OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 7 No. 1 2025 where χi is the coefficient of the cutting force decomposition on the i-axis of tool deformation. The deformation movement model of the tool tip is as follows: 2 11 12 13 11 12 13 2 2 21 22 23 21 22 23 2 2 31 32 33 31 32 33 2 ; ; , f p c d x dx dy dz m h h h c x c y c z F dt dt dt dt d y dx dy dz m h h h c x c y c z F dt dt dt dt d z dx dy dz m h h h c x c y c z F dt dt dt dt + + + + + + = + + + + + + = + + + + + + = (14) where m [kg∙s2/mm]; h [kg∙s/mm]; c [kg/mm] are the inertia, dissipation, and stiffness matrices, respectively. The differential equation describing the temperature transfer through back surface of the tool from the previous spindle revolution to the current contact zone between the tool and the workpiece is given by: 2 1 2 1 2 2 ( ) h h h d Q dQ T T T T Q kN dt dt + + + = , (15) where 3 1 2 1 2 , c h T T V λ = = α α are time constants; 3 1 2 Q c k h k V λ = α α is the transmission ratio; α1, α2 are identifiable dimensionless scaling parameters of the integral operator; λ is the thermal conductivity coefficient; kQ is the coefficient characterizing the conversion of irreversible transformation power into temperature; ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) z v v ñ v ñ v v ñ h dz t T dz t T N F t T V F t T F t T V dt dt − − = − − = χ − + − − is the force of irreversible transformations. Thus, the system of equations (3) through (15) constitutes a virtual mathematical model for the digital twin of the metalworking process on a metal-cutting machine. Research Results and Discussion The most promising approach for parametric identification of virtual digital twin models is the use of data acquired from a vibration monitoring system [20]. Here, from the complete dataset, we propose focusing on so-called scattering ellipses, which can be obtained directly from the recorded vibration acceleration signal. Let us consider the scattering ellipses for the cases presented in Fig. 8, identifying the parameters of the digital twin model from these data. The results of comparing these two methods for calculating scattering ellipses, for equal values of cutting wedge wear, are shown in Fig. 20. As seen in Fig. 20, the span of the scattering ellipse axes coincides almost exactly. However, the vibration dispersion for the real machine is significantly higher. This is attributed to the presence of vibration activity not only from the cutting process but also from other system carriers within the real machine. Modeling all supporting systems in conjunction with the cutting system remains an unresolved task. To evaluate the quality of the surface obtained during cutting, only the vibration activity coordinate of the cutting tool along the y-axis was analyzed. This choice is based on the fact that vibrations at the cutting tool tip along the z- and x-axes will be attenuated during longitudinal turning, which forms the basis for modeling the cutting system. However, in other metalworking operations, the vibration signal in other directions can also be analyzed. To analyze the quality of the surface obtained by cutting, we needed to process the data to remove any constant component, considering only the vibrations within a defined time range. For this purpose, we developed a program that allows us to analyze the quality of the surface obtained during cutting, both within the digital twin system and based on data acquired by the vibration monitoring system. This resulted in a sufficiently powerful computational tool that can subsequently be used for parametric identification of virtual digital twin models on real metalworking machines.
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