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

123

2

analysis provided for example by Randall at al. (2001), or Antoni at al. (2004). The cyclostationary analysis involves the calculation of the cyclic auto-correlation function and its two-dimensional Fourier transforms, searching for correlated harmonics, the calculation of coherence functions and their integrating. Than the so-called “informative frequency band” is detecting and applied the various some “end in view” procedures as in Borghesani at al. (2013), Obuchowski at al. (2014), Sawalhi at al. (2017), Wang at al. (2019). The effective techniques for discovering of the hidden periodicities of the first and second order of vibrations developed in Javorskyj at al. (2012) provide the determination of the period for the deterministic oscillations and the time variation of the moment functions of the second order for each individual realization. It enables the estimation of the respective amplitude spectra, which can be used as a basis for assessing the machinery conditions. Now we detail briefly the main stages of the PCRP covariance analysis. Thus, the time-averaged power of the signal ( ) 0 0 R at the point 0  = is equal to the sum of the modulation powers ( ) 0 pp r :

1 =− =  L p L

( )

( ) 0

0 0

R

r

.

pp

1

The amplitude of the individual harmonic for order k is the total characteristic for the correlations of the spectral harmonics whose frequencies are shifted by 0  k where 0 is estimated value of the basic cyclic frequency. The non-zero th components of ( )  k R characterize the summary cross-correlations of the modulating processes dependent from time lag  , whose numbers differ by k . Consequently, we obtain the correlations of the spectrum components of frequencies, shifted on distance between spectrum periodical components (orders). These correlations are described by the spectral components

 

1

( ) 

( )  R e d .   − i

f

=

(1)

k

k

2



−

0  = are their total characteristics in the time domain:

The values of the covariance components at the point



( ) 0

( )

= 

R

f

  d .

(2)

k

k

−

The time-averaged power of stochastic oscillations, which is determined by ( ) 0 0  R as the damage grows. Thus, we can expect that the indicator, formed on the basis of all variance Fourier coefficients, will be as sensitive as possible to changes in the gear pair conditions. It follows from (2), that the variance time variations in general are not localized in the frequency domain. The maximum frequency distance between correlated harmonics is determined by the highest number of the variance harmonic 2 L and is equal to 2 0  L . This means that the bandwidth for the filtering of the raw signal cannot be narrower that 2 0  L and must be carried out over the whole signal frequency band. If this condition is not satisfied, the filtering will results in the decrease of amplitudes values for variance harmonics as well as evaluation of their number. This non-stationarity property of vibration signal must be taken into account when the so-called “informative frequency band” is selected. It has to be applied to the purely random part of the signal; therefore, the deterministic components must be extracted. When using Hilbert transform it is said that the modulus of the analytic signal is a low-frequency deterministic function describing the signal envelope - Smith and Randall (2015). However, theoretical analyses of the Hilbert transform of PCRP and the corresponding analytic signal show that this judgment is incorrect. The sum of squares of the accordingly filtered signal and its Hilbert transform is not a low-frequency deterministic function describing the squared envelope. On the contrary, it is a square of the pure stochastic high-frequency random signal ( ) 0 0 R , increases by value