Synergetic approach to improve the efficiency of machining process control on metal-cutting machines

OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. No. 3 2021 1 2,1 3,1 1 0 2 2 2,2 3,2 2 0 2 (0) 0 2 3 2,3 3,3 3 0 2 1 4,4 2 1 0 2 0 0 ( ) χ χ r     χ χ r   = r   χ χ r   χ + χ r     P c c V T c c V T t V T c c V T c l V T 1 X ∆ ; 1,1 1 0 2 2,1 3,1 1 1,2 2 0 2 2,2 3,2 2 (0) 0 2 1,3 3 0 2 2,3 3,3 3 1 0 2 1 0 0 + χ r χ     + χ r χ   = r   + χ r χ   χ r χ   P c V T c c c V T c c t V T c V T c c V T Y ∆ . The speed 2 V is considered averaged over the periods of rotation of the workpiece in determining 2 2 ( ) V L , and its variation of 2 V should not exceed the allowable values based on the requirements for surface roughness. At the second stage , asymptotically stable trajectories are selected from this set 2 2 ( ) V L . For this purpose it is necessary to calculate the equilibrium points 1 2 1 1 { , , , } ∗ ∗ ∗ ∗ ∗ = T X X X Y X and 0 ∗ F , and after substitution of variables ∗ = + X(t) X x(t) and 0 0 ( ) ( ) ∗ = + F t F f t determine the equation in variations, linearize it and investigate by known methods [55–58]. The linearized equation in variations corresponding to (5) and (6) in the neighborhood of equilibrium is given in (11) , + + = 2 2 Md z(t) / dt Hdz(t) / dt Cz(t) 0 (11) where { } 1 2 3 ( ), ( ), ( ), ( ), ( ) = T x t x t x t y t f t z(t) ; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         =         m 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) 1 (0) (0) 1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 exp( ) 0 P P P P h h h h h h h h h h kS t X t X S V V ∗ ∗           =       −   −r − ςµ −ς     H ; { } { } { } 1,1 2,1 3,1 1 1,2 2,2 3,2 2 1,3 2,3 3,3 3 4,4 2 1 (0) (0) (0) 3 3 1 3 0 0 0 0 0 0 ( ) 1 exp( ) 1 exp( ) 0 1 exp( ) 1 P P P ñ ñ ñ ñ ñ ñ ñ ñ ñ ñ L V S V t X V S ∗ −χ     −χ     −χ =     −χ       r + µ −ς r + µ −ς − r + µ −ς     Ñ . The synthesis procedure is illustrated by the example of turning a fuel system nozzle shaft made of steel 45. Length (0) 2 120 mm = L , diameter 20 mm, modes: (0) 2.0 mm ≤ p t , 3 2.0 mm/s = V . The parameters of the matrix of velocity coefficients and the dynamic coupling are given in Table 1 and Table 2, respectively. The generalized masses: 3 2 0.5 10 kg s /mm − = ⋅ ⋅ m , generalized mass 0 m and coefficient 4,4 h change when