral peak is calculated by the 5-sigma method.Te red line in Figure 12(d) shows the determination thresholds at 5 times the standard deviation, above which the BES spectrum has a signifcant peak.To reduce the probability of misclassifcation of normal sample as fault, the sample is determined as a fault sample only when the characteristic frequencies all exceed the threshold.Figure 12(     following phenomena can be seen from the fgure: (1) envelope spectrums have a similar scale; (2) ofset of the original FTF spectral line with high bottom noise and low energy is removed; (3) the BSF spectral line with high energy but no obvious fault characteristics is suppressed; (4) the BPFO spectral line with less energy but the clearest Eigen frequency achieves the largest amplitude.In conclusion, spectral lines are generally clean and spectral peaks are clear.
In addition, three groups of inner race bearing fault data (REC3532-REC3537) and three groups of outer race bearing fault data (REC3477-REC3479) are used to verify the accuracy rate of the proposed method in inner and outer ring fault diagnosis.Ten segments are selected randomly during speed variation of each data set.Te proposed method and the original method are applied to conduct 100 experiments on each segment, respectively.Te comparison of process and diagnostic accuracy between the proposed method (MBAFD) and the original method (OM) [31] is shown in Figure 13.
It can be seen from Figure 13(a) that the original method has the following shortcomings.On the one hand, only the vibration signals are used for ridge search.On the other hand, the fltering parameters need to be set manually and the amplitudes of signals after fltering are not unifed.However, the proposed MBAFD fts more signals to avoid information loss and ridge search failure, simultaneously using Protrugram and BES to complete automatic directional fltering and unifed amplitudes.Te above steps can be automatically completed with a given bearing size.Te points in Figure 13(b) represent diagnostic accuracies of corresponding signals' segment.Te histograms represent the average accuracy of ten signal segments.As shown in the fgure, fault diagnosis accuracy of the proposed method is much higher than that of the original method in every group.When the proposed method is applied to inner ring fault diagnosis, the average accuracy rate is 82.9%, which is 22.1% higher than the original method.When the proposed method is applied to

Experiment Case 2: Bearing Experiment
In order to further verify the accuracy of MBAFD diagnosis, a test bench is built as shown in Figure 14   Select 5-second deceleration segments, respectively, to verify diagnostic accuracy of the proposed and original method.Te verifcation method is randomly selecting ten 2-second segments from 5-second segments to repeat the experiment.We repeat the experiment for 100 times for every 2-second segment and calculate the accuracy of the MBAFD and original method.
Te experimental results are shown in Figure 16.Hereinto, points represent fault diagnosis accuracy of different methods for each 2-second signal segment and histograms represent the mean of accuracy rates.As shown in the fgure, fault diagnosis accuracy of the proposed method is much higher than the original method in every group.When the proposed method is used for rolling element failure diagnosis under variable speed conditions, the average accuracy rate is 66.4%, which is 7.9% higher than the original method.

Conclusion
Tis paper proposed a new method MBAFD to diagnose fault of rolling bearings automatically under variable speed conditions.Te method consists of two modules.On the one hand, the Multispectral Lossless Preprocessing Module is designed to eliminate the infuence of variable speed conditions and avoid the loss of information.On the other hand, the Balanced Envelope Demodulation Module is designed to realize automatic fault diagnosis by Protrugram and Balancing Envelope Spectrum (BES).Experiment of simulated signals, SQV dataset, and bearing experiment were used to verify the efciency and accuracy of the proposed method.Several conclusions can be drawn from the above results.
(1) Multispectral Lossless Preprocessing Module can resample vibration signals of rolling bearings under variable speed conditions without the loss of fault information.Tis module obtains the instantaneous phase based on the original spectrum, envelope spectrum, and velocity information and then implements resampling and order tracking.(2) Balanced Envelope Demodulation Module is designed to diagnose faults of bearings automatically.In this module, the Protrugram-based algorithm can select the optimal fltering center frequency point automatically and BES can unify amplitudes of fltered spectrums.Te results show that this method is able to extract a clear characteristic frequency to diagnose the type of fault automatically.(3) Compared with neural network-based automatic fault diagnosis methods and traditional signal processing methods, the proposed method has better scientifc and engineering value.From a scientifc perspective, the generalization and interpretability of the proposed method are better than neural network-based methods.From an engineering perspective, neural network-based methods require a large amount of data to train the model, and the lack of actual fault data limits their application.Te proposed method only needs to know the size of the bearing in advance to diagnose the type of fault automatically.Meanwhile, the proposed MBAFD has higher accuracy than traditional signal processing methods.
In conclusion, the proposed MBAFD with the Multispectral Lossless Preprocessing Module and Balanced Envelope Demodulation Module integrates information scattered in diferent frequency spectrums and uses reasonable flter to complete automatic diagnosis of rolling bearings under variable speed conditions.

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Fitted values of the instantaneous frequencies are obtained by deconvolving time domain signals of the original signals and envelope signals.Te power spectrums after IGFT obtained in the second step are normalized to calculate the weight of the two estimated values.Ten, the Module.In Section 2.1, the Multispectral Lossless Preprocessing Module is proposed to eliminate the frequency modulation caused by speed changes.Based on the research in Section 2.1, it is necessary to further analyze the resampled signals for fault diagnosis.Terefore, the Balanced Envelope Demodulation Module is constructed to automatically flter resample signals and diagnose fault types.Te most critical step in constructing a flter is to set the center frequency point and bandwidth.Te adaptive method for setting flter parameters optimally realized based on Protrugram.Protrugram inherits the advantage of resistance to noise from the spectral kurtosis method.Meanwhile, directional fltering can be completed according to diferent fault frequencies.Te input of the algorithm is spectrum f(x), iteration step s, and bandwidth b w .Hereinto, f(x) is the spectrum of resampling signals.In order to make the fltered signals contain three harmonics, b w is equal to 3.5 times the fault characteristic frequency and let s � 100Hz.Te method looks for location in signals' spectrum where spectral line harmonics are clearest and uses the location as the central frequency point of flter.
Four diferent fault characteristic frequency values can be obtained from the formula of fault characteristic frequency.Ten, b w is obtained, and four directional fltering is performed.After that, four segments' fltered signals are obtained after fltering according to four diferent fault characteristic frequency Structural Control and Health Monitoring (b), where second harmonic signals are mapped to constant frequency signals.After bandpass fltering of constant frequency signals, the second harmonic is obtained and interference of the frst and third harmonics is excluded.Te results are shown in Figure 3(c).Ten, IGFT is performed on fltered signals, and results of power spectrum are shown in Figure 3(d).Te information of envelope signals is shown in Figure 4, where (a) shows original signals, (b) shows signals after the Hilbert transform, and (c) is the envelope spectrum of original signals.For envelope signals obtained by the above transformation, results of the ridge search on power spectrum are shown in Figure 5(a).It is clear that signal characteristics in the low frequency band are enhanced.Terefore, the second harmonic frequency is ftted by ridge search results of enveloped signals.Remaining processing steps are all similar to the steps of original signals' processing.Because envelope signals are mapped to constant frequency signals by GFT, signals of doubled frequency are obtained by flter and IGFT, and results are shown in Figures 5(b)-5(d).Te next step is calculating confdence coefcients of original and envelope signals.Te two power spectrums of original and envelope signals are normalized according to their maximum values.Consequently, the confdence coefcients are obtained according to energy share.

Figure 2 :
Figure 2: Simulation signal: (a) simulated shock signals; (b) superposition of simulated shock signals and harmonic signals; (c) superposition of simulated shock signals, harmonic signals, and noise; (d) instantaneous frequency and confdence range.

Figure 6 :
Figure 6: Comparison between actual values and estimated values: (a) frequency estimated from the original signals; (b) frequency estimated from the envelope spectrum; (c) ftted instantaneous phase values.
(a), the test bench based on laboratory simulation for variable-speed spectra quest (VSQ) is composed of a servomotor, rotors, an acceleration sensor, a CoCo80 data-gathering device, and load imposed by a tensioned belt.A total of six types of faults are simulated, including the inner race fault (IF) and outer race fault (OF) under three diferent damage degrees, respectively.Photos of faulty bearings are shown as Figures 8(b) and 8(c).A group of outer race fault data (REC3495) is used to verify the feasibility of MBAFD.In

Figure 7 :Figure 8 :
Figure 7: Te results of the multispectral lossless preprocessing module: (a) the signals before resampling; (b) the results of resampling; (c) order spectrum.
Figure 12(b) shows the envelope spectrums of signals after fltering according to the optimal center frequency point calculated by Protrugram.As shown in the fgure, the envelope spectrum obtained by analyzing BPFO as the characteristic frequency shows the clearest characteristic frequency spectrum peak.Te Protrugram aimed at BPFI, which has a similar center frequency point and a wider bandwidth to the BPFO flter, also shows the fault characteristic frequency to some extent.
e) synthetically shows the balanced envelope spectrum.Te

Figure 11 :
Figure 11: Results of the multispectral lossless preprocessing module: (a) original signals; (b) the results of resampling; (c) order spectrum after resampling.

Figure 12 :
Figure 12: Filtering process and results: (a) Protrugram kurtosis spectrum and optimal center frequency; (b) Protrugram; (c) each characteristic frequency is scaled according to the frequency conversion energy; (d) threshold and fault determination; (e) balanced envelope spectrum.

Figure 13 :Figure 14 :
Figure 13: Comparison of MBAFD and original method (OM) by the SQV dataset: (a) the process of two methods; (b) diagnostic accuracies of two methods.

Figure 16 :
Figure 16: Comparison of diagnostic accuracy of MBAFD and original method (OM) about bearing experiment.

Table 1 :
Details of experiment case 1.

Table 2 :
Details of experiment case 2.
directions of test bearings and large bearing pedestal.Photoelectric sensor is used to collect bearing speed pulse signals.Sensor installation diagram is shown in Figure 14(b).Eight channel signals were collected in this set of experiments, and the fnal fle had 9 columns of data.Te sampling frequency is 25600 Hz.Six bearings were tested until they fail, and three bearings with only rolling element failure are selected for analysis.Te details of the experiment are summarized in Table2.
(a).Te bearing model used in the experiment is ER-16K.Te accelerometer PCB 356A15 is installed to collect vibration signals in z direction of the small bearing pedestal and x, y, z