PSI - Issue 59

Viktor Kovalov et al. / Procedia Structural Integrity 59 (2024) 779–785

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4

V. Kovalov et al. / Structural Integrity Procedia 00 (2019) 000 – 000

Р0/(t)= - λ Р0(t)+ μ ; Р1/(t)= λ Р0(t) - μ Р1(t). If the system at t=0 is in operation, the initial conditions are: P0(0)=1, P1(0)=0. When the system at t=0 is in repair, then the initial conditions will be: P0(0)=0, P1(0)=1 . Applying the Laplace transform to the equations and considering the initial conditions p0(0)=1, p1(0)=0 , we obtain: s Р 0 (s)-1+ λ Р 0 (s)- μ Р 1 (s)=0, s Р 1 (s)- λ Р 0 (s)+ μ Р 1 (s)=0 and after bringing similar terms :

(s+ λ ) Р0 (s)- μ Р1 (s)=1, - λ Р0 (s)+ (s+ μ ) Р1 (s)=0.

To solve this system of equations, we introduce the determinant D, whose elements are the coefficients at Pi(s) . In addition, we introduce the determinant Di , which is formed by replacing the i -th column by the column of coefficients of the right-hand side of the equations of the system. Then

       s s 0 1



    s

 

 

Р i (s)= D i / D and therefore

or

.

s P 0

s P 0





 s s

  

s

   

The readiness function, which we denote by A(t), is the inverse Laplace transform for Р 0 (s) , i.e.





T       e ) (

0 ( )  

As F P t



    s

  At L P 0 1  

or

(3)



 



The average serviceability time А ( Т ct ) of the turning cutter for some finite time interval T can be determined by summing P 0 (t) over the entire interval and dividing by it:

  T 1

ct ( )

P t dt ( )

AT

(4)

0

T

0

For this case



T    2 )



T       e ) (

ct ( )

AT







(5)

(  







Consider the reliability of a prefabricated turning cutter of a more complex design, consisting of a body, a block insert on which the cutting insert is mounted. A prefabricated turning cutter restores its serviceability in two ways. When the first failure occurs (failure of the cutting insert), partial recovery is carried out - turning or replacing the insert and the system restores its functionality. After occurrence of the second failure (failure of the whole unit or fixing elements) the unit is replaced. In this case the system restores its full functionality. Let  1 mean the failure