# Incipient Fault Feature Enhancement of Rolling Bearings Based on CEEMDAN and MCKD
## 3 Based on CEEMDAN-MCKD Bearing Fault Feature Enhancement Method  
This work proposes a vibration signal rolling bearing failure feature augmentation approach based on CEEMDAN and MCKD, which can successfully reduce noise inter ference and enhance SNR. The length L, deconvolution period T, and shift number M of the parametric filter have a direct impact on the enhancing effect of MCKD. L and M are optimized by the PSO method for the parameters while T is computed using the bearing parameters. Figure 1 shows the flow diagram of the CEEMDAN-MCKD method, and the specific flow is as follows:  
1. Initial noise reduction of the original signal using the CEEMDAN method.
2. The optimal IMF component is selected based on kurtosis-correlation coefficient selection criterion, and the fault signal is reconstructed.
3. Determining the optimization range of parameters L and M and performing parameter optimization for the MCKD algorithm.  
The envelope spectral peak factor [22], a dimensionless indicator, is taken into account when using the particle swarm algorithm for the optimization search to take into account both the period and the intensity of the shock component of the signal, whereas the kurtosis is sensitive to the shock signal. As a result, the fitness function is the multiple of the envelope spectral peak factor and the kurtosis.  
\[Ec=\frac{\max(X(z))}{\sqrt{\frac{\sum_{z}X(z)^{2}}{/Z}}} \tag{15}\]  
where \(Ec\) is the envelope spectrum peak factor, signal envelope spectrum amplitude sequence for \(X(z)(z=1,2,\cdots,Z)\); for the envelope spectrum in the \([f_{r^{\prime}},\xi f_{r^{\prime}}]\) frequency, the range of the amplitude is \(X(z)\), the value of \(f_{r^{\prime}}\) is greater than the rotation frequency of the shaft where the faulty bearing is located, \(f_{r^{\prime}}\) value should be greater than the maximum fault frequency within the bearing drive system, \(\xi\) value of 4-8.  
Figure 1: Flow chart of CEEMDAN-MCKD method.  
The expression for kurtosis is shown as follows:  
\[K=\frac{\frac{\frac{1}{N}\sum\limits_{n=1}^{N}x(n)^{4}}{X_{rms}^{4}}}{X_{rms}^{4}} \tag{16}\]  
\[X_{rms}=\sqrt{\frac{1}{N}\sum\limits_{n=1}^{N}x(n)^{2}} \tag{17}\]  
where \(K\) is kurtosis value, \(x(n)\) is vibration signal, \(N\) is sampling number, and \(X_{rms}\) is root mean square value of signal \(x(n)\).  
Combining the kurtosis and envelope spectral peak factors, the fitness function is defined as the product of the two:  
\[KE=K\times Ec \tag{18}\]  
\(KE\) is selected as the fitness function of PSO algorithm optimization, which can not only consider the existence of the impact component in the signal, but also consider the periodicity and intensity of the impact component in the signal so that more accurate optimization results can be obtained. Before the optimization search, the ranges of L and M are determined to be [50; 700] and [1; 7], respectively, and the deconvolution period T can be calculated according to the equation \(T=f_{s}/f_{fault}\). The feature frequencies of the failure features of the inner and outer rings of the bearings are calculated as shown in Equations (19) and (20). The optimal combination of the influencing parameters is determined by using PSO for MCKD on the reconstructed signal.  
\[f_{fault}=\frac{r}{60}\times 0.5\times n\bigg{(}1+\bigg{(}\frac{d}{D}\bigg{)} \cos\alpha\bigg{)} \tag{19}\]  
\[f_{fault}=\frac{r}{60}\times 0.5\times n\bigg{(}1-\bigg{(}\frac{d}{D}\bigg{)} \cos\alpha\bigg{)} \tag{20}\]  
where \(n\) is the number of rolling elements, \(D\) is the bearing diameter, \(d\) is the rolling element diameter, \(r\) is the speed of the bearing, and \(\alpha\) is the contact angle.  
Deconvolution of the reconstructed signal by bringing the optimal parameters L, M, and T into the MCKD algorithm to weaken the effect of noise on the signal and enhance the impact characteristics.  
To achieve the feature extraction of the bearing, envelope processing is implemented on the enhanced signal.