Determination of optimal coordinates for switching processing cycles on metal-cutting machines

OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. No. 1 2021 From where it follows ( ) ( ), , 1, 2, 3, ..., i i s s V l V l i s n   . (4) Condition (4) enables simplifying the calculation of coordinates, as well as physically implementing a system that meets the condition Ç min  . Moreover, this condition takes into account the physical and economic requirements for optimality. The optimality condition is suf fi cient if, additionally from (2), n is determined, with Ç min  . Earlier in [10], an approximation of the speed change along the cutting path was proposed in the form of an exponential function ( ) ( ) 0 0 4 ( ) exp[ ], (0, ), ( ) i i i i i i i i V l V l l V l      , (5) where 0 V is the initial value of the speed in m/s; i  is the parameter in m -1 . Further, the approximation (5) is considered to be fair without violating the generality. Papers [8–10] demonstrate that the trajectories (5) determined by the evolutionary properties of the system are sensitive to small variations in the initial parameters and uncontrolled disturbances, for example, beats. Therefore, with the initial speed 0 V unchanged, the parameter i  may vary. It is considered as a Gaussian random variable with mathematical expectation  [ ] i M    and variance   . Then, according to the mathematical expectation  Ç( ) n for (2) taking into account (5), there is the dependence of the given costs for manufacturing a batch of parts on the number of switches n :     1 2 0 ( ) ( ) exp 1 ( 1) ñ L ñ n V                     Ç n n n (6) Obviously (6), varying n has a minimum which corresponds to the optimal minimum speed  0,1 0 0 exp L V V n          . Here 0 n is the number of switches that correspond to a minimum (6). If the proven necessary optimality condition (4) is considered, the optimal value 0 n and the corresponding optimal speed are calculated from (6) 0,1 V . Then i l is calculated, which correspond to   ( 3 , 3 ) i           that characterize the set  (l) . They are determined by an obvious dependence 0,1 0,1 0 1 ln , i i o V l V V l V          . Importantly, all i l  (l) correspond to the speed 0,1 const V  . Fig. 2. Diagram explaining the intersection of a hyperplane and a hypersurface 1 2 ( , ,..., ) n l l l 