The Dynamics of Contact Interaction during the Cutting Process

At parts manufacturing using metal-cutting machine tools, the process conditions eliminating high vibration levels are among the most important factors ensuring safe operation of the said metal-cutting machine tools. To solve the task, specific features typical for creation of the dynamic model of mechanical processing in the space of condition variables at contact interaction in the cutting zone based on the piecewise linear approximation were discussed. The contact interaction process was considered as the double-phase one envisaging sequence of retention at adhesion and sliding at adhesion bonds rupture. The cutting tool’ and the workpiece’ contact interactions are presented in the form of rheological models. Conditions of retention and sliding phases sequence are formed by the system itself that is the self-excited one. The set of research performed allowed considering contact interaction of the machined surface with the rear surface of the tool and the of the moving chip with the anterior surface of the tool with the anterior surface of the tool as factors largely defining the conditions of self-sustained oscillations. The contact interaction’ double-phase nature ensures selforganised dosing and selectiveness of the dynamic contours at interaction in the autonomous dynamic process system. KeywordsProcess system, Cutting process dynamic model, Rheological equations, Sliding, retention, Piecewise linear approximation.


Introduction
At parts manufacturing using metal-cutting machine tools, the process conditions eliminating high vibration levels are among the most important factors ensuring safe operation of the said metalcutting machine tools. Vibrations lead to loss of precision, premature failure of the equipment and cutting tool wear. To solve the problem, the set of studies aimed at investigating dynamics of the mechanical treatment' process system (Kudinov, 1967;Elyasberg, 1993;Vasilkov et al., 2004;Maksarov, 2015;Olt et al., 2016;Temraz. 2018;Skeeba and Ivancivsky, 2018;Maksarov and Efimov, 2018). Analysis of processes taking place at contact interaction of chippings with the anterior surface of the tool and of the machined surface with the rear surface of the tool allowed presenting the process as the double-phase one: with the retention and sliding phases (Vasilkov et al., 1997;Maksarov et al., 2017, Vasilkov, 2018. As compared to the conventional approach to the equilibrium conditions at the cutting wedge and the treated material interaction a new approach has been introduced. Specifically, molecular processes occurring in the chip formation zone are presented in the form of rheological models connecting subsystems of the workpiece and the cutting tool (Olt and Maksarov, 2015;Olt et al., 2016b). In the model, the discussed processes are presented as viscoelastic model of Voigt, Maxwell's visco-hereditary model, or as more complex media modified relative to the discussed task. Conditions of interaction of elastic-dissipative and inertial characteristics of the technological system and viscoelastic-plastic characteristics in the chipformation zone are modelled.
Conditions ensuring sequence of the retention and sliding phases at interaction of the anterior surface of the tool and the rear surface of the tool with, respectively, the moving chip and the machined surface are formed by the system itself that potentially is self-excited one (Zhukov et al., 2016;Skvortsova and Nurulin, 2018;Völkers et al., 2018).
The principal idea underlying the discussed approach is piecewise linear approximation of the contact interaction in the technological system (TS) directly within the cutting process. The said idea consists of considering the contact interaction at cutting as sequence of states either of which can be presented by its rheological models complex. At that, sequence functions are formed that determine conditions for transfer from one state to another (Vasilkov et al. 2018).
Differential equations system describing dynamic processes taking place in the TS in the space of condition variables is expressed as follows: where u is condition variables vector; D is transfer matrix with the constant coefficients; S is vectorfunction of the piecewise linear type.
It should be noted that at contact interaction, the processes taking place on the anterior and rear surfaces of the tool are connected. Let us discuss a double-loop TS as an example ( Figure 1). It includes two joint coordinates y, z. Chip-formation zone 1 is considered as the connected dynamic subsystem that, in turn, has two double-loop subsystems: on the side of the the anterior and rear surfaces of the tool (Khoromskij and Repin, 2015; Abushawashi et al., 2017).
Elastic-dissipative characteristics of the TS are defined by the following parameters: cy, cz, by, bzcoefficients of stiffness (factor of rigidity) and dissipation in the direction of y, z axes (Garshin et al., 2017;Mirsaidov et al., 2018).
Description of metal movement in the chip-formation zone using rheological models makes it possible to substitute differential equations in partial derivative with the ordinary differential equations. It clearly simplifies mathematics of modelling dynamic interactions at cutting

Materials and Methods
In accordance with the rheological presentation from the direction of the rear surface of the tool, viscous-elastic interaction with the machined surface occurs. Rheology of the near-surface zone of the part' material from the direction of the rear surface of the tool (RST) is modelled by RYR and RZR ( Figure 2). From the direction of the anterior surface of the tool (AST) viscoelastic-plastic interaction takes place and is modelled by elements RYA and RZA. Variants of subsystem formation from the direction of the tool' anterior surface exist where front angle (positive or negative), slip bands, etc. are considered.

Vector-function ()
Su has the following distinct from zero components: is static deformation of the elastic element with factor of stiffness coefficient у с . The fact that vectorfunction S(u) is piecewise linear function is specific feature of the differential equations system (1).

Discussion of the Results
The discussed dynamic system in the space of condition variables is nonlinear one of the piecewiselinear type. Based on the said dynamic system, it is possible to investigate frequency content and oscillations amplitude level at wide variability of the model' parameters (Dencker et al., 2016;Bulyanitsa et al., 2017).
To assess the movements nature, conditions of the computing experiment shall be formulated. The workpiece specifications: diameter d=90 mm; material -NiCr20TiAl. The tool specifications: rear International Journal of Mathematical, Engineering and Management Sciences Vol. 4, No. 5, 1218-1227, 2019https://dx.doi.org/10.33889/IJMEMS.2019.4.5-096 1223 angle α = 80; anterior angle γ = -80; main angle in plan φ = 700; auxiliary angle in plan φ1 = 200; cutting head material -M101S; toolholder cross-section -40х20; free length of the console tool holder -95 mm. Cutting modes: working feed of the tool -S=0.19 mm/turn; cutting depth -t=1.0 mm; cutting speed -V = 30 … 220 m/min. Treatment without cooling. Relative vibration displacement along the normal to the forming point of the tool in the direction y (upper curve in Figure 4, a) was produced based on calculation according to the model (1). It gives clear understanding of the periodic solution that is formed as a result of sliding and setting phases sequence. According to the diagrams, transfer from the setting phase to the sliding one is accompanied by the displacement pike у (upper curve in Figure 4, a). At that, phase trajectory escape to the limit cycle with 2-5 µm amplitude is observed on the phase-plane portrait (the curve in Figure  4, b).
At the moment of the retention phase transition into the sliding phase, typical displacement of the phase trajectory from the limit cycle with 3-12 µm amplitude followed by returning to the limit cycle. The said characteristic behavior of the dynamic system is observed within the cutting speed

Conclusions
The set of studies performed gives grounds for considering contact interaction of the machined surface with the rear surface of the tool and the moving chip with the anterior surface of the tool as factors largely defining conditions for self-induced vibrations generation. Double-phase nature of the contact interaction process self-organises dosing and selectivity of the dynamic contours at interaction in the autonomous dynamic TS. Pulsating nature of movement is manifested in the vibratory displacements' timing diagrams, at turning in particular, which proves the accepted schematization of the dynamic processes at cutting. Parametrisation of TS' dynamic model is performed based on the traditional model solutions for force interaction at machining transformed into the dynamic characteristics within the range of rheological presentations of adhesive-deformation contact interaction.