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

128

7

2  L

2







  

  

ˆ

( c R jh k ˆ

ˆ

(

)

)

(

)

, , R nh jh

, cos 

, sin 

+ k nh R jh s

k nh ,

=



(11)

k





1

k

=



      

      

cos 2

k nh

( ( c R jh R jh k s k ˆ ˆ , ,

) )

    

    

 

2

=−  K n K



(

)

(

)

( ) nh m nh ( ) ˆ  −

(

)

(

)

ˆ

   

 

+ − + n j h m n j h



=

 

, (12)

2

2 1 + K



sin

k nh



The detection of the periodic time variation of the vibration residual variance is the first test procedure of this approach. The detection was carried out using statistics (10) and (11) for 1, 9 = k . The estimator of the covariance function period is found as the point of the maximum of the statistics (10) with respect to test period  . The aliasing effects of the first and the second kinds are absent if the sampling step satisfies the inequality 1 2 4 1.  + h P L The estimator for variance ( ) 1 ˆ ˆ , 0, R t P defines the time variations of the power of the vibration’s stochastic component and the estimator of the zero th covariance component at the point 0 = j ( ) 0 ˆ 0 R that is a time-averaged value of this power. The quantities ( ) ( ) 2 2 1 1 1 2 ˆ ˆ ˆ 0, 0, ˆ        = +         c s k k V k R P R P P , ( ) ( ) 1 1 1 ˆ 0, 2 ˆ ˆ 0,     =     s k c k R P k arctg P R P , 2 1, = k L , (13) can be considered as the amplitude and phase spectrum of the power time variations respectively. As follows from the previous the vibration signal generated by a rotating machine acquires properties of the periodical non-stationarity of the second order as the fault initiates. This means that the time variations of the variance and the covariance function estimators are the test indication of machine damage. Thus, indicators formed on the basis of covariance components are sensitive to the appearance of a fault. Knowing the values of 0 ˆ f , on the basis of expressions (12) and (13) we calculate the amplitude spectrum of the variance time variation, which are represented in the form of the diagrams in Fig. 6.

(a)

(b)

(c)

Fig. 6 The amplitude spectrum of the variance’s periodical variation

The variance amplitude spectrum ( ) 0 ˆ ˆ V kf slowly decays as the frequency increases. The rate of decay decelerates within the frequency range 45 – 945 Hz and it is significant for both last stages of damage. The Fourier coefficient values of numbers larger than twelve are negligible. As it follows from (1) the correlations of the spectral components, the frequency interval between which is greater than 280 Hz, are weakly correlated. So, we can conclude that the low-frequency and the high-frequency modulations are non-correlated. The ratio of the current value of the zero th covariance component ( ) ( ) 0 0 c R and the initial ( ) ( ) 0 0 i R component essentially differs for different gear damage stages. To take this feature into account, we form the indicator ( ) ( ) ( ) ( ) 2 0 0 0 1 ˆ ˆ ˆ ˆ 0 0 =   =  +      L i k I R V kf R ,