OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 6 No. 2 2024 it is accurately formed by the trajectories L(Φ). Then observation and/or calculation of L(Φ) allows to accurately predict (L, R). If vector L is defined and its accuracy is ensured by the CNC system, then to determine C(L, X) it is necessary to calculate X. For this purpose we can use the developed mathematical models [22, 23, 45, 46, 54, 55, 58, 61]. Then [ ] { } { } (0) 1 2 3 (0) (0) (0) (0) 3 3 1 2 2 { , , } ; / 1 exp ( ) ( ) ( ) ( ), T t P t T F T dF dt F V v t X V v d - + + = = χ χ χ + = ρ + µ -ζ - - ξ - ξ ξ ∫ 2 2 d X dX m h cX F(L, V, X); dt dt F (2) where m, h, c are symmetric and positively definite matrices, i.e. potential matrices; ρ is a chip pressure on the leading edge of the tool; T(0) is a chip time constant, that takes into account transients in the cutting zone; μ, ζ are parameters determining the dependence of forces on cutting speed; χi, i = 1, 2, 3 is an angular coefficient of cutting force orientation; (0) P t is a depth of cut without elastic deformations; T is a workpiece turnover time, i.e. { } ( ) 3 ( ) 3 3 3 3 ( , ) ( ) ( ) L L D d T v V v Φ Φ -π ξ Ω = ξ - ξ ∫ . (3) System (2) is valid for small deformations in the vicinity of equilibrium, when the forces acting on the auxiliary flanks of the tool can be neglected.At large deviations of coordinates from equilibrium, it is necessary to take into account all nonlinear relationships and to introduce the interactions between the auxiliary flanks of the tool and the workpiece into the forces, as suggested in our earlier studies [22, 23, 45, 46, 54–61]. Adequacy of the “base” model Let us first consider the adequacy of the deformations mapping X in a “base” model in which the forces are perturbed by a “white” noise of low intensity φ(t). The case when X* is asymptotically stable is analyzed here. The point X* corresponds to F(0, *). In view of the smallness of φ(t) it is sufficient to consider the linearized system (2) in variations with respect to equilibrium. For this purpose, let’s make a substitution: X(t)=X*+x(t), F(0) (t)= F(0, *)+f(t). The equation in variations relative to const ∗ = X , (0, ) F const ∗ = , and is obtained. Then the linearized Laplace equation in images is obtained + + = ϕ 2 ( ), mp z hpz cz p (4) where 1 21 3 { ( ), ( ), ( ), ( )} T x p x p x p f p = ( ) z p ; ) {0, 0, 0, ( )} p ϕ = ϕ (p ; p is a Laplace transform symbol; (0) 1 P P t t X ∗ ∗ = - ; 0 0 0 0 0 0 0 0 0 0 0 0 0 m m m = m ; 1,1 2,1 3,1 1,2 2,2 3,2 1,3 2,3 3,3 (0) (0) 0 0 0 0 0 P P h h h h h h h h h t S T ∗ = ρµζ h ; 3 3 1,1 2,1 3,1 1 1,2 2,2 3,2 2 1,3 2,3 3,3 3 (0) 1 1 0 1 V V P P c c c c c c c c c e S e t -ζ -ζ ∗ -χ -χ = -χ ρ + µ ρ + µ c .
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