# Fault Diagnosis for a Bearing Rolling Element Using Improved VMD and HT  
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morphology, Hilbert-Huang transform, empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), auto regression moving average method (ARMA), and so on [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. The EMD method and local mean decomposition (LMD) were proposed for adaptive signal decomposition [50]. The EMD method can adaptively decompose the signal from high frequency to low frequency according to the time scale characteristics of the signal, which are adaptive and multi-scale. However, if there is an abnormal event in the fault signal, modal aliasing in the decomposition process occurs and the EMD method is inefficient. In order to suppress the modal aliasing of the EMD method, the EEMD method was proposed to eliminate the noise of the original signal by adding random Gaussian white noise [51; 52; 53]. [PERSON] et al. [54] proposed a fault diagnosis method based on the EEMD method for rolling bearing, and the fault frequency was successfully extracted. However, the EEMD method only overcomes the modal aliasing phenomenon of the EMD method to a certain extent, and still has the problems of the modal aliasing phenomenon and low efficiency.  
At present, there is much research being conducted on the faults of inner and outer rings of motor rolling bearings, and the proposed diagnosis methods have achieved good diagnosis results. However, few research has studied fault diagnosis of the rolling element of motor rolling bearings. Moreover, the characteristic frequency of the rolling bearing in the envelope analysis spectrum is regarded as the judgment basis for misdiagnosing the fault. Variational mode decomposition (VMD) [55] is a new adaptive signal processing method. It can determine the frequency center and band width of each component by iteratively searching for the optimal solution of the variational model in the process of obtaining the decomposition component. The frequency domain division of the signal and the effective separation of each component can be adaptively realized. Compared with the recursive \"screening\" modes of the EMD method and EEMD method, the VMD method transforms the signal into non-recursive and variational mode decomposition modes, which have a solid mathematical theoretical basis. It shows better noise robustness, and can effectively separate two pure harmonic signals with similar frequencies. However, when the VMD method decomposes the signal, the decomposition effect is seriously affected by the number of decomposition components [56; 57; 58; 59; 60]. Some other methods have been proposed to realize signal analysis and fault diagnosis in recent years [61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80].  
Therefore, on the basis of studying the VMD method, a new center frequency method of the multi-threshold is proposed in order to determine the decomposition modal number, \(K\), in the VMD method. Then, an improved VMD (MTCFVMD) algorithm based on the center frequency method of the multi-threshold is proposed to decompose the vibration signal into a series of intrinsic modal functions (IMFs). Then, the Hilbert transformation is used to calculate the envelope signal of each IMF component, and the maximum frequency value of the power spectral density is calculated, the fault characteristic frequency is obtained, and the fault diagnosis is realized. The rolling element vibration data of the motor bearing is used to test the effectiveness of the proposed methods.