OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 6 No. 2 2024 plastic deformation in the cutting zone increases. If we compare the coherence functions shown in fig. 5 and fig. 6, we can conclude that the topologies C(L, X) and (L, X) are adequate. With the given mathematical tools it is possible to estimate only macro geometric characteristics.As far as surface roughness is concerned, a frequency range of up to 5.0 kHz, depending on the accuracy qualification, should be considered when forming it. In addition, molecular-mechanical interactions, plastic deformation processes and the dynamics of chip formation itself are of greater importance. If we follow GOST 25142-82, then irregularities within the length of the reference surface are in the frequency range in which 1 2 , ( ) 0.7 X R K ω 〉 are located only in rough machining. To confirm this, it is sufficient to analyze the auto spectrum (fig. 7, a) calculated from the profile function measurement and the corresponding machined surface morphologies, which are obtained at three cutting speeds (fig. 7, b, c). The geometrical topologies determined with the Contour ELITE microscope clearly show additional fine irregularities in the surface topography, which are formed in the vicinity of the tool tip trace obtained at low cutting speed (speed of 0.75 m/s). Such fine irregularities are practically absent at a cutting speed of 3.0 m/s. At the same time, the relief spectrum changes in the high-frequency region (fig. 7, a). In addition, the transition from relief at 3.0 m/s to relief at 0.75 m/s is characterized by instability of formation of additional deviations of relief different from the trace left by the tool. Conclusions When creating a digital twin of the cutting process, one of the problems is to create a mathematical toolkit that can be used to reconstruct the geometry of the surface formed by cutting. The study considers the adequacy of the reconstructed geometrical topology C(L, X), obtained by calculating and/or measuring the trajectories of formative motions L(Φ)(t), as well as the real topology (L, X). The real topology is represented as a function of the profile in the direction of cutting speed. The reconstructed topology C(L, X) is based on the trajectory of forming motions, which represent the unity of the TMAE L(t) defined by the CNC program and the trajectories of deformation displacements of the tool tip X(t) relative to the workpiece. Two cases when the trajectory X(t) is measured or calculated are considered. In order to analyze the adequacy, the main attention is paid to the coherence function between the formative motions and the surface topography along the direction of tool motion. Examples of shaft surface morphology obtained by turning the shaft under different machining conditions and modes are also considered. The studies have shown, first, that the frequency range in which the reconstructed topology C(L, X) adequately represents the real topology (L, X) is limited by the bandwidth of the adopted finitedimensional model of the dynamic cutting system. In the examples under consideration, this band is 0,0 (0, ) ω∈ ω . Here, the upper frequency ω0,0 depends not only on the bandwidth of the subsystems interacting through the cutting process, but also on the process modes. In the example under consideration, this range is limited to frequencies, at best in the range of 200–300 Hz. Under the conditions of the performed studies, there is a tendency of some expansion of the frequency range of adequate representation of the reconstructed topology of the real topology with increasing cutting speed. The 0,0 (0, ) ω∈ ω range decreases as tool wear develops and the amount of plastic deformation of the material in the cutting zone increases. By comparing the topology reconstructed from the measured vibration sequences C(L, X) and the real topology (L, X) the frequency range 0,0 (0, ) ω∈ ω can be extended to 500 Hz. However, even in this case, only macro geometrical properties of the surface formed by cutting in the unity of dimensional accuracy and waviness can be adequately evaluated in the reconstructed topology. When estimating microrelief, more complex statistical estimations and more accurate measuring devices are required, which allow to significantly expanding the frequency range of modeled and measurable vibration sequences. The performed analysis of elementary surface morphologies has shown that, when cutting speed decreases, additional deviations are formed in the vicinity of the trajectory formed by the tool tip, the physical nature of which is related to the plastic deformation of micro areas in the contact of auxiliary flanks
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