DETECTION OF DEFECTS IN A BEARING BY ANALYSIS OF VIBRATION SIGNALS

This work presents the analysis of vibration signals by an approach consists of several mathematical tools more elaborate such as the Hilbert transform, kurtogram, which allows the detection of vibration defects in a simple and accurate way. The steps or methods inserted in the process one complementary to the other as scalar indicators generally used in monitoring to follow the evolution of the functioning of a machine when an abnormal functioning it must make a diagnosis to detect the failing element through the use of a process. The determination of the defective organs at an optimal time is a very important operation in the industrial maintenance, which keeps the equipment in a good condition and ensures the assiduity of work. To see the effectiveness of fault detection by the proposed approach by analyzing the real vibration signals of a bearing type 6025-SKF available on the Case Western Reserve University platform


List of Symbols/Acronyms
-SNR (Signal to Noise Ratio);

INTRODUCTION
Conditional maintenance consists of several techniques applied to the industrial environment, but the vibration signal analysis is a better technique for diagnosing bearings, gears, and other components of rotating machines [1]. Bearing a critical component used to support the load and responsible for 40 to 45% of machine failures, bearing defects are characterized as inner ring defects, outer ring defect, defect of the rolling element and defect of the cage [2]. Each defect creates a pulse in the vibratory signal spectrum, the pulse produced in the bearing at a specific frequency called the default frequency depends on the rotation speed and geometric parameters of the bearing [2].
The vibration produced when the bearing subjected to internal forces [3]. And the vibration magnitude signal measured with a measuring chain consisting of a sensor, a data system, for the distinguished sensors three types: the accelerometer that measures acceleration is used in case of high operating speed, speed sensor used when average operating speed, motion sensor worn in case of low speed [4].
The vibration signal often processed in the time domain, frequency domain, and time-frequency domain, the temporal analysis performed by the calculation of scalar indicators such as RMS, kurtosis, and Crest factor [5]. And the frequency analysis made by the Fourier transform which allows to see the variation of a signal according to the DIAGNOSTYKA, Vol. 24, No. 2 (2023) Bouaouiche K, Menasria Y, Khalifa D: Detection of defects in a bearing by analysis of vibration signals 2 frequency also the time-frequency analysis examined the variation of the frequency according to the time is accomplished by transformations such as the transform of Hilbert Huang, the transform of Fourier to short term [6].
Monitoring and detecting faults in a bearing requires the acquisition and processing of signals through mathematical tools [7] such as signal decomposition by CEEMDAN or VMD, methods that analyse the non-stationary and non-linear signal [8]. Envelope analysis is a more efficient method to demodulate vibratory signals, and is composed by Hilbert transform and band pass filtering, demodulation of signals intended for the elimination of high frequency components [9].
In this study, we proposed a new approach to the treatment of vibratory signals, the integrated approach of several methods organized in the form of stages each complementary to the other which finally ensure the determination of the frequency of defects of the bearing components. The methods of signal processing are known, but the organization of the methods to achieve an effective diagnostic result must be tested and differently for each author.

METHODS AND MATERIALS
The proposed approach to represent in the flowchart (Fig 1) and consist of four steps. The vibration signal is an input parameter to the proposed approach, the signals measured by a measurement chain that allows the storage of vibration data in the form of MATLAB files. -Step 1: calculation of scalar indicators when the values differ from the values of the healthy state (vibration signature) deduces the exceeding danger thresholds, using a set of indicators represented as follows [10]: -Root Mean Square : (1) -Kurtosis : -Crest factor : Step 2: signal decomposition by two methods variational mode decomposition (VMD) and Complete ensemble empirical mode decomposition with adaptive noise CEEMDAN, there are several methods, but we will choose two algorithms the first CEEMDAN and an improvement of the EMD algorithm illustrated as follows [8]: ..., m) realization to get the first mode : 2. Calculate the first residual for k=1 as in the equation The components obtained after the decomposition of the signal called intrinsic mode function (IMF), for empirical mode decomposition (EMD) method, each function (IMF) must satisfy the following two conditions [11]: -All maxima are positive, and all minima are negative. -The local average is approximately equal to zero.
The second VMD algorithm processes the signal decomposition in variational model and translates it into a solution with the constraint that the sum of the IMF be equal to the input signal [12]: -The variational model is expressed by the following equation: After the decomposition of the vibratory signal realizes the selection of the optimal intrinsic mode function, defining two criteria for the selection the first is the value of cross correlation between the original signal and the IMF, the second criterion is the value of kurtosis.
The optimal IMF indicates a maximum crosscorrelation value as well as a kurtosis value greater than 3.
The kurtosis used to define the impulsiveness of a signal when the value of kurtosis superior to three, signals it of impulse form and in the case the kurtosis inferior to three signals, it almost sinusoidal to the causes all IMF of kurtosis inferior to three are eliminated [13].
The cross correlation function between two signals (x, IMF) defined by the following equation [14]: -Step 3: calculate the kurtogram of the vibratory signal for defining the resonance frequency that will used in the band pass filtering. The kurtogram is a graphical representation of spectral kurtosis values as a function of frequency f and bandwidth (∆ ) [15].
The spectral kurtosis ( ) of a nonstationary signal ( ) defined as the fourthorder normalized spectral momentum [15]: ( ) : Orthogonal spectral increment. ( , ) : It is the complex envelope of ( ) at frequency ( ), can estimated by the short term Fourier transform [15]. The initial and final value of the interval with maximum spectral kurtosis values in the kurtogram is the filter pass frequency. -Step 4: determining the envelope spectrum of the selected IMF and filtered signal, Hilbert transform and Fourier transform define the envelope spectrum of a signal ( ). The Hilbert transform defined by the formula follows [16]: This transform is a convolution between ( ) and the signal [16]. The analytical signal associated with ( ) noted [16]: The absolute value | ( )| is the envelope ( ) of the signal ( ) and ( ) is the phase modulation [16]: According to the proposed approach in obtaining three envelope spectra, one of the filtered signal is the other of the optimal IMF obtained by the CEEMDAN algorithm and the last one of the IMF obtained by VMD. In interpreting the envelope spectra, the relationship between the peak frequency and the defect frequency of the bearing components is sought.
The Case Western Reserve University (CWRU) platform consists of a test on the of ball bearing type 6025-SKF supports the shaft of a rotating electric motor and subjected to different speeds and loads as well as the actual signals for the bearing available in the form of MATLAB files.
Taking the signals of the healthy state and the inner race defects when the bearing is rotating at a speed of 1750 rpm and subjected to a load of 1491.4 watt, the defect size is 0.1778 mm [17].
Frequencies of defects of bearing components defined by mathematical formulas depend on geometrical parameters [10]  Finally, on applying the proposed approach on the selected signals to see the efficiency of the fault detection.

RESULTS AND DISCUSSIONS
Vibratory signals measured by accelerometers placed at the level of the electric motor, have distinguished two bearings, the first replaces next to DIAGNOSTYKA, Vol. 24, No. 2 (2023) Bouaouiche K, Menasria Y, Khalifa D: Detection of defects in a bearing by analysis of vibration signals 4 the motor driver and the other next to the fan, we'll take the bearing signals next to the trainer and the sampling frequency 12 kHz [17].
The CWRU bearing data set is long, varied and complex [18].
The signals inserted in the MATLAB code are: -99. mat: healthy state to identify the signature. -107. mat: defect of the inner race of size 0.1778 mm.

Vibration signature
The three scalar indicators used indicate the following values: -Root Mean Square : RMS=0.0644 -Kurtosis: Ku=2.925 -Crest factor: Fc=3.484 The time signal and the spectrum of the healthy state represented on  The information of the healthy state of the bearing called the vibration signature and all the vibratory defects modify this signature.  From the resonance frequency and width determined by the kurtogram on defining the two pass frequencies ( ) of the Buttrworth type band pass filter: 1 = 3000 , 2 = 4500 , the filtered vibratory signal is represented in the Fig. 8.

Selection of optimal IMF
Optimal IMF obtained by CEEMDAN: the first function (IMF=1) is optimal since it consists of a maximum cross-correlation value (911.3) and the kurtosis = 5.31 > 3. Optimal IMF obtained by VMD: third function (IMF=3) is optimal since it consists of a maximum cross-correlation value (271.51) and the kurtosis = 3.7 > 3.

Spectrum of envelope
Finally obtained three envelope spectra represented in Figs 11, 12 and 13, the envelope spectra show the same peak amplitude at frequency = 157.5 , this value very close to the fault frequency of the inner race = 157.9 . Fig. 11. Envelope spectrum of the filtered signal

CONCLUSION
In this paper, by proposing a set of methods organized in the form of processing steps of vibration signals to detect the defects of the bearing, we analyzed the signal of the bearing in a healthy state and in faulty state.
The Hilbert and Fourier transform is very important in the diagnosis of the bearing, which allows the simplicity of the form of the vibration signal.
Signal decomposition is a very significant step that allows the division of a signal into several functions, and the selection of useful functions according to criteria such as cross-correlation ensures the removal of undesirable functions and finally improves the quality of the useful signal. Fig. 12. Envelope spectrum of the optimal IMF obtained by CEEMDAN Fig. 13. Envelope spectrum of the optimal IMF obtained by VMD The healthy state information is a reference value used for fault detection and all vibratory faults change the healthy state data.