PSI - Issue 40

V.N. Syzrantsev et al. / Procedia Structural Integrity 40 (2022) 411–417 V.N. Syzrantsev et al./ Structural Integrity Procedia 00 (2022) 00 – 000

414

the use of which does not lead to the problems of finding the functional maximum J ( h n ). Moreover, it is for this kernel function for value h n an estimate close to the optimal one * n h has been obtained

0,2    .

(6)

* n n h D n

n D is a sample variance calculated based on the values sample  i

1, i n  by formula:

Here,

2

1

1



  

n

n

i    

.

(7)

D

2

1 n    i i



n

1   1 i 

n

As a result, to estimate the unknown density function f  (  ) with kernel (5) and smoothing parameter (6), opening (2), we have the expression:



   

2

    

1

n



.

(8)

exp 0,5 

( ) 

f



 

i

 



*

h

*

2

n h

   

 



1

i



n

n

After the implementation of the described procedure, function f  (  ) shown on the histogram in Fig.2 in the form of a line has been determined on the basis of experimental data in Fig.1. Using function f  (  ) , we consider the algorithm for building a loading cyclogram. The cyclogram is a finite number ( m ) of steps - blocks of stresses ( , 1, j j m   ) - the sum of relative durations of which ( , 1, j t j m  ) is equal to one 1 1 m j j t    . The integral of function f  (  ) is also equal to one. Using sample i  1, i n  we determine a range of stress variations: min min( ) i i    and max max( ) i i    . This range is divided into m intervals of width h m :   max min / m h m     . (9)

Fig. 2. The results of recovery of function f  (  )

In each j -th interval we calculate its midpoint:   min / 2 1 j m m h j h j         .

(10)