OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 7 No. 1 2025 If N(t) is not equal to N0 = const, and taking into consideration that N0, N1, …, Nn−1 due to the small size ΔA = Ak − Ak−1 are values close to constants, we obtain the following approximate sum describing the integral operator (5): ( ) 1 1 1 1 2 ( ) 1 3 0 1 0 2 1 1 ( ) ( ) ... A A A A A h N e N N e N N e α − −α α − β = − − − − − − α 1 1 ( ) 1 2 1 ( ) n A A n n n N N e N − α − − − − − − + + 2 2 1 2 2 ( ) ( ) 2 0 1 0 2 1 2 ( ) ( ) ... A A A A A N e N N e N N e α −α − −α − β + − + − + − + α 2 1 ( ) 1 2 1 ( ) , n A A n n n N N e N − −α − − − − + − − (6) where 1 β , 2 β , 1 α , 2 α are the parameters identified in the wear calculation model, based on the results of preliminary experiments. The graph of the cutting tool wear development along the back edge based on the calculated data is shown in Fig. 19. h, mm L, m 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 800 1600 2400 3200 4000 4800 сritical wear zone run-in area stabilization zone Fig. 19. Calculated cutting tool wear curve A comparative analysis of Fig. 8 and 19 shows that the critical wear zone begins earlier in Fig. 8 than in Fig. 19. This discrepancy is attributed to the chip formation process. During machining, we deliberately allowed cutting to occur with strong vibrations resulting from the accumulation of flow chips. Consequently, these vibrations caused changes in the contact between the tool back face and the workpiece, leading to the appearance of new elemens in the wear zone. As can be seen in Figs. 8 and 19, the wear development curve of the cutting tool along the workpiece is divided into three distinct areas: – the run-in area, where initial tool wear is established; – the area of wear stabilization, where the rate of wear increases slowly; – the area of formation of critical wear, where the rate of tool wear increases rapidly. As the fundamental virtual model for a digital twin, we will adopt a system of nonlinear equations. The of the cutting force on the temperature-speed coefficient of cutting will be represented as a decreasing exponential dependence:
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