Rolling bearing fault diagnosis based on improved complete ensemble empirical mode of decomposition with adaptive noise combined with minimum entropy deconvolution

2.1. The EMD algorithm

The EMD method decomposes a signal into several intrinsic mode functions (IMF) and a residual, where the IMF should meet the following two conditions [1-5]:

1) The number of extremes and the number of zero crossings must be equal or differ at most by one.

2) The average value of the envelope of local maxima and local minima is zero.

For a given signal $x$, the EMD algorithm is described in the following steps:

Step 1: define all the local extrema of $r_0=x$, and connect all the local maxima (resp. minima) by a cubic spline line to find the upper envelope $e_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (resp lower envelope $e_{\mathrm{m}\mathrm{i}\mathrm{n}}$).

Step 2: calculate the mean envelope: $m=(e_{\mathrm{m}\mathrm{i}\mathrm{n}}+e_{\mathrm{m}\mathrm{a}\mathrm{x}})/2$.

Step 3: calculate the difference between the signal $r_k$ and $m$: $h_k=r_k-m$.

Step 4: if $h_k$ is an IMF?

Yes: save $h_k$ as $IM{F}_k$ and compute the residue $r_{k+1}=x-{\sum }_{i=1}^k{h}_i$, do $k=k+\text{1}$, then treat $r_{k+1}$ as input data in step 2.

No: treat $h_{k+1}$ as input data in step 2.

Step 5: Continue until the final residue $r_k$ satisfies predefined stopping criterion.

The stopping condition (SD) is calculated from the two consecutive sifting results, namely $h_{k-1}$ and $h_k$ as:

The process (step 2 to 5) requires certain number of iterations called the sifting process to extract each mode.

The sifting process is terminated when the SD value is greater than a certain threshold.

The signal can be expressed as follows:

where $r_n$ the final residue, $n$ the number of IMF.

2.2. The EEMD method

The Problem of EMD is the appearance of mode mixing effect. Wu and Huang [6] propose ensemble empirical mode decomposition (EEMD) to avoid mixing mode, thus algorithm is described as follows [6, 11].

Step 1. Generate $x^i=x+\epsilon {\omega }^i$, where ${\omega }^i$ ($i=$ 1, …, $I$) is a zero mean unit variance white noise realization, and noise standard deviation $\epsilon >$ 0.

Step 2. Decompose $x^i$ ($i=$ 1, …, $I$) by EMD to obtain the modes ${IMF}_k^i$, where $k=$ 1, …, $K$.

Step 3. Compute the $k$th mode $\stackrel{-}{{IMF}_k}$ of $x$ by averaging the corresponding modes:

The relationship among the amplitude of added white noise and the number of ensemble trials is given by:

where$\epsilon$ is the amplitude of the added noise, and ${\epsilon }_n$ is the final standard deviation of error.

It ca be seen some disadvantages in the EEMD such as [8]:

1) The decomposition is not complete.

2) The different realizations of the noisy signal can produce different number of modes.

2.3. The complementary EEMD

In EEMD, the problems of white noise are not totally eliminated, and the disadvantages of this technique are the interaction between the signal and the noise. The complementary EEMD has been presented to solve this problem [7]. The idea of this technique is to add positive and negative white Gaussian noises to the original signal to generate two sets of IMF [8]. The complementary EEMD algorithm is summarized in the following steps [12-14]:

Step 1: Add in pairs the white noise to the original signal $x$ and construct two new data sets $x^{+}$ and $x^{-}$:

Step 2: decompose $x^{+}$ and $x^{-}$ by EMD and obtain ${IMF}_i^{+}$, ${IMF}_i^{-}$ from those positive mixtures and negative mixtures respectively.

Step 3: Repeat the above step $I$ times to obtain two sets${IMF}_{ij}^{+}$, ${IMF}_{ij}^{-}$, the $i$th IMF obtained with addition of the $j$th positive mixtures and negative mixtures respectively.

Step 4: the final IMF is calculated by:

The original signal can be expressed by:

2.4. Complete ensemble empirical mode of decomposition with adaptive noise (CEEMDAN)

The problem of decomposition complementary EEMD is the high computational cost and there is no guarantee that $x^{+}$ and $x^{-}$ will produce the same number of modes. To solve these problems and retain the characteristics of complementary EEMD (elimination of mode mixing) [7, 12], an algorithm called CEEMDAN was presented in the following steps [8, 10]:

Step 1: Decompose by EMD $I$ realizations of $x+{\epsilon }_0{\omega }^i$ ($i=$ 1, …, $I$) to obtain its first modes and calculate:

Step 2: Calculate the first residue: $r_1=x-\stackrel{-}{{IMF}_1}$.

Step 3: Decompose by EMD $r_1+{\epsilon }_1{E}_1\left({\omega }^i\right)$, $i=$ 1, …, $I$ to obtain its first modes and define the second CEEMDAN mode as:

Step 4: for $k=$ 2, …, $K$ calculate the $k$th residue:

Step 5: Decompose the realizations $r_k+{\epsilon }_k{E}_k\left({\omega }^i\right)$$i=$ 1, …, $I$ by EMD and define the ($k+1$)th CEEMDAN mode as follows:

where $E_k\left(.\right)$ is a function to extract the $k$th IMF decomposed by EMD.

Step 6: Go to step 4 for the next $k$.

Step 7: Iterate the steps 4 to 6 until the obtained residue cannot be further decomposed by EMD. The final residue is given by:

So, the given signal can be expressed as:

where $n$ the total number of modes, ${\epsilon }_k$ the amplitude of the added white noise, and $\omega$ is the white noise with unit variance.

7.1. Case 1: bearing with inner race defect

In the first case, the vibrations signals of the bearing with inner race fault are studied. The measured vibrations of defect diameter 0.007 inches are displayed in Fig. 3(a). The improved CEEMDAN is applied to decompose the vibrations data and the first four IMFs are plotted in Fig. 3(b-d) and (e)). After calculating the correlation coefficient between the original signal and its IMFs (Fig. 3(f)), it is can be seen that the first two IMFs have a higher correlation coefficient, so these are relevant modes, then we applied the MED technique to selected modes and the results is shown in Fig. 4(a). Table 3 compares the Kurtosis values of the measured signals, the reconstructed signals for each bearing condition by EMD-MED method [20] and the proposed method. For the bearing with an inner race defect, the kurtosis of the measured vibrations is 6.10, 7.40, 7.39 and 5.95 for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively, the kurtosis of obtained signals by EMD-MED method are 8.70, 9.84, 10.65 and 10.60 for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM. These values increase by the application the proposed method to 12.52, 17.01, 13.17 and 12.91 for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively. We can see that the effect of the proposed method on kurtosis is obviously superior to the other method.

Fig. 3Signals of bearing inner race defect with diameter 0.007 inches: a) the measured signal, b) IMFs signal with speed motor 1797, c) IMFs signal with speed motor 1772, d) IMFs signal with speed motor 1750, e) IMFs signal with speed motor 1730, f) The coefficient correlation between the signal original and its IMFs

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The envelopes spectrums are plotted in Fig. 4(b). The characteristic frequencies 162.44 Hz, 160.18 Hz, 158.2 Hz and 156.36 Hz for motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively can be clearly observed which is very close to the calculated inner race pass frequency in Table 2. Therefore, it is identified as the inner race defect. Fig. 13 represents the reconstructed signal obtained by the EMD-MED method and their envelope spectrums of vibration signal of bearing defect with diameter 0.007 inches and speed motor 1797 RPM. As we see, the EMD-MED method also identifies the frequency of the inner race defects at 162.4 Hz, whereas the proposed technique shows these frequencies more clearly and with higher amplitudes.

Similar observations are made for the results obtained by applying the proposed method to measured vibration signals of bearing with inner race defect with diameter 0.014 and 0.021 inches. Fig. 5(a), 6(a) shows the measured signals of bearing with inner race defect with diameter 0.014 and 0.021 inches respectively and the results are presented in Fig. 5(b), 6(b). The kurtosis values of the vibrations signals with the defect diameter 0.014 inch and 0.021 inches are presented in Table 4 and Table 5 respectively for different motor speeds. We thus see that the proposed method is better than the EMD-MED method to improve the value of kurtosis and thus the early detection of defects.

Fig. 4The reconstruction signals by the proposed method of bearing with inner race defect with diameter 0.007 inches: a) time signals, b) envelopes spectrums

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Fig. 5The signals of bearing with inner race defect with diameter 0.014 inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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The envelopes spectrums are showed in Fig. 5(c), 6(c) for fault diameter 0.014 and 0.021 inches respectively. It can be clearly seen for the two cases of diameter the frequency of motor rotation and its harmonics and also other frequencies can be observed at 162.44 Hz, 160.18 Hz, 158.2 Hz and 156.36 Hz for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively which are equals the fault frequency at the inner ring calculated in Table 2, so it can be concluded that the defect has been detected at the inner ring.

Fig. 6The signals of bearing with inner race defect with diameter 0.021 inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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7.2. Case 2: bearing with outer race defect

In the second case, the measured vibrations of the bearing with a defect diameter 0.007 inches located on the outer race are displayed in Fig. 7(a) and the first four IMFs obtained by the improved CEEMDAN decomposition are presented in Fig. 7(b-d and e) for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively. The curves in Fig. 7(f) represent the values of the correlation coefficient between the original signal and its IMFs for different motor speeds. The first tow IMFs have higher correlation coefficients, so they are selected as relevant modes. The results of applying the MED technique to the selected IMFs are shows in Fig. 8(a). The kurtosis values of the measured vibrations are 4.25, 4, 4.36 and 4.03 for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively, after application of the proposed method; the kurtosis values increased remarkably to 13.61, 11.47, 15.74 and 11.01 for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively. As see in Table 3, the proposed method gives the best results for enhancement of sensitivity of kurtosis and therefore better detection than EMD-MED.

The envelope method (Fig 8(b)) distinguishes the harmonics at frequencies 107.10 Hz, 105.61 Hz, 104.30 Hz and 102.9 Hz for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively, these frequencies are similar to the calculated values of the outer race pass frequencies (Table 2); therefore, it can be judged that the fault is located at the outer race of bearing. For a comparison, the signal of bearing defect with diameter 0.007 inches and motor speed 1797 RPM is treated by EMD-MED method and the result is shown in Fig. 13. We can see more interfering noise component than the corresponding result obtained by the proposed method. This makes the proposed method better than EMD-MED method for the diagnosis of defects.

Fig. 7Signals of bearing with outer race defect with diameter 0.007 inches: a) the measured signal, b) IMFs signal with speed motor 1797, c) IMFs signal with speed motor 1772, d) IMFs signal with speed motor 1750, e) IMFs signal with speed motor 1730, f) The coefficient correlation between the signal original and its IMFs

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For fault diameter 0.014 and 0.021 inches, the measured vibrations are showed in Fig. 9(a), 10(a) respectively and the result of the proposed method are plotted in Fig. 9(b), 10(b). As shown in Table 4, the kurtosis values of the vibratory signals of the bearing having a defect diameter of 0.014 inches are 3.13, 3.21, 3.04, and 3.19 for motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively, these values are increased by the EMD-MED method to 10.26, 9.95, 4.79 and 6.03 for motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM. When we apply the proposed method the values of kurtosis become 10.31, 10.09, 7.94, and 10.10 for motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively. Therefore, it can be found the effect in kurtosis of the proposed method is obviously superior to the other method.

Similar observations for the signals of bearing with fault diameter 0.021 inches; the kurtosis increases to a remarkable degree when the proposed method is applied (Table 5) and gives better results than the EMD-MED method. The envelopes spectrum of obtained signals by proposed method are showed in Fig. 9(c), 10(c) for signals of fault diameter 0.014 and 0.021 inches respectively. We can be clearly detected in two case of fault diameter the frequencies 107.10 Hz, 105.61 Hz, 104.30 Hz and 102.9 Hz for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively, which are equal to the theoretical values of frequency of outer race fault calculated in Table 2. Therefore, we locate the defect on the outer ring.

Fig. 8The reconstruction signals by the proposed method of bearing with outer race defect with diameter 0.007 inches: a) time signals, b) envelopes spectrums

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Fig. 9The signals of bearing with outer race defect with diameter 0.014 inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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7.3. Case 3: bearing with ball defect

The measured signals are plotted in Fig. 11(a), 12(a) for different motor speeds and defects diameter 0.007, 0.014 inches respectively. The results obtained by the proposed method are displayed in Fig. 11(b), 12(b) for different defect diameter and the kurtosis values of measured signals, EMD-MED method and proposed method are presented in Table 6. We can see that there is a significant increase in kurtosis according to the proposed method than the EMD-MED method and consequently a better detection.

Fig. 10The signals of bearing with outer race defect with diameter 0.021 inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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Fig. 11The signals of bearing with ball defect with diameter 0.007 inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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The envelopes spectrums of the signals obtained by the proposed method are shown in Fig. 11(c), 12(c) for different motor speeds and defect diameters. We note the existence of the frequencies 139.8 Hz, 137.9 Hz, 136.2 Hz, 134.6 Hz which are equal to the theoretical frequencies of ball defect calculated in the Table 2 and these mean that the defect is located at the ball of bearing. By comparing these results with those obtained by the EMD-MED method, we note that the peak of the fault frequency is unclear and with amplitude less than the proposed method.

Fig. 12The signals of bearing with ball defect with diameter 0.014inches: a) the measured signals, b) the reconstructed signal, c) the envelopes spectrum

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7.4. Case 4: healthy bearing

The measured vibration signals of healthy bearing are displayed in Fig. 14(a) for different motor speeds and the results of proposed method are showed in Fig. 14(b). As we see in Table 7, we notice that the kurtosis remains almost to the same value before and after the application of the proposed method which means there is no defect in the bearing.

The envelopes spectrums clearly show (Fig. 14(c)) the rotation frequencies and its harmonics 29.9 Hz, 29.5 Hz, 29.1 Hz, 28.3 Hz for the motor speeds 1797 RPM, 1772 RPM, 1750 RPM and 1730 RPM respectively. We note that there is no other peak and therefore there is no fault in the bearing.

Fig. 13The reconstruction signals by EMD-MED method of bearing with defect diameter 0.007 inches and speed motor 1797 RPM: a) original signals, b) reconstructed signals, c) envelopes spectrums

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Fig. 14The reconstruction signals by the proposed method of healthy bearing: a) original signals, b) reconstructed signal, c) envelopes spectrums signals

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