OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 4 No. 4 2022 where m = [ms], ms = m, s = 1, 2, 3 kgs 2/mm; h = [h s,l], kgs 2/mm, c = [c s,l] kg/mm; s,l = 1,2,3 – are symmetric, positively determined matrices of inertial, velocity, and elastic coeffi cients. The subsystem of the workpiece has stiffness in the direction of its rotation axis by an order of magnitude greater than in other directions. In the plane Y1–Y2 it has full symmetry. Therefore, in this plane any orthogonal coordinate system is the main one. Then the force F1 corresponds to the deformations only in the direction Y1 [48]. The force F [4] can be represented in the form (0) 1 2 3 1 2 3 { , , } { , , } F F F F F . It is convenient to consider the following representation of modes (feed SP(p), depth tP(p) and cutting speed VP(p)): 2 2 2 ( ) ( ) ( ) ( ) ; ( ) t P X Y P t T S t V v v d V t 3 2 1 1 1 ( ) ( ); ( ) / 2 ( ), X Y P D v t v t t t D L X Y (2) where T = (Ω)–1 is turnaround time, sec; D – diameter, m. If X = 0, Y = 0, then (0) P S , (0) P t , and (0) 3= D P V V , are traditional modes. Then the model of relation of the force F(0) with the system coordinates has the equation [46, 47]: 3 2 (0) (0) (0) (0) 3 1 1 2 / 1 exp ( ) ( ) ( ) , t X X P t T T dF dt F V v t X Y V v d (3) where ρ – chip pressure on the leading edge of the tool, kg/mm2; μ – dimensionless coeffi cient; ζ – steepness parameter of forces, sec∙m-1; T(0) – chip time constant, sec. If L, V are set, then systems (1)–(3) allow to determine X, Y, and L(F). If (F) (F) L , then L, V determine CNC program. Otherwise, it is necessary to ensure the asymptotic stability of the resulting path and to choose L, V or available variations of the parameters so that the condition (F) (F) L or (F) (F) 0 L L is satisfi ed. To calculate L, V, at which (F) (F) L is provided, let us use the principle of motions breaking down. First, a set of paths (F) (F) L in “slow” time are defi ned; then in this set we choose those for which the trajectories of “fast” motions are asymptotically stable, and among them those for which the wear intensity is minimal. To determine the desired path of the “slow” motions, consider the values L, V averaged over the period of rotation of the workpiece. For this purpose, (1) and (3) are considered in the slow discrete time T = (Ω) – 1: C(iT) Z(iT) = F(iT) (4) where Z(iT) = {X1(iT), X2(iT), X3(iT), Y1(iT)} T; F(iT) = ρ 0tP (0)S P (0)(iT){χ 1,χ2,χ3} T; (0) (0) 1,1 0 1 2,1 3,1 0 1 (0) (0) 1,2 0 2 2,2 3,2 0 2 (0) (0) 1,3 0 3 2,3 3,3 0 3 (0) (0) ( ) 0 1 0 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 ( ) P P P P P P Y P P c S iT c c S iT c S iT c c S iT iT c S iT c c S iT S iT c S iT Ñ , 0 3 1 exp( ) V In (4) 0 1 i s i iT L (Fig. 1). From (4) 0.5ΔD(iT) = X1(iT) + Y1(iT) is calculated. During processing it is required to ensure the condition ΔD(iT) = const [51–55]. Let us determine ΔD(iT) from (4):
RkJQdWJsaXNoZXIy MTk0ODM1