PSI - Issue 16

Ihor Javorskyj et al. / Procedia Structural Integrity 16 (2019) 205–210 Ihor Javorskyj et al./ Structural Integrity Procedia 00 (2019) 000 – 000

207

3

  

  

  

  

 

 

1

1

  k

  k

  t 

 





 





c B u

s B u

Im , f

0  u du k t

0  u du k t

sin( ) cos 

sin( ) sin 

,





(2)

k

k

2

2









It follows from formulas (1) and (2), that band, in which spectral density is time-varying, is determined by cosine Fourier transformation of the cosine and sine covariance components. Therefore for the determination both of a spectral band of the time variation and the period of such variation the statistics                 ˆ , cos 1 1 cos ˆ 2 sin ,                                    c k s k f k t h u t t u dt u du k t f                   (3) were used. In (3)   h u – is a covariance window which has the following properties:   0 1 h  ,     h u h u     0 h u  if   , m m u u u   and   0 h u  if   , m m u u u   . Value   is unknown, herewith 2 /      , where  is so-called test period. It is shown that extremum points of functional (1) are asymptotically unbiased and asymptotically consistent estimators of a basic frequency of PNRP. The first approximations for systematic and mean squared error of estimation depending on length of realization and parameters of signal were obtained on the base of solutions of nonlinear equations that defines necessary conditions of extremum existence using a small parameter method. 3. The spectral analysis of the rotary machinery vibrations Developed method of spectral analysis was used to detect frequency bands of periodic non-stationary of vibration for defective rotary unit. The first stage of analysis is separation of the deterministic and stochastic parts. In order to reach this purpose we used we used coherent method (Javorskyj I. et al. (2017, 2018)) for estimation period, mean function and its Fourier coefficients. The values of estimators of harmonic amplitude are shown in Fig. 1. in the form of a diagram ( ˆ k m – estimation of mean function Fourier coefficients). As it can be seen from figure, amplitude spectrum is quite wide, it contains almost 40 harmonics, that correspond to frequency of almost 2,5 kHz.

Fig. 1. The amplitude spectrum of the deterministic part of vibration.

The fluctuation part of the vibration primarily was investigating on the basis of the stationary approximation characteristics. The graph of spectral density power which was calculate using formula       0 ˆ ˆ 2     L i n u n n L u f h n u B n u e    ,