Diagnosis of rotating machine defects by vibration analysis


ACTA IMEKO 
ISSN: 2221-870X 
March 2023, Volume 12, Number 1, 1 - 6 

 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 1 

Diagnosis of rotating machine defects by vibration analysis 

Karim Bouaouiche1, Yamina Menasria1, Dalila Khalfa1 

1 Electromechanical Engineering Laboratory, Badji Mokhtar University - Annaba B.P.12  23000, Algeria  

 

 

Section: RESEARCH PAPER  

Keywords: Vibration signal; kurtosis; FMD; Hilbert transform; filtering  

Citation: Karim Bouaouiche,Yamina Menasria, Dalila Khalfa, Diagnosis of rotating machine defects by vibration analysis, Acta IMEKO, vol. 12, no. 1, article 26, 
March 2023, identifier: IMEKO-ACTA-12 (2023)-01-26 

Section Editor: Francesco Lamonaca, University of Calabria, Italy  

Received December 27, 2022; In final form March 16, 2023; Published March 2023 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Funding: This work was supported by electromechanical engineering laboratory funding, Algeria. 

Corresponding author: Karim Bouaouiche, e-mail: karimbouaouiche@gmail.com  

 

1. INTRODUCTION 

Rotating machines such as engine and gearbox produce a 
vibration which characterized by a set of parameters such as 
amplitude, velocity and acceleration [1]. The measurement of the 
parameters depends on the frequency of the vibration such as 
displacement sensor used at low frequency and velocity sensor at 
medium frequency and accelerometer at high frequency, 
vibration monitoring is a technique more used in maintenance, 
which allows the detection of defects [1]. 

The method of machine monitoring divided into two types: 
the first is the methods of fault detection based on the creation 
of an analytical model to correlate the signature and detect the 
damage parameters such as hidden Markov model (HMM), 
artificial neural network (ANN) [1]. The second method is the 
processing and extraction of the characteristics of the vibration 
signal in the time domain, frequency domain, and time-frequency 
domain [1]. 

Data acquisition is an important step in the diagnosis of 
machine faults and the information collected divided into two 
types: data from events dependent on installation, repair, but 
monitoring data collected by measurements such as vibration 
data, the next step after the acquisition is the analysis of the data 
through models, algorithms, tools, which ensures a better 
interpretation [2]. 

The algorithms of decomposition of a signal are the most 
used in data analysis, such as EMD, EEMD, CEEMD, empirical 
mode decomposition (EMD) created by Huang in 1998 but this 
method presents the problem of mixing modes [3]. Through its 
EMD improved by Wu in 2009 to ensemble empirical mode 
decomposition (EEMD) and in 2011 improved by Torres to 
complete ensemble empirical mode decomposition (CEEMD) 
[3]. The last decomposition method created by Yonghao Miao in 
2022 is called feature mode decomposition (FMD) [4]. 

In this work, we propose an approach to analyse the real 
signals of the ball bearing 6025-SKF, the experimental signals 
available on the platform (CWRU) in the form of MATLAB files, 
see Figure 1. 

2. METHOD 

In this study, we propose an approach that allows the 
processing of vibratory signals of a machine in an efficient way 
and gives a result that directly indicates the kinematic frequency 

 

Figure 1. Parts of the proposed approach.  

ABSTRACT 
This work presents the analysis of the vibration signals of a ball bearing, the signals available on the platform Case Western Reserve 
University in the form of MATLAB files. By proposing an approach for the diagnosis of any rotating machine and detects the failed 
components in a simple way, the proposed approach consists of several methods and steps each complementary to the other. First  on 
decomposing the signal by the feature mode decomposition (FMD) method and then, according to the Kurtosis values by selecting the 
useful signals after the selection by computing the sum of the useful signals called the residual of the original signal. The simplicity of 
the form of the residual makes through the application of band pass filtering and Hilbert transform, but the analysis of the peaks 
represents the frequency of the failed components.  



 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 2 

of the failed components. The approach consists of two parts, 
the first is the experimental part of signal acquisition by a 
measurement chain and the second part is the processing of the 
vibration signal by a MATLAB code compatible with a proposed 
approach that provides the frequencies of defects: 

Figure 2 illustrates the proposed approach that consists of 
several steps each complementary to the other: 

- Step1: After insertion of the signal in fact the 
decomposition by FMD through the use of a finite 
impulse response (FIR) filter bank the signal decomposes 
into different modes, the algorithm of FMD consists of 
the following steps [4]: 

- Enter the number of modes 𝑛 and the length of the 
filter 𝐿. 

- Initialization of filter banks by a Hanning window 

using 𝐾 filter. 
- Obtain the modes (filtered signals) by the following 

equation: 

𝑢𝑘
𝑖 = 𝑥 ∗ 𝑓𝑘

𝑖  (1) 

𝑢𝑘
𝑖 : modes, 𝑥: signal, 𝑓𝑘

𝑖  : filter coefficient, 𝑖 : 
iteration, ∗  : convolution. 

- Updating the filter coefficients by using the signal 𝑥 
and the modes 𝑢 and the estimated period 𝑇, which 
chosen as the point where the autocorrelation 

spectrum reaches the maximum value 𝑅𝑘
𝑖  after the 

point zero crossing. 

- Judges the number of iterations 𝑖 if it reaches the 
pre-interaction number by going to the next step. 

- Calculation of correlation coefficient 𝐶𝐶, both 
modes build a matrix 𝐶𝐶𝐾×𝐾 , then calculating the 
correlated kurtosis 𝐶𝐾 using the estimated period. 

- Judges if the number of modes 𝑛 reaches the 
specified number. 

- Obtain the final modes. 

Figure 3 shows the flowchart of the FMD method [4]: 
The input parameters to the FMD algorithm are the 

number of modes 𝑛, the length of the filter 𝐿 and the 
number of frequency band segments 𝐾: 
- The filter length 𝐿 has an influence on the filtering 

performance, the FMD performance test consists of 

the variation of the correlated Kurtosis 𝐶𝐾 values 
of the filtered signals as a function of the filter 

length 𝐿, and the test result defines the optimal 
FMD filter length range, 𝐿 ∈ [30; 100], [4]. 

- The parameter 𝐾 must be greater than 𝑛 to ensure 
decomposition, the FMD performance test which 

consists of the variation of 𝐶𝐾 values as a function 
of 𝐾 defines the optimal parameter interval 𝐾 ∈
[5; 10], [4]. 

However, we in your study propose criteria to define the 
values of the parameters of FMD: 
- The number equals half the number of modes 

obtained by EMD: 

𝑛(FMD) =
𝑛(EMD)

2
 . (2) 

- The parameter 𝐾 is half of the optimal interval: 

𝐾 =
5 + 10

2
≅ 8 . (3) 

- The number (𝐿) defined according to the variation 
of cross correlation (𝐶) between the modes 
obtained by FMD and the original signal, the case 

that represents a maximum total value (𝑇𝐶) of 
cross correlation (𝐶) chosen. 

 

Figure 2. The signal processing approach.  

 

Figure 3. FMD algorithm.  



 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 3 

The cross correlation function applied to measure 
the similarity between two functions and calculated 
by the following formula [5]: 

𝐶(𝜏) = ∫ 𝑥(𝑡) 𝐼𝑀𝐹𝑖 (𝑡 − 𝜏) d𝑡
+∞

−∞

 (4) 

𝑖 = 1, 2, … , 𝑛(FMD): the number of modes, 𝐼𝑀𝐹: 

the modes, 𝜏: time shift parameter. 

𝑇𝐶 = ∑ 𝐶𝑖 (𝜏)

𝑛(FMD)

𝑖=1

 . (5) 

- Step2: the kurtosis is a moment of order 4 and used to 
identify the non-periodic impacts and operation of a 
machine, the value there is a threshold in the detection 
of faults when the kurtosis greater than three the signal 
is impulsive [6]. As the case of bearing in good condition, 
the kurtosis is less than three and the distribution of the 
Gaussian signal [7]. 

𝑘𝑢 =
1

𝑁
∑ (

𝑥𝑖 − �̅�

𝜎
)

4𝑁

𝑖=1

 , (6) 

�̅�: the mean, 𝜎: standard deviation. 
- Step3 : the sum of the IMF that illustrates a kurtosis value 

greater than three called the signal residual 𝑥res [8], after 
the determination of residual on calculating the 
kurtogram to define the two pass frequencies of the 
digital band pass filter, the limits of the interval that 

illustrate a high value of spectral kurtosis 𝐾(𝑓) is the pass 
frequencies. 
The kurtogram represents the variation of spectral 

kurtosis values as a function of frequency 𝑓 and width 
∆𝑓, [9]. 
𝐾(𝑓) is calculated by the following formula, [9]: 

𝐾(𝑓) =
< 𝐻4 (𝑡, 𝑓) >

< 𝐻2(𝑡, 𝑓) >2
− 2 (7) 

𝐻(𝑡, 𝑓): complex envelope of the signal at frequency 𝑓, 
estimated by the short-term Fourier transform [9]. 

- Step 4: calculation of the envelope spectrum of the 

filtered signal 𝑥resf by the Hilbert 𝐻[𝑥resf(𝑡)] and 
Fourier transform, the absolute value 𝐸(𝑡) of the 
analytical signal 𝑧(𝑡) is the envelope of 𝑥resf and 𝐸(𝑓) is 
the spectrum, 𝐸(𝑡) and 𝐸(𝑓) calculated by the following 
equations [10]: 

𝐻[𝑥resf(𝑡)] = 𝑥resf(𝑡) ∗
1

π 𝑡
 (8) 

∗: convolution product. 

𝑧(𝑡) = 𝑥resf + 𝑗 𝐻[𝑥resf(𝑡)] (9) 

𝑧(𝑡) = 𝐸(𝑡) 𝑒𝑗 𝜑(𝑡) (10) 

𝑥resf(𝑡) = 𝑅{𝑧(𝑡)} = 𝐸(𝑡) cos (𝜑(𝑡)) (11) 

𝐸(𝑡) = |𝑥resf(𝑡) + 𝑗 𝐻[𝑥resf(𝑡)]| (12) 

𝐸(𝑓) = ∫ 𝐸(𝑡)𝑒 −𝑗 2 π 𝑓 𝑡  d𝑡
+∞

−∞

 . (13) 

- Step 5: the peak is the maximum amplitude value of a 
signal found by the following formula [11]: 

𝑃𝑒𝑎𝑘 = max|𝐸(𝑓)| (14) 

The relationship between the peak frequency and the 
kinematic frequency of the components of a machine is 
the result of the diagnosis that ensures the detection of 
the faulty component. 

3. VIBRATION SIGNALS 

In this section, analysing the real vibratory signal of ball 
bearing type 6025-SKF that supports the shaft of an electric 
motor at the drive end and subjected to different speeds and 
loads, among the signals available in the platform CWRU taking 
two signals of defects, the characteristics of each signal 
represented in Table 1, [12]. 

Fault frequencies of bearing components are the multiple of 
the rotation frequency in Hz with a coefficient as shown in 
Table 2 [12]. 

4. RESULTS AND DISCUSSIONS 

4.1. Case 1 

By applying the proposed approach on the inner race fault 
signal (107.mat), starting with the estimation of the parameters 
of the FMD algorithm: 

- The number of modes obtained by applying the EMD 

algorithm on the signal 107.mat equals 13 so 𝑛FMD ≅ 6. 
- The variation of 𝑇𝐶 as a function of 𝐿 is represented in 

Figure 4. 

The maximum value of cross-correlation 𝑇𝐶 = 893.8 
has 𝐿 = 32, the step 𝑃 of variation of the interval 𝐿 
equals two, 𝑃 = 2. 

The decomposition of the vibration signal (107.mat) by the 
FMD method gives six modes represented in Figure 5, and the 
spectrum of each mode illustrated in Figure 6. 
The kurtosis values for each mode shown in the Table 3. 

The original signal residual is the sum of the modes (2, 3, … , 6): 

𝑥res(𝑡) = ∑ 𝐼𝑀𝐹𝑖 (𝑡)

6

𝑖=2

 . (15) 

The spectrum and the temporal signal of 𝑥res(𝑡) are represented 
in Figure 7. 

Table 1. Vibration signals. 

Signal Defect diameter Sampling frequency 

Inner race (107.mat) 
Load (1491.4 N m/s) 
Speed (1750 rpm) 

0.1778 mm 12 kHz 

Inner race (211.mat) 
Load (1491.4 N m/s) 
Speed (1750 rpm) 

0.5334 mm 12 kHz 

Table 2. Defect frequencies. 

Components Coefficients Frequency (Hz) 

Inner race 5.4152 157.94 

Outer race 3.5848 104.55 

Cage 0.39828 11.61 

Rolling element 4.7135 137.47 



 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 4 

The residual spectrum is complex and rich with information 

as shown in Figure 7, the interpretation of 𝑥res(𝑓) is difficult for 
this reason, in fact the pass band filtering through the Buttworth 
filter, the two pass frequencies of the filter defined from 

kurtogram of 𝑥res(𝑡). 

The kurtogram in Figure 8 shows a maximum value of 

spectral kurtosis 𝐾max = 0.5 at the centered frequency 𝑓c =
3750 Hz with a width 𝐵𝑤 = 500 Hz at the level 𝑙 = 3.5 so the 
two passing frequencies 𝐹𝑝 = [3250 ;  4250] (Hz). 

The spectrum of the envelope 𝐸(𝑓) of the filtered signal 
𝑥resf(𝑡) in Figure 9, illustrates a peak max|𝐸(𝑓)| = 0.661 at 
frequency 𝑓 = 157.5 Hz, the value of 𝑓 closer to the fault 
frequency of the inner race (𝐹inner race = 𝑓). 

4.2. Case 2 

In this case by analysing the vibration signal (211.mat) by the 
proposed approach, but summarizes the different steps: 

 

Figure 4. The variation of TC.  

 

Figure 5. The temporal variation of the modes.  

 

Figure 6. The spectrum of the modes. 

 

Figure 7. The residual.  

 

Figure 8. Kurtogram of 𝑥𝑟𝑒𝑠 (𝑡). 

Table 3. Kurtosis values. 

IMF Kurtosis 

1 2.1 

2 5.8 

3 6.3 

4 8.8 

5 5.7 

6 5 



 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 5 

- The EMD decomposition of the signal yields 14 modes, 

the maximum cross-correlation value 𝑇𝐶 = 329.55 at 
𝐿 = 30 as shown in Figure 10, the estimation of FMD 
parameters according to the proposed criteria is: 

𝑛FMD = 7;   𝐿 = 30;  𝐾 = 8. 
- The residual is equal to the sum of all modes (1, 2, … ,7) 

as shown in the Table 4. 

- We observe on the spectrum 𝑥res(𝑓) (Figure 11) a very 
high amplitude peak at the inner race fault frequency. 

𝑥res(𝑡) = ∑ 𝐼𝑀𝐹𝑖 (𝑡)

7

𝑖=1

 . (16) 

From the kurtogram of 𝑥res(𝑡) find the two frequencies of 
passage 𝐹𝑝 = [5375 ; 5875] Hz. 

On the envelope spectrum 𝐸(𝑓) (Figure 12), a high amplitude 
peak is found in the rotation frequency (29 Hz) and the second 
peak at the inner race fault frequency (157.5 Hz). 

5. CONCLUSION 

In this paper, we analyse two fault vibration signals of the 
inner ring of the ball bearing by a new proposed approach, which 
consists of a set of signal processing methods organized in the 
form of steps. 

From the final result of the proposed approach, we conclude 
that the new criteria formulated to estimate the input parameter 
of the FMD algorithm show a good efficiency since the number 
of modes and the length of the filter are two essential parameters 
to ensure the signal decomposition by FMD. As well as the 

 

Figure 9. 𝐸(𝑡) and 𝐸(𝑓).  

 

Figure 10. 𝑇𝐶 as a function of 𝐿 

 

Figure 11. 𝑥res(𝑡) and 𝑥res(𝑓).  

 

Figure 12. 𝐸(𝑡) and 𝐸(𝑓).  

Table 4. The values of kurtosis. 

IMF Kurtosis 

1 3.5 

2 9.7 

3 10 

4 10.6 

5 10.2 

6 10.2 

7 16 



 

ACTA IMEKO | www.imeko.org March 2023 | Volume 12 | Number 1 | 6 

parameters guarantee the decrease of the equality constraint 
between the original signal and the set of modes obtained after 
the decomposition. 

The selection of useful modes through the use of parameters 
such as kurtosis is a very important step to extract the 
information depends on defects. 

The envelope analysis is the last step of the proposed 
approach that represents the peaks of high amplitudes at the 
default frequency, envelope analysis is a combination of band 
pass filtering and Hilbert transform and Fourier transform, in 
your study the two pass frequencies of the filter defined by the 
maximum values of spectral kurtosis in the kurtogram. 

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