OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology In addition to these processing factors, one more important temperature factor can be taken into account – this is the linear expansion of metal with the increase in the contact temperature. It should be noted that along with the thermal expansion of metal, there is the increase in the pushing force, which will lead to restructuring of the entire cutting system, and to a possible loss of stability [9]. With regard to simulation the dynamics of the cutting process, the maximum resistance of the cutting wedge coincides with the maximum stability margin of the dynamic cutting system. To paraphrase the statement of A.D. Makarov, it can be said that in a dynamic cutting system there is some optimal, by the margin of this system stability, cutting mode, directly related to the speed (temperature) of cutting and the amount of the cutting wedge wear. In connection with all mentioned above, the purpose of the paper is to consider the statement of A.D. Makarov about the existence of an optimal cutting mode, from the point of view of dynamics stability of metal turning. The above given reasoning allows formulating two hypotheses; the verification of compliance with the position of A.D. Makarov will be the objectives of the study: The first hypothesis: the optimal value of the cutting speed (cutting temperature), when simulating the dynamics of the cutting process, is determined by a combination of the following factors: the incident characteristic of the cutting force (according to N.N. Zorev) and the minimum coefficient of friction associated with the transition of friction from adhesive to diffusion nature. The second hypothesis: the optimal value of the cutting speed (cutting temperature), when simulating the dynamics of the cutting process, is determined by a combination of the following factors: the incident characteristic of the cutting force (according to N.N. Zorev), the minimum coefficient of friction due to the transition of friction from adhesive to diffusion nature and the dependence of the force pushing the tool on the preheating of the processing zone. Testing the first hypothesis Formulation of a mathematical model of the cutting control system When formulating a mathematical model, the cumulative force reaction of the cutting process to the forming motions of the tool, relying on the synergetic concept and the mechanical-thermodynamic approach is considered. The implementation of the synergetic concept in the synthesis of a mathematical model consists in the fact that the forces are described in the coordinates of the process state, and the mechanicalthermodynamic approach consists in the fact that in addition to the coordinates of the cutting system state, the force response includes the dynamics of production and temperature dispersion during processing. For the convenience of formulating a mathematical model, let’s consider the main axes of the deformation coordinates, along which the equations of motion will be written (see Figure 1). Fig. 1. Diagram of the coordinates and forces of the model
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