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 internal dynamics of the controlled system is effective for managing complex interconnected systems. Concerning machine tools processing, this principle means coordination of the machine’s executive units trajectories (MEUT) with a dynamic cutting system (DCS) [8–10], whose properties change depending on the development of the cutting tool wear and the produced power in the cutting zone [8–14]. Two circumstances are taken into account. Firstly, the properties of DCS change depending on the wear of the tool and the energy supplied to cutting [8–10]; secondly, evolutionary changes lead not only to a dynamic restructuring of the cutting properties but also change the technological modes with minimal tool wear intensity. The geometric topology of the workpiece surface also changes. These changes follow, fi rst of all, from the thermodynamic nature of wear and the dependence of the wear rate on the energy of the mechanical system introduced into the cutting zone [18–24]. Such features of the cutting process led to the creation of a different class of processing control systems on metal-cutting machines [25–34]. For example, if the choice of the optimal cutting speed relies on the optimal temperature in the contact zone of the tool’s planes with the workpiece, the optimal power of the energy consumed in the cutting zone should correspond to the optimal temperature. Consequently, a monotonous decrease in the cutting speed should correspond to ensuring optimal power as wear develops. Besides, to stabilize elastic deformations the feed rate should decrease, i.e. the feed rate should decrease even faster along the cutting path [8–10]. A MEUT decrease along the cutting path leads to the fact that further processing at low cutting speeds and feeds becomes impractical. Therefore, a new problem is formulated to determine the coordinates in which the tool needs to be replaced; this problem is close to the synthesis of optimal performance systems, which is solved, for example, based on using the L. Pontriagin maximum principle. [35, 36]. A similar problem was solved by the authors for drilling deep holes of small diameter [37]. However, in the case of processing, it has certain features the article deals with. The purpose of the article is to develop mathematical algorithms and techniques that allow determining these coordinates. Research Methodology Mathematical formulation We will limit ourselves to the consideration of longitudinal turning on lathes. The obtained results can be easily generalized to other types of processing: milling, drilling, including drilling deep holes [37]. The paths are set: the general path of the tooltip movement L , which is determined by the sum (Fig. 1) 1 i n i i L l     . (1) The path L is the same for a batch of workpieces. We set the task of determining the coordinates ( ) i l along the tool movement trajectory L ( (1) 1 l l  , (2) 1 2 l l l   , (3) 1 2 3 l l l l    ,... ( 1) n n l L l    ), at which the cost of manufacturing a batch of parts is minimal. They are determined by the costs for the actual cutting and the replacement of the tool and its changeover. Additionally, along the trajectory, there are set speeds ( ) i V l that change depending on the current wear of the tool. Initial speed value 0 ñînst. V  The trajectory of the cutting speed ( ) ð ( ) i i V l along the path is calculated in such a way that the wear intensity of the tool is minimal [10]. The calculation method is based on the hypothesis that the wear intensity is related to the power of irreversible energy transformations. The optimal power corresponds to the optimal temperature; this is the transition region from the prevailing adhesive to diffusion wear of the tool [7, 24]. The speed ( ) ð ( ) i i V l corresponds to the feed ( ) i V l rate, which is limited by the value that affects surface roughness formed by cutting. However, the feed rate must be reduced in the course of evolution due to tool wear and the associated increase in the volume of plastic deformation in the cutting zone. Therefore, the feed ( ) i i V l rate along the cutting trajectory decreases faster than along the trajectory of the cutting speed [10].