PSI - Issue 16

Ihor Javorskyj et al. / Structural Integrity Procedia 00 (20 9) 0 0 – 00

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Ihor Javorskyj et al. / Procedia Structural Integrity 16 (2019) 205–210

whereby u is the lag sampling interval, L is integer number which equals to m u u and m u is point of correlogram cut-off, is shown in Fig. 2. This quantity describes spectral structure of the stochastic vibration. The functional that is the integral sum of the transformation (3) was used for the determination of the frequency band in which the vibration properties vary periodically in time. The dependency of this functional on test period at the frequency that is the point of the maximum of the estimator   0 ˆ f  is shown in Fig. 3. The point of the maximum was chosen as the estimator of the non-stationary period. The estimators of the spectral components which were calculated using (3) for 0 ˆ    are shown in Fig. 4. It was obtained that the power of fluctuation component is concentrated in the band 0,5 2,5  kHz and using functional (1) it was shown that periodically variable in time part of spectrum belongs to the range 1,0 2,3  kHz. Spectrum shifting to the higher frequency band is explained by the fact that harmonics with shaft rotation frequency are modulated by high frequency narrow band processes (Fig. 5 and Fig. 6). The mathematical model which describes the vibration properties can be represented in the form           0 0 0 1 ( ) cos sin          L c s k k k t t t k t t k t       . The auto- and cross-covariance function of the jointly stationary random processes have the form of the decreasing oscillations. The frequency of these oscillations is defined by the resonance frequency of the individual mechanism. The investigations showed that we have dealing with distributed faults if 10 L  and local faults if 30 L  . The diagnostic indicators which are defined by formulas     0 1 1 0 0 ˆ , ˆ ,    L k m k m f I f     ,     2 1 2 1 0 1 2 0 0 ˆ , ˆ ,        M L k m r M k M m r M f r I f r     , whereby max min 1 2 , M M       and   min max ,   is the found frequency band. It turned out, that these indicators are sensitive to fault parameters change than the indicators formed on the basis of the covariance characteristics.

Fig. 2. The correlogram estimators of the power spectral density