OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology where 0t k – some constant minimum value of the friction coefficient, t k ∆ – the value of the increment of the friction coefficient when the temperature changes in the contact zone, 1 f K and 2 f K – coefficients determining the steepness of the fall and growth characteristics of the friction coefficient. Thus, generalizing the description of the force response from the cutting process to the forming motions of the tool, we obtain the following equations describing the force response: ( ) 1 ( ) 2 ( ) 3 , , . x f h y p h z c h F F F F F F F F F = χ + = χ + = χ + (7) In addition to the power and thermodynamic subsystems of the cutting system, in the general structure of the control system (see Figure 1), there is a subsystem of deformation motions of the tool tip, which was indirectly included in our reasoning, but is not directly represented by the model. Taking into account the dependencies of response forces proposed by Equation 6, as well as relying on the approach to modeling the dynamics of the deformation motion of the tool used in the scientific school of V.L. Zakorotny [12], we assume that the model of a tool tip deformations will take the following form: 2 11 12 13 11 12 13 2 2 21 22 23 21 22 23 2 2 31 32 33 31 32 33 2 , , . f p c d x dx dy dz m h h h c x c y c z F dt dt dt dt d y dx dy dz m h h h c x c y c z F dt dt dt dt d z dx dy dz m h h h c x c y c z F dt dt dt dt + + + + + + = + + + + + + = + + + + + + = (8) where [ 2 s / kg mm ⋅ ]; h[ s/ kg mm ⋅ ]; ñ[ / kg mm] – matrices of inertia coefficients, dissipation coefficients and stiffness coefficients, respectively. As a result of the cutting wedge evolution, a contact area is formed along the flank, the length of which determines the interaction time, as well as the interaction time is determined by the cutting speed. The conversion of cutting power to temperature requires a preliminary formalized description of the most instantaneous cutting power, which is conveniently represented by the following expression: ( ) ( ) 3 z ñ ñ ñ h dz dz N F V F F V dt dt = − = χ + − . (9) Using the approach proposed in [8], let’s synthesize a differential equation describing the thermodynamic component of the system as: 2 1 2 1 2 2 ( ) z z z d Q dQ T T T T Q kN dt dt + + + = , (10) where 1 1 T λ = α , 3 2 2 2 h ñ T h T V = = α α , 3 1 2 Q ñ k h k V λ = α α – transmission ratio, α1, α2 – dimensionless scaling parameters, λ – the coefficient of thermal conductivity, Q k – the coefficient characterizing the conversion of irreversible transformations power released in the tool/workpiece contact zone into temperature. Thus, the system of equations (8)–(10) will be the mathematical model of the cutting system. Mikhailov criterion and linearization of the system of equations To assess the stability of the control systembased on the Mikhailov criterion, the characteristic polynomial of the transfer function of the control system is used:
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