# Diagnostics 101: A Tutorial for Fault Diagnostics of Rolling Element Bearing Using Envelope Analysis in MATLAB

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## Abstract

**:**

## 1. Introduction

## 2. Bearing Diagnostics

#### 2.1. Fault Signal Given by Amplitude Modulation

#### 2.2. Simulation of Fault Signal

## 3. Diagnosis Using Simulation Data

#### 3.1. Problem Definition (Lines 1–7)

`bearFreq`denote the sampling rate in Hz, shaft rotation speed in rpm and bearing fault frequency. Note that there are four fault frequencies in the bearing, which are calculated by the Equations (1)–(4) based on the shaft frequency in rpm and bearing specifications. In the simulation, however, a single fault frequency is directly assigned, which is ${f}_{f}=10$ Hz as shown in Figure 3a. Therefore, the rpm is just set at 60 here. The last two parameters

`maxP`and windLeng are to control the performance of the algorithm:

`maxP`is the maximum order of the AR model, which is given by 300 from the ratio of the

`sampRate`

`= 3000`and

`bearFreq = 10`. This is based on the criteria that the model order should not be greater than the number of data within a fault interval [23]. The parameter

`windLeng`is a row vector of the lengths in the window function of STFT. In this example, four different window lengths

`[2^4 2^5 2^6 2^7]`are used. More details about how to set optimum window length can be referred to in Ref. [24].

#### 3.2. Discrete Signal Separation (Lines 8–22)

`sampRate`per seconds. The AR model is then constructed to predict the next value in a time series by using a series of previous observations, which is described by the following equation [22] (line 15):

`maxP = 300`(lines 10–18). The result is in Figure 7a, where the maximum is at p = 82, hence, is used for

`optP`(line 19). Then, the AR model is constructed again using this, which is to repeat lines 12–15. In this case, however, it is implemented by the MATLAB function filter, which does the same but in more simple way (line 21). Residual signal thus obtained can be plotted by entering

`plot(rawData.t (optP+1:end),e)`in the MATLAB command window, and is shown in Figure 7b along with its kurtosis value. Additionally, the raw signal and its kurtosis are given in Figure 7c for comparison. Kurtosis is higher by about four times after removing the discrete signals, reflecting the fault presence more clearly. For more advanced results, however, another criteria as mentioned above may be applied to determine the model order p and choose the one with better performances after comparison. This may be beyond the scope of this paper.

#### 3.3. Demodulation Band Selection (Lines 23–49)

_{0}~ t

_{w}, from which the local signals are made as shown in the figures below. Then, one obtains the bottom figure by taking the Fourier transform for each signal. Among various window functions, this paper employs the Hann window as follows with length L [24] (line 27):

^{7}= 128; for example, the first signal at ${t}_{0}$= 0 in the middle figure resides in the range of 0$\le t\le 0.0427\text{}(1\le n\le 128$), and the second at ${t}_{1}=0.0213\left(n=65\right)$, in the range of $0.0213\le t\le 0.0640\text{}(65\le n\le 192$), etc.

`maxP`. In practice, however, since the signal is not so clear, several window lengths are usually attempted [14], which are in this example given arbitrarily by

`[2^4 2^5 2^6 2^7]`(line 7). Upon obtaining the SK with each window lengths (lines 26–42), they are plotted in superposition (line 45), from which the frequency band for higher SK is visually determined (lines 47). The result is shown in Figure 10a, where demodulation bands are set between

`[500 700]`for the range of higher kurtosis in common. This should be input in the command window as shown in Figure 10b. Then, the residual signal goes through band-pass filtering to result in the signal in Figure 10c of which the kurtosis increases by twice to 23.1865 from 10.5885.

#### 3.4. Envelope Analysis (Lines 50–64)

## 4. Diagnosis of Korea Aerospace University (KAU) Bearing Faults

`sampRate`and

`rpm`are set at

`51.2e3`and

`1200`, respectively. Bearing fault frequencies are assigned in

`bearFreq`, which is

`[4.4423 6.5577 0.4038 5.0079]`for BPFO, BPFI, FTF, and BSF, respectively, based on Table 3. The maximum order of the AR model

`maxP`is set at

`390`based on the ratio of the sampling frequency versus the highest fault frequency (51200/(6.5577*1200/60)) as mentioned before. Vector of the window lengths

`windLeng`for STFT are given by

`[2^4 2^5 2^6 2^7 2^8]`under the constraint of maximum length being smaller than

`maxP`(390). As a result, the commands in the Problem definition (lines 1–7) are modified as follows:

- -
- Line 2:
`rawData = load('bearing1')`; - -
- Line 3:
`sampRate = 51.2e3`; - -
- Line 4:
`rpm = 1200`; - -
- Line 5:
`bearFreq = [4.4423 6.5577 0.4038 5.0079]*rpm/60`; - -
- Line 6:
`maxP = 390`; - -
- Line 7:
`windLeng = [2^4 2^5 2^6 2^7 2^8]`.

`optP`being 318 for the AR model by maximizing the kurtosis of the residual signal. Then, the resulting residual signal becomes Figure 14 and its kurtosis is 44.0245, roughly a nine-fold increase. Demodulation band selection (lines 23–49) is then carried out using the four sets of SK made with five window lengths, which are given in Figure 15a. The proper frequency band for higher SK is selected by visual examination of this figure, which is given arbitrarily as

`[1.8e4 2.3e4]`by the authors. The residual signal after applying band-pass filter is then obtained as shown in Figure 15b, of which the kurtosis increases to 208.849, five times greater than the previous one.

## 5. Diagnosis of CWRU Bearing Faults

`vib`for vibration data, and

`t`for measuring time in

`rawData`) at https://www.kau-sdol.com/. Further details regarding the data and test rig can be found in references [36,37].

- -
- Line 2:
`rawData = load (‘bearing270’)`; - -
- Line 3:
`sampRate = 12e3`; - -
- Line 4:
`rpm = 1796`; - -
- Line 5:
`bearFreq = [3.053 4.947 0.382 1.994]*rpm/60`; - -
- Line 6:
`maxP = 82`; - -
- Line 7:
`windLeng = [2^4 2^5 2^6]`.

`[5.3e3 5.9e3]`Hz is selected in the command window for the demodulation band selection. After band-pass filtering by this interval, the results of the envelope analysis are shown in Figure 17b, which shows a dominant peak at BPFI (148.08 Hz) and sidebands with harmonics at the interval of shaft frequency (29.93 Hz), convincing that the bearing has inner race fault.

`aX = hilbert(rawData.vib); %=== (a) Raw signal``envel = abs(aX);``envel = envel-mean(envel);``fftEnvel = abs(fft(envel))/Ne*2;``fftEnvel = fftEnvel(1:ceil(N/2));``freq = (0:Ne-1)/Ne*sampRate;``freq = freq(1:ceil(N/2));``figure(3)``stem(freq,fftEnvel,’LineWidth’,1.5); hold on;``[xx,yy] = meshgrid(bearFreq,ylim);``plot(xx(:,1),yy(:,1),’*-’,xx(:,2),yy(:,2),’x-’,xx(:,3),yy(:,3),’d-’,xx(:,4),yy(:,4),’^-’)``legend(‘Envelope spectrum’,’BPFO’,’BPFI’,’FTF’,’BSF’);``xlabel(‘Frequency [Hz]’); ylabel(‘Amplitude[g]’); xlim([0 max(bearFreq)*1.8])``aX = hilbert(e); %=== (b) Residual signal``envel = abs(aX);``envel = envel-mean(envel);``fftEnvel = abs(fft(envel))/Ne*2;``fftEnvel = fftEnvel(1:ceil(N/2));``figure(4)``stem(freq,fftEnvel,’LineWidth’,1.5); hold on;``[xx,yy] = meshgrid(bearFreq,ylim);``plot(xx(:,1),yy(:,1),’*-’,xx(:,2),yy(:,2),’x-’,xx(:,3),yy(:,3),’d-’,xx(:,4),yy(:,4),’^-’)``legend(‘Envelope spectrum’,’BPFO’,’BPFI’,’FTF’,’BSF’);``xlabel(‘Frequency [Hz]’); ylabel(‘Amplitude [g]’); xlim([0 max(bearFreq)*1.8])`

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## 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])

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**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 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 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 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 15.**Demodulation band selection for data of bearing 1: (

**a**) spectral kurtosis, and (

**b**) vibration signal after band-pass 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.

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 |

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 |

Frequency Name | Value | Frequency Name | Value |
---|---|---|---|

BPFO | $4.4423\times {f}_{r}$ | FTF | $0.4038\times {\mathrm{f}}_{\mathrm{r}}$ |

BPFI | $6.5577\times {f}_{r}$ | BSF | $5.0079\times {\mathrm{f}}_{\mathrm{r}}$ |

Parameter Name | Value | Parameter Name | Value |
---|---|---|---|

Pitch diameter (D) | 1.122 inch | BPFO | 3.0530${f}_{r}$ |

Ball diameter (d) | 0.2656 inch | BPFI | 4.9469${f}_{r}$ |

Number of rolling element (n) | 13 | FTF | 0.3817${f}_{r}$ |

Contact angle ($\mathsf{\alpha}$) | 0 | BSF | 1.994${f}_{r}$ |

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## Share and Cite

**MDPI and ACS Style**

Kim, S.; An, D.; Choi, J.-H.
Diagnostics 101: A Tutorial for Fault Diagnostics of Rolling Element Bearing Using Envelope Analysis in MATLAB. *Appl. Sci.* **2020**, *10*, 7302.
https://doi.org/10.3390/app10207302

**AMA Style**

Kim S, An D, Choi J-H.
Diagnostics 101: A Tutorial for Fault Diagnostics of Rolling Element Bearing Using Envelope Analysis in MATLAB. *Applied Sciences*. 2020; 10(20):7302.
https://doi.org/10.3390/app10207302

**Chicago/Turabian Style**

Kim, Seokgoo, Dawn An, and Joo-Ho Choi.
2020. "Diagnostics 101: A Tutorial for Fault Diagnostics of Rolling Element Bearing Using Envelope Analysis in MATLAB" *Applied Sciences* 10, no. 20: 7302.
https://doi.org/10.3390/app10207302