OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 2 4 Sp (0), V p (0) are the nominal technological cutting modes set by the CNC machine control system (depth of cut, feed rate and cutting speed). Following the works [27, 30], spatial formative tool paths Φ(t) = {Φ1, Φ2, Φ3} T are defined as a set of motions specified by the machine control system U(t), i.e., technological modes tp (0), S p (0), V p (0), and motions R(t) caused by the elastic and deformation properties of the tool subsystem, as well as by perturbing influences, i.e., deviations from the tool trajectory specified by the control: ( )= ( ) ( ) ( ) ( ) ( ) R Φ + = - + ∆ t U t t U t t t X x , (2) where Δx(t) = {Δx1, Δx2, Δx3}, Δxi(t) = 1 sin( ) n k k k t = ω ∑ Á are tool vibrations in the cutting zone; Ak, ωk are the parameters of the frequency components in the tool vibration signal, considered as harmonic functions with constant orientation along the axes X1, X2, X3, with a set of frequency components ωk, Hz, and its corresponding amplitudes Ak, k = 1, 2, 3. Taking into account R(t), equation (1) is transformed into the form (3) where (0) 1 1 ( ) ( ) sin( ) n p p i i i t t t X t t = = - + ω ∑ Α , mm; { } (0) 2 2 1 ( ) cos ( ) / t n p i i i i t T S t t t V dX dt dt = - = ω ω + - ∑ ∫ Α , mm/rev; 1 ( ) T - = Ω is a part turnover time, s; (0) 3 3 1 ( ) / cos ( ) n ð i i i i V t V dX dt t t = = - + ω ⋅ ω ∑ Α , m/s. Equation (3) presents a model of a perturbed dynamic system. Technological modes depend on tool deformation displacements and forced oscillations. Cutting forces, which are the link between the tool and workpiece subsystems, are represented as { } (0) (0) (0) / 1 exp [ ( )] ( ) ( ), ð P p T dF dt F V t t t S t + = ρ + µ -ς ⋅ (4) where ρ is the chip pressure on the tool cutting face, kg/mm2; μ is a dimensionless parameter; ς is the slope coefficient of non-linear change of cutting forces with speed Vp(t), s∙m-1; T (0) is the chip formation time constant, which takes into account transients in the cutting zone, s. a b Fig. 1. Orientation of the deformation axes and cutting forces: a – diagram of deformation axes in the plane X1-X2; b – diagram of force axes and deviations of tool path in the plane X1-X3 ( , , , ) P P P X t S V Σ + + = 2 2 d X dX m h cX F dt dt ,
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