To extract efficient feature from mechanical vibration signal for fault diagnosis

a feature extraction method of double-scale fractal dimension based on variational mode decomposition (VMD) for vibration signal was proposed. Variational mode decomposition decomposed multi-component vibration signal into several intrinsic mode functions (IMFs) in different time scale by solving variational model iteratively. In a multidimensional measure space

the space occupied by a multivariate time series within a certain period can be measured by a multidimensional super body volume. Due to IMFs produced by VMD were regarded as multivariate time series

the multidimensional super-body volume was defined and calculated utilizing IMFs in multidimensional measure space. Then the log-log curve with time scale and super-body volume for vibration signal was acquired. According to fractal theory and the abrupt point in log-log curve

the log-log curve was segmentally fitted by least-squares linear fitting. Then

the double-scale fractal dimension feature for vibration signal was defined and acquired. The simulated signal results showed that the average relative error of fractal dimension estimation using VMD method was 4.71%

which improved the accuracy of fractal dimension estimation. The experimental results of planetary gearbox vibration signal indicated that the double-scale fractal dimension feature based on VMD could describe the fractal feature of mechanical vibration signal efficiently

and the accuracy of planetary gearbox fault diagnosis had reached 100%.