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 energy transformations 1 2 3 { , , } = T N N N N(t) are considered. The phase trajectory ( À (t) – ) N(t) is the generator of all evolutionary changes in the cutting process [55–58]. Model of interactions A system is defined if the interactions between the subsystems are disclosed. The interactions between the tool and the workpiece through the medium formed by the cutting process are formed by the intersection of the tool body and the workpiece. Parameters, characterizing the intersection, are modes (feed, depth and cutting speed): { ( ), ( ), ( )} T P P P S t t t V t = (t) Τ . It is related to V(t) and X v (t) by the relations 2 ,2 3 ,3 1 ,1 ,4 0 ( ) ( ) ( ) ; ( ) ( ) ( ); ( ) / 2 ( ) ( ) ( ) , − = ξ − ξ ξ     = − = − ξ − ξ − ξ ξ     ∫ ∫ t P X t T t P X P X X S t V v d V t V t v t t t d V v v d (4) where 1 ( ) ñonst T − = W = – time of a turn in [ ] S . We suppose that there is no torsional deformation of the workpiece and the workpiece of constant diameter is being processed, and the system is undisturbed. It fol- lows from (4) that interactions only exist when the tool moves relative to the workpiece. For example, the feed rate ( ) P S t is determined by an integration operator of the total feed rate in the time window determined by the frequency of W . If in (4) there are no deformations and all velocities are constant, we will use the values for the modes: (0) Ð S , (0) Ð t , (0) Ð V . We will rely on the studies [55, 56] to determine the deformations, ( , , ), Σ + + = P P P S t V 2 2 d X dX m h cX F dt dt (5) where , , , [ ], , : , 0, : , , 1, 2, 3 s k s k s k m m m m s k m s k s k = = = = ≠ = ïðè ïðè , 4,4 0 ( ) = m m L в 2 kgs / mm , , [ ], = s k h h ,4 4, 0, 1, 2,3 = = = s s h h s в kgs/mm , , [ ], , 1, 2,3, 4 = = s k c c s k , ,4 4, 0, 1, 2,3 = = = s s c c s in kg/mm are symmetric, positively determined matrices of inertial, velocity, and elastic coefficients. The parameters of the workpiece subsystem ( 4,4 0 ( ) = m m L , 4,4 ( ) h L , 4,4 ( ) c L ) depend on L . The force projections in space (3) ℜ are determined by the coefficients χ i satisfying the conditions of 3 2 1 ( ) 1 = = χ = ∑ i i i . In this case 4 1 = − F F . When machining a part of complex geometry, the coefficients χ i and matrices c change depending on the trajec- tory. The possibility of considering the deformation displacements of the workpiece as a scalar model is justified by the fact that its cross section is circular. Then any orthogonal system of coordinates, normal to the axis of rotation, is the main one. We will take into account the dependence F on the area S , the cutting speed ( ) P V t and take into account the delay of forces in relation to variations of S . Then (0) 1 2 3 1 ( ){ , , , } , = χ χ χ χ T F t F(t) (6) where ( ) { } { } (0) (0) (0) (0) 3 ,3 1 2 ,2 / 1 exp ( ) ( ) ( ) ; −   + = r + µ −ζ − − − ξ − ξ ξ       ∫ t X X P t T T dF dt F V v t t X Y V v d r – pres- sure in 2 [kg/mm ] ; ζ – steepness parameter of forces in [s/mm] ; µ – dimensionless coefficient; (0) T – pa- rameter that determines the delay of forces.