Vol. 25 No. 1 2023 3 EDITORIAL COUNCIL EDITORIAL BOARD EDITOR-IN-CHIEF: Anatoliy A. Bataev, D.Sc. (Engineering), Professor, Rector, Novosibirsk State Technical University, Novosibirsk, Russian Federation DEPUTIES EDITOR-IN-CHIEF: Vladimir V. Ivancivsky, D.Sc. (Engineering), Associate Professor, Department of Industrial Machinery Design, Novosibirsk State Technical University, Novosibirsk, Russian Federation Vadim Y. Skeeba, Ph.D. (Engineering), Associate Professor, Department of Industrial Machinery Design, Novosibirsk State Technical University, Novosibirsk, Russian Federation Editor of the English translation: Elena A. Lozhkina, Ph.D. (Engineering), Department of Material Science in Mechanical Engineering, Novosibirsk State Technical University, Novosibirsk, Russian Federation The journal is issued since 1999 Publication frequency – 4 numbers a year Data on the journal are published in «Ulrich's Periodical Directory» Journal “Obrabotka Metallov” (“Metal Working and Material Science”) has been Indexed in Clarivate Analytics Services. Novosibirsk State Technical University, Prospekt K. Marksa, 20, Novosibirsk, 630073, Russia Tel.: +7 (383) 346-17-75 http://journals.nstu.ru/obrabotka_metallov E-mail: metal_working@mail.ru; metal_working@corp.nstu.ru Journal “Obrabotka Metallov – Metal Working and Material Science” is indexed in the world's largest abstracting bibliographic and scientometric databases Web of Science and Scopus. Journal “Obrabotka Metallov” (“Metal Working & Material Science”) has entered into an electronic licensing relationship with EBSCO Publishing, the world's leading aggregator of full text journals, magazines and eBooks. The full text of JOURNAL can be found in the EBSCOhost™ databases. WEB OF SCIENCE
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 4 EDITORIAL COUNCIL EDITORIAL COUNCIL CHAIRMAN: Nikolai V. Pustovoy, D.Sc. (Engineering), Professor, President, Novosibirsk State Technical University, Novosibirsk, Russian Federation MEMBERS: The Federative Republic of Brazil: Alberto Moreira Jorge Junior, Dr.-Ing., Full Professor; Federal University of São Carlos, São Carlos The Federal Republic of Germany: Moniko Greif, Dr.-Ing., Professor, Hochschule RheinMain University of Applied Sciences, Russelsheim Florian Nürnberger, Dr.-Ing., Chief Engineer and Head of the Department “Technology of Materials”, Leibniz Universität Hannover, Garbsen; Thomas Hassel, Dr.-Ing., Head of Underwater Technology Center Hanover, Leibniz Universität Hannover, Garbsen The Spain: Andrey L. Chuvilin, Ph.D. (Physics and Mathematics), Ikerbasque Research Professor, Head of Electron Microscopy Laboratory “CIC nanoGUNE”, San Sebastian The Republic of Belarus: Fyodor I. Panteleenko, D.Sc. (Engineering), Professor, First Vice-Rector, Corresponding Member of National Academy of Sciences of Belarus, Belarusian National Technical University, Minsk The Ukraine: Sergiy V. Kovalevskyy, D.Sc. (Engineering), Professor, Vice Rector for Research and Academic Affairs, Donbass State Engineering Academy, Kramatorsk The Russian Federation: Vladimir G. Atapin, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Victor P. Balkov, Deputy general director, Research and Development Tooling Institute “VNIIINSTRUMENT”, Moscow; Vladimir A. Bataev, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Vladimir G. Burov, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Aleksandr N. Gerasenko, Director, Scientifi c and Production company “Mashservispribor”, Novosibirsk; Aleksandr N. Korotkov, D.Sc. (Engineering), Professor, Kuzbass State Technical University, Kemerovo; Evgeniy A. Kudryashov, D.Sc. (Engineering), Professor, Southwest State University, Kursk; Dmitry V. Lobanov, D.Sc. (Engineering), Associate Professor, I.N. Ulianov Chuvash State University, Cheboksary; Aleksey V. Makarov, D.Sc. (Engineering), Corresponding Member of RAS, Head of division, Head of laboratory (Laboratory of Mechanical Properties) M.N. Miheev Institute of Metal Physics, Russian Academy of Sciences (Ural Branch), Yekaterinburg; Aleksandr G. Ovcharenko, D.Sc. (Engineering), Professor, Biysk Technological Institute, Biysk; Yuriy N. Saraev, D.Sc. (Engineering), Professor, Institute of Strength Physics and Materials Science, Russian Academy of Sciences (Siberian Branch), Tomsk; Alexander S. Yanyushkin, D.Sc. (Engineering), Professor, I.N. Ulianov Chuvash State University, Cheboksary
Vol. 25 No. 1 2023 5 CONTENTS OBRABOTKAMETALLOV TECHNOLOGY Ryaboshuk S.V., Kovalev P.V. Analysis of the reasons for the formation of defects in the 12-Cr18-Ni10-Ti steel billets and development of recommendations for its elimination............................................................... 6 Lapshin V.P., Moiseev D.V. Determination of the optimal metal processing mode when analyzing the dynamics of cutting control systems................................................................................................................... 16 Gimadeev M.R., Li A.A., Berkun V.O., Stelmakov V.A. Experimental study of the dynamics of the machining process by ball-end mills.................................................................................................................. 44 Bratan S.M., Chasovitina A.S. Simulation of the relationship between input factors and output indicators of the internal grinding process, considering the mutual vibrations of the tool and the workpiece................... 57 EQUIPMENT. INSTRUMENTS Podgornyj Yu.I., KirillovA.V., Skeeba V.Yu., Martynova T.G., Lobanov D.V., Martyushev N.V. Synthesis of the drive mechanism of the continuous production machine......................................................................... 71 Lobanov D.V., Rafanova O.S. Methodology for criteria analysis of multivariant system................................ 85 MATERIAL SCIENCE Sokolov A.G., Bobylyov E.E., Popov R.A. Diffusion coatings formation features, obtained by complex chemical-thermal treatment on the structural steels............................................................................................ 98 Filippov A.V., Khoroshko E.S., Shamarin N.N., Kolubaev E.A., Tarasov S.Yu. Study of the properties of silicon bronze-based alloys printed using electron beam additive manufacturing technology................... 110 Lysykh S.A., Kornopoltsev V.N., Mishigdorzhiyn U.L., Kharaev Yu.P., Tikhonov A.G., Ivancivsky V.V., Vakhrushev N.V. The effect of borocoppering duration on the composition, microstructure and microhardness of the surface of carbon and alloy steels............................................................................................................. 131 EDITORIALMATERIALS 149 FOUNDERS MATERIALS 159 CONTENTS
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 TECHNOLOGY Determination of the optimal metal processing mode when analyzing the dynamics of cutting control systems Victor Lapshin a, *, Denis Moiseev b Don State Technical University, 1 Gagarin square, Rostov-on-Don, 344000, Russian Federation a https://orcid.org/0000-0002-5114-0316, lapshin1917@yandex.ru; b https://orcid.org/0000-0002-7186-7758, denisey2003@mail.ru Obrabotka metallov - Metal Working and Material Science Journal homepage: http://journals.nstu.ru/obrabotka_metallov Obrabotka metallov (tekhnologiya, oborudovanie, instrumenty) = Metal Working and Material Science. 2023 vol. 25 no. 1 pp. 16–43 ISSN: 1994-6309 (print) / 2541-819X (online) DOI: 10.17212/1994-6309-2023-25.1-16-43 ART I CLE I NFO Article history: Received: 15 December 2022 Revised: 12 January 2023 Accepted: 21 January 2023 Available online: 15 March 2023 Keywords: Dynamics Tool vibration Cutting stability Processing speed Wear and tear Regenerative effect Acknowledgements Research were partially conducted at core facility “Structure, mechanical and physical properties of materials”. ABSTRACT Introduction. In numerous experimental studies of metal cutting processes on metal-cutting equipment, the existence of some optimal processing mode is noted, which was most vividly formulated by A.D. Makarov in his point on the existence of an optimal cutting temperature (processing speed). Here, by the authors from Russia, the emphasis is on the description of the optimality of cutting processes related to the properties of the processed material and the properties of the tool used in this process. However, there is another opinion in the Western scientific literature, which is generally based on the regenerative nature of vibrations in cutting dynamics. Vibration regeneration is associated with the dynamics of the cutting process, which is significantly affected by a lagging argument reflecting the variability of the cut layer. The connection of these two approaches is seen through the analysis of the stability domain of the dynamic cutting system in the parameter space: cutting speeds and tool wear values. Subject. Based on this, the paper considers the question of the relationship between the optimal according to A.D. Makarov the processing mode and the dynamics of the cutting process, including the regeneration of tool vibrations during metal turning. To do this, two research hypotheses are formulated and numerical modeling is performed in order to determine its reliability. Purpose of the work: to consider the position of A.D. Makarov on the existence of an optimal cutting mode, from the point of view of the stability of the dynamics of metal turning. For this purpose, two hypotheses are put forward in the work to be analyzed. The paper investigates: a mathematical model describing the dynamics of vibration oscillations of the cutting wedge tip, taking into account the dynamics of the temperature formed in the contact zone and its influence on the forces that prevent the forming motions of the tool. Research methods: a series of field experiments was carried out on a metalworking equipment using the capabilities of the measuring stand STD.201-1, the purpose of which was to determine the effect of the thermal expansion of metals on the value of the buoyant force. Based on numerical simulation of the initial nonlinear mathematical models, as well as simulation of models linearized in the vicinity of the equilibrium point, an analysis of the stability of the cutting system with variations in the cutting speed and the amount of tool wear along the flank is conducted. The results of the work. The results of field experiments are presented, which showed a significant linear increase in the force pushing out the tool with an increase in temperature in the contact zone of the tool and the workpiece. The results of simulation of the state and the corresponding phase trajectories when the cutting wedge is embedded in the workpiece, as well as the forces decomposed along the axis of deformation of the tool, are presented. The results of modeling the Mikhailov vector hodograph for a linearized model of the dynamics of the cutting process are presented. Conclusions: The research results have shown that only the second hypothesis put forward by the authors makes it possible to adequately interpret the point put forward by A.D. Makarov. The main addition to the description of the point of A.D. Makarov, the authors consider it necessary to take into account changes in the pushing force with an increase in the temperature of the contact zone of the tool and the workpiece. For citation: Lapshin V.P., Moiseev D.V. Determination of the optimal metal processing mode when analyzing the dynamics of cutting control systems. Obrabotka metallov (tekhnologiya, oborudovanie, instrumenty) = Metal Working and Material Science, 2023, vol. 25, no. 1, pp. 16–43. DOI: 10.17212/1994-6309-2023-25.1-16-43. (In Russian). ______ * Corresponding author Lapshin Viktor P., Ph.D. (Engineering), Associate Professor Don State Technical University, 1 Gagarin square 344000, Rostov-on-don, Russian Federation Tel.: 8 (900) 122-75-14, e-mail: lapshin1917@yandex.ru
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 Introduction The quality of machining, as well as the cutter power, is associated with the stability of the vibration dynamics of the cutting process. Vibrations of the tool tip during cutting depend on two groups of factors, the first group is vibration activity unrelated to processes occurring in the cutting zone, and the second group is vibration activity caused by processes occurring in the cutting zone. In this regard, the development of technologies and methods for minimizing vibration activity, the nature of which is determined by the cutting process, is of great importance. The vibrations of the tool accompanying the cutting process are largely determined by the regeneration of vibrations when cutting along the “trace”, which is called the regenerative effect [1–4]. In these papers, it is noted that the main factor influencing the regenerative effect is the so-called “time delay” [5–7], it determines the stability of the process dynamics. In addition to the regenerative nature of the self-excitation vibration dynamics of the cutting system, the stability of the cutting tool vibrations is affected by: the temperature in the contact zone of the tool and the workpiece [8]; changes in the force response from the cutting process to the forming motions of the tool [9]; the value characterizing the degree of the cutting wedge wear, etc. [10]. However, the developing synergetic concept describing the processes occurring in machines [11–12] allows us to speak not about every individual factor, but about a certain set of interrelated and interacting factors that determine the mechanism of self-excitation of the cutting control system [13–14]. The most important factor in the complexity of the mathematical description of cutting processes dynamics is, already mentioned earlier, the “time delay”, which determines the regenerative nature of the self-excitation of the cutting system. It should be noted that in the process of linearization of a system of integro-differential equations describing the complex, nonlinear, delayed dynamics of the cutting process, one will have to deal with an element containing a lagging argument. Such an element will not allow an analysis of cutting control system differential equations using a linearized model in the vicinity of the equilibrium point based on algebraic criteria, such as Hurwitz criterion [15], or Raus criterion [16]. The solution to this problem is the use of frequency stability criteria, such as the Nyquist criterion [17-18], or its Soviet counterpart, the Mikhailov criterion [19-21]. The Nyquist criterion itself, applied to mathematical models of metal cutting control systems, is well considered in research papers of V.L. Zakorotny [11, 12], but Mikhailov’s criterion, well-known and long-known in the American engineering school [21], is not widely used yet. The purpose of such modeling is to determine some best cutting mode; such a mode, in which selfexcitation factor of the cutting control system will be minimized. It has already been experimentally proved that such a mode exists, and it is related to the cutting speed [22, 23]. In these research papers, the best mode is understood as the mode that provides the minimum roughness of the processed surface and the maximum dimensional stability of the cutting tool. For example, A.D. Makarov, in his monograph [22], formulates the following statement: “the most important factor determining the characteristics of the cutting process is the average contact temperature determined by the cutting mode (cutting speed).” In this and other papers, the tool contact temperature is determined by the current power released during cutting and converted into heat, which linearly depends on the cutting speed. However, in the paper [8] it was shown that when the wear of the cutting wedge along the flank is formed, an additional thermodynamic feedback is formed, which prewarms the cutting zone for the period up to the current moment of cutting. In the future, this will lead to a thermal expansion of the workpiece material, which will increase the value of the force pushing the tool. It should be noted that this factor, the restructuring of the force reaction, which is confirmed by experimental studies [9], was not previously taken into account when forming mathematical models of cutting systems. A.D. Makarov himself, in his reasoning, relied on the following well-known factors: 1) the falling nature of the cutting force (temperature-speed factor of processing), identified and presented as a graphical characteristic in the works of N.N. Zorev [24]; 2) the existence of “favorable conditions” in the cutting zone, interpreted by the transition from the adhesive nature to the diffusion nature of friction [22].
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology In addition to these processing factors, one more important temperature factor can be taken into account – this is the linear expansion of metal with the increase in the contact temperature. It should be noted that along with the thermal expansion of metal, there is the increase in the pushing force, which will lead to restructuring of the entire cutting system, and to a possible loss of stability [9]. With regard to simulation the dynamics of the cutting process, the maximum resistance of the cutting wedge coincides with the maximum stability margin of the dynamic cutting system. To paraphrase the statement of A.D. Makarov, it can be said that in a dynamic cutting system there is some optimal, by the margin of this system stability, cutting mode, directly related to the speed (temperature) of cutting and the amount of the cutting wedge wear. In connection with all mentioned above, the purpose of the paper is to consider the statement of A.D. Makarov about the existence of an optimal cutting mode, from the point of view of dynamics stability of metal turning. The above given reasoning allows formulating two hypotheses; the verification of compliance with the position of A.D. Makarov will be the objectives of the study: The first hypothesis: the optimal value of the cutting speed (cutting temperature), when simulating the dynamics of the cutting process, is determined by a combination of the following factors: the incident characteristic of the cutting force (according to N.N. Zorev) and the minimum coefficient of friction associated with the transition of friction from adhesive to diffusion nature. The second hypothesis: the optimal value of the cutting speed (cutting temperature), when simulating the dynamics of the cutting process, is determined by a combination of the following factors: the incident characteristic of the cutting force (according to N.N. Zorev), the minimum coefficient of friction due to the transition of friction from adhesive to diffusion nature and the dependence of the force pushing the tool on the preheating of the processing zone. Testing the first hypothesis Formulation of a mathematical model of the cutting control system When formulating a mathematical model, the cumulative force reaction of the cutting process to the forming motions of the tool, relying on the synergetic concept and the mechanical-thermodynamic approach is considered. The implementation of the synergetic concept in the synthesis of a mathematical model consists in the fact that the forces are described in the coordinates of the process state, and the mechanicalthermodynamic approach consists in the fact that in addition to the coordinates of the cutting system state, the force response includes the dynamics of production and temperature dispersion during processing. For the convenience of formulating a mathematical model, let’s consider the main axes of the deformation coordinates, along which the equations of motion will be written (see Figure 1). Fig. 1. Diagram of the coordinates and forces of the model
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 In the diagram shown in Figure 1, deformations are decomposed into three main axes: x-axis is the axial direction of deformations (mm), y-axis is the radial direction of deformations (mm) and z-axis is the tangential direction of deformations (mm). Along the same axes, the force response decomposes from the cutting process into the forming motions of the tool (Ff , Fp, Fc (N)), Vf and Vc (mm/s) feed and cutting speeds, respectively, ω is the angular velocity of the spindle (rev/s). The description of the cutting force is generalized, based on the point of proportionality of its area of the cut layer, in the form: i p i F a S = ρ χ , (1) where i χ – a certain expansion coefficient of the general vector of response forces on the i-axis of the tool deformation; it should be noted that this approach is widely used within the scientific school of V.L. Zakorotny [12], the depth of processing also depends on the deformations of the tool and the workpiece 0 p p a a y = − , where 0 p t technologically specified processing depth without taking into account deformations of the tool and the workpiece, the amount of feed per revolution – S. The feed value can be represented as the following integral: V t f t T dx S V dt dt − = − ∫ , (2) where TV – revolution period of the workpiece. The most important component of the cutting force is the force component, which is formed not in the zone of primary deformation and friction of the cuttings on the tool face, but on the tool flank, where the pushing force and friction force are formed in the direction of the primary motion. This component of the force depends on the wear of the tool along the flank, therefore, based on the approach proposed in the work of V.L. Zakorotny [22], we describe the force formed here as: h K x h h F S e − = σ , (3) where σ – compressive strength of the processed metal in (kg/mm2); h K – the coefficient of the increase steepness in force, h S – the contact area of the tool and the workpiece along the flank of the cutting wedge, which is defined as: 3 h p S h t = , h K – the coefficient determining the steepness of the nonlinear increase in the contact area of the tool and the workpiece when the tool and the workpiece approach. Through the side cutting edge angle – φ, we decompose the force reaction on the x and y axes of the formation, as follows: ( ) ( ) cos sin x h h y h h F F F F = ϕ = ϕ . (4) The force response in the direction of z coordinate is, in essence, nothing more than the friction force, which can be represented as: ( ) z h t h F k F = , (5) where t k – coefficient of friction. Using the approach proposed by A.D. Makarov [22], we identify the characteristic of friction coefficient by the following expression: 1 2 0 / 2 f f K Q K Q t t t k k k e e − = + ∆ + , (6)
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:
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 D (p) = a0p n + a 1p n–1 + …+ a n-1p + an (11) where n – the degree of polynomial and it is also the order of the differential equation, for the case represented by the expression (8), (10), n = 8 . Assuming p = jω , we transform the characteristic polynomial into a complex frequency polynomial: D (jω) = a0(jω) n + a 1(jω) n-1 +…+ a n-1(jω) + an . In case of stable systems, the hodograph of the Mikhailov vector has the property of starting from the point U(0) = an, V(0) = 0. As ω increases from zero to infinity, the point M(U,V) moves to the left so that the curve tends to cover the origin, while moving away from it. If we draw the radius vector from the origin to the point M(U,V), it turns out that the radius vector rotates counterclockwise, continuously increasing. The Mikhailov criterion itself is formulated as follows: when the frequency changes from zero to infinity, the Mikhailov hodograph begins on the real axis at point an, sequentially passes counterclockwise n quadrants of the complex plane without passing through zero, and goes to infinity in the nth quadrant, the system is stable. In case of unstable systems, the curves do not cover the origin, while if the hodograph starts from the origin or passes through the origin, the system is on the stability boundary. To assess the stability of the control system by the Mikhailov method, it is necessary to determine the characteristic polynomial of the control system, described by the system of equations (8), (10). As this system is nonlinear, the first thing that is required is to linearize this system of equations in some vicinity of the equilibrium point, which is done below. 2 11 12 13 11 1 3 2 2 12 1 3 13 1 1 2 21 22 23 21 2 3 2 2 (1 )( ) cos( ) ( ) cos( ) 0, [ (1 )( ) sin( ) ] [ v v jT p p p p jT p d x dx dy dz m h h h x c e h t dt dt dt dt y c S h c z Q t S d y dx dy dz m h h h x c e h t dt dt dt dt y c − ω − ω + + + + + χ − ρ + ρµ + ϕ σ α + + + χ ρ + ρµ + ϕ σ + + χ ρµα = + + + + + χ − ρ + ρµ + φ σ α + + 22 2 3 23 2 1 2 31 32 33 31 3 0 3 2 2 32 3 0 3 33 3 1 1 2 3 ( ) sin( ) ] 0, (1 )( ) ( ) ( ) ( ) ( ) 0, 2 ( v p p p jT t t p t p t t p p f f p S h c z Q t S d z dx dy dz m h h h x c e k k h t dt dt dt dt k y c S k k h c z Q t S t h T − ω + χ ρ + ρµ + φ σ + + χ ρµα = + + + + + χ − ρ + ρµ + + ∆ σ α + ∆ + + χ ρ + ρµ + + ∆ σ + + χ ρµα + α − α σ = 2 1 2 1 2 0 3 0 0 3 2 1 3 1 0 3 3 0 0 3 2 0 3 1 0 3 2 ) ( ) ( ) ( ) ( ) ( ) (1 )( ) ( ) (1 )( ) ( ) v v v p p t t p jT p p t t p jT c t t p c jT c t t d Q dQ dz T T T k t S k k k h t dt dt dt dz k t S k k k h t e dt x e V k k k h t V k x e V k k k h t − ω − ω − ω + + + χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ + + χ − ρ + ρµ + + ∆ σ α + + χ − ρ + ρµ + + ∆ σ α 1 3 0 3 3 1 0 1 3 3 1 1 2 3 3 1 1 1 2 3 1 ( ) ( ) ( ) ( ) 1 ( ) 2 ( ) 0, 2 v v v jT p c jT p c t t c p c t t c t p p c f f p c jT t p p c f f p c V k e y S kV k k kV h y S k V k k k V h e k Q t S kV t h kV k Qe t S k V t h k V − ω − ω − ω + + χ ρ + ρµ + + ∆ σ + χ ρ + ρµ + + ∆ σ + ∆ + + χ ρµα + α − α σ + ∆ + χ ρµα + α − α σ = (12)
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology The system (12) v jT e− ω is like a lagging argument, where v T is the revolution period of the workpiece. For the subsequent analysis of the control system, let’s move on to the operator form of the system (12), it means, let’s implement the Laplace transform, assuming that the initial conditions are zero (p = d/dt), the following formulas are obtained. 2 11 12 13 11 1 3 2 12 1 3 13 1 1 2 21 22 23 21 2 ( ) ( ) ( ) ( ) ( ) (1 )( ) cos( ) ( )[ ( ) cos( ) ] ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) (1 )( ) v v jT p p p p jT mp x p h px p h py p h pz p x p c e h t y p c S h c z p Q p t S mp y p h px p h py p h pz p x p c e − ω − ω + + + + + + χ − ρ + ρµ + ϕ σ α + + + χ ρ + ρµ + ϕ σ + + χ ρµα = + + + + + + χ − ρ + ρµ 3 2 22 2 3 23 2 1 2 31 32 33 31 3 0 3 2 32 3 0 3 sin( ) ( ) ( ) sin( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) (1 )( ) ( ) ( ) ( ) ( ) v p p p p jT t t p p t t h t y p c S h c z p Q p t S mp z p h px p h py p h pz p x p c e k k h t y p c S k k h c − ω + ϕ σ α + + + χ ρ + ρµ + ϕ σ + + χ ρµα = + + + + + + χ − ρ + ρµ + + ∆ σ α + + + χ ρ + ρµ + + ∆ σ + 33 3 1 1 2 3 2 1 2 1 2 0 3 0 0 3 1 3 1 0 3 3 0 0 ( ) ( ) ( ) 0, 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1 )( ) ( v jTv t p p f f p p p t t p jT p p t t p c t z p k Q p t S t h T T p Q p T T pQ p pz p k t S k k k h t pz p k t S k k k h t e x p e V k k k − ω − ω + ∆ χ ρµα + + α − α σ = + + + χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ + + χ − ρ + ρµ + + ∆ 3 2 0 3 1 0 3 2 1 3 0 3 3 1 0 1 3 3 1 1 2 ) ( ) (1 )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 2 v v v t p c jT jT c t t p c p c t t c jT p c t t c t p p c f f h t V k x p e V k k k h t V k e y p S kV k k kV h y p S k V k k k V h e k Q p t S kV t − ω − ω − ω σ α + + χ − ρ + ρµ + + ∆ σ α + + χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ + ∆ + + χ ρµα + α − α 3 3 1 1 1 2 3 1 ( ) ( ) 0. 2 v p c jT t p p c f f p c h kV k Q p e t S k V t h k V − ω σ + ∆ + χ ρµα + α − α σ = (13) It is convenient to consider the system (13) in matrix-vector form: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) ( ) 0, ( ) ( ) a a a a x p a a a a y p a a a a z p a a a a Q p = (14)
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 where the coefficients have the following values: 2 11 11 11 1 3 2 12 12 12 1 3 13 13 13 14 1 1 21 21 21 2 3 2 2 22 22 22 2 3 23 (1 )( ) cos( ) , ( ) cos( ) , , , (1 )( ) sin( ) , ( ) sin( ) , v v jT p p p p jT p p a mp h p c e h t a h p c S h a h p c a t S a h p c e h t a mp h p c S h a − ω − ω = + + + χ − ρ + ρµ + ϕ σ α = + + χ ρ + ρµ + ϕ σ = + = χ ρµα = + + χ − ρ + ρµ + ϕ σ α = + + + χ ρ + ρµ + ϕ σ = 23 23 24 2 1 31 31 31 3 0 3 2 32 32 32 3 0 3 2 33 33 33 34 3 1 1 2 3 41 3 0 0 3 , , (1 )( ) ( ) , ( ) ( ) , , ( ) , 2 (1 )( ) ( ) v jTv p p jT t t p p t t t p p f f p c t t h p c a t S a h p c e k k h t a h p c S k k h a mp h p c k a t S t h a e V k k k h − ω − ω + = χ ρµα = + + χ − ρ + ρµ + + ∆ σ α = + + χ ρ + ρµ + + ∆ σ = + + ∆ = χ ρµα + α − α σ = χ − ρ + ρµ + + ∆ σ 2 0 3 1 0 3 2 1 42 3 0 0 0 3 3 1 0 1 3 43 0 3 0 0 3 1 3 (1 )( ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) ( ) v v v p c jT jT c t t p c p c t t c jT p c t t c p p t t p p p t V k e V k k k h t V k e a S k V k k k V h S k V k k k V h e a p k t S k k k h t p k t S − ω − ω − ω α + + χ − ρ + ρµ + + ∆ σ α = χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ = χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ 1 0 3 2 44 1 2 1 2 3 1 0 1 2 3 0 3 1 1 1 2 3 1 ( ) , ( ) ( ) 1 ( ) 2 ( ) . 2 v v jT t t p t p p c f f p c jT t p p c f f p c k k k h t e k a T T p T T p t S k V t h k V k e t S k V t h k V − ω − ω + + ∆ σ ∆ = + + + + χ ρµα + α − α σ + ∆ + χ ρµα + α − α σ (15) Subsequently, it is necessary to move to the time domain by replacing p = jω, and the characteristic polynomial of the control system is nothing more than the determinant of the matrix A presented in equation (14). 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) det( ) . a a a a a a a a D j A a a a a a a a a ω = = (16) Thus, the equation (16) is the characteristic polynomial of the control system that needs to be researched for behavior on the complex plane when the frequency of ω changes from zero to infinity.
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology Simulation results and discussion of the first hypothesis For the convenience of representing the behavior of the system, simulation was carried out in the Matlab/Simulink 2014 package, where a nonlinear system (8), (10) was directly modeled in Simulink, and the characteristic polynomial (16) was calculated by a cycle in Matlab itself, where at every step of the cycle a determinant for a specific frequency value was considered, and the resulting value was deposited on the complex plane, then everything was repeated. In general, the value for ω was calculated from zero to 2,000 Hz in increments of 0.01 Hz. To assess the stability of the cutting control system by the Mikhailov method, the variants of the control system, the variant of a stable and the variant of an unstable (at the boundary of stability) system were considered. The factor affecting the stability of the cutting process was a tool wear along the flank; the second factor is the processing speed factor. Here there is a possibility of checking the A.D. Makarov statement. In total, 29 high-speed cutting modes were considered, in each of which a stable, unstable and at the boundary of stability cutting mode was studied. Let’s consider the set of parameters of the cutting control system, which includes a processing speed of 1,600 rpm and a wear value of 0.22 mm. The results of modeling the coordinates of the system state and the corresponding phase directions are presented in Figure 2. 15 x, mm (10-3) y, mm, (10-3) z, mm t, s a) b) c) 6 4 2 0 0 0.05 0.1 0.15 0.35 0 -5 5 10 0 0.02 0.04 0.05 0 0 0.2 0.25 0.3 -2 t, s 0.01 0.03 t, s -4 0.05 0.35 0.3 0.25 0.2 0.15 0.1 0.35 0.3 0.2 0.1 0.05 0.25 0.05 dx/dt, mm/s d) e) f) 2 -4 0 -4 -2 10 5 0 -2 0 2 -5 5 10 00 0.01 2 z, mm*10-3 y, mm*10-3 dy/dt, mm/s dz/dt, mm/s x, mm *10-3 4 6 15 0 4 -2 0.03 0.04 0.05 0 4 -4 0.02 Fig. 2. For the case of wear h = 0.22: a – deformations along the x coordinate; b – deformations along the y coordinate; c – deformations along the z coordinate; d – phase trajectory along the x coordinate; e – phase trajectory along the y coordinate; f – phase trajectory along the z coordinate As can be seen from Figure 2, the system is stable, in addition, it can be seen from the phase trajectories that there is a constant restructuring of the control system, where the phase trajectory is compressed. However, each cycle of adjustment and subsequent compression is associated with cutting along the “trace” that was formed when the tool was embedded on the first turn. The stability assessment according to the Mikhailov criterion, for the considered variant of the parameters of the cutting control system, is shown in Figure 3. а d e f
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 The Mikhailov vector hodograph, shown in Figure 3, confirms the conclusion about the stability of the cutting control system, which is made by analyzing the coordinates of the deformation motions of the tool tip earlier. A feature of the Mikhailov vector hodograph is a constant increase in the amplitude of the hodograph with the increase in the frequency of modeling, therefore, the description of the hodograph has to be done in two figures, in the first one we see the movement of the hodograph through the first five quadrants, and in the second the hodograph covers the point zero through three more quadrants. The stability limit for the case of processing at a speed of 1,600 rpm, occurs at a wear value of 0.473 mm, the results of modeling the Mikhailov vector hodograph for this variant of modeling the cutting control system are shown in Figure 4. Fig. 3. The hodograph of the Mikhailov vector, a stable system: a – the beginning of the Mikhailov vector; b – the end of the Mikhailov vector а b Fig. 4. The hodograph of the Mikhailov vector, the system on the boundary of stability: a – the beginning of the Mikhailov vector; b – the end of the Mikhailov vector Re Jm Точка касания Re Jm а b
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology Ta b l e 1 The boundary of the cutting system stability h3, mm 0.3 0.32 0.34 0.35 0.37 0.38 0.4 0.41 0.42 0.43 n, rev/m 300 340 400 440 500 600 660 700 760 820 h3, mm 0.436 0.44 0.445 0.4455 0.455 0.46 0.46 0.473 0.483 0.491 n, rev/m 880 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 As it can be seen from Figure 4, the hodograph of the Mikhailov vector touches the imaginary axis and returns back to the first quarter, if this characteristic crosses the imaginary axis and returns back, it will be the mechanism for displaying the loss of stability of the system. Subsequently, with the increase in the amount of the cutting wedge wear, this is exactly what happens. The results of all studies are summarized in one general table, which is given below. Graphically, the area of stable dynamics of the cutting process, corresponding to the data given in Table 1, is shown in Figure 5. n, rev/m h, mm The area of stable dynamics of the cutting system The area of unstable dynamics of the cutting system Fig. 5. Areas of stable and unstable behavior of the cutting system As shown by Figure 5, the stability limit of the cutting system tends to grow indefinitely, while there is no pronounced maximum of this characteristic. In other words, the area of stable dynamics of the cutting system does not have a local extremum that would reflect the maximum dimensional stability of the tool corresponding to the statement, put forward by A.D. Makarov. As the reason for such a strange behavior of the cutting control system, it can be indicated that the control system model given in this section does not actually include the structural adjustment of the force response, which is identified and described in [9]. To clarify this assumption, consider the variant of force responses at a cutting speed of 1,600 rpm, which are shown in Figure 6.
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 As it can be seen from Figure 6, despite a significant change in the wear of the cutting wedge along the flank, the stationary components of the response forces to the forming motions have hardly changed. It is also clear from the figure that the relations between these forces do not change either, which clearly contradicts the assumption about the restructuring of response forces. Thus, the first hypothesis put forward at the beginning of the paper cannot be used as an objective for the scientific position put forward by A.D. Makarov. In other words, the combination of factors: the incident characteristic of the cutting force (according to N.N. Zorev) and the minimum coefficient of friction associated with the transition of friction from adhesive to diffusion nature is not sufficient to ensure the optimality of the cutting system according to the statement of A.D. Makarov. First of all, this is due to the lack of adjustment of the force response on the part of the cutting process to the forming motions of the tool with the increase in the wear of the cutting wedge along the flank. Testing the second hypothesis Correction of the mathematical model of the cutting control system Let’s consider the second statement put forward as an objective of the scientific position of A.D. Makarov, it is important to note that it is necessary to supplement the mathematical model describing the response of the cutting system to the forming motions of the tool with an additional element displaying the Fx, kgF Fy, kgF Fz, kgF t, s a) b) c) t, s t, s Fx, kgF Fy, kgF Fz, kgF t, s d) e) f) t, s t, s Fig. 6. Reaction forces for the option with a processing speed of 1,600 rpm: a – Fx for h = 0.22; b – Fy for h = 0.22; c – Fz for h = 0.22; d – Fx for h = 0.49; e – Fy for h = 0.49; f – Fz for h = 0.49 а b c d e f
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology dependence of the pushing force on the contact temperature of the tool and the workpiece. It is convenient to interpret this through the introduction of the dependence – the strength limit of the processed metal under compression on the corresponding contact temperature. To determine this dependence, a series of experiments was carried out using the experimental setup STD.201-1, which involves adjusting the weighting coefficients to calculate the cutting temperature based on the values of the removed natural thermal EMF. The STD.201-1 installation was developed by American engineers to solve the problems of analyzing the dynamics of cutting processes on machines of the turning group. The setup allows measuring the forces and vibrations of the tool, spread out along the axes of deformation (see Figure 1), as well as the temperature in the cutting zone. The research methodology provides for a whole setup procedure, which includes a double measurement of the contact temperature, a measurement using a natural thermal EMF and measurements carried out next to the contact with a calibrated thermocouple. An example of connection for the case of measuring the effect of contact temperature on the expulsive force, for the case of Steel 45, is shown in Figure 7. As can be seen from Figure 7, the measuring unit, in addition to the STD201-1 stand, contains a specially prepared shaft made of Steel 45, which has a radiator working area, heated by a special circular heater that is put on the shaft. In addition, a tool is brought to the shaft with some effort, next to the contact, which has a thermocouple inserted into the shaft material that measures the real value of the contact temperature. The measurement results are presented by the system’s software interface, the appearance of which is shown in Figure 8. As shown in Figure 8, the interface of the stand STD201-1 provides for the possibility of the measuring subsystem calibration based on the measurement of the natural thermal EMF of the cutting zone. It was the calibration mode of this measuring stand that was used for the experiment to determine the dependence of the expulsive force on the contact temperature of the tool and the workpiece. Fig. 7. Experimental setup prepared to assess the effect of contact temperature on the expulsive force
OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 Ta b l e 2 Dependence of Fy on the contact temperature Q, °C 30 40 50 60 70 80 90 100 110 120 130 Fy, kgF 9.5 9.7 10.3 10.9 11.5 12.2 12.8 13.4 13.8 14.3 14.6 The results of experiments to assess the influence of the contact temperature on the value of the force pushing the tool out of the cutting zone, for the case of processing Steel 45, are shown in the table below (Table 2). A graphical representation of the experiment results is shown in Figure 9. As is shown in Figure 9, the expulsive force almost linearly depends on the contact temperature, which is quite understandable from the point of view of the linear nature of metals expansion with the increase in its temperature. The average coefficient of linear increase in the expulsive force with the increase in the contact temperature F Q k , for the case of Steel 45, was 0.05625. Thus, the experiments carried out showed that the expulsive force linearly depends on the temperature of the workpiece, for Steel 45 the coefficient of amplification of the expulsive force was 0.05625. Let’s imagine the dependence of the cutting force on the temperature-speed factor of cutting in the form of a falling exponential dependence of the coefficient ρ on the actual cutting speed, as it is presented in the figure below. 1 0 1 c dz V dt e −α − ρ = ρ + µ , (17) Fig. 8. Interface of the stand STD201-1
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology where 0 ρ – some minimum value of the coefficient ρ; µ – the coefficient showing the increase of the value ρ to some maximum value; 1 α – the steepness coefficient of the value drop ρ; c dz V dt − – instantaneous cutting speed. Taking into account expression (17), as well as relying on equation (1), the cutting force will be interpreted as: 1 0 1 ( ) c v dz t V dt p f t T dx F e t y V dt dt −α − − = ρ + µ − − ∫ , (18) where ( ) p t y − – instantaneous cutting depth, v t f t T dx V dt dt − − ∫ – real feed. Based on the feed transformation presented in the equation (18), we obtain the equation for calculating the feed in the following form: [ ] 0 ( ) ( ) (1 ) v v t jT f f v v t T dx V dt V T x t x t T S x e dt − ω − − = − − − = − − ∫ , (19) where 0 S – the technological feed set by the CNC program, а (1 ) v jT x e− ω − – the deformation motion of the cutting tool along the feed axis transformed through the delay link. Equation (18), taking into account (19), will describe the cutting force, which after the linearization procedure in the vicinity of the equilibrium point, will take the following value: 0 0 0 0 1 0 (1 ) c v c V jT V p p dz F y S x e t t S dt − ω = − ρ − − ρ + ρ µα , (20) where ( ) 0 0 1 1 (1 ) c V c V ρ = ρ + µ − α . Fy, kgF Q, oC 30 40 50 8 60 70 100 9 80 90 110 120 10 11 12 13 14 15 Fig. 9. Results of the experiment on Steel 45
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
OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology 3 0 2 1 0 3 ( ) ( ) v jT h h Q p t t f f t p Q k h t k k K K k h t e V − ω + + ∆ + − ∆ σ − ñ 3 0 0 0 3 0 ( ) . c v V jT p t t dz S h t k k e dt − ω − χ ρ + σ + ∆ (28) The general system of equations describing the dynamics of the cutting system is given below. 1 0 2 11 12 13 11 12 13 2 2 21 22 23 21 22 23 2 2 31 32 33 31 32 2 1 ( ) , , , c v dz t V dt p f t T f p dx F e t y V dt dt 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 dt dt dt dt −α − − = ρ + µ − − + + + + + + = + + + + + + = + + + + + + ∫ ( ) 1 2 33 ( ) 1 ( ) 2 ( ) 3 0 3 ( ) ( ) ( ) 0 2 1 2 1 2 2 ( ) 3 , , , , , cos , sin , , / 2, ( ) ( ), ( ) ( ) ( h f z f z c x f h y p h z c h K x F h Q h p x h h y h h z h t h K Q K Q t t t h h h v z v v h c z F F F F F F F F F F F k Q h t e F F F F F k F k k k e e d Q dQ T T T T Q kN t T dt dt N t T F t T F t T − − = = χ + = χ + = χ + = σ + = ϕ = ϕ = = + ∆ + + + + = − − = χ − + − ( ) ( ) ) . v v dz t T V dt − − ñ (29) The same system, but already in a linearized form, in the vicinity of the equilibrium point and after switching to the operator form of recording, will take the following form:
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