OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 Let’s consider, in the light of the conducted field experiments, the component of the force that depends on the wear of the tool along the flank, described in the previous section of the paper in the equation (2), which, taking into account the revealed linear dependence, can be conveniently considered as: 0 3 ( ) ( ) h K x F h Q h p F k Q h t y e − = σ + − , (21) where 0 σ – the ultimate strength of the processed metal under compression in [kg/mm2], at the contact temperature along the tool flank h Q and the workpiece at zero degrees. The linearized value of the force along the flank in the vicinity of the equilibrium point will have the following form: 0 3 0 3 3 h h h p h Q p F xK h t y h Q k h t = − σ − σ + . (22) In the description of the cutting force along z coordinate, there is a coefficient of friction described by the equation (5), a linearized version of this coefficient is given below: 0 1 2 (1 ) / 2 (1 ) / 2 t t t f h t f h k k k K Q k K Q = + ∆ − + ∆ + . (23) Taking into account (22 and (23), the linearized value ( ) z h F in the vicinity of the equilibrium point will take the form: 0 3 0 0 3 0 ( ) ( ) z h h p t t t t F xK h t k k y h k k = − σ + ∆ − σ + ∆ + 3 0 2 1 0 3 ( ) ( ) h h Q p t t f f t p Q k h t k k K K k h t + + ∆ + − ∆ σ . (24) The contact temperature of the tool flank and the workpiece is defined as the solution of the following differential equation: 2 1 2 1 2 2 ( ) ( ) h h h d Q dQ T T T T Q kN t T dt dt + + + = − , (25) where ( ) ( ) 3 ( ) ( ) ( ) ( ) z v v v v h dz t T N t T F t T F t T V dt − − = χ − + − − ñ . (26) The equation (26) will take the form: ( ) 3 ( ) ( ) ( ) z v v v h N t T F t T V F t T V − = χ − + − − ñ ñ ( ) 3 ( ) ( ) ( ) ( ) z v v v v h dz t T dz t T F t T F t T dt dt − − −χ − − − . (27) The linearized value of irreversible transformations power in the vicinity of the equilibrium point will take the form: 3 0 0 3 0 3 0 1 0 0 3 0 0 3 0 ( ) (1 ) ( ) ( ) v v v v v v jT V jT jT V jT v p p jT jT h p t t t t dz N t T y e S V x e e t V e t S V dt xK h t k k e V y h k k e V − ω − ω − ω − ω − ω − ω − = − χ ρ − χ − ρ + χ ρ µα − − σ + ∆ − σ + ∆ + c c ñ ñ ñ ñ ñ
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