OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 2 4 The trajectories L(Φ) and V(Φ) take into account the deformations of the tool tip in addition to the TMAE. In the present study, we will focus on the case of machining an absolutely rigid workpiece by longitudinal turning. Then ) ) = - = - (Ô (Ô ( ); ( ), L L X L V V v L (1) where 1 2 3 { , , } T X X X = ∈ ℜ(3) X X , 1 2 3 { , , } T v v v = ∈ ℜ(3) X v are the vectors of deformations X and its velocities = / v dX dt considered in the moving coordinate system given by L (fig. 1, a). Let’s consider turning the shaft const = D with constant cutting modes: 1 { / 2, L d = = L 2 2 3 , } T L V t L d = = π Ω . Thirdly, it is necessary to study the pattern of transformation L(Φ) and V(Φ) into geometrical surface topology (L, R). Geometrical topology is a mapping of shaft surface irregularities into a bounded plane. In the plane, the length of the shaft circumference along the L3 axis, the length of the shaft along the longitudinal feed direction along the L2 axis, and the variations of the radius of the vector R in the form of irregularities along the L1 axis. If the deformations are X = 0, then (L, R) is the plane «L2-L3». Let us also consider a point mapping of the tool vertex constructed in a similar way. Let us call such a mapping a “skeletal” geometric topology C (L, X). Fig. 1, c shows its example for the vector X. The function X1(t) characterizes the height irregularities (fig. 1, с), X2(t) and X3(t) are stepwise. If the dynamical system is stable and unperturbed, the deformations are 1 2 3 { , , } T X X X ∗ ∗ ∗ = ∈ ℜ(3) * X X . This is the equilibrium point. It is shifted relative to L by a constant value X*. If 1 2 2 3 { / 2, , } T L d L V t L d = = = = π Ω L , then C (L, X) represents the plane parallel to «L2-L3», and shifted upward along the axis L1 by the value const = * 1 X . If the geometrical topology (L, R) formed by cutting along a point contour is C (L, X), then a b c Fig. 1. Scheme for the formation of state space coordinates (a), and an example of “skeleton” geometrical topology (c) for variations of deformation displacements with frequency equal to the workpiece rotation frequency (b)
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