PSI - Issue 59

Viktor Kovalov et al. / Procedia Structural Integrity 59 (2024) 779–785 V. Kovalov et al. / Structural Integrity Procedia 00 (2019) 000 – 000

781

3

fraction of time during which the system is ready for use. In the following we will consider the technological system "machine - tool- workpiece - machinist" from the point of view of its reliability, in the centre of which there is a cutting tool, the operating regulations of which influence the state of the whole system. Fig. 1. shows an example of machining parts on heavy machine tools.

Fig. 1. А n example of a heavy machine tool.

To determine the availability factor of the prefabricated tool, let us consider two states in which it can be in the process of operation: state 0, when it is operable, and state 1, when the machine is idle (replacement of the tool and rotation of the polyhedral insert). The matrix of transitions of the system (prefabricated tool) from one state to another has the form:

1

 

P

 

1



 

и

, The finite-difference equations describing the stochastic behavior of this system can be composed from the following considerations: the probability that the system is in state 0 by time t+dt is derived from the probability that this system was in state 0 at time t and did not fail over the interval t-t+dt , or that it was in state 1 at time t and returned to state 0 over the interval t - t+dt . Therefore, we obtain where

) P t dt P t (

( )(1    

) ( ) dt P t dt  

0( ) dt



0

0

1

(1)

The probability of the system being in state I is determined in a similar way:

( )(1 dt Pt dt Pt dt Pt        ( ) ) (

) 0( ) dt

1

0

1

(2)

The term 0 (dt ) represents an infinitesimal quantity since it is the probability of two incompatible events of the ordinal flow. Using the transition at dt → 0 and defined:     P dt P t dt P t i i i dt / 0 lim       0 0 lim 0   dt dt i dt ,