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

OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS The model of forces Ф ( t ) is presented in state coordinates. The modulus of these forces depends on the convergence of the back faces to the workpiece, that is, on the back angle ( ) Σ α t (Fig. 1). This angle is defined by the sum of ( ) ( ), Σ α = α + ∆α t t (7) where α is the value of the back angle; its increment 2 ,2 3 ,3 ( ) ( ) àrctg −   ∆α =   −   X X V v t V v t . Similarly, the angle between the trailing edge and the workpiece (1) (1) (1) ( ) ( ) Σ α = α + ∆α t t changes. Since ( ) Σ α t and (1) ( ) Σ α t are small values, it is true for the forces t ( ) Φ . (1) 1 0 2 ,2 1 (0) 2 0 1 2 3 1 2 ( ) exp ( ) ; ( ) ( ) exp ( ); [ ], t X t T p T V v t dt t t X t Y t t k Σ − Σ      Φ = r − α α    ∫             Φ = r − − α α     Φ = Φ + Φ  (8) where 1 α , 2 α is a slope coefficient; 0 r is a parameter, which has the sense of stiffness; T k – coefficient of friction in the contact of the tool face with the workpiece. System (8) augmented by (2–6) allow to study X , F , Ф and the power of irreversible energy transformations. It changes if the parameters of the dynamic coupling and V(t) vary. The dependences also make it possible to calculate the power 2 ( ) N t and work of the forces [ ] 1 ,1 2 2 ,2 1 2 4 ,3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) . X X T X N t t v t t V v k t t V v t = Φ + Φ − + Φ + Φ −         (9) Consideration of power 2 ( ) N t is necessary for predicting back-edge wear, changes in dynamic coupling parameters and evolutionary restructuring of cutting dynamics. In this paper, we limited ourselves to considering the power of irreversible transformations in the interface area of the back face of the tool and the workpiece, because when cutting with carbide tools, the prevailing wear is observed exactly at its back face. Alignment of trajectories with varying stiffness In the paper, the problem of synergistic matching of machine TEEs with an a priori specified law of variation 0 ( ) m L , 4,4 ( ) h L , 4,4 ( ) c L is considered. The problem is solved in three stages. At the first stage , a set of phase trajectories 2 2 ( ) V L is defined at which the diameter deviation 1 2( ) const ∆ = + = D X Y . The function 4,4 2 [ ( )] ñ L t is assumed constant within the impulse response of the system. Then to determine the relation ∆ D and 2 V we can use the system 1 const, Σ ∆ + ∆ + = = ∆ = ∆ X Y 1 X Y (10) where 1,1 1 0 2 2,1 3,1 1 0 2 1,2 2 0 2 2,2 3,2 2 0 2 1,3 3 0 2 2,3 3,3 3 0 2 1 0 2 4,4 2 1 0 2 0 0 ( ) + χ r χ r     + χ r χ r   ∆ =   + χ r χ r   χ r + χ r     c V T c c V T c V T c c V T c V T c c V T V T c L V T ; [ ] { } 0 3 1 exp ( ) r = r + µ −ζ V ;