Appendix A. Bearing Fault Simulation
%% Bearing fault simulation signal
% Parameter setting ===================================================
fr = 600; % Carrier signal
fd = 13; % discrete signal
ff = 10; % Characteristic frequency(Modulating signal)
a = 0.02; % Damping ratio
T = 1/ff; % Cyclic period
fs = 3e3; % Sampling rate
K = 50; % Number of impulse signal
t = 1/fs:1/fs:2; % Time
A=5; % Maximum amplitude
noise = 0.5;
%=====================================================================
for k = 0 : K-1
for i = 1 : length(t)
if t(i)-k*T>=0
x1(i) = A*exp(-a*2*pi*fr.*(t(i)-k*T));
x2(i) = sin(2*pi*fr.*(t(i)-k*T));
x3(i) = x1(i).*x2(i);
end;end;end
x5 = normrnd(0,noise,1,length(x3));
x4 = 2*sin(2*pi.*t.*fd);
vib = x3 + x4 + x5;
save(‘Simulation’,’vib’,’t’)
Appendix B. Bearing Fault Diagnosis
1 %======================PROBLEM DEFINITION ===========================
2 rawData = load(‘Simulation’); % Data load
3 sampRate = 3e3; % Sampling rate (Hz)
4 rpm = 60; % Shaft rotating speed
5 bearFreq = [10]*rpm/60; % BPFO, BPFI, FTF, BSF
6 maxP = 300; % Maximum order of AR model
7 windLeng = [2^4 2^5 2^6 2^7]; % Window length of STFT
8 %==============Discrete signal separation (AR model) ==================
9 x=rawData.vib(:); N=length(x);
10 for p = 1 : maxP
11 if rem(p,50)==0; disp([‘p=‘ num2str(p)]); end
12 a = aryule(x,p); % aryule returns the AR model parameter, a(k)
13 X = zeros(N,p);
14 for i = 1 : p; X(i+1:end,i) = x(1:N-i); end
15 xp = -X*a(2:end)’;
16 e = x-xp;
17 tempKurt(p,1) = kurtosis(e(p+1:end));
18 end
19 optP = find(tempKurt==max(tempKurt)); %==== Optimum solution
20 optA = aryule(x,optP);
21 xp = filter([0 -optA(2:end)],1,x);
22 e = x(optP+1:end) - xp(optP+1:end); % residual signal
23 %============Demodulation band selection (STFT & SK) =================
24 Ne = length(e);
25 numFreq = max(windLeng)+1;
26 for i = 1 : length(windLeng)
27 windFunc = hann(windLeng(i )); %==== Short Time Fourier Transform
28 numOverlap = fix(windLeng(i)/2);
29 numWind = fix((Ne-numOverlap)/(windLeng(i)-numOverlap));
30 n = 1:windLeng(i);
31 STFT=zeros(numWind,numFreq);
32 for t = 1 : numWind
33 stft = fft(e(n).*windFunc, 2*(numFreq-1));
34 stft = abs(stft(1:numFreq))/windLeng(i)/sqrt(mean(windFunc.^2))*2;
35 STFT(t,:) = stft’;
36 n = n + (windLeng(i)-numOverlap);
37 end
38 for j = 1 : numFreq %==== Spectral Kurtosis
39 specKurt(i,j) = mean(abs(STFT(:,j)).^4)./mean(abs(STFT(:,j)).^2).^2-2;
40 end
41 lgd{i} = [‘window size = ‘,num2str(windLeng(i))];
42 end
43 figure(1) %==== Results
44 freq = (0:numFreq-1)/(2*(numFreq-1))*sampRate;
45 plot(freq,specKurt); legend(lgd,’location’,’best’)
46 xlabel(‘Frequency[Hz]’); ylabel(‘Spectral kurtosis’); xlim([0 sampRate/2]);
47 [freqRang] = input(‘Range of bandpass filtering, [freq1,freq2] = ‘);
48 [b,a] = butter(2,[freqRang(1) freqRang(2)]/(sampRate/2),’bandpass’);
49 X = filter(b,a,e); % band-passed signal
50 %=======================Envelope analysis ============================
51 aX = hilbert(X); % hilbert(x) returns an analytic signal of x
52 envel = abs(aX);
53 envel=envel-mean(envel); % envelope signal
54 fftEnvel = abs(fft(envel))/Ne*2;
55 fftEnvel = fftEnvel(1:ceil(N/2));
56 figure(2) %==== Result plot
57 freq = (0:Ne-1)/Ne*sampRate;
58 freq = freq(1:ceil(N/2));
59 stem(freq,fftEnvel,’LineWidth’,1.5); hold on;
60 [xx,yy]=meshgrid(bearFreq,ylim);
61 plot(xx(:,1),yy(:,1),’*-’)
62 % ,xx(:,2),yy(:,2),’x-’,xx(:,3),yy(:,3),’d-’,xx(:,4),yy(:,4),’^-’)
63 legend(‘Envelope spectrum’,’BPFO’,’BPFI’,’FTF’,’BSF’);
64 xlabel(‘Frequency [Hz]’); ylabel(‘Amplitude [g]’); xlim([0 max(bearFreq)*1.8])
Figure 1.
Bearing geometry and impact signal.
Figure 1.
Bearing geometry and impact signal.
Figure 2.
Amplitude modulation and envelope signal: (a) modulating signal, (b) carrier signal, (c) AM signal, (d) envelope signal, in the time domain; (e) modulating signal, (f) carrier signal, (g) AM signal, (h) envelope signal, in the frequency domain.
Figure 2.
Amplitude modulation and envelope signal: (a) modulating signal, (b) carrier signal, (c) AM signal, (d) envelope signal, in the time domain; (e) modulating signal, (f) carrier signal, (g) AM signal, (h) envelope signal, in the frequency domain.
Figure 3.
Modulation of simulation fault signal: (a) fault signal, (b) resonance signal, (c) AM signal, in the time domain; (d) fault signal, (e) resonance signal, (f) AM signal, in the frequency domain.
Figure 3.
Modulation of simulation fault signal: (a) fault signal, (b) resonance signal, (c) AM signal, in the time domain; (d) fault signal, (e) resonance signal, (f) AM signal, in the frequency domain.
Figure 4.
Simulation of bearing fault signal: (a) discrete signal, (b) white noise, (c) raw signal, in the time domain; (d) discrete signal, (e) white noise, (f) raw signal, in the frequency domain.
Figure 4.
Simulation of bearing fault signal: (a) discrete signal, (b) white noise, (c) raw signal, in the time domain; (d) discrete signal, (e) white noise, (f) raw signal, in the frequency domain.
Figure 5.
Procedure for bearing fault diagnosis.
Figure 5.
Procedure for bearing fault diagnosis.
Figure 6.
Flow chart on how the residual signals are obtained.
Figure 6.
Flow chart on how the residual signals are obtained.
Figure 7.
Kurtosis variation in terms of model order and corresponding signals: (a) model order selection based on maximum kurtosis, (b) residual signal for p = 82, and (c) raw signal.
Figure 7.
Kurtosis variation in terms of model order and corresponding signals: (a) model order selection based on maximum kurtosis, (b) residual signal for p = 82, and (c) raw signal.
Figure 8.
Concept of STFT.
Figure 8.
Concept of STFT.
Figure 9.
Results of STFT and SK: (a) STFT S(t, f) and (b) SK K(f).
Figure 9.
Results of STFT and SK: (a) STFT S(t, f) and (b) SK K(f).
Figure 10.
Demodulation band selection: (a) spectral kurtosis for different window lengths, (b) frequency band selection input in MATLAB, and (c) vibration signal after band-pass filtering.
Figure 10.
Demodulation band selection: (a) spectral kurtosis for different window lengths, (b) frequency band selection input in MATLAB, and (c) vibration signal after band-pass filtering.
Figure 11.
Result of envelope analysis: envelope signal in (a) time domain (b) frequency domain of faulty bearing, in (c) time domain and (d) frequency domain of normal bearing.
Figure 11.
Result of envelope analysis: envelope signal in (a) time domain (b) frequency domain of faulty bearing, in (c) time domain and (d) frequency domain of normal bearing.
Figure 12.
Korea Aerospace University (KAU) bearing test rig.
Figure 12.
Korea Aerospace University (KAU) bearing test rig.
Figure 13.
Raw vibration signals (bearing 1).
Figure 13.
Raw vibration signals (bearing 1).
Figure 14.
Residual signal after the AR filtering.
Figure 14.
Residual signal after the AR filtering.
Figure 15.
Demodulation band selection for data of bearing 1: (a) spectral kurtosis, and (b) vibration signal after band-pass filtering.
Figure 15.
Demodulation band selection for data of bearing 1: (a) spectral kurtosis, and (b) vibration signal after band-pass filtering.
Figure 16.
Frequency domain of envelope analysis.
Figure 16.
Frequency domain of envelope analysis.
Figure 17.
Results obtained from the code: CWRU bearing data (bearing280.mat) (a) Spectral kurtosis for bearing 270, and (b) Envelope spectrum after SK filtering.
Figure 17.
Results obtained from the code: CWRU bearing data (bearing280.mat) (a) Spectral kurtosis for bearing 270, and (b) Envelope spectrum after SK filtering.
Figure 18.
Envelope spectrum reflecting the raw and AR filtered signals: (a) envelope spectrum of the raw signal, and (b) envelope spectrum after AR filter.
Figure 18.
Envelope spectrum reflecting the raw and AR filtered signals: (a) envelope spectrum of the raw signal, and (b) envelope spectrum after AR filter.
Table 1.
Bearing dimensions.
Table 1.
Bearing dimensions.
Parameter Name | Value | Parameter Name | Value |
---|
Bearing type | NJ 2306 | Width (w) | 27 mm |
Inner diameter (id) | 30 mm | Dynamic load rating | 51,500 N |
Outer diameter (od) | 72 mm | Static load rating | 51,000 N |
Number of rollers (n) | 11 | | |
Table 2.
Operating conditions.
Table 2.
Operating conditions.
Data Name | RPM | LOAD |
---|
bearing 1 | 1200 rpm | 0 N |
bearing 2 | 1200 rpm | 200 N |
bearing 3 | 1000 rpm | 0 N |
bearing 4 | 1000 rpm | 200 N |
Table 3.
Bearing fault frequencies.
Table 3.
Bearing fault frequencies.
Frequency Name | Value | Frequency Name | Value |
---|
BPFO | | FTF | |
BPFI | | BSF | |
Table 4.
Bearing information.
Table 4.
Bearing information.
Parameter Name | Value | Parameter Name | Value |
---|
Pitch diameter (D) | 1.122 inch | BPFO | 3.0530 |
Ball diameter (d) | 0.2656 inch | BPFI | 4.9469 |
Number of rolling element (n) | 13 | FTF | 0.3817 |
Contact angle () | 0 | BSF | 1.994 |
Table 5.
Open Data for Fault Diagnosis and Prognosis of Bearings.
Table 5.
Open Data for Fault Diagnosis and Prognosis of Bearings.
Data Set Name | Diagnosis | Prognosis | Failure Type | Operating Condition |
---|
FEMTO [38] | X | O | Natural | Different load and speed |
MFPT [39] | O | X | Artificial | Different load |
IMS [40] | X | O | Natural | Single condition |
WT [41] | X | O | Natural | Single Condition |
Paderborn [42] | O | X | Artificial, Natural | Different operating condition |