PSI - Issue 36

Ihor Javorskyj et al. / Procedia Structural Integrity 36 (2022) 122–129 Ihor Javorskyj et al. / Structural Integrity Procedia 00 (2021) 000 – 000

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4

density. It follows from the obtained results that the spectrum of vibration is located within the frequency range of 0…10 kHz, but the dominant power portion belongs to the band limited by 3 kHz. The chart of the spectral density estimator for this frequency domain is shown in Fig. 2.

(a) (c) Fig. 1 The parts of vibration signals for different (a, b, c) stages of tooth damage (b)

(a) (b) Fig. 2 The estimators of the power spectral densities of the stationary approximation for the raw signal: (a) – original; (b) – zoomed

The graphs on this chart have the form of a comb with different amplitudes and bandwidths. The estimator takes the peak values at the points which coincide with the mesh frequency and its multiples, the pinion gear rotation frequency and its multiples and their combinations. We also highlight the frequency bands which correspond to powerful resonances, i.e.   , 1.8 m m f f and   2.2 , 3 m m f f . The powers of the spectral components which correspond to wheel rotation (approximately 6.4 Hz) and its multiples are negligible. Hence, we can assume that the deterministic and the stochastic modulations caused by PCRP oscillations of the input rotation period are negligible too, and formally can analyze the present data as a segment of the PCRP realization of the output period. Further, we are focusing on the analysis of the signal properties at frequencies less than 1.8 m f . Estimators of the covariance function and of the spectral density for the stationary approximation of the filtered signals corresponding to the three stages of the pinion tooth damage are given in Figs. 3. The undamped tail is a distinctive feature of the covariance function estimators. As it follows from the covariance function of the PCRP stationary approximation the undamped tail includes cosine oscillations:

( )  = = +  L 1 0 1 2 k 1



( )  R R

2

cos 2

m k



,

(6)

k

P

with amplitudes, corresponding to the power of each deterministic harmonic. At the point 0  = , the expression (6) defines an aggregate power of the deterministic and the stochastic oscillations, P is a period of deterministic oscillations. At the point  = r rP , where r is a natural number, for which ( ) 0 0  R rP , we obtain the value of