OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 6 No. 2 2024 [5–10]. However, tool trajectories are determined not only by technological cutting modes, but also by deformation displacements of the tool relative to the workpiece, as well as vibrations, the source of which are oscillations caused by the regenerative effect [11–15]. Complex techniques for solving the problem of ensuring a given part surface quality and its prediction have been considered in many empirical and analytical studies. Attempts have been made to analyze and predict surface roughness based on regression models and response surface methodology (RSM) [16–18]. A great deal of current research is aimed at predicting surface roughness by means of artificial intelligence systems and simulation models [19–23]. In the works [21–23] Y. Altintas et al. consider the creation of simulation models of the dynamics of the cutting process, which are based on the analytical representation of the mutual influence of cutting parameters on the dynamics of machining and experimental identification of the coefficients of dynamic cutting forces in the resulting patterns. National studies consider a neural network model of the cutting tool kinematics, which allows the calculation of the optimal cutting speed according to the criterion of minimizing the tool wear intensity [24]. The complex simulation of the milling process with the estimation of the trajectory of formative tool motions is presented in [25]. The analytical dependence of the surface roughness on the elastic deformation displacements of the tool relative to the workpiece is presented [26]. In [27], a methodology for constructing the geometrical topology of the workpiece surface is proposed to evaluate the influence of tool deformation displacements on the geometrical profile of the workpiece based on stroboscopic Poincaré mapping. Taking into account the dependence of the dynamic link parameters on vibrations in the mathematical description of the dynamics of the cutting process is a necessary condition for simulating and predicting the output characteristics of part machining [28–31], since as a result of vibrations, a real change in the trajectory of the forming motions of the tool relative to the workpiece is observed. The analysis of the research has shown the relevance and attention of scientists to the issues of creating various methods for evaluating and predicting the surface roughness of a part during machining. This paper proposes to consider the analytical dependence of cutting forces on technological modes, taking into account disturbance influences and tool deformation displacements. The geometrical topology of the workpiece surface is considered as a pointwise representation of the tool tip in the workpiece space, taking into account vibrations and deformation displacements. The aim of the work is to evaluate the influence of the properties of the dynamic characteristics of the cutting process on the geometry of the workpiece surface using simulation study of the dynamics of the cutting process. Research methodology Determination of the formative tool path Let’s consider a dynamic system of longitudinal turning of a non-deformable workpiece. The system model is represented in the form of a spatial finite-dimensional model of the motion of the tool tip interacting with the workpiece. The interaction is described by cutting forces, which are functions of tool deformation displacements X = {X1, X2, X3} T∈(3) and technological cutting modes (fig. 1, a). Therefore, by considering the total cutting force FS(X) in the system model, a feedback between the dynamics of the cutting process and the tool subsystem is formed. This feedback can either stabilize the cutting process or lead to a loss of stability. ( ) (0) (0) (0) P P P X, t ,S ,V + + = S 2 2 d X dX m h cX F dt dt (1) where m, kgs2/mm; h, kg2/mm; c, kg/mm are symmetric, diagonal positive definite inertia, dissipation and stiffness matrices with dimension [3×3]; X = {X1, X2, X3} T∈(3) is the vector of elastic deformation displacements of the tool relative to the machine tool bearing system in space (3), system state coordinates; FΣ(X, tp (0), S p (0), V p (0)) are the total cutting forces acting in space on the cutting tool F Σ = F (0){χ 1, χ2, χ3} T, where χi is orientation coefficient of cutting forces in space, satisfying the normalization conditions; tp (0),
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