OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 4 4 2 The space (3) R is defi ned by the directions of TEE motion, which are set by the CNC system. The shaping motion path is understood as the sum of the paths of the TEE, vector (3) 1 2 3 / , , P L L L L R , and the paths of the deformation displacements of the tool – vector (3) 1 2 3 / , , P X X X X R , and the workpiece – vector (3) 1 2 3 / , , P Y Y Y Y R Consequently (F) – – L L X Y . The paths L and (3) 1 2 3 dL , , dt P V Y Y Y R are defi ned by the NC program. The deformation displacements X and Y are considered in the motion coordinate system given by TEE. If X = 0, Y = 0, then (F) L L . The velocities ( ) 1 2 3 , , P x X X X dX v v v v dt and ( ) 1 2 3 , , P Y Y Y Y dY v v v v dt are also considered. Thirdly, such coordination of the terminal path with the paths of the state space is provided, under which L, X, Y are asymptotically stable; – in this case 0 L F is an attractor. The difference between the synergistic paradigm of NC program synthesis and the traditional one is its defi nition based on the mutual agreement of all subsystems and the provision (F) (F) 0 L L of the property of attraction of the entire state space. In addition, the conditions (F) L , dictated by the quality requirements of the parts, should be met. And the dynamics of the system is taken into account as a whole. Therefore, we rely on research in the fi eld of cutting dynamics [24–43] to develop a synergistic approach to the machining processes control. There is a far from complete list of works on the dynamics of cutting. In spite of many works on the dynamics of cutting, it considers some particular models of the representation of cutting forces in the coordinates of an elastic system. The following ones are analyzed: buckling collapse conditions, formation of different attractors of deformation displacements of the tool and workpiece. It is necessary to analyze the dynamic system as a whole when solving the problem of synergistic synthesis, including determination of the desired path (F) (F) L and corresponding paths L(F), X and Y. Here ℵ(F) – is the set of admissible variations of L(F). The paper discusses all the stages of synergistic control of turning parts, the stiffness parameters of which change along the tool path: the methodology of creation (F) 0 L and its asymptotic stability. An analysis of the effectiveness of synergistic control is presented on a specifi c example of manufacturing a “basic” part, the drawing of which is shown in Fig. 1, b. Research methodology Determining the desired path of the forming motion. When analyzing the system, it is advisable to use the principle of motions breaking down into “slow” ones, lying within the bandwidth of the motors of the machine’s executive elements, and “fast” ones, determined by the dynamic properties of the tool and workpiece subsystems. Its use is based on the asymptotic properties of nonlinear differential equations with small parameters in the derivatives [44, 45]. Moreover, the subsystem of “fast” motions is considered in variations with respect to the paths of “slow” motions. If the subsystems are asymptotically stable, the path of the “slow” motions becomes an attractor. Typical for practice is the case when TEE are given and controllable within the servomotor bandwidths. Then we have the following equation of dynamics [40]: 2 ( ) ( ) ( ) (0) 1 1 1 1 2 ( ; ( ) , Y Y Y d Y dY m h c Y F dt dt 2 2 d X dX m h cX F L, V, X, Y) dt dt L, V, X, Y (1)
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