OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology informative, as the loss of stability is displayed at the beginning of the hodograph; from these considerations, in the future, the second part (the end of the hodograph) will not be given. Let’s consider the analysis of the cutting control system stability at 0.36 mmwear, the results of modeling the Mikhailov vector hodograph are shown in Figure 12. As shown by Figure 12, the beginning of the Mikhailov vector hodograph is still far from the origin of coordinates, but here the regenerative effect is clearly manifested, which in the linearized system of equations is described by the delay operator v jT e ω − . The influence of this operator, in the initial section, becomes more significant with the increasing cutting speed, which leads to the increase in the fluctuations of the Mikhailov vector hodograph in the initial segment of the characteristic. As a result of such a change in the characteristics of the Mikhailov vector hodograph, the loss of stability may be associated with the entry of the Mikhailov vector hodograph into the second quadrant and subsequent return to the first quadrant. The point, where the hodograph is closest to the second quadrant (the convergence point on the graph) will determine the stability margin of the cutting system. As is clear from Figure 12, the hodograph of the Mikhailov vector leaves the first quadrant by cutting due to the influence of the delay operator on it. Let’s consider this point more closely in Figure 12 on the right, from where it can be seen that the mechanism of reflection of the stability loss in the cutting system on the Mikhailov vector hodograph is associated with the intersection of the imaginary axis by the hodograph in the direction of the second quadrant of the complex plane. Such a change in the behavior of the hodograph is associated with the increase in the effect of the cutting system self-excitation, which in the English-language scientific literature, is commonly called the regenerative effect. For the further analysis of the cutting control system, we will form into one table all the data obtained in previous parts of work on the upper limit of the stability of the cutting control system according to the Mikhailov criterion (see Table 3). Table 3 shows that the maximum of the stability area of the cutting control system, in the space of the parameters of the cutting speed and the amount of the cutting wedge wear, is observed at a cutting speed of 1,620 rev/min. At this point, the amount of wear, allowed with regard to ensuring the stability of the machining process, was 0.47 mm, which is significantly higher than the average value for the sample, which was h ≈ 0.39 mm. a b Fig. 11. The hodograph of the Mikhailov vector, a system with h = 0.24: a – the beginning of the Mikhailov vector; b – the end of the Mikhailov vector Re Jm Re Jm
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