OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology Simulation results and discussion of the first hypothesis For the convenience of representing the behavior of the system, simulation was carried out in the Matlab/Simulink 2014 package, where a nonlinear system (8), (10) was directly modeled in Simulink, and the characteristic polynomial (16) was calculated by a cycle in Matlab itself, where at every step of the cycle a determinant for a specific frequency value was considered, and the resulting value was deposited on the complex plane, then everything was repeated. In general, the value for ω was calculated from zero to 2,000 Hz in increments of 0.01 Hz. To assess the stability of the cutting control system by the Mikhailov method, the variants of the control system, the variant of a stable and the variant of an unstable (at the boundary of stability) system were considered. The factor affecting the stability of the cutting process was a tool wear along the flank; the second factor is the processing speed factor. Here there is a possibility of checking the A.D. Makarov statement. In total, 29 high-speed cutting modes were considered, in each of which a stable, unstable and at the boundary of stability cutting mode was studied. Let’s consider the set of parameters of the cutting control system, which includes a processing speed of 1,600 rpm and a wear value of 0.22 mm. The results of modeling the coordinates of the system state and the corresponding phase directions are presented in Figure 2. 15 x, mm (10-3) y, mm, (10-3) z, mm t, s a) b) c) 6 4 2 0 0 0.05 0.1 0.15 0.35 0 -5 5 10 0 0.02 0.04 0.05 0 0 0.2 0.25 0.3 -2 t, s 0.01 0.03 t, s -4 0.05 0.35 0.3 0.25 0.2 0.15 0.1 0.35 0.3 0.2 0.1 0.05 0.25 0.05 dx/dt, mm/s d) e) f) 2 -4 0 -4 -2 10 5 0 -2 0 2 -5 5 10 00 0.01 2 z, mm*10-3 y, mm*10-3 dy/dt, mm/s dz/dt, mm/s x, mm *10-3 4 6 15 0 4 -2 0.03 0.04 0.05 0 4 -4 0.02 Fig. 2. For the case of wear h = 0.22: a – deformations along the x coordinate; b – deformations along the y coordinate; c – deformations along the z coordinate; d – phase trajectory along the x coordinate; e – phase trajectory along the y coordinate; f – phase trajectory along the z coordinate As can be seen from Figure 2, the system is stable, in addition, it can be seen from the phase trajectories that there is a constant restructuring of the control system, where the phase trajectory is compressed. However, each cycle of adjustment and subsequent compression is associated with cutting along the “trace” that was formed when the tool was embedded on the first turn. The stability assessment according to the Mikhailov criterion, for the considered variant of the parameters of the cutting control system, is shown in Figure 3. а d e f
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