OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 6 No. 2 2024 model of the machining process on the machine tool, based on the intellectual approach to model building, which reveals the relationship between technological modes and output parameters of the cutting process, considered in the unity of the quality of parts manufacturing and machining efficiency [17–21], has also been formed. At the same time, the content of these transformations is not disclosed. Efficiency is also evaluated on the basis of determining the cutting speed, at which the tool wear intensity is minimal. When solving this issue, the choice of technological modes is considered, for example, according to the criterion of optimal cutting temperature [22–25]. The quality of the models depends on the depth of consideration of all the factors affecting the process. The important problem in building a virtual model is to ensure the required motion trajectory of the tool tip relative to the workpiece, taking into account its elastic deformations, as well as its transformation into the geometric topology of the surface formed by cutting. Its solution is based on studies of the dynamic cutting system (DCS), the study of which has been carried out since the mid-50s-60s of the last century [26–28]. The idea of a DCS consisting of subsystems of the tool and the workpiece, which are united by a link formed by cutting [29–32], was formed. This coupling is a model of the forces represented in the state coordinates. The modeling of forces takes into account the regeneration of the trace on the workpiece left by the deformations in the previous revolution [33–36]. A bifurcation analysis of the stability of the cutting process in “trace” machining and a process analysis based on finite element modeling are given [37–40]. The lag of force variations during changes in the cutting area is taken into account [29, 41–46]. Nonlinear dependences of cutting and friction forces on velocities and displacements are taken into account [47–54]. Parametric self-excitation is considered [55–57]. This list does not exhaust the publications on DCS. It considers the stability of trajectories and the formation of various attracting sets of deformations (limit cycles, invariant tori, chaotic attractors, etc.). However, the problem of its transformation into the geometry of the part formed by cutting remains open. The purpose of this paper is to investigate the mechanism of transformation of deformation displacements of the tool into the geometry of the workpiece taking into account vibration disturbances in the dynamics of the turning cutting process in various machining conditions and modes. The paper evaluates the adequacy of the deformations calculated by the simulation model and measured during the real experiment, as well as its transformation into the geometrical topology of the workpiece. The adequacy is determined based on the proximity of the spectra as well as coherence functions. The research allows us to determine the adequacy of part geometry formation by forming motion trajectories (FMT), which are the unity of trajectories of machine actuators and deformation displacements of the tool relative to the workpiece. Research technique Mathematical description of the dynamic system The following transformations in the cutting system should be considered as the basis for building a numerical model. The first is the transformation of the trajectories defined in the CNC system in the form of the control vector 1 2 3 { , , } T U U U = ∈ ℜ(3) U U into the trajectories of the machine actuating elements (TMAE). The TMAE space for a lathe is given by the vector 3 1 2 3 { , , } T L L L = ∈ ℜ( ) L (fig. 1, a). Here L 1(t) and L2(t) are movements of transverse and longitudinal calipers; 0 ( ) ( ) t D d = π Ω ξ ξ ξ ∫ ( ) 3 L t is the movement of the workpiece along the direction L3. The velocities are also set as 3 1 2 3 { , , } T V V V = = ∈ ℜ( ) / ( ) dL dt V t . The transformation ℜ ⇒ ℜ (3) (3) U is not considered in this study. The trajectories L and V are assumed to be set within the bandwidths of the servomotors. Thus, the trajectories L and V describe the ideal contour of the workpiece. Secondly, it is necessary to find out the transformation of the trajectories L and V into forming motion trajectories (FMT) { } ( ) ( ) ( ) Ô Ô Ô (3) 1 2 3 , , T L L L = ∈ ℜ Ô L( ) and { } ( ) ( ) ( ) Ô Ô Ô ) ) (3) 1 2 3 / , , . T V V V = = ∈ ℜ ( ( V dL dt Ô Ô
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