OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 6 No. 2 2024 System (3), (4) analytically defines the relationship between forces, deformations, vibration disturbances and control parameters of the dynamic cutting system. The ideal case of longitudinal turning of a workpiece is considered, i.e. in the absence of tool deformation, displacements and disturbing vibrations R(t) = 0. Then, the trajectory of formative motions of the tool Φ(t) at each moment of time τi is determined only by the values of the parameters of technological cutting modes and forms a set of segments of the tool path relative to the workpiece for each revolution ( ) 1 i n U i L V iT = S = = ∑ , where this L(U) is the tool path relative to the workpiece at constant values of cutting speed, depth and feed. In this case, the tool tip traces on the part surface form the reference geometrical topology of its surface L(U) ⸦ (0) (fig. 1, a). However, taking into account vibrations and deformations, the tool path will differ from that specified by the machine control program, and the geometrical topology of the workpiece surface will be determined by the characteristics and properties of the cutting process L(Φ) ⸦ (D) (fig. 2). Fig. 2. Example of the tool trajectories of the forming motions L(Φ) taking into account tool vibrations and the corresponding geometrical topology of the workpiece surface (D) According to (2) L(Φ) is a pointwise representation of the tool path relative to the workpiece in each direction of its mobility and is represented as { } ( ) ( ) ( 1 2 ) ( ) 3 , , T L L L L Φ Φ Φ = Φ ∈ (D). The function takes into account the height L1 (Φ) and step L 2 (Φ), L 3 (Φ) irregularities left by the tools during cutting. These are deviations of the geometrical surface of the tool from its reference shape L(U) ⸦ (0). Then (2)–(4) is a basic mathematical model for simulation study of the dynamics of cutting process, estimation of roughness and waviness of the obtained surface. These models allow for a comprehensive study of the dynamics of the cutting process, taking into account various forced oscillations of the tool relative to the workpiece and predicting the output characteristics of the cutting process when changing the technological modes. The basic model is valid in the case of small deviations of the motion trajectory of the machine tool actuators from the equilibrium point in the system. For practitioners, the adequacy of simulation modeling corresponds to workpiece machining in the period of normal tool wear or quasi-constant rate of dimensional wear. Simulation study of the dynamics of the cutting process is carried out in several stages. Firstly, it is necessary to find out the parameters of the vibration characteristics of the machine from the spindle side
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