Application of tempered stable distribution for selection of optimal frequency band in gearbox local damage detection

Local damage detection methods based on the vibration signal analysis are widely discussed in the literature. One of the most popular approach is signal enhancement by filtering focused on extraction of informative content at so called informative frequency band (IFB). In such approach the vibration signal first is decomposed into time-frequency representation and then the measures of impulsivity are applied to the appropriate sub-signals. Till now, kurtosis was preferred as criterion for IFB search. However, for some cases more robust criteria should be used. Jablonski proposed to calculate kurtosis from envelope spectrum, Urbanek suggested MID (Modulation Intensity Distribution), set of alternative selectors as extension of spectral kurtosis (SK) have been defined by Obuchowski. Further extension was recently proposed by Zak, namely statistics related to α-stable distribution were used as IFB indicators. This distribution is especially important in modeling data with outliers so it was reasonable to apply this approach in the considered problem. As it was shown, the α-stable based methodology for some vibration signals more clearly indicates the IFB in contrast to the classical ones. However, in case of early stage of fault development or developed fault with high level of noise the mentioned criteria (including stability index as a measure of impulsivity) may be insufficient. It is related to the fact that cyclic impulses related to damage are often hidden in the noise (even after time-frequency decomposition). In this paper we propose to apply (instead of kurtosis or stability index) the parameters of tempered stable distribution. This distribution is an extension of the α-stable one, however, it possesses many properties of Gaussian systems. It is associated with two parameters which may indicate properly the IFB in case of poor signal-to-noise ratio. In this paper we remind the approach of IFB selection based on the kurtosis and α-stable distribution and describe how to localize the information of the fault by using tempered stable approach. The real vibration signal from gearbox is analyzed in the context of presented methodology. Finally, we examine two examples of vibration signals for which the proposed approach is superior with respect to the classical methodology.