Research On Fractal Dimension Characteristics And Methodology Of Fractal Fault Diagnosis

Fractal, which is defined as “shape made of parts similar to the whole in someway”, is an active branch in the study of modern mathematics and nonlinear science.As an essential parameter in fractal theory, fractal dimension has been widely used inmany areas of science, including the capability of quantitative measurement for fractalcharacteristics of nonlinear systems and the ability of space-filling capacitymeasurement for signals. In mechanical fault diagnosis, the influences of nonlinearfactors for mechanical vibration signals are different under diverse fault status. Fractaldimension describes the fault features of mechanical system and recognizes the faultstatus of mechanical device effectively. Based on the fractal theory, the investigationon the fractal dimension features and fractal fault diagnosis methodology was carriedout.As the fractal box dimension is not sensitive to noise, the anti-noise property ofbox dimension was studied. According to influences of different noise strength, theanti-noise behavior of fractal box dimension was analyzed by changing signal to noiseratio (SNR). The fractal box dimension varies with SNR and its curve can be dividedinto two parts with a point which is defined as the frontier point between them.Through the tendency of each part of the curve, its anti-noise performance wasunveiled.Utilizing the fractal box dimension as characteristic parameter, the single fractalfault diagnosis method was discussed. The result showed that the fractal boxdimension had the ability of quantitative measurement for mechanical vibration signaland recognition of the fault status. In several measures, a nonlinear signal can bedescribed entirely and exactly by multi-fractal, especially for unbalanced property andsubsection attribute. Based on sample sequence, the application range of fractal faultdiagnosis is enlarged by multi-fractal using relationship to recognize the device status.On the basis of multi-fractal fault diagnosis, relating to the signal decomposition,the fractal fault diagnosis characteristic parameter was extended from single fractaldimension, generalized dimension to fractal matrix. Moreover, a new correlationcoefficient calculation method, adapting to fractal characteristic parameter matrix,was presented based on improvement of generalized dimension correlation coefficientmethod. In order to make the fractal matrix construction out of limited to any signaldecomposition methods, wavelet, wavelet packet analysis and empirical mode decomposition were discussed and used in constituting the fractal matrix. In addition,the selection method of composition signals based on correlation coefficient waspresented for diversity of signal decompositions. The experiment showed that thefractal characteristic parameter matrix obtained through this selection method hadmore efficient for mechanical fault recognition and diagnosis.As is well known, high frequency noise, which is not conducive to fractal faultdiagnosis, need to be restrained by means of filtering method. Therefore, cascadedbistable stochastic resonance with well filtering property, combining with generalizeddimension, was applied to mechanical fault diagnosis in high frequency noisebackground. The power can be transferred from high frequency domain to lowfrequency domain by cascaded bistable stochastic resonance in order to not onlyremove the high frequency noise but also enhance the low frequency signal.Furthermore, for the filtered signals, it is easier to distinguish between different faultstatuses by fractal dimension so as to advance the effect of fractal fault diagnosis.