OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 4 where ρ0 is a force coefficient, reduced to the cutting blade contact length in [kg/mm]; ς is a parameter depending on the back angle α and tool wear; kT is a coefficient of friction; kΦ is a dimensionless coefficient of elastic recovery. Equations (1)–(3) characterize the tool subsystemwith nonlinear feedback. Since the system is nonlinear, its response depends on frequency and amplitude. Let us first analyze the frequency response at small perturbations. Linearized system reactions. The linearized representation is valid for small perturbations S of the forces and variations of the shear area (Fig. 1) in the vicinity of equilibrium. Then the dynamics of the system perturbed by the forces f(t), can be represented solely as a function of frequency. Moreover, it is convenient to consider the force perturbations as “white” noise. In this case, one can use the Laplace transform methods. For small deformations in the vicinity of equilibrium, the forces acting on the auxiliary flanks can be neglected. Then instead of (2) it is true that [ ] { } d d (0) (0) (0) (0) (0) (0) (0) 1 1 2 2 / ( ) ( ) ( ) p P P P P T F t F t S X k X t T S X t X t T t + = ρ - - - - - - + ε , (7) where ( ) { } { } 3 3 1 3 0 0 ( ) 1 exp ( ) 1 exp[ ] const X f t F t V V d V - - Ω + µ -ς - ξ ρ = Ω ρ ξ ⇒ ρ + µ -ς = ∫ , because 2 ( ) 0 X V t → ; [ ] 1 2 2 ( ) ( ) ( ) 0 1 p X k X t T X t X t T ε = - - - - = , since ε is the product of small quantities. Instead of (1) and (7) in the Laplace images there will be [ ] { } 0 0 (0) (0) (0) (0) 1 2 (0) (0) ( ) ( ) ( ), 1, 2,3; ( ) 1 exp( ) ( ) 1 exp( ) , (1 ) i i F X p P P P P X p W p F p i t S X p S k Tp X p Tp t F T p = = ρ - - - - - - = + (8) where p is a Laplace image symbol; 0, ( ) ( ) / ( ), 1, 2,3 i i F X X W p p p i = ∆ ∆ = ; 2 1,1 1,1 2,1 2,1 3,1 3,1 2 1,2 1,2 2,2 2,2 3,2 3,2 2 1,3 1,3 2,3 2,3 3,3 3,3 ( ) ( ) ( ) ( ) ( , ) ( ) ( , ) ( , ) ( , ) ( ) mp h p c h p c h p c p h p c mp h p c h p c h p c h p c mp h p c + + + + ∆ = + + + + + + + + ; 1 1 2,1 2,1 3,1 3,1 2 2 2,2 2,2 3,2 3,2 2 3 2,3 2,3 3,3 3,3 ( ) ( ) ( ) ( ) ( , ) ( , ) ( ) X h p c h p c p mp h p c h p c h p c mp h p c χ + + ∆ = χ + + + χ + + + ; 2 2 1,1 1,1 1 3,1 3,1 1,2 1,2 2 3,2 3,2 2 1,3 1,3 3 3,3 3,3 ( ) ( ) ( ) ( , ) ( , ) ( , ) ( ) X mp h p c h p c p h p c h p c h p c mp h p c + + χ + ∆ = + χ + + χ + + ; 3 2 1,1 1,1 2,1 2,1 1 2 1,2 1,2 2,2 2,2 2 1,3 1,3 2,3 2,3 3 ( ) ( ) ( ) ( , ) ( ) ( , ) ( , ) X mp h p c h p c p h p c mp h p c h p c h p c + + + χ ∆ = + + + χ + + χ .
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