# Fault Diagnosis for a Bearing Rolling Element Using Improved VMD and HT
## 3 Improved VMD Method Based on the Center Frequency Method of the Multi-Threshold  
Each IMF component is obtained by using the VMD method, in which the decomposition corresponds to a central frequency, \(cf\), and the centre frequency of the obtained IMF component under each scale is distributed from low frequency to high frequency. A center frequency method of the multi-threshold is proposed to determine the model number, \(K\), of the VMD method (MTCFVMD). On the basis of the center frequency, the multi-threshold of \(T_{1}\), \(T_{2}\), and \(T_{3}\) are set. The threshold, \(T_{1}\), is used to measure the difference between the maximum values of the central frequency under different scales and the maximum value of the center frequencies of all modes. For the central frequency method, it is known that the center frequency of the obtained IMF component under a certain scale is distributed from low frequency to high frequency. If the difference is greater than \(T_{1}\), it indicates that the maximum component of the central frequency in the original signal is not decomposed. The threshold, \(T_{2}\), is used to measure the difference between the value of the central frequency of the \(L-th\) IMF component and the value of the central frequency of the \(L-1-th\) IMF component under the same decomposition scale, \(K\) (\(K=L\)). The VMD method can effectively decompose the two pure harmonic signals with a similar frequency, and the IMF components with a similar frequency are retained according to the threshold, \(T_{2}\). The threshold, \(T_{3}\), is used to measure the difference between the value of the central frequency of the \(L-1-th\) IMF component and the value of the central frequency of the \(L-2-th\) IMF component under the same decomposition scale, \(K\) (\(K=L\)). The maximum value, \(K_{max}\), of the decomposition scale, \(K\), is then set. The signal is decomposed under the decomposition scale, \(K=1\), \(2\),...,  
IMF component and the value of the center frequency of the \(L-1-th\) IMF component under the decomposition scale, \(K\), is calculated (\(K=L\)). If the difference value is less than the threshold, \(T_{2}\), the difference between the value of the center frequency of the \(L-1-th\) IMF component and the value of the center frequency of the \(L-2-th\) IMF component under the decomposition scale, \(K\), is calculated (\(K=L\)). If the difference value is less than the threshold, \(T_{3}\), the value of \(K\) that meets the above conditions is regarded as the optimal mode number of the VMD decomposition method. The expressions of \(T_{1}\), \(T_{2}\), and \(T_{3}\) are described as follows:  
\[T_{1}=a*cf\_max_{IMF} \tag{14}\]  
\[T_{2}=b*cf\_max_{IMF} \tag{15}\]  
\[T_{3}=c*cf\_max_{IMF} \tag{16}\]  
The flow of the improved VMD decomposition method based on the center frequency method of the multi-threshold is shown in Figure 1. The left half of the flow chart can determine the number of data modals of the inner and outer rings, and the right half can determine the number of data modals of the rolling element, indicating that the method has wide adaptability. The implementation steps of the improved VMD method based on the center frequency method of the multi-threshold are described as follows:  
1. Determine the maximum value, \(K_{max}\), of the decomposition scale. 2. The VMD method is used to decompose the original signal into a series of IMF components under different scales, \(IMF_{K}\) (\(K=1,2,\ldots,K_{max}\)). The center frequency value of each IMF component is calculated in order to obtain the set, \(cf_{IMF_{K}}(K=1,2,\ldots,K_{max})\). 3. Calculate the maximum value of the center frequency of the set, \(cf_{IMF_{K}}\), under different scales, which is regarded as the \(cf\_{max_{IMF}}\) (\(K=1,2,\ldots,K_{max}\)). 4. Calculate the maximum value of the center frequency of the set, \(cf\_{max_{IMF_{K}}}\), which is regarded as the \(cf\_{max_{IMF}}\) (\(K=1,2,\ldots,K_{max}\)). 5.