GEODESIC WINDING OF CORD OF TIRE TAKING INTO ACCOUNT HER FORM ON-LOADING. I. Tabakova *

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1 GEODESIC WINDING OF CORD OF TIRE TAKING INTO ACCOUNT HER FORM ON-LOADING I. Tabakoa * Summary. A method oer of geometrical design of geodesic trajectories is in-process brought on a surface that approaches the surface of tire taking into account her geometrical form on-loading. Keyords: a geodesic trajectory, motor-car tire, process of inding of cord, periodicity of location of geodesic. Formulation of the problem. Making of the reinforced materials a inding method is one of perspectie technologies of strengthening of ares. The special alue she has at making of motor-car tires, hen inding comes true by a metallic (sometimes kelar) cord [1]. For realization in practice of inding it is necessary to kno the la of distribution of coils on frameork. The geodesic trajectories of inding are mostly used. Proided in this case, in the certain understanding, maximal durability of good. In addition, for proiding of reliability and longeity of tires it is necessary to take into account the dynamics of the mutual moing of coils of cord in points their mutual crossing as a reaction on the change of loading on a tire. The marked specifies on actuality of theme of ork. Analysis of recent research. One of technologies of making [1] motor-car tires consists in that the reticulated cylindrical shell is transformed like as torus surface by tires. The process of making of motorcar tire is based on the operation of transformation of shell-pureyance in eentual good. After transformation the la of change of corners of reenforcement is described by the so-called "bus geometry". Modern technologies enisage inding of cylindrical shell-pureyance after geodesic trajectories (fig.1). In-process [] a method oer of calculation of geodesic trajectory of cord tire is brought taking into account the size of pantograph corners beteen contiguous cords (fig.). But in-process [] mainly empiric dependences are used, and the construction of geodesic trajectories of cord is not based on the decision of the system of differential equalizations. * Superisor - Professor Tkachenko V.P.

2 Fig. 1. Location cord on the surface of the tire. Fig.. Pantograph model at points of entry. Formulation of article purposes. To ork out the method of geometrical design of geodesic trajectories on the surface of rotation, that approaches the surface of tire taking into account her form on-loading. Basic part. Logically to consider that in the process of exploitation of motor-car tire the corners of pantograph model must insignificantly change at the change of loading. It must decrease the risk of breakage of tire "from middle" due to the mutual moing of cords. Will consider the surface of rotation, that reminds the surface of motor-car tire approximately. Conditionally ill name her a like tire surface equalization of that it is suggested to choose in a kind X [ R ( ), R sin( ), u ]. (1) Here u is a parameter along the meridian of surface, is a parameter along the parallel of surface, R is a radius of base circle. The feature of description (1) consists in the presence of parameter of, that regulates the form of chiropody surface depending on her state - free or loaded. Yes, in the free state of alue of parameter of must be elected ithin the limits of = 1.4, and in the loaded state ithin the limits of = All sizes are in conditional units. The presence of parameter of ill allo in the future to atch the change of pantograph of corner beteen inding cords in case of change of loading on a tire. At the stoage of the system of differential equalizations it is necessary to calculate for description of geodesic on a surface tire [3,4] coefficients of the second quadratic form and other accompanying formulas.

3 D R R R 3 ( 1) E ; F ; G R ; R (3) E u () 4 R (1) (1) R (1) / ; (3) sin(1 ) ; E ; F ; F ; u sin( u) R Gu ; G. As a result obsessed system of differential equalizations for description of geodesic on a chiropody surface in a kind: d GEu d GGu d u( t) u( t) ( t) ; () dt D dt D dt d EG d d ( t) u u( t) ( t). dt D dt dt The system of equalizations () as untied by a method Runge-kut. Thus a test ariant as calculated ith initial conditions u / and and by the limits of change of parameters / 4 u / 4 and ; R=3. The amount of points on a geodesic line equals 15. The corners of "exit" of geodesic changed only from an initial point. On fig. 3 results oer of calculations of geodesic trajectory are brought no-load at = 4 (on the left) and ith loading at = 16 (on the right) for u, 43. It as succeeded to find an initial u, 43self, the method of "animation" iterations that proides acceptable periodicity of geodesic trajectories on a surface regardless of presence or absence of loading. he considered geodesic inding of cord it follos to consider the first step design of technology, in fact the geometrical form of tire onloading and in the free state must change not "globally" - id est simultaneously on all surface of tire, but "locally" - only in the zone of contact of tire ith the surface of road.

Fig. 3. Chiropody surface for a parameter u, 43.

form of chiropody surface only in the zone of contact ith a road depending on the state of loading of tire.

Walking aay from the geometrical form of surface of rotation causes complication of decision of this task.

4 Fig. 3. Chiropody surface for a parameter u, 43. The next step of design ill be taking into account of change of geometrical form of tire only in the zone of contact ith the surface of road. For this purpose in equalization (1) the constant of ill substitute by a function 16 1 H( ), (3) hat must regulate the form of chiropody surface only in the zone of contact ith a road depending on the state of loading of tire. In a formula (3) is a central corner of arc of contact, Н is a function of Heayside. Walking aay from the geometrical form of surface of rotation causes complication of decision of this task. At the stoage of the system of differential equalizations it is as () necessary to take into account already unzero alues F, E, F u and F. A stoage and decision of the system of differential equalizations are carried out in the enironment of mathematical package of Maple. Will point the type of the system of differential equalizations

4 an offer geometrical model of chiropody surface is represented ith the zone of contact ith the surface of road. On fig.

5 At a decision, the left parts of equalizations (4) must be diided into D. The coefficients of the second quadratic form and other accompanying formulas hae a too bulky kind and that is hy not led here. On fig. 4 an offer geometrical model of chiropody surface is represented ith the zone of contact ith the surface of road. On fig. 5 a geometrical model oer of chiropody surface is brought ith a fe coils of geodesic inding at =,8. Notedly, that geodesic inding takes into account the geometrical form of bulge of chiropody surface. On fig. 6 it is represented 15-torns of geodesic inding of chiropody surface for a parameter u, 43. Fig. 4. Chiropody surface ith the zone of contact ith a road. Fig. 5. Geodesic inding chiropody surfaces.

6 Conclusions. The brought method oer of construction of geodesic trajectories on a surface that approaches the surface of motor-car tire taking into account her form on-loading allos in the first approaching to analyse the location of family of geodesic ith an aim in further to analyse time-histories of pantograph corners. Fig. 6. Geodesic inding chiropody surfaces. Literature 1. Иванов А. М. Основы конструкции автомобиля [Учебник для ВУЗов.] / А.М. Иванов, А.Н. Солнцев, В.В. Гаевский. М.: ООО «За рулём», 5, - 14 c.. Koutny F. Geometry and mechanics of pneumatic tires / F. Koutny. Zlin: CZE 7, 14 р. 3. Погорелов А. И. Дифференциальная геометрия. / А.И. Погорелов М.: Наука, 1974, с. 4. Голованов Н.Н. Геометрическое моделирование. / Н.Н. Голованов. М.: Издательство Физико-математической литературы,. 47 с. 5. Смирнов, В.И. Курс высшей математики. / В.И.Смирнов. - 6-е изд., перераб. М. : Наука, с. 6. Жукова Н.И. Геодезические линии на поверхностях: учеб. пособие / Н.И. Жукова, А.В.Багаев. Н. Новгород: Издательство Нижегородского госуниверситета, с.

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