OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 4 where ( ) diag m = m , , [ ] S k h = h , , [ ] S k c = c , , 1, 2,3 = s k are positively defined symmetric matrices of inertial, velocity and elastic coefficients; (3) 1 2 3 { , , } T X X X X = ∈ ℜ X is a vector of tool deformations considered in the moving coordinate system of the trajectories of the machine actuating elements (TMAE); Ô F F Σ = + is a vector-function of forces on the main F and auxiliary Φ flanks formed in coordinates of the DCS state; (3) 1 2 3 { , , } T X F F F = ∈ ℜ F ; (3) 1 2 3 Ô {Ô ,Ô ,Ô } T X = ∈ ℜ . TMAE are represented by displacements (3) 1 2 3 { , , } T L L L L = ∈ ℜ L and velocities (3) 1 2 3 { , , } T L V V V = = ∈ ℜ d / d L t V . In addition, let us introduce the deformation velocities (3) ,1 ,2 ,3 { , , } T X X X X V V V = = ∈ ℜ d / d X V X t . Therefore, (3) L ℜ is the working space of the TMAE and the elastic deformation space (3) X ℜ is movable. It is defined by the trajectories L and V (fig. 1). In the following, we will rely on the motion separation method [56, 57], which allows independent consideration of “slow” motions lying within the servomotor bandwidth. It also includes displacements of the equilibrium point of elastic deformations. In real systems, the frequency range of “slow” motions is limited from above to a frequency not exceeding 10.0 Hz. This is the frequency range in which the tool tip motions are TIEC controlled. “Fast” motions are considered in variations relative to “slow” motions [58]. It lies within the bandwidth of the tool subsystem. This is the range between 10.0 Hz and 2.0 kHz. These motions are not controllable by TIECs, but it is possible to control its properties. Fluctuations lying in this range are considered as the VAE of the cutting process. Let us also consider “superfast” motions lying outside the bandwidth of subsystem (1). Such vibrations are characterized as acoustic emission. The subsystems of “fast” and “superfast” motions are to be considered. In studying the relationship between “fast” motions and wear, the frequency response of DCS are considered. It changes during the development of wear. When studying “superfast” motions, the force emission signal as a random impulse sequence (RIS) of force actions is considered. First, let us consider the subsystem of “fast” motions. The system (1) has a priori specified and unchanging parameters. Therefore, the frequency characteristics of deformations “highlight” the natural frequencies of the tool subsystem. As the frequency of force perturbations in system (1) increases, peaks at natural frequencies and damping at antiresonances are observed. The properties of the subsystem of “fast” motions change if the forces are expressed through the state coordinates as follows [17, 24, 25] { } { } 3 2 (0) (0) (0) (0) 3 1 1 2 / ( , ) ( ) ( ) ( ) , t X p X P t T T dF dt F V V t X k X t T V V d - + = ρ - - - ξ - ξ ξ ∫ (2) where { } 3 3 0 1 exp ( ) X V V ρ = ρ + µ -ς - is a chip pressure 2 [kg/mm ]; 0 ρ is a pressure in the area of low cutting speeds; µ is a dimensionless parameter; ς is a steepness factor 1 [s m ] - ⋅ ; (0) T is a chip formation time constant [s]; p k is a trace regeneration coefficient, dimensionless (0 1) p k 〈 〈〈 . Technological modes determining the CNC program are the following: { } { } 2 1 2 3 (0) 1 1 2 1 2 3 ( ) ( ) ( ) ( ) ; ( ) ( ) ( ) ; ( ) ( ) ( ) , ( ) ( ) , ( ) ( ) , P p P t P X t T T P X X X t t t t X t k X t T S t V V d V t Mod V t V t V t V t V t V t - = - - - = ξ - ξ ξ = - - - ∫ (3) where ( ) P t t , ( ) P S t , ( ) P V t are depth, feed rate and cutting speed; (0) 1 ( ) ( ) ( ) P t t R t L t = - .
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