Vol. 25 No. 2 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.
OBRABOTKAMETALLOV Vol. 25 No. 2 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 Aff airs, 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. Korotkov, D.Sc. (Engineering), Professor, Kuzbass State Technical University, Kemerovo; 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. 2 2023 5 CONTENTS OBRABOTKAMETALLOV TECHNOLOGY Kisel’ A.G., Churankin V.G. Predicting the coolant lubricating properties based on its density and wetting eff ect.................................................................................................................................................................... 6 Berezin I.M., Zalazinsky A.G., Kryuchkov D.I. Analytical model of equal-channel angular pressing of titanium sponge.............................................................................................................................................. 17 EQUIPMENT. INSTRUMENTS Kuts V.V., Chevychelov S.A. Theoretical study of the curvature of the treated surface during oblique milling with prefabricated milling cutters....................................................................................................................... 32 Skeeba V.Yu., Zverev E.A., Skeeba P.Yu., Chernikov A.D., Popkov A.S. Hybrid technological equipment: on the issue of a rational choice of objects of modernization when carrying out work related to retrofi tting a standard machine tool system with an additional concentrated energy source................................................ 45 MATERIAL SCIENCE Vorontsov A.V., Filippov A.V., Shamarin N.N., Moskvichev E.N., Novitskaya O.S., Knyazhev E.O., Denisova Yu.A., Leonov A.A., Denisov V.V. In-situ analysis of ZrN/CrN multilayer coatings under heating................................................................................................................................................................. 68 Kornienko E.E., Gulyaev I.P., Kuzmin V.I., Tambovtsev A.S., Tyryshkin P.A. Structure and properties of WC-10Co4Cr coatings obtained with high velocity atmospheric plasma spraying.................................... 81 Balanovsky A.E., Nguyen V.V., Astafi eva N.A., Gusev R.Yu. Structure and properties of low carbon steel after plasma-jet hard-facing of boron-containing coating............................................................................. 93 Emurlaeva Yu.Yu., Lazurenko D.V., Bataeva Z.B., Petrov I.Yu., Dovzhenko G.D., Makogon L.D., Khomyakov M.N., Emurlaev K.I., Bataev I.A. Evaluation of vacancy formation energy for BCC-, FCC-, and HCP-metals using density functional theory................................................................................................ 104 EDITORIALMATERIALS 117 FOUNDERS MATERIALS 127 CONTENTS
OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 Theoretical study of the curvature of the treated surface during oblique milling with prefabricated milling cutters Vadim Kuts a, *, Sergey Chevychelov b Southwest State University, 94, 50 let Oktyabrya str., Kursk, 305040, Russian Federation a https://orcid.org/0000-0002-3244-1359, kuc-vadim@yandex.ru, b https://orcid.org/0009-0006-8958-2191, tschsa@yandex.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. 2 pp. 32–44 ISSN: 1994-6309 (print) / 2541-819X (online) DOI: 10.17212/1994-6309-2023-25.2-32-44 ART I CLE I NFO Article history: Received: 13 March 2023 Revised: 29 March 2023 Accepted: 15 April 2023 Available online: 15 June 2023 Keywords: Form-generating method Interlocking side mill Oblique milling Surface curvature The main radii of curvature Acknowledgements Research were partially conducted at core facility “Structure, mechanical and physical properties of materials”. ABSTRACT Introduction. The paper discusses the methods of processing large parts having curved convex surfaces with a rectilinear guide on multi-coordinate CNC machining centers using the touch method with a discrete motion of the tool feed along the profile of the part. It is shown that the main disadvantages of this method are lower productivity, which is due to the presence of discrete tool motions between cycles of its translation mode, where the value of discrete tool motion for a given processing accuracy depends on the curvature of the surface being processed. To improve processing performance, it is proposed to use prefabricated disc cutters equipped with replaceable polyhedral inserts (RPI) with rectilinear cutting edges. Its installation in the cutter body with non-zero angles of inclination of the main cutting edge, in combination with an additional rotation of the cutter, during processing, along the direction of the translational feed movement, allows you to obtain a concave surface and ensure a tighter fit of the producing surface of the tool and the machined surface of the part. The aim of the work is to reduce the error of approximation of the profile when it is processed using the touch method with discrete motion of prefabricated disc cutters along the profile and, consequently, to ensure workpiece the possibility of increasing the step of tool movement along the profile being formed to improve processing performance. Research methods: geometrical theory of designing metal-cutting tools. Results and discussion. The regularities established in the work made it possible to create a method for determining the angle of inclination of the main cutting edge of the RPI milling cutter and the angles of rotation of the milling cutter along the direction of translational feed movement during line-by-line processing of extended sections of parts with a curved profile on multi-coordinate CNC machines by turning the milling cutter to ensure the best fit of its producing surface to the surface being processed at the point of its contact, to reduce the approximation error processed profile and improve processing performance, due to the possibility of increasing the tool movement step. For citation: Kuts V.V, Chevychelov S.A. Theoretical study of the curvature of the treated surface during oblique milling with prefabricated milling cutters. Obrabotka metallov (tekhnologiya, oborudovanie, instrumenty) = Metal Working and Material Science, 2023, vol. 25, no. 2, pp. 32–44. DOI: 10.17212/1994-6309-2023-25.2-32-44. (In Russian). ______ * Corresponding author Kuts Vadim V., D.Sc. (Engineering), Associate Professor Southwest State University, 94, 50 let Oktyabrya str., 305040, Kursk, Russian Federation Tel.: 8 (903) 639-94-01, e-mail: kuc-vadim@yandex.ru Introduction Currently, machining of large pieces that have curved convex surfaces with a linear guide within the conditions of individual, small-series and repair production is performed with the help of multi-axis CNC machining centers due to the economic inexpediency of using special equipment. In this case, the formation of the parts’ surface can be performed using the touch method with a continuous feed movement of the tool along the profile of the part (fig. 1, a) or with a discrete movement of the tool (fig. 1, b). For example, when milling parts with a thickness less than the height of the cutter, machining can be performed using the touch method with a continuous tool feed movement along the profile of the part (see fig. 1, a); and while milling parts with a larger thickness, the touch method with a discrete feed movement along the profile of the part can be applied (see fig. 1, b), where the milling cutter performs cyclic recipro-
OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 2 2023 cal feed movements perpendicular to the profile of the part and shifts with each cycle along the part profile at a given discrete value depending on the required machining accuracy. An example of such parts is spur coarse pitch gears with pitch values greater than 9 mm and a face width greater than 50 mm, for which the machining according to the first method is effortful. The machining of parts by the touch method with a continuous tool feed movement along the profile of the part has become widespread; there is a large number of works devoted to this problem [1–6]. The issues of machining by the touch method with discrete feed movement along the profile of the part due to the lower prevalence of shaped parts with heavy thickness are less studied [7–10]. The main disadvantage of this method is lower machining speed, which is due to the presence of discrete tool movement between the cycles of reciprocal movements. Moreover, the value of discrete tool movement ΔΩ for a set machining accuracy depends on the value of the surface curvature being formed (fig. 2), which leads to a lower machining speed. To increase the machining speed in this case, it is advisable to use milling cutters having a concave shape of the generating surface, which ensures its tighter fit to the surface to be machined. The generating surface is the one formed by the shape-generating cutting edge of the milling cutter due to its primary motion, i.e. the motion that determines the cutting speed [11]. However, with regard to the designs of prefabricated side milling or face milling cutters equipped with replaceable polyhedral inserts (RPI), it can be said that there are no RPIs of a standard design with a concave cutting edge. In [12–16], it has been found that when a milling cutter with a linear RPI cutting edge a b Fig. 1. The formation of the surface of the part by the touch method with the feed movement of the tool: a – continuous; b – discrete a b Fig. 2. The dependence of the value of the discrete tool movement on the curvature of the profile of the surface being processed: a – at low curvature; b – at high curvature
OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 mounted in its housing is rotated at an angle of inclination of the main cutting edge that is different from zero (l ≠ 0), the generating surface takes a concave shape, and with an increase in the angle l the curvature of this surface becomes larger. Moreover, the works [15, 16] show that the curvature of the generating surface of such cutters can be increased by turning the cutter along the direction of translation feed movement by the angle x, thereby implementing a scheme of oblique milling, for which it is assumed to use five-axis machining centers. Therefore, it is advisable to develop a milling cutter design by selecting the inclination angle of RPI’s main cutting edge, in which the balance of the generating surface curvature and the least curvature of the convex surface profile being machined will be reached. By rotating the milling cutter for the calculated angle x while milling, this balance will also be reached along the entire profile. The implementation of this approach requires studying the influence of the cutter parameters (diameter, angle of inclination of the main cutting edge) and the angle of inclination of the cutter along the direction of the translational feed movement x on the change in the curvature of the machined surface (principal radii of curvature). From the foregoing, the purpose of this study can be formulated as reducing the error in approximating the profile of the workpiece when it is machined by the touch method with discrete movement of interlocking side or end mills along the profile and, as a result, providing the possibility of increasing the tool approach increment along the formed profile to increase processing productivity. The objective is to carry out a theoretical study of changes in the curvature of the part surface machined during oblique milling with interlocking side mills equipped with RPIs, as well as to develop a method for determining the inclination angle of RPI main cutting edge of a milling cutter and the rotation angles of a milling cutter along the direction of translation feed movement, ensuring the best fit between the producing surface of the cutter and the surface of the part in its points of contact. Methodology To carry out this study, a model of a prefabricated milling cutter with a nominal diameter (d), consisting of one RPI installed in the mill body with a taper lead angle (j) and the side rake angle (l) was developed (fig. 3). а b c Fig. 3. Simulation scheme of sequential installation of the RPI in the milling cutter body: а – setting the taper lead angle; b – setting the side rake angle; c – installation according to a given diameter in the milling cutter body Х2
OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 2 2023 The equation of RPI cutting edge which determines the generating surface of the milling cuter under study is described in its own coordinate system (X1Y1Z1), as 1( ) [ 0 0 1] , r t t Τ = (1) where t is a length parameter of RPI cutting edge. The equation of RPI cutting edge (1) is sequentially converted into the coordinate system of the mill body (X4Y4Z4), taking into account the specified taper lead angle (fig. 3, position 1), the side rake angle (fig. 3, position 2) and the diameter of the cutter (fig. 3, position 3). ( ) { } { } { } 2 5 6 4 1 43 32 21 ( 2) ( ) ( 2 ) ( ) r t A d A A r t = l p − j ⋅ , (2) where { } 2 43 ( 2) A d is the matrix that determines the installation of RPI on a specified diameter of the cutter d in the coordinate system of the mill body(X4Y4Z4): { } 2 43 1 0 0 0 0 1 0 2 ( 2) 0 0 1 0 0 0 0 1 d A d = ; { } 5 32 ( ) A l is the matrix specifying the rotation of RPI relative to the axis OX3 of the coordinate system (X3Y3Z3) to provide a set side rake angle: { } 5 32 cos 0 sin 0 0 1 0 0 ( ) sin 0 cos 0 0 0 0 1 A l l l = − l l ; { } 6 21 ( 2 ) A p − j is the matrix that determines the rotation of RPI relative to the axis OX2 of the coordinate system 2 2 2 X Y Z to attain a set major cutting edge angle: { } 6 21 cos ( 2 ) sin ( 2 ) 0 0 sin ( 2 ) cos ( 2 ) 0 0 ( 2 ) 0 0 1 0 0 0 0 1 A p − j − p − j p − j p − j p − j = . By specifying the rotation of RPI cutting edge (2) relative to the tool axis, we obtain the equation of the generating surface of the milling cutter under consideration { } { } 5 4 4 5 54 ( , ) ( 2) ( ) ( ) f f r t A A r t q = p q (3) where q is the angular parameter of the milling cutter generating surface; { } 5 5( 2) f A p is the matrix specifying the rotation of the coordinate system of the working tool to align axis Zf with the axis of the milling cutter body: { } ( ) 5 5 cos( 2) 0 sin( 2) 0 0 1 0 0 2 sin( 2) 0 cos( 2) 0 0 0 0 1 f A p p p = − p p ; { } 4 54 ( ) A q is the matrix that specifies the rotation of RPI cutting edge profile 4( ) r t by the angle q.
OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 Fig. 4 shows the result of developing the milling cutter generating surface with parameters d = 30 mm, j = 90° and l = 20° according to equation (3). As a result of specifying the angle l ≠ 0, the generating surface of the milling cutter is a unipolar hyperboloid of rotation, characterized by the variability of the values of the principal radii of the surface curvature along the rotation axis, where its minimum value is reached at points (0, q), for all q ∈ [0; 2p]. The equation of the surface machined with the translation tool feed movement along axis X0 of the part and a set value of the milling cutter rotation by the angle ξ (during oblique milling) is developed on the basis of the shaping equation { } { } { } 1 5 6 0( , , ) ( ) ( ) ( ) ( , ) f r x t A x A A r t q = x −q q (4) where x is the parameter of the milling cutter travel along axis X0; { } 1 ( ) A x is the matrix that specifies the travel of the milling cutter along axis X 0: { } 1 1 0 0 0 1 0 0 ( ) 0 0 1 0 0 0 0 1 x A x = ; { } 5 ( ) A x is the matrix of the milling cutter rotation along the direction of translation feed movement by angle value ξ: { } 5 cos 0 sin 0 0 1 0 0 ( ) sin 0 cos 0 0 0 0 1 A x x x = − x x ; { } 6 ( ) A −q is the matrix that specifies the rotation of the milling cutter: { } 6 cos ( ) sin ( ) 0 0 sin ( ) cos ( ) 0 0 ( ) 0 0 1 0 0 0 0 1 A −q −q −q −q −q = . To take into account the connection of the envelope of the form q = q(x,t), based on equation (4) we compose and solve the equation with respect to parameter q: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) 0 ( , , ) ( , , ) ( , , ) r x t r x t r x t i j k x x x r x t r x t r x t i j k t t t r x t r x t r x t i j k ∂ q ∂ q ∂ q ∂ ∂ ∂ ∂ q ∂ q ∂ q = ∂ ∂ ∂ ∂ q ∂ q ∂ q ∂q ∂q ∂q (5) which makes it possible to represent equation (4) as a bivariate function: { } { } { } ( ) ( ) 1 5 6 0( , ) ( ) ( ) ( , ) , ( , ) f r x t A x A A x t r t x t = x −q q . (6) Fig. 4. Milling cutter generating surface
OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 2 2023 Fig. 5 shows the graphs of function q(x, t) at x = 0 for the milling cutter with parameters d = 30 mm, j = 90°, l = 20°and the angle of the milling cutter rotation ψ being equal to 0° (line 1) and 20° (line 2). Fig. 6 shows the results of modeling the milling cutter generating surface when it is rotated by the angle of y = 20° and the nominal machined surface. Fig. 5. Function graphs θ(x, t) at x = 0 We estimate the curvature of the surface machined by calculating its main curvatures (k1 и k2), which are the solution of the equation 2 2 0 k H k K − ⋅ + = , (7) where 2 2 2( ) LG FM EN H ÅG F − + = − , (8) 2 2 LN M K EG F − = − , (9) where E, F, G are the coefficients of the first quadratic form (g) of the machined surface (6), described by the equation: 2 2 2 g E dx F dx dt G dt = ⋅ + ⋅ ⋅ + ⋅ , (10) where 2 0( , ) r x t E x ∂ = ∂ ; 0 0 ( , ) ( , ) r x t r x t F x t ∂ ∂ = ∂ ∂ ; 2 0( , ) r x t G t ∂ = ∂ , (11) L, M, N are the coefficients of the second quadratic form of the machined surface (6), described by the equation: 2 2 2 q L dx M dx dt N dt = + + , (12) where 0 2 ( , ) ( , ) r x t n x t L x EG F ∂ = ∂ − ; (13) 0 2 ( , ) ( , ) r x t n x t M x t EG F ∂ = ∂ ∂ − ; (14) Fig. 6. Modeling of the milling cutter’s producing surface and the nominal machined surface
OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 0 2 ( , ) ( , ) r x t n x t N t EG F ∂ = ∂ − , (15) where ( , ) n x t is the normal to the machined surface; 0 0 0 0 ( , ) ( , ) ( , ) ( , ) ( , ) r x t r x t x t n x t r x t r x t x t ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ . (16) For the convenience of perception, in the future, instead of (k1 и k2), we will consider the principal radii of the machined surface curvature 1 1 1 R k − = and 2 2 2 R k − = . Studying the principal radius of the machined surface curvature in cross section (fig. 7) for the milling cutter with d = 30 mm, j = 90°, l = 20° and x = 0° confirmed that it reaches the lowest value at the point of the surface being formed by the middle of the RPI cutting edge (t = 0) and increases as it moves away from the middle (curve 1). The study also showed (fig. 7) that an increase in the angle of rotation of the cutter (at x = 20°, curve 2) leads to a decrease in the principal radius of curvature. Fig. 8 shows graphs of the change in the main curvature of the machined surface in the cross section (at the point t = 0) at different angles of rotation of the cutter with parameters x ∈ [0;45°] with parameters d = 30 mm, j = 90° and angle l = 10° (line 1) and l = 20° (line 2). Fig. 7. Change of the main radius of curvature (R) on different sections of the processed surface in cross-section Fig. 8. Change of the main radius of curvature (R) of the processed surface in cross-section from the angle x
OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 2 2023 The use of the obtained equations makes it possible to calculate the inclination angle of RPI side rake angle and the rotation angles of the milling cutter along the direction of the translational feed movement that will provide the best fit of the generating surface of the milling cutter and the part surface at its contact points. It is advisable to perform this calculation in the following sequence: 1) for a specified diameter of the milling cutter, using equation (7), we calculate the minimum value of the angle λ (at x = 0°) that provides the best fit of the generating surface of the milling cutter at the point of the surface with the least curvature (the largest principal radius of curvature) according to the condition: max ( , ) d R R ≈ l x , (17) where max d R is the largest radius of the profile curvature of the surface being formed; ( , ) R l x is the principal radius of the machined surface curvature by the milling cutter at a specified angle λ and the angle of the cutter rotation ξ. 2) at the specified value of angle l, we calculate the inclination angles of the milling cutter x at the remaining points of the profile of the surface being formed according to condition (17). In case when the surfaces to be machined have a large value range of the principal radius of curvature, it may not be possible to achieve strict equality (17) at all points. Then for these points it is necessary to take angle x equal to the largest possible value (x = 45°). Results and discussion The practical application of the constructed models and established regularities will be considered through the example of machining the involute surface of a spur coarse pitch gear (fig. 9, a) with pitch of 9 mm, 21 teeth and a face width equal to 50 mm and with the following equation: [ ] 0 0 ( , ) (cos sin ) (sin cos ) 1 , d r u v R u u u R u u u v Τ = + − (18) where R0 is the radius of the generating circle of the gear wheel; for our wheel R0 = 197.3 mm. The calculation of the principal radius of curvature of this surface in the across-track direction has shown that its size at u ∈ [0; 0,61] varies from 0 to 120.5 mm. Taking the diameter of the milling cutter as 30 mm, the calculation of the minimum angle value l (at x = 0°) is performed to attain the best fit of the generating surface of the milling cutter to the surface point with the least curvature (the largest principal radius of curvature) u = 0.61 according to the condition (17). By specifying the incremental step of angle l equal to 30’ and the angle x = 0°, it is found that condition (17) is fulfilled at l = 19° with R(19°) = 126.5 mm. Further, with the set the angle l = 19°, the calculation of inclination angles of the milling cutter ξ is performed for the profile points of the surface formed (fig. 10). a b Fig. 9. Spur gear: a – a geometric model; b – the result of modeling the surface of the teeth according to (17)
OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 2 3 Fig. 10. Change of angle x at ξ = 19° and u ∈ [0; 0.61] Fig. 11 demonstrates the position of the milling cutter during the formation of different sections of the wheel tooth surface being machined in accordance with the calculated rotation angles of the milling cutter (fig. 10). In fig. 11, position 1 corresponds to the point of the tooth surface profile u = 0 rad, position 2 corresponds to the point of the tooth surface profile u = 0.44 rad, position 3 corresponds to the point of the tooth surface profile u = 0.61 rad. It follows from fig. 11 that with increase in the curvature of the surface being machined, the rotation angle of the milling cutter becomes larger. Conclusion The established regularities of changing the principal radius of curvature of the surface machined in cross-section in case of the lineby-line machining of extended sections of parts with a curved profile (in particular, convex surface sections of the parts) on multi-axis CNC machines by rotating the milling cutter to ensure the best fit of its generating surface to the machined surface at its point of contact. It also ensures a decrease in an approximation error of the surface profile of the machined surface and improves the processing productivity due to the possibility of increasing the tool approach increment along the formed profile. Fig. 11. Installation of the milling cutter across points of the surface being formed References 1. Wei P.M. Povyshenie effektivnosti konturnoi obrabotki na stankakh s ChPU putem korrektsii traektorii i rezhimov rezaniya. Avtoref. kand. tekhn. nauk [Improving the efficiency of contour machining on CNC machines by correcting the trajectory and cutting modes. Author’s abstract of PhD eng. sci. diss.]. Moscow, 2014. 22 p. 2. Petrakov Y., Shuplietsov D. Contour milling programming technology for virtual basing on a CNC machine. Eastern-European Journal of Enterprise Technologies, 2019, vol. 2, no. 1 (98), pp. 54–60. DOI: 10.15587/17294061.2019.162673. 3. Petrakov Y., Korenkov V., Myhovych A. Technology for programming contour milling on a CNC machine. Eastern-European Journal of Enterprise Technologies, 2022, vol. 2, pp. 55–61. DOI: 10.15587/17294061.2022.255389. 4. Dumitrache A., Borangiu T., Dogar A. Automatic generation of milling toolpaths with tool engagement control for complex part geometry. IFAC Proceedings Volumes, 2020, vol. 43, pp. 252–257. DOI: 10.3182/20100701-2-pt4011.00044. 5. Timiryazev V.A., Khostikoev M.Z., Danilov I.K., Datsko A.G. Upravlenie tochnost’yu konturnoi obrabotki kontsevymi frezami [Precision control of contour machining with end mills]. STIN = Russian Engineering Research, 2020, no. 12, pp. 22–26. (In Russian).
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