OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 4 4 2 The path SP (0)(L 2) in Fig. 2, b corresponds to the speed of longitudinal motions of the slide V2L2 = SP (0) (L2)(T) –1. The paths shown in Fig. 2, c characterize the attracting set of deformational displacement along the whole path of the tool motion in the case when the subsystem of “fast” motions is asymptotically stable. The paths in Fig. 2 are obtained by assuming that V3 = 1.5 m/s = VP = const. Varying V3 and tP (0) will lead to a displacement of the path (Fig. 2). Let us consider the problem of asymptotic stability of deformational displacements for the subsystem of “fast” motions. Quasi-permanent paths of deformational displacements Xs (*)(iT),s = 1,2,3; Y(*)(iT); force F(*,0)(iT) and velocities V 2 (*)(iT) correspond to the curves (Fig. 2). X s (*)(iT), Y(*)(iT), and F(*,0)(iT) are slowly changing state coordinates. Equation in variations with respect to the paths of “slow” motions is obtained after replacing of Xs(t) – Xs (*)(iT) = x s(t); Y(t) – Y (*)(iT) = y(t); and F(0)(t) – F(0,*)(iT) = f(t); Its linearization in the vicinity, Xs (*)(iT),s = 1,2,3; Y(*)(iT); and F(*,0)(iT) leads to a system of linear equations with lagged arguments. The analysis of stability of such systems on the basis of algebraic criteria, as well as the Mikhailov criterion, is not fair [47, 48]. The simulation of forces in state coordinates allows us to interpret the forces as feedbacks in the system. Therefore, Nyquist stability criteria are used, for which c Fig. 2. An example of matching the path of changing the radial stiffness of the workpiece (a) with the value of the feed per revolution (b), changing the diameter variations (c): 1 – complete matching along the path; 2 – processing with a constant feed; 3 – linear interpolation of feed change over four nodal points (A-B-C-D) а б
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