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

206

2

1. Introduction

Methods of periodically non-stationary random processes (PNRP) and their generalizations discover new possibilities for faults detection in rotating mechanisms at early stage of development, Antoni J. (2007). The higher efficiency of this methods is provided by using such characteristics of processes, which describes nonlinear interaction between recurrence and stochasticity arised in the case of damaged mechanism units. The analysis of nonlinear effects caused a spectrum shifting, its time variability is important to identify the sources and types of faults. Using of covariance methods of detecting and analysis of hidden periodicities allow to estimate only total power of a nonlinear interaction without specifying its spectral structure. Investigation of properties of a periodic nonstationarity in frequency domain allows to study the nature of mechanism element faults more detailed.

2. The spectral functional for the detection of the non-stationarity

In PNRP model of the hidden periodicities (Antoni J. (2007), Antoni J. (2009), Javorskyj I. et al. (2017), Javorskyj I. (2013)) the vibration properties are described by time periodical mean function   m t , which represents the deterministic oscillation, the covariance function   , b t u and its Fourier transform – the instantaneous spectral density

 

1

  , t 

  , b t u e du   iu

f



.

2





The instantaneous spectral density can be represented in the form of Fourier series:     0 ,    ik t k k f t f t e  

whereby

 

1

  

  B u e du   iu

f



,

k

k

2





2



  k B u are the Fourier coefficients of the covariance function.

, T is the non-stationarity period,

0  

where

T

  0 B u and zero spectral coefficient

  0 f  are the characteristics of the

The zero covariance coefficient

  0 f  which describes the spectral structure of the stochastic

stationary approximation of PNRP. The function

vibration. The spectral density   , f t  is a complex-valued function:   , f t 

  

  t 

Re , f

t i 

Im , f



. Herewith

0  

1

  t 

  0

Re , f

B u

u du

cos( ) 







  

  

  

  

 

 

1

1

  k

  k

 





 





c B u

s B u

0  u du k t

0  u du k t

cos( ) cos 

sin( ) sin 

,





(1)

k

k

2

2







