Copyright © 2019 A.V. Zubiekhina-Khaiiat, D.D. Marchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Problems of Tribologyy, 24 (3/93) (2019) 45-50 Problems of Tribology Website: http://tribology.khnu.km.ua/index.php/ProbTrib E-mail: tribosenator@gmail.com DOI: https://doi.org/10.31891/2079-1372-2019-93-3-45-50 Mathematical modeling of the process of rolling body rolls with needed rollers A.V. Zubiekhina-Khaiiat*, D.D. Marchenko* Nikolaev national Agricultural University, Nikolaev, Ukraine *E-mail: zubehinakhayat@gmail.com *E-mail: marchenkodd@mnau.edu.ua Abstract The processes of needle roller rolling with the wide cavity and archimedean worms are investigated. A method of rolling in threads and worms with large angles of elevation of the coil line by means of flexible needle rollers is suggested. Key words: needle rollers,rolling screw surfaces needle rollers, curvature of a screw surface, a roughness. Introduction The processes of needle roller rolling with the wide cavity and archimedean worms are investigated. A method of rolling in threads and worms with large angles of elevation of the coil line by means of flexible needle rollers is suggested. Research methodology We have conducted a study of modeling the process of rolling of screw surfaces with needle rollers in order to expand the nomenclature of threads and worms that can be rolled up by cylindrical rollers and to increase their durability, reliability, extend the service life of parts; while maintaining the equality of curvature of the contacting bodies and the rate of slippage in the deformation zone. Establish the ability to roll with needle rollers with deformation along the entire depth of the Archimedes Worms profile. For modeling, archimedean worms with 10-24 mm modules, which have large angles of elevation of the coil lines, are taken. Trapezoidal and thrust threads have significantly smaller angles of elevation of the coil line and less deep recesses that have not been modeled. The nomenclature of threads and worms that can be rolled up by cylindrical rollers with a straight line is limited by the limiting magnitude of the curvature of the screw surface in the plane of the rollers. This curvature depends on the diameter, the angles of elevation of the coil line and the thread profile. Large threads are threaded by self-adjusting cylindrical rollers of small diameter [1]. When rolling threads with large lift angles, the plane of the rollers is rotated about an axis passing through the middle of the trough to the angle p , in General, different from the angle of lift  . In addition, the plane of the rollers is displaced relative to the axial section of the workpiece by an amount sufficient h to form an angle between the plane of the rollers and the thread  in the middle diameter of the cutting. This 06  creates a force component that pushes the rollers to the housing of the device. Shift and reversal of the roller plane lead to the appearance of a final (positive or negative) curvature of the thread profile in the roller plane. The nomenclature of threads and worms, which can be rolled up by cylindrical rollers, based on the possibility of their deformation along the entire depth of the profile, was determined experimentally by the rolling of models. The rolling of the screw surfaces with positive curvature in the plane of the rollers was simulated by the rolling of the cones, and of the surfaces with negative curvature by the rolling of the hyperboloids. Problems of Tribology 46 When running models, the following process parameters are preserved: a) curvature of the running surface in the plane of the rollers; b) the amount of relative sliding in contact of the roller with the workpiece. The radius of curvature  BR of the cone  surface in the plane of the roller ( 2X , 2Z , Fig. 1), obtained by turning the angle of the axial section of the cone around the perpendicular to its formation, will be determined by Euler formula [2]: 2sin sin cp K K r R     , (1) where crr – is the average radius of the cone; K – the angle at the base of the cone. а b Fig. 1. Threading with large lift angles Given the condition of relative sliding in the direction perpendicular to the axis of the roller, we have: sin 2 cos cp K cp D r     , (2) where cpD – average diameter and angle of thread profile. The angle of the cone K , given (1) and (2), 1/K BR K is determined: Problems of Tribology 47 2arctg 2 cos sin cp B K D K        . (3) The results From the solution of the geometric problem of intersection of the cylinder with the generating and conical surfaces, we determine the values of the lengths of contact of the roller with the surface of the thread and the layout, the ratio of these values is equal to the coefficient of refinement  УK . Taking into account [1] the equations of the screw surface in the coordinate system are written 3X , 3Y , 3Z , (Fig. 1, a) where 3X is the axis of the roller. The distance 1 and 2 between the points 1M and 2M the screw surface, respectively, on the outer and inner diameter of the thread and the surface of the roller at the point contact of the roller with the surface of the thread on the average diameter of the thread (Fig. 1, b) is determined by:     3 3 3 sin cos P Z NX r NZ    , (4) where Pr – roller radius; 3NX and 3NZ – respectively, the angles between the axes 3X and 3Z and the normal N to the helical surface passing through the points 1M and 2M and 3X the axis. The magnitude of the indentation of the roller in the direction of the axis until 3Z the contact of the surface of the roller points 1M and 2M the screw surface: 1 2 1 2 1 2 cos coscp q         , (5) where the angles 1 and 2 are determined by equality: 3 3 tg Y Z   . (6) The length of contact of the roller  Pl with the helical surface is defined as the difference of the abscissa 3X points 1M and 2M . The coordinates ZYX 333 ,, of the points 1M and 2M are determined by the joint solution of equations (3), (4), (5), (6) of the screw surface and the normal N equation. The length of contact of the roller with the conical surface of the maket  Pl , measured in the direction of the axis of the roller, is calculated by a joint solution of the cylinder and cone equations in the coordinate system 3X , 3Y , 3Z , related to the axis of the roller, provided that the roller is pressed into the cone by the magnitude cpq (Fig. 2) and is equal to the difference between the coordinates 3Z of the points 1M and 2M the conical surface. Problems of Tribology 48 Fig. 2. Scheme to calculate the contact length of the roller with the conical surface of the layout A hyperboloid, which is a model of a screw surface with a negative curvature in the plane of the rollers, is formed by machining a straight cutting edge of the cutter. The cutting edge, initially coinciding with the generating cone with the angle at the base 2 , rotates around the perpendicular to the generating conus by the angle P (Fig. 3). When the workpiece is rotated, the conical surface becomes a medium-radius hyperboloid by cutting the allowance 2r . Fig. 3. Scheme of conical surface conversion into a hyperboloid The plane of the roller when rolling hyperboloid forms with its axial section angle 06  . The curvature of the intersection of the hyperboloid with the plane of the roller at a point M according to [2] will be determined by: 2 2 1 2 1 cos sin R R R     , (7) where 1R – is the radius of curvature of the generating hyperboloid; 2R – is the radius of curvature of the intersection, normal to the formation. Applying formula (7) to the cutting edge line, we obtain: 2 2 2 2 1 2 cos sin 0 R R     . (8) Problems of Tribology 49 Solving together equations (7) and (8) with given 22 sin r R   , we obtain 1 BKR  the formula for calculating the angle of rotation of the cutting edge of the cutter: 2 2 2 2 arctg tg cos sin B P r R         . (9) The values 2r in formula (9) should be equal to the values of the radii cpr calculated by the formula (2) at 02 60K    . The geometric dimensions of the models Kr , 2r , and K the refinement coefficient УK were calculated on a PC. Modeling was performed by a device with a self-adjusting cylindrical roller-cam on a lathe [10]. As a result of this work, it is shown that cylindrical rollers can run almost all threads with a lift angle of not more than 100. Conclusions Features of rolling of worms and worms with a wide hollow are investigated. The modeling of rolling of the screw surfaces by needle rollers was carried out. The convex surfaces were modeled by cone-self, concave by unipolar hyperboloids. In doing so, the curvature of the contacting bodies and the rate of slippage in the deformation zone were maintained. It has been found that needle rollers can be deformed with the entire depth of the profile by archimedean worms with a rotation angle of 10   . It is proposed to roll Archimedean worms with 10   flexible needle rollers, a device for this purpose was granted a patent of Ukraine for the invention and a patent of Ukraine for a utility model. In this way, the nomenclature of threading and archimedean worms was expanded. Literature 1. Babej Yu.I. Poverhnostnoe uprochnenie materialov / Yu.I.Babej, B.I.Butakov, V.G. Sysoev. – Kiev: Nauk. dumka, 1995. – 104 s. 2. Bronshtejn I.N. Spravochnik po matematike / I.N. Bronshtejn, K.A. Semendyaev – M., Tehizdat, 1956. 3. Odincov L.G. Uprochnenie i otdelka detalej poverhnostnym plasticheskim deformirovaniem / L.G Odincov. – M.: Mashinostroenie, 1987. – 328 s. 4. Papshev D.D. 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Patent 101718 Ukraina MPK V24V 39/04 (2006.01), B21H 3/00. Pristrij dlya obkatuvannya krupnih rizb i arhimedovih cherv’yakiv rolikami / B.I. Butakov, A.V. Zubehina, zayavnik i patentovlasnik B.I. Butakov.; zayavl. 18.07.2011, nomer zayavki: a201108944; opubl. 25.04.2013, Byul. № 8, 2013. 11. Butakov B. Volnistost poverhnosti pri obkatyvanii tel vrasheniya rolikami / B. Butakov // Motrol, Motoryzacja I energetyka rolnictwa. Lublin. – Vol15, No2., 2013 – S. 15 – 22. Problems of Tribology 50 Зубєхіна-Хайят О.В., Марченко Д.Д. Математичне моделювання процесу обкатування тіл обертання голчастими роликами. Досліджено процеси обкатування голчастими роликами різьб з широкою впадиною і архімедо- вих черв’яків. Запропоновано спосіб обкатування різьб і черв’яків з великими кутами підйому лінії витку за допомогою гнучких голчастих роликів. Ключові слова: голчасті ролики, обкатування гвинтових поверхонь голчастими роликами, кривизна гвинтової поверхні, шорсткість.