OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 7 1 5 As shown in Fig. 18, the roughness indicators begin to increase with increasing wear. At the initial stage, the dependence between Ra, Rz on wear is decreasing, which is due to the running-in of the cutting tool. As the tool approaches the point of catastrophic wear, the increase in roughness becomes more pronounced, and the dependence becomes nonlinear. Virtual models for the digital twin The synthesis of virtual models for the digital twin begins with constructing a cutting tool wear curve using a Volterra integral operator of the second kind [20]. Here, it is necessary to consider the actual path traveled by the tool, including the path deviations resulting from vibrational movements of the tool tip. This path is calculated taking into account the processed data received from the vibration monitoring system. That is, the path traveled by the tool will be determined as the sum of the calculated path, denoted as L0, which is determined by the cutting speed and feed rate, and the virtual path traveled by the tool: 2 2 2 v L x y x = + + . (1) The simulation results of the digital twin’s equation systems allow for the calculation of the predicted cutting tool wear. Modeling must account for the fact that the cause of these evolutionary transformations is related to the power and action of cutting forces, specifically the energy of irreversible transformations in the machining zone. Volterra integral operators of the second kind are used to model these evolutionary changes, having the following structure [21‑22]: 0 ( ) ( ) 3 A h k w t N d = − ξ ξ ξ ∫ , (2) where ( ) w t − ξ is the kernel of integral operator; ( ) N ξ is the phase trajectory of the power of irreversible transformations based on perfect work А is the work of cutting forces. The wear depends on the intensity of irreversible transformations and its history, which is captured by the integral operator’s kernel. The kernel of the integral operator is expressed as: ( ) 1 2 ( ) ( ) 3 1 2 0 ( ) A A A h e e N d α ξ− α −ξ = β + β ξ ξ ∫ , (3) where ( ) 1 2 ( ) ( ) 1 2 A A e e α ξ− α −ξ β + β is the sum of the kernels of an integral operator, where 1( ) 1 A eα ξ− β is the kernel that determines the running-in processes of the tool; 2( ) 2 A eα −ξ β is the kernel that determines the processes of wear; 1 β , 2 β , 1 α , 2 α are the parameters to be identified; N is the power of irreversible transformations; A is the performance. The power of irreversible transformations is defined as 2 2 N R Vf V = + , where R is the cutting force. We define the working process in the form of the following integral: 0 ( ) t A N t dt = ∫ . In case N = N0 = const, (2) the following equation is used: ( ) 1 2 ( ) ( 3 0 0 ) A A A h N e e d α ξ− α −ξ = + ξ ∫ . (4) The solution for (4) will be as follows: ( ) ( ) 1 2 3 0 0 1 2 1 1 1 1 A A h N e N e −α α = − + − α α . (5)
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