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

OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 1 At the speci fi ed speeds ( ) i V l , the processing costs З are determined by the formula 1 2 1 2 1 0 ( , , ..., ) ( 1) ( ) i l i n n i i d ñ V            Ç ñ l l l n , (2) where 1 ñ is the cost of the machine minute in RUB/min; 2 ñ is the cost of replacing the tool and its adjustment. Drilling deep holes of small diameter with spiral drills has a similar cost structure, but in this case, the switching of the processing cycle corresponds to the cost of removing the tool from the cutting zone to clean it from chips [37]. The solution to the problem is reduced to the calculation i l for which Ç min  . Necessary optimality conditions. Calculation algorithm Due to the mentioned circumstances, the speed is a monotonically decreasing function. In this case, the optimal coordinates of switching cycles correspond to equal cutting speeds. Moreover, these speeds are equal for various monotonically varying functions ( ) i i V l  0 . The set  0 is a set of the phase trajectories of speeds at which the condition of a minimal wear intensity and ensuring a given quality of manufacturing is simultaneously met. Solving the problem requires, fi rstly, fi xing the number of switches n and for the given n and L determining the coordinates of the switches at which in (2) 1 2 ( , , ..., ) min n l l l   , provided that the requirement (1) is additionally ful fi lled. If the conditions (1) are not considered and fi xed when determining the optimal coordinates n , the task does not make sense. If the condition 1 2 ( , , ... ) n i l l l    is set, the resulting surface can intersect with a hyperplane 1 2 1 ( , , ..., ) i n n i i L l l l l     (Fig. 2). If i  is reduced, the lines of intersection of the surface 1 2 ( , , ... ) i n l l l  with the hyperplane represent convex closed trajectories that degenerate into a point 0 1 2 ( , , ..., ) l l l  that corresponds to the desired coordinates. The surfaces 1 2 ( , , ..., ) i n l l l  are convex since as the wear develops, the speed change function is monotonically decreasing. Therefore, the only optimal point at which the condition min Ç  is met is the condition of touching the hypersurface 1 2 ( , , ..., ) n l l l  and the hyperplane 1 2 ( , , ..., ) n L l l l (Fig. 2). Therefore, it is true for the point 0 1 2 ( , , ..., ) n l l l  1 2 1 2 ( , , ..., ) / ( , , ..., ) / . n i n i l l l l L l l l l      (3) Fig. 1. Scheme for determining the coordinates of processing switching cycles