OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 4 No. 4 2022 from (1) taking into account (3) the transfer function WP(p) of the system in the open state is obtained for the linearized system in variations: 1 2 (0) 0 (0) (0) (0) ( ) ( ) 1 exp( ) 1 ( ) 1 exp( ) ( ) 1 exp( ) , P P FX FX FY P P W p S g p Tp T p t g p Tp S g p Tp (8) where 1 1 ( ) / FX X g p ; 2 2 ( ) / FX X g p ; ( ) / FY Y g p ; 1,1 2,1 3,1 1,2 2,2 3,2 1,3 2,3 3,3 ( ) 0 0 0 0 0 0 Y c c c c c c c c c c ; 1 1 2,1 3,1 2 2,2 3,2 3 2,3 3,3 ( ) 1 0 0 0 0 0 X Y c c c c c c c ; 2 1,1 1 3,1 1,2 2 3,2 1,3 3 3,3 ( ) 1 0 0 0 0 0 X Y c c c c c c c ; 1,1 2,1 3,1 1 1,2 2,2 3,2 2 1,3 2,3 3,3 3 1 . 0 0 0 c c c c c c c c c The equation (8) includes technological parameters tP (0) and S P (0), and indirectly V P, because T (0) mainly depends on the cutting speed, and spindle speed. Parameters tP (0), S P (0) and Ω are defi ned by the CNC program. Here is an example of changing the stability area in the plane of two varying parameters “VP – ρ” (Fig. 3). Above the fi gurative lines, the system is unstable. The curves are typical when evaluating the effect of spindle speed on the stability of elastic deformational displacements of the tool and workpiece. Periodic bursts of admissible values ρ explained by the regenerative effect of the integral feed forming operator. Analysis of the properties of the system at t < T shows that the cutting speed primarily changes the parameter T(0). Its increase causes two opposite trends. On the one hand, an increase of T(0) leads to additional phase shifts between forces and strains, which contributes to the loss of stability. On the other hand, an increase of T(0) contributes to an increase in the damping effect of the cutting process on oscillations, which stabilizes the equilibrium. Therefore, there is a minimum when VP is increased. Fig. 3. An example of changing the region of stability in a plane “VP – ”: 1 – с(Y) = 400 kg/mm; 2 – с(Y) = 150 kg/mm
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